Research ArticleTraffic Intensity Estimation in Finite MarkovianQueueing Systems

Frederico R B Cruz 1 Maacutercio A C Almeida 2

Marcos F S V DrsquoAngelo 3 and Tom vanWoensel 4

1Departamento de Estatıstica Universidade Federal de Minas Gerais 31270-901 Belo Horizonte MG Brazil2Pro-Reitoria de Planejamento e Desenvolvimento Universidade Federal do Para 66075-110 Belem PA Brazil3Departamento de Ciencia da Computacao Universidade Estadual de Montes Claros 39401-089 Montes Claros MG Brazil4Department of Industrial Engineering amp Innovation Sciences Eindhoven University of Technology5600 MB Eindhoven Netherlands

Correspondence should be addressed to Frederico R B Cruz fcruzestufmgbr

Received 5 January 2018 Revised 10 April 2018 Accepted 16 May 2018 Published 26 June 2018

Academic Editor Jason Gu

Copyright copy 2018 Frederico R B Cruz et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

In many everyday situations in which a queue is formed queueing models may play a key role By using such models which areidealizations of reality accurate performance measures can be determined such as traffic intensity (120588) which is defined as the ratiobetween the arrival rate and the service rate An intermediate step in the process includes the statistical estimation of the parametersof the proper model In this study we are interested in investigating the finite-sample behavior of some well-known methods forthe estimation of 120588 for single-server finite Markovian queues or in Kendall notation 1198721198721119870 queues namely the maximumlikelihood estimator Bayesian methods and bootstrap corrections We performed extensive simulations to verify the quality of theestimators for samples up to 200The computational results show that accurate estimates in terms of the lowest mean squared errorscan be obtained for a broad range of values in the parametric space by using the Jeffreysrsquo prior A numerical example is analyzed indetail the limitations of the results are discussed and notable topics to be further developed in this research area are presented

1 Introduction

Queueing models are idealizations of many real systemsHowever they enable the accurate determination of perfor-mance measures as long as a previous step has been fulfilledthat step is the statistical estimation of its parameters [12] It is impossible to discuss inference in queues withoutmentioning the pioneering work of Clarke in the 1950swho describes maximum likelihood estimators for the arrivalrates and service times in simple queues and the workof Schruben and Kulkarni in the 1980s who consider theproblem of bias in queue estimation It is also importantto mention the Bayesian papers on the subject includingthat of Muddapur in the 1970s who published one of thefirst results that extended Clarkrsquos methodology the series ofpapers from Armero and Bayarri [3 4] Armero and Conesa[5ndash8] Choudhury and Borthakur [9] and more recently

Chowdhury and Mukherjee [10 11] Cruz et al [12] andQuinino and Cruz [13] These are only a few examples of thepapers in this important research area

The purpose of this paper is to evaluate the behaviorof a traffic intensity estimator (120588) which is defined as theratio between the arrival rate (120582) and the service rate (120583)specifically for single-server finite Markovian queues whichmodel various real systems [14]The type of queue researchedin this paper is typically seen in many service-orientedsettings where there is a finite queue in front of a serverThink of a gas station where cars can queue up for a limitedamount of space and the traffic intensity should be nottoo high as in this case cars might not join the queueAn alternative example is observed within a supermarketcontext where customers line up for the check out Having agood understanding of the traffic intensity is crucial in thesesituations Good estimations are needed to properly design

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 3018758 15 pageshttpsdoiorg10115520183018758

2 Mathematical Problems in Engineering

MLERoMM1Kltminusfunction (Ksamp) loglikefltminusfunction (rho K n sumxi)

nlowastlog (1minusrho)+sumxilowastlog (rho)minusnlowastlog (1minusrhoand(K + 1)) EpsMLEltminus1eminus06nltminuslength (samp)sumxiltminussum (samp)resltminusoptimize (loglikefc(EpsMLE1minusEpsMLE) K n sumxi maximum=TRUE tol=EpsMLE)return (res$maximum)

Listing 1 MLE for 120588

the system (eg to install the gas pump buffer sizes needed)or to properly manage the system (eg to set waiting spacesfor the store clerks at check out)

In summary previous results obtained for infiniteMarko-vian queues are extended here for finite Markovian queuesTo reach this goal this paper combines some techniquesand approaches from the work of Almeida and Cruz [15](ie Bayesian inference and Monte Carlo simulation forevaluation of estimators under finite samples) with otherclassical tools (eg Gibbs sampling and bootstrapping)

The remainder of the paper is organized as followsSection 2 details the queue equations and estimators for 120588The computational results are presented and discussed inSection 3 followed by Section 4 which concludes the textwith some final remarks and topics for future research in thearea

2 Material and Methods

When you have Poisson arrivals exponential service timesa single server and limited waiting space you have an1198721198721119870 queue in Kendall notation 119870 represents thenumber of customers simultaneously allowed in the queueingsystemThe probability of a number119883 of users of the systemfor 119909 = 0 1 119870 is given by [14]

119875 (119883 = 119909) equiv 119901119909 =

120588119909 (1 minus 120588)1 minus 120588119870+1 for 120588 = 11119870 + 1 for 120588 = 1 (1)

where 120588 is the traffic intensity Estimating traffic intensityis important as it is a key design parameter in productionnetwork design routing of products and so on

21 Maximum Likelihood Estimator Maximum likelihoodestimation for the truncated geometric model is knownsince Thomasson and Kapadia [16] Consider the stationaryprobability distribution given by (1) Next consider a randomsample of size 119899 x = (1199091 1199092 119909119899)T where 119909119894 is the numberof customers an outside observer finds in the system Inthis case a maximum of 119870 customers are allowed in the

queueing system at once Therefore the likelihood functionis

119871 (120588 x) = 1205881199091 (1 minus 120588)1 minus 120588119870+1 times sdot sdot sdot times 120588119909119899 (1 minus 120588)

1 minus 120588119870+1= 120588119910 times (1 minus 120588)119899

(1 minus 120588119870+1)119899 (2)

where 119910 = sum119899119894=1 119909119894 Note that the likelihood is a function oftraffic intensity 120588 and sample x although only its size 119899 andits sum 119910 which is a sufficient statistic for 120588 are necessary

Needless to say that for the implementation of maximumlikelihood estimator (MLE) any bounded optimization algo-rithm could be used However for the sake of simplicity theimplementation used in this study was encoded in R [17] andcan be seen in Listing 1 For convenience the logarithm of thelikelihood function was considered because it allows prod-ucts to be turned into sums Maximizing the log-likelihood isdone numerically through an R internal function Howevertests (not shown)were conductedwith the original likelihoodfunction they indicated that the results did not changesignificantly in terms of accuracy or computational effort

A nice feature about the MLEs is their invariance totransformations [18] in such a way that if 120579 is the MLE of 120579and 120572 = ℎ(120579) is a function of 120579 then = ℎ(120579) is the MLEof 120572 Thus the expected number of customers in the systemand the average queue length have respectively the followingMLEs [14]

= 1205881 minus 120588 minus (119870 + 1) 120588119870+11 minus 120588119870+1 (3)

119871119902 = 1205881 minus 120588 minus 1205881 + 1198701205881198701 minus 120588119870+1 (4)

in which 120588 is the MLE for 12058822 Bayesian Inference One of the alternatives for makinginferences is the BayesianmethodOne of itsmain differencesfrom the classical method is that Bayesian inference allowsthe incorporation of some a priori information into themodelof the unknown parameters Unlike the classical methodthe Bayesian method considers these parameters randomvariables ie associates them with probability distributions

Mathematical Problems in Engineering 3

require (HI)EBaRoMM1Kltminusfunction (K samp a b)

logpostfltminusfunction (rho K n sumxi a b) (sumxi+a minus 1)lowastlog (rho)+(n+b minus 1)lowastlog (1minusrho)minusnlowastlog (1minusrhoand(K + 1))

sSizeltminus5000nltminuslength (samp)sumxiltminussum (samp)resltminusarms(runif (1) logpostf function (x K n sumxi a b)((x gt 0)lowast(x lt 1)) sSize K nsumxi a b)

return (mean(res))Listing 2 Bayes estimator for 120588 for beta prior

Therefore the knowledge that the manager has regarding agiven unknown parameter can be considered Two differentprior distributions are described as follows However otheralternatives are possible such as those proposed in the studyby Armero and Bayarri [3] Lingappaiah [19] also considereda Bayesian approach to this distribution

221 Beta Prior Therefore the inference process for estimat-ing 120588 starts with (2) and assumes an a priori beta distributionthat is 119901(120588) sim Beta(119886 119887) which has been successfully used ininference in other Markovian queues [12 13] and results inthe following a posteriori distribution

1199011 (120588 | 119883) prop 119871 (120588 x) times 119901 (120588)prop 120588119910 (1 minus 120588)119899

(1 minus 120588119870+1)119899times Γ (119886 + 119887)Γ (119886) Γ (119887)120588(119886minus1) (1 minus 120588)(119887minus1)

prop 120588119910+119886minus1 (1 minus 120588)119899+119887minus1(1 minus 120588119870+1)119899

(5)

Because the a posteriori distribution of 120588 (5) is not aknown distribution it is necessary to use an approximationmethod to generate samples from the distributionThe imple-mentation of the Bayesian estimator for beta prior is shownin Listing 2 Note that this is a bounded one-dimensionalproblem and numerical integrations could be used as wellto find the a posteriori distribution which should be followedby another numerical integration to compute the estimatorAn open research question is whether or not one methodcould be consistently superior to the other depending on thesample sizes in Gibbs sampling and the precision of the twonumerical integrations

A sample is extracted from the a posteriori distribution byusing the function arms (for size 5000) which is available inRrsquos HI package [20] The a posteriori distribution was repre-sented by the logarithm of the probability density function(without the normalization constant) Because a quadraticloss function is considered the Bayes point estimator issimply the average of the sample Examples of a priori betadistributions are shown in Figure 1

Beta(1525)Beta(1010)Beta(2515)

00

10

15

02 04 06 08 1000휌

05p(휌)

Figure 1 Beta prior distributions for traffic intensity 120588

222 Jeffreys Prior Someone might argue that it would bemore natural to use a noninformative Jeffreysrsquo prior distribu-tionwhich is defined in terms of the Fisher information givenby

119868 (120588) = 119864[minus1205972 log119901 (119883 | 120588)1205971205882 ] (6)

Thus the following prior distribution for 120588 can beobtained

119901 (120588) prop [119868 (120588)]12 (7)

Thus from (1) a 119901(120588) can be found as follows Thelogarithm of 119901(119883 | 120588) from (1) is given by

log119901 (119883 | 120588) = 119883 log 120588 + log (1 minus 120588) minus log (1 minus 120588119870+1) for 120588 = 1 (8)

First and second derivatives are given respectively by

120597 log119901 (119883 | 120588)120597120588 = 119883120588 minus 11 minus 120588 + (119870 + 1) 1205881198701 minus 120588119870+1 (9)

4 Mathematical Problems in Engineering

require(HI)EJeRoMM1Kltminusfunction (K samp)

logJefltminusfunction (rho K n sumxi) logpostltminus(sumxi)lowastlog (rho)+nlowastlog(1minusrho)minusnlowastlog(1minusrhoand(K+1))Irholtminus (1rhoand2)lowast(rho(1minusrho)minus(K+1)lowastrhoand(K+1)(1minusrhoand(K+1))) minus

1(1minusrho)and2 minus (K+1)lowast(Klowastrhoand(minusKminus1)+1)(rhoand(minusK)minusrho)and2if (( is nan(Irho)) ampamp (Irhogt0))

logpostlt minuslogpost+05lowastlog(Irho) return(logpost)

sSizeltminus1000nltminuslength(samp)sumxiltminussum(samp)resltminusarms(runif(1) logJef function(x K n sumxi)((x gt 0)lowast(x lt 1)) sSize K n sumxi)return(mean(res))

Listing 3 Bayes estimator for 120588 for Jeffreysrsquo prior

1205972 log119901 (119883 | 120588)1205971205882 = minus1198831205882 + 1

(1 minus 120588)2

+ (119870 + 1) (119870120588minus119870minus1 + 1)(120588minus119870 minus 120588)2

(10)

It follows that

119868 (120588) = 119864(minus1205972 log119901 (119883 | 120588)1205971205882 )

= 119864[1198831205882 minus 1(1 minus 120588)2 minus

(119870 + 1) (119870120588minus119870minus1 + 1)(120588minus119870 minus 120588)2 ]

= 119864 (119883)1205882 minus 1(1 minus 120588)2 minus

(119870 + 1) (119870120588minus119870minus1 + 1)(120588minus119870 minus 120588)2

(11)

The expectation 119864[119883] is given by [14]

119864 [119883] = 1205881 minus 120588 minus (119870 + 1) 120588119870+11 minus 120588119870+1 (12)

Then

119901 (120588) prop [119868 (120588)]12 = [ 11205882 ( 1205881 minus 120588 minus (119870 + 1) 120588119870+11 minus 120588119870+1 )

minus 1(1 minus 120588)2 minus

(119870 + 1) (119870120588minus119870minus1 + 1)(120588minus119870 minus 120588)2 ]

12

(13)

Thus combining the likelihood (2) and Jeffreys priorgiven by (13) it is possible to find the following posteriorprobability distribution for 120588

1199012 (120588 | x) prop 119871 (120588 x) times 119901 (120588) prop 120588119910 times (1 minus 120588)119899(1 minus 120588119870+1)119899

times [ 11205882 ( 1205881 minus 120588 minus (119870 + 1) 120588119870+11 minus 120588119870+1 ) minus 1(1 minus 120588)2

minus (119870 + 1) (119870120588minus119870minus1 + 1)(120588minus119870 minus 120588)2 ]

12

(14)

in which 0 lt 120588 lt 1Note that unlike the case shown earlier (5) the posterior

distribution given by (14) is fixed Indeed it is not possibleto vary Jeffreysrsquo prior which assumes a specific form Theimplementation of the Bayesian estimator for Jeffreysrsquo prioris shown in Listing 3 Because the logarithm of the posteriordistribution is used variable Irho as defined in Listing 3must be such that 0 lt Irho lt infin

23 Bootstrap Correction Among the bias correction meth-ods that are commonly used for estimators the bootstrapmethod is widely used [21] In its nonparametric versionthe method consists of making several (usually 119861 = 1 000)resamplings xlowast (with replacement)The parameter of interestis reestimated by some (usually biased) method Θlowast Thenthe average is calculated Θlowast and the bias is estimated using

bias = Θlowast minus Θ (15)

where Θ is the estimate obtained from the original sampleThen the following corrected version of the estimator isobtained

Θ119861 = 2Θ minus Θlowast (16)

The procedure is illustrated in Figure 2 The R code [17]for the bootstrap corrected estimator is shown in Listing 4The parameter 119870 is the maximum number of users simul-taneously (in service and waiting) in the 1198721198721119870 queueNote that 1000 bootstrap replicates are used and that thecorrection occurs as part of the MLE (see Listing 1)

Mathematical Problems in Engineering 5

x x1 x2 middot middot middot middot middot middotxn xlowast(1)

xlowast(2)

xlowast1(1) xlowast2(1) xlowastn(1)

middot middot middotxlowast1(2) xlowast2(2) xlowastn(2)

xlowast(B) middot middot middotxlowast1(B) xlowast2(B) xlowastn(B)

Θlowast(1)Θ

Θlowast(2)

Θlowast(B)

Θlowast=

sumBi=1 Θ

lowast(i)

B

Figure 2 The bootstrap method

EBoRoMM1Kltminusfunction (K samp) Bltminus1000summltminus0for (i in 1B) resampltminussample (samp replace=T)estmltminusMLERoMM1K (K resamp)summltminussumm+estm

estmStar=summBreturn (2lowastMLERoMM1K(K samp)minusestmStar) Listing 4 Bootstrap corrected estimator

Besides bias correction the bootstrap method has beenused by many researchers in the past with good results inconfidence interval building and hypothesis testing [22] Asan example an empirical bootstrap confidence interval isused in this work as a simple way of interval estimation for 120588If the distribution of 120575 = Θ minus Θ was known then the criticalvalues 1205751205722 and 1205751minus1205722 could be found where 120575120574 is its 120574100thpercentile and then

Pr (1205751205722 le Θ minus Θ le 1205751minus1205722 | Θ) = (1 minus 120572) lArrrArrPr (Θ minus 1205751205722 ge Θ ge Θ minus 1205751minus1205722 | Θ) = (1 minus 120572) (17)

which gives an (1 minus 120572)100 confidence interval of

CIΘ(1minus120572)100 = [Θ minus 1205751minus1205722 Θ minus 1205751205722] (18)

The bootstrap makes it possible to estimate the distribu-tion of 120575 by the distribution of 120575lowast = Θlowast minus Θ where Θlowast isthe estimate obtained from an empirical bootstrap sample asexplained earlier

24 Simulation MM1K of Queues The number of userspresent in an1198721198721119870 queue follows the distribution givenby (1) To efficiently generate randomvariables fromadiscrete

distribution several methods are widely used in the literatureincluding function sample from R [17] The method usedhere is the discrete analog of the inverse transformationmethod in which it is necessary to generate numbers 119877 simUnif(0 1) ie from a uniform distribution between 0 and 1and to know the probabilities of interest 119875119883 = 119895 = 119901119895 forall119895Therefore to simulate a discrete random variable 119883 with theprobability function

119875 119883 = 119895 = 119901119895 119895 = 0 1 sumforall119895

119901119895 = 1 (19)

it is necessary to compute

119883 =

0 if 119877 le 11990101 if 1199010 lt 119877 le 1199010 + 1199011119895 if

119895minus1sum119894=0

119901119894 lt 119877 le 119895sum119894=0

119901119894

(20)

because

119875 119883 = 119895 = 119875119895minus1sum119894=0

119901119894 lt 119877 le 119895sum119894=0

119901119894 = 119901119895 (21)

and119883 follows the required probability distributionFor 1198721198721119870 queues and from (1) the following must

hold119899sum119894=0

119901119894 = 1 minus 120588119899+11 minus 120588119870+1 119899 = 0 1 119870 (22)

Setting 119888 = 1 minus 120588119870+1 it follows that119883minus1sum119894=0

119901119894 lt 119877 le 119883sum119894=0

119901119894 997904rArr

6 Mathematical Problems in Engineering

Table 1 Mean estimates for 120588 and the mean squared error (MSE in parentheses) for 119870 = 5Estimate 120588 119899

10 20 50 100 200

MLE

001 000928 (000084) 000960 (000045) 001000 (000019) 001003 (000010) 001007 (000005)010 009172 (000715) 009614 (000384) 009845 (000160) 009907 (000081) 009964 (000041)020 018449 (001214) 019235 (000644) 019700 (000258) 019833 (000130) 019921 (000065)050 048492 (001946) 049422 (000932) 049824 (000358) 049884 (000180) 049952 (000088)090 087146 (001703) 088854 (000953) 089839 (000467) 090021 (000271) 090044 (000142)099 092503 (001488) 094638 (000734) 096396 (000303) 097225 (000159) 097854 (000083)

Beta

001 009043 (000699) 005378 (000226) 002864 (000052) 001953 (000018) 001487 (000007)010 015663 (000793) 013021 (000396) 011251 (000160) 010617 (000080) 010321 (000041)020 023444 (000989) 021739 (000567) 020690 (000244) 020327 (000126) 020167 (000064)050 050855 (001688) 050669 (000913) 050295 (000357) 050116 (000180) 050067 (000088)090 081770 (001385) 085318 (000654) 088214 (000283) 089387 (000178) 089954 (000114)099 085425 (002303) 089298 (001185) 092818 (000491) 094668 (000248) 096038 (000121)

Jeffreys

001 005000 (000215) 003182 (000084) 001931 (000026) 001476 (000012) 001244 (000005)010 011630 (000497) 010928 (000314) 010422 (000149) 010202 (000078) 010113 (000040)020 019457 (000908) 019512 (000520) 019804 (000229) 019907 (000123) 019963 (000063)050 049320 (002081) 050148 (001058) 050337 (000377) 050230 (000175) 050121 (000084)090 081749 (001400) 085312 (000655) 088216 (000283) 089387 (000178) 089954 (000115)099 085415 (002308) 089294 (001186) 092819 (000491) 094668 (000249) 096037 (000121)

Bootstrap

001 000992 (000096) 001000 (000049) 001018 (000020) 001013 (000010) 001012 (000005)010 009746 (000791) 009970 (000409) 010009 (000164) 009993 (000082) 010007 (000041)020 019446 (001290) 019837 (000665) 019976 (000262) 019977 (000131) 019994 (000065)050 049647 (001868) 050011 (000901) 050065 (000353) 050007 (000178) 050012 (000088)090 089517 (001990) 090321 (001177) 090473 (000574) 090256 (000314) 090074 (000152)099 095119 (001453) 096522 (000742) 097635 (000317) 098100 (000170) 098475 (000090)

rmm1klt minusfunction (ssizerhoK) simulateRltminusrunif(ssize01)cltminus1minusrhoand(K+1)logrholtminuslog(rho)xltminuslog(1minusRlowastc)logrhoreturn(x)gtgt set seed (13579)gt sampltminusrmm1k (ssize=10 rho=020 K=5)gt samp[1] 0 1 0 1 0 0 0 2 1 0

Listing 5 Sample generation for1198721198721119870 queues

1 minus 120588119883119888 lt 119877 le 1 minus 120588119883+1119888 997904rArrlog (1 minus 119877119888)log (120588) minus 1 le 119883 lt log (1 minus 119877119888)

log (120588) (23)

Therefore

119883 = lceil log (1 minus 119877119888)log (120588) minus 1rceil (24)

MtCaRoMM1Kltminusfunction(ssize rho K fEst) repltminus10000sampltminusnumeric(ssize)estltminusnumeric(rep)for (i in 1rep) sampltminusrmm1k(ssize rho K)oldseedltminusGlobalEnv$Randomseedest [i]ltminusfEst (K samp )GlobalEnv$Randomseedltminusoldseed

return(c(mean(est) var(est)))Listing 6 Monte Carlo simulation

where lceil119910rceil is the ceiling function that is its value is the leastinteger that is not inferior to 119910

Listing 5 presents the R implementation [17] used in thisstudy with a sample call

3 Computational Results

To analyze the performance of the estimators 10000 MonteCarlo replications were made using the R code [17] shownin Listing 6 Note that fESt(K samp ) can be any of the

Mathematical Problems in Engineering 7

Table 2 Mean estimates for 120588 and the mean squared error (MSE in parentheses) for 119870 = 20Estimate 120588 119899

10 20 50 100 200

MLE

001 000928 (000084) 000960 (000045) 001000 (000019) 001003 (000010) 001007 (000005)010 009158 (000710) 009609 (000383) 009844 (000159) 009906 (000081) 009964 (000041)020 018345 (001178) 019186 (000631) 019684 (000255) 019827 (000128) 019919 (000064)050 047371 (001424) 048785 (000666) 049543 (000256) 049750 (000127) 049885 (000062)090 089316 (000326) 089758 (000151) 089922 (000057) 089956 (000029) 089979 (000014)099 097283 (000156) 097979 (000073) 098507 (000030) 098728 (000017) 098877 (000009)

Beta

001 009009 (000692) 005374 (000226) 002864 (000052) 001953 (000018) 001487 (000007)010 015477 (000743) 012978 (000388) 011242 (000159) 010616 (000080) 010320 (000041)020 022893 (000850) 021530 (000529) 020632 (000237) 020303 (000124) 020157 (000063)050 047818 (001132) 048902 (000599) 049557 (000245) 049753 (000125) 049885 (000062)090 088374 (000296) 089484 (000142) 089880 (000058) 089938 (000029) 089970 (000014)099 094835 (000246) 096296 (000107) 097492 (000037) 098068 (000017) 098467 (000008)

Jeffreys

001 005049 (000219) 003182 (000084) 001931 (000026) 001476 (000012) 001244 (000005)010 011678 (000489) 010981 (000320) 010425 (000149) 010203 (000078) 010113 (000040)020 019205 (000801) 019570 (000503) 019866 (000233) 019927 (000123) 019970 (000063)050 046191 (001540) 048175 (000769) 049434 (000302) 049828 (000152) 050063 (000074)090 088365 (000296) 089481 (000142) 089880 (000058) 089938 (000029) 089970 (000014)099 094829 (000247) 096295 (000107) 097493 (000037) 098068 (000017) 098467 (000008)

Bootstrap

001 000993 (000096) 001000 (000049) 001018 (000020) 001013 (000010) 001012 (000005)010 009763 (000795) 009976 (000409) 010010 (000164) 009994 (000082) 010007 (000041)020 019462 (001281) 019846 (000661) 019982 (000260) 019982 (000130) 019997 (000065)050 049385 (001404) 049896 (000659) 050019 (000254) 049995 (000127) 050008 (000062)090 090117 (000310) 090065 (000147) 090021 (000056) 090004 (000028) 090002 (000014)099 098215 (000147) 098590 (000076) 098872 (000035) 098963 (000020) 099023 (000012)

implemented estimation functions for 120588 (MLE Bayesian andbootstrap corrected MLE) and that the state of the randomseed GlobalEnv$Randomseed was stored immediatelybefore the function fESt(K samp ) which can be stochas-tic and was reloaded immediately after its call to ensure thatthe samples generated for the estimates were the same for allestimators

31 Simulation Results Samples were generated from (24)and Listing 5 for sizes 119899 isin 10 20 50 100 200 and trafficintensities 120588 isin 001 010 020 050 090 099 In thesescenarios averages of 10000 Monte Carlo replications werecomputed (via code from Listing 6) and point estimates of 120588were computed by

(i) the MLE via numerical maximization of the likeli-hood function (2) and code from Listing 1

(ii) the Bayesian method via an a priori Beta(10 10)distribution and an average of 5000 samples from thea posteriori distribution (5) obtained by the functionarms [23] from Rrsquos HI package [20] via code fromListing 2

(iii) the Bayesian method via Jeffreys prior distributionand an average of 1000 samples from the a posterioridistribution (14) obtained by the function arms [23]from Rrsquos HI package [20] via code from Listing 3

(iv) the bootstrap corrected MLE (16) and code fromListing 4 Additionally mean squared errors (MSEs)were calculated

The results are shown inTables 1 2 and 3 and summarizedin Figures 3 4 and 5 In Figures 3 4 and 5 the averageestimation error and the mean squared error (MSE) definedas MSE(120588) = Var120588(120588) + Bias2(120588 120588) are shown as functions ofboth the traffic intensity 120588 (averaged over all sample sizes)and the sample size 119899 (averaged over all traffic intensities)

For queues with capacity 119870 = 5 (Figure 3) an approx-imately constant average estimation error is observed forthe MLE and the bootstrap corrected MLE except when120588 asymp 10 (Figure 3(a)) The Bayesian estimators (beta priorand Jeffreysrsquo prior) did not show equivalent performancetheir estimates tended to overestimate the true value (positiveerror) when 120588 lt 05 and to underestimate otherwiseRegarding the sample size 119899 all of the estimators showeda monotonic decrease in error (Figure 3(c)) From the MSEside bothBayesian estimators presented themselves generallyas the best alternative because they presented the lowestvalues most of the time Although the bootstrap correctedMLE presented the lowest bias the method achieves thisperformance at the cost of high variability as reflected by itshighest MSEs

When the queue capacity was increased slightly to119870 = 20(Figure 4) a very similar behavior was noted However the

8 Mathematical Problems in Engineering

Table 3 Mean estimates for 120588 and the mean squared error (MSE in parentheses) for 119870 = 80Estimate 120588 119899

10 20 50 100 200

MLE

001 000928 (000084) 000960 (000045) 001000 (000019) 001003 (000010) 001007 (000005)010 009158 (000710) 009609 (000383) 009844 (000159) 009906 (000081) 009964 (000041)020 018345 (001178) 019186 (000631) 019684 (000255) 019827 (000128) 019919 (000064)050 047366 (001422) 048783 (000666) 049543 (000255) 049750 (000127) 049885 (000062)090 089001 (000144) 089551 (000057) 089835 (000020) 089915 (000010) 089960 (000005)099 098737 (000016) 098903 (000008) 098982 (000003) 098995 (000002) 098998 (000001)

Beta

001 009009 (000692) 005374 (000226) 002864 (000052) 001953 (000018) 001487 (000007)010 015477 (000743) 012978 (000388) 011242 (000159) 010616 (000080) 010320 (000041)020 022892 (000850) 021530 (000529) 020632 (000237) 020304 (000124) 020157 (000063)050 047779 (001125) 048892 (000597) 049556 (000245) 049753 (000125) 049885 (000062)090 088221 (000180) 089167 (000065) 089680 (000021) 089837 (000010) 089921 (000005)099 098149 (000017) 098570 (000007) 098856 (000003) 098953 (000002) 098990 (000001)

Jeffreys

001 005225 (000231) 003334 (000089) 001998 (000027) 001506 (000011) 001253 (000005)010 011744 (000482) 011002 (000316) 010425 (000149) 010203 (000078) 010113 (000040)020 019224 (000794) 019572 (000502) 019866 (000233) 019927 (000123) 019970 (000063)050 046147 (001532) 048167 (000767) 049430 (000302) 049826 (000151) 050064 (000074)090 088217 (000180) 089165 (000065) 089680 (000021) 089837 (000010) 089921 (000005)099 098148 (000017) 098570 (000007) 098856 (000003) 098953 (000002) 098990 (000001)

Bootstrap

001 000993 (000096) 001000 (000049) 001018 (000020) 001013 (000010) 001012 (000005)010 009763 (000795) 009976 (000409) 010010 (000164) 009994 (000082) 010007 (000041)020 019462 (001281) 019846 (000661) 019982 (000260) 019982 (000130) 019997 (000065)050 049390 (001405) 049897 (000659) 050019 (000254) 049995 (000127) 050008 (000062)090 089910 (000118) 089987 (000051) 090008 (000019) 090002 (000009) 090003 (000005)099 099000 (000016) 099042 (000009) 099031 (000004) 099012 (000002) 099002 (000001)

difficulty of estimation for traffic intensities120588 asymp 10 seemed todecrease and the highest MSEs occurred when 120588 asymp 05 TheBayesian estimators maintained the performance presentedearlier for 119870 = 5 that is the estimates have positive biasfor 120588 lt 05 and negative bias otherwise The errors of all ofthe estimators converged to zero as the sample size grewThebootstrap corrected MLE presented an average estimationerror near zero for samples 119899 ge 50 However from the pointof view of the MSE the smaller values were again obtainedusing the Bayesian methods (beta and Jeffreysrsquo prior)

Finally for queues with 119870 = 80 (Figure 5) the observedbehavior could be considered for practical purposes as beingequal to that of an infinite Markovian queue in terms of theaverage estimation error and the MSE This behavior wasobserved for infinite Markovian queues (1198721198721 queues)[15] This finding is merely evidence of the correctness of ourimplementations and the quality of the computational resultspresented

Additionally computational experiments were per-formed for the estimators for 119871 (3) and for 119871119902 (4) and fortheir bootstrap corrected versions with 1000 resamplingsand 119870 = 20 Table 4 and Figure 6 show the resultsobtained for 119871 isin 05 1 2 4 8 16 for sample sizes119899 isin 10 20 50 100 and averages of 10000 Monte Carloreplications Similarly for 119871119902 the results are presented inTable 5 and Figure 7 In summary at an extra cost of thebootstrap method and without inflation of the MSEs the

researcher may achieve with samples of size 119899 = 10 estimatesfor 119871 and 119871119902 with the same average error of the MLE forsamples of size 119899 = 100 a reduction that is relevant inpractical terms because it may lead to reduction in time andcost to obtain the estimates Note that the bootstrap methodalways provides smaller errors and MSEs than the MLEmethod for all estimates of 119871 and 119871119902 even when 120588 gt 1 Alsonote the jump up and down in the errors and MSEs when 120588transitions from 120588 lt 1 to 120588 gt 1

Finally to illustrate the ease of use of the bootstrap in theinterval estimation of the traffic intensity 120588 computationalexperiments were performed The length and coverage ofempirical bootstrap intervals computed from (18) and froma normal distribution approximation (ie Θ plusmn 1199111205722120590Θ where119911120574 is the 120574100th percentile of the standard normal distri-bution and the standard deviation 120590Θ was estimated also bybootstrapping) were evaluated for 120588 isin 010 020 050 090for sample sizes 119899 isin 10 20 50 100 200 averages of 10000Monte Carlo replications and 119870 = 20 The satisfactoryperformance of the bootstrapwas demonstrated as presentedin Table 6 with the coverages approaching the nominalconfidence of 95 (that is 1 minus 120572 = 095) as the sample sizesincrease

32 Numerical Example To better illustrate an application ofthe method a numerical application based on the data con-sidered in Table 7 collected in a large supermarket network

Mathematical Problems in Engineering 9

MLEBeta

JeffreysBootstrap

minus0080

minus0060

minus0040

minus0020

0000

0020

0040

Error

010 020 050 090 099001흆

(a) Average error for 120588

MLEBeta

JeffreysBootstrap

0000000010000200003000040000500006000070000800009000100

MSE

010 020 050 090 099001흆

(b) Average MSE for 120588

MLEBeta

JeffreysBootstrap

minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0000

Error

20 50 100 20010n

(c) Average error for 119899

MLEBeta

JeffreysBootstrap

00000

00020

00040

00060

00080

00100

00120

00140

MSE

20 50 100 20010n

(d) Average MSE for 119899

Figure 3 Performance of estimators for 120588 and119870 = 5

Table 4 Mean estimates for 119871 and the mean squared error (MSE in parentheses) for 119870 = 20Estimator 119871 120588 119899

10 20 50 100

MLE

050 033333 045010 (001698) 047586 (000806) 049098 (000303) 049523 (000150)100 050000 090008 (002353) 095255 (000877) 098190 (000286) 099007 (000137)200 066714 180441 (004755) 190696 (001287) 196438 (000283) 198120 (000112)400 080959 372455 (008082) 387956 (001669) 395673 (000268) 397720 (000092)800 094574 788487 (001625) 797002 (000230) 799390 (000057) 799583 (000029)160 123520 162965 (009872) 161779 (003702) 160743 (000747) 160377 (000238)

Bootstrap

050 033333 048573 (001586) 049686 (000777) 050033 (000299) 050009 (000149)100 050000 097570 (001460) 099583 (000661) 100076 (000254) 099982 (000127)200 066714 196026 (001062) 199245 (000416) 200066 (000154) 199975 (000076)400 080959 397317 (000515) 400156 (000204) 400475 (000081) 400113 (000039)800 094574 796978 (000350) 800755 (000138) 800792 (000059) 800277 (000027)160 123520 161315 (003011) 160376 (000699) 160043 (000188) 160037 (000095)

10 Mathematical Problems in Engineering

MLEBeta

JeffreysBootstrap

010 020 050 090 099001흆

minus0030

minus0020

minus0010

0000

0010

0020

0030

0040

Error

(a) Average error for 120588

MLEBeta

JeffreysBootstrap

00000

00010

00020

00030

00040

00050

00060

MSE

010 020 050 090 099001흆

(b) Average MSE for 120588

MLEBeta

JeffreysBootstrap

minus0015

minus0010

minus0005

0000

0005

0010

0015

0020

Error

20 50 100 20010n

(c) Average error for 119899

MLEBeta

JeffreysBootstrap

00000

00010

00020

00030

00040

00050

00060

00070

00080

MSE

20 50 100 20010n

(d) Average MSE for 119899

Figure 4 Performance of estimators for 120588 and119870 = 20

Table 5 Mean estimates for 119871119902 and the mean squared error (MSE in parentheses) for 119870 = 20Estimator 119871119902 120588 119899

10 20 50 100

MLE

050 050000 042637 (001897) 046470 (000776) 048647 (000272) 049256 (000132)100 061818 086547 (002882) 093515 (000918) 097534 (000248) 098726 (000108)200 073449 177618 (005718) 189501 (001417) 196115 (000266) 197942 (000099)400 084574 374340 (006999) 389252 (001340) 396186 (000214) 397930 (000077)800 097126 794718 (000577) 799972 (000141) 800478 (000056) 800250 (000028)160 132722 161536 (003518) 161286 (002392) 160683 (000800) 160254 (000210)

Bootstrap

050 050000 048185 (001434) 049688 (000660) 050057 (000254) 049986 (000127)100 061818 097175 (001146) 099492 (000493) 100110 (000185) 100049 (000092)200 073449 196521 (000787) 199479 (000303) 200236 (000113) 200027 (000056)400 084574 397906 (000409) 400426 (000174) 400494 (000069) 400067 (000033)800 097126 796303 (000394) 800653 (000136) 800732 (000058) 800362 (000028)160 132722 161202 (003015) 160862 (001687) 160226 (000418) 160228 (000229)

Mathematical Problems in Engineering 11

MLEBeta

JeffreysBootstrap

minus0015minus0010minus000500000005001000150020002500300035

Error

010 020 050 090 099001흆

(a) Average error for 120588

MLEBeta

JeffreysBootstrap

00000

00010

00020

00030

00040

00050

00060

MSE

010 020 050 090 099001흆

(b) Average MSE for 120588

MLEBeta

JeffreysBootstrap

minus0015

minus0010

minus0005

0000

0005

0010

0015

0020

0025

Error

20 50 100 20010n

(c) Average error for 119899

MLEBeta

JeffreysBootstrap

00000

00010

00020

00030

00040

00050

00060

00070

MSE

20 50 100 20010n

(d) Average MSE for 119899

Figure 5 Performance of estimators for 120588 and 119870 = 80

Table 6 Average length (L) and coverage (C) of 95 confidence intervals for 120588 and119870 = 20Method 120588 119899 = 10 119899 = 20 119899 = 50 119899 = 100 119899 = 200

L C L C L C L C L C

Empirical Bootstrap

010 0189 0626 0185 0612 0143 0885 0106 0906 0077 0933020 0308 0619 0265 0797 0186 0902 0135 0923 0097 0940050 0402 0836 0294 0892 0191 0924 0137 0938 0097 0943090 0207 0835 0148 0911 0094 0958 0066 0953 0046 0952

Normal Approximation

010 0219 0636 0200 0865 0146 0890 0107 0911 0077 0936020 0335 0868 0274 0835 0188 0909 0136 0934 0098 0942050 0410 0885 0298 0916 0193 0934 0137 0942 0098 0947090 0213 0906 0149 0937 0094 0953 0066 0950 0046 0950

in a region of interest [12] is providedThe goal is to evaluatetraffic intensity which for managerial reasons should notexceed 87 if it does users may leave The data comprise200 random observations of the number of customers in thesystem at random times sufficiently spaced and previouslydefined by the person responsible for collecting the data to

avoid correlationThe observed values (V) and frequency (F)are presented inTable 7 For instance of the 200 observationsat 8 times no customers were found in the system at 21 timesonly 1 customer was found and so on

The estimates are shown in Table 8 they were calculatedusing the MLE (Listing 1) and the Bayesian estimators

12 Mathematical Problems in Engineering

MLEBootstrap

100 200 400 800 160050L

minus0150

minus0100

minus0050

0000

0050

0100

0150

0200

Error

(a) Average error for 119871

MLEBootstrap

00000

00050

00100

00150

00200

00250

00300

00350

00400

MSE

100 200 400 800 160050L

(b) Average MSE for 119871

MLEBootstrap

minus0080minus0070minus0060minus0050minus0040minus0030minus0020minus001000000010

Error

20 50 10010n

(c) Average error for 119899

MLEBootstrap

0000000050001000015000200002500030000350004000045000500

MSE

20 50 10010n

(d) Average MSE for 119899

Figure 6 Performance of estimates for 119871Table 7 Observed values (V) and frequency (F) for a sample of size119899 = 200 with 119870 = 14 [12]V F0 81 212 273 394 295 286 167 158 69 510 211 114 3

(Listings 2 and 3) based on an a prioriBeta(10 10) and 5000samples and Jeffreysrsquo distribution and 1000 samples from

Table 8 Point estimates for the numerical example

Estimate MLE Bayes Jeffreys Bootstrap Corrected120588 08405396 08403625 08403625 08405683

the a posteriori distribution the bootstrap corrected MLE(Listing 4) was based on 1000 resamplings The completescript is shown in Listing 7

According to the results presented in the previous sectionthe Bayesian estimates should be the most reliable It isnotable that the system utilization seemed to be below thetarget (120588 = 08405396 lt 087) Note that the analysis is basedonly on counts of the number of users in the system It is notnecessary to estimate the arrival and service rates separatelyto determine 1205884 Conclusions and Final Observations

The problem of traffic intensity estimation in finite Markovqueues (1198721198721119870 queues) is presented as quite challengingIn fact no estimator is absolutely superior to another in all

Mathematical Problems in Engineering 13

MLEBootstrap

minus0150

minus0100

minus0050

0000

0050

0100

0150

Error

100 200 400 800 160050푳풒

(a) Average error for 119871119902

MLEBootstrap

00000

00050

00100

00150

00200

00250

MSE

100 200 400 800 160050푳풒

(b) Average MSE for 119871119902

MLEBootstrap

minus0120

minus0100

minus0080

minus0060

minus0040

minus0020

0000

0020

Error

20 50 10010n

(c) Average error for 119899

MLEBootstrap

00000

00050

00100

00150

00200

00250

00300

00350

00400

MSE

20 50 10010n

(d) Average MSE for 119899

Figure 7 Performance of estimates for 119871119902

read samplesampltminusc(rep(0 8) rep(1 21) rep(2 27) rep(3 39) rep(4 29) rep(5 28) rep(6 16)rep(7 15) rep(8 6) rep(9 5) rep(10 2) rep(11 1) rep(14 3)) MLE estimateKltminus14hatrhoMLEltminusMLERoMM1K(K samp) Bayesian estimatealtminus10bltminus10setseed(13579)hatrhoBayesltminusEBaRoMM1K(K samp a b) Bayesian Jeffreys estimatesetseed(13579)hatrhoJeltminusEJeRoMM1K(K samp) Bootstrap corrected estimatesetseed(13579)hatrhoBootltminusEBoRoMM1K(K samp)c(hatrhoMLE hatrhoBayes hatrhoJe hatrhoBoot)gt [1] 08405396 08403625 08403625 08405683

Listing 7 Estimations from real data

14 Mathematical Problems in Engineering

parametric space Although the estimates of the MLE andthe bootstrap corrected MLE exhibit less bias the Bayesianestimates (beta and Jeffreysrsquo prior) present the lowest MSEin general Perhaps due to the skewness of the a posteriordistribution the Bayesian estimators do not present low biasIn general for sample size 119899 = 50 and queues with 119870 ge 20the average estimation error is less than 0005 this value wasonly exceeded by the Bayesian estimator for queues with totalcapacity 119870 = 5

In regard to the behavior of the average estimation errorand the average MSE as functions of the traffic intensity 120588the major errors are observed when the sample size is small(119899 le 20) and the traffic intensities are 120588 asymp 10 unlike inthe case of the 1198721198721 queues which exhibit higher biaseswhen 120588 asymp 05 Perhaps due to the truncation of the numberof users to the maximum queue length119870 systems with hightraffic intensities require more computational effort and arethe most difficult to estimate

Finally it is important to note that for queues withcapacity 119870 = 80 the systemrsquos behavior is similar to that of aninfinite Markovian queue (an 1198721198721 queue) as expectedThat is the average estimation error is greater and the MSEis highest when 120588 asymp 05

Future work in this area includes testing other Bayesianpoint estimators (eg the median because of the asym-metry of the a posterior distribution) developing intervalestimators hypothesis testingmethods or even Kernel-basedmethods [24]

Data Availability

The data used to support the findings of this study areincluded within the article

Disclosure

The Brazilian government funding agencies mentioned hadno role in the study

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

Special thanks are due to Gabriel and Carolina for helpingwith the algebra This work was supported by the Brazil-ian agencies CNPq (Conselho Nacional de DesenvolvimentoCientıfico e Tecnologico of the Ministry for Science andTechnology) [Grant nos 3046712014-2 3058412016-5] andFAPEMIG (Fundacao de Amparo a Pesquisa do Estado deMinas Gerais) [Grant nos CEX-PPM-00564-17 APQ-02119-15 and BIP-00106-16]

References

[1] D Ma D Wang Y Bie F Sun and S Jin ldquoA method for queuelength estimation in an urban street network based on roll time

occupancy datardquo Mathematical Problems in Engineering vol2012 p 12 2012 httpdxdoiorg1011552012892575

[2] S Zhao S Liang H Liu and M Ma ldquoCTM based real-timequeue length estimation at signalized intersectionrdquo Mathemat-ical Problems in Engineering vol 2015 Article ID 328712 12pages 2015

[3] C Armero and M J Bayarri ldquoBayesian prediction in 1198721198721queuesrdquo Queueing Systems vol 15 no 1-4 pp 401ndash417 1994

[4] C Armero and M J Bayarri ldquoQueuesrdquo in Proceedings of theInternational Encyclopedia of the Social Behavioral Sciences J DWright Ed pp 784ndash789 Oxford UK 2015

[5] C Armero and D Conesa ldquoInference and prediction in bulkarrival queues and queues with service in stagesrdquo AppliedStochastic Models in Business and Industry vol 14 no 1 pp 35ndash46 1998

[6] C Armero and D Conesa ldquoPrediction in Markovian bulkarrival queuesrdquo Queueing Systems vol 34 no 1-4 pp 327ndash3502000

[7] C Armero and D Conesa ldquoStatistical performance of a mul-ticlass bulk production queueing systemrdquo European Journal ofOperational Research vol 158 no 3 pp 649ndash661 2004

[8] C Armero and D Conesa ldquoBayesian hierarchical models inmanufacturing bulk service queuesrdquo Journal of Statistical Plan-ning and Inference vol 136 no 2 pp 335ndash354 2006

[9] A Choudhury and A C Borthakur ldquoBayesian inference andprediction in the single server Markovian queuerdquo MetrikaInternational Journal for Theoretical and Applied Statistics vol67 no 3 pp 371ndash383 2008

[10] S Chowdhury and S P Mukherjee ldquoEstimation of trafficintensity based on queue length in a single MM1 queuerdquoCommunications in StatisticsmdashTheory and Methods vol 42 no13 pp 2376ndash2390 2013

[11] S Chowdhury and S P Mukherjee ldquoBayes estimation inMM1 queues with bivariate priorrdquo Journal of Statistics andManagement Systems vol 19 no 5 pp 681ndash699 2016

[12] F R Cruz R C Quinino and L L Ho ldquoBayesian estimationof traffic intensity based on queue length in a multi-serverMMS queuerdquo Communications in StatisticsmdashSimulation andComputation vol 46 no 9 pp 7319ndash7331 2017

[13] R C Quinino and F R B Cruz ldquoBayesian sample sizesin an MM1 queueing systemsrdquo The International Journal ofAdvanced Manufacturing Technology vol 88 no 1-4 pp 995ndash1002 2017

[14] D Gross J F Shortle and J M Thompson Fundamentals ofQueueing Theory Wiley-Interscience New York NY USA 4thedition 2009

[15] M A C Almeida and F R B Cruz ldquoA note on Bayesianestimation of traffic intensity in single-server MarkovianqueuesrdquoCommunications in Statistics -SimulationComputationhttpdxdoiorg1010800361091820171353614

[16] R LThomasson and C H Kapadia ldquoOn estimating the param-eter of a truncated geometric distributionrdquo Annals of theInstitute of Statistical Mathematics vol 20 pp 519ndash523 1968

[17] R Core Team R A Language and Environment for StatisticalComputing R Foundation for Statistical Computing ViennaAustria 2017 httpwwwR-projectorg

[18] N Mukhopadhyay Probability and Statistical Inference MarcelDekker New York NY USA 2000

[19] G S Lingappaiah ldquoBayes inference in right truncated geo-metric distributionrdquo Malaysian Mathematical Society BulletinSecond Series vol 15 no 2 pp 61ndash67 1992

Mathematical Problems in Engineering 3

require (HI)EBaRoMM1Kltminusfunction (K samp a b)

logpostfltminusfunction (rho K n sumxi a b) (sumxi+a minus 1)lowastlog (rho)+(n+b minus 1)lowastlog (1minusrho)minusnlowastlog (1minusrhoand(K + 1))

sSizeltminus5000nltminuslength (samp)sumxiltminussum (samp)resltminusarms(runif (1) logpostf function (x K n sumxi a b)((x gt 0)lowast(x lt 1)) sSize K nsumxi a b)

return (mean(res))Listing 2 Bayes estimator for 120588 for beta prior

Therefore the knowledge that the manager has regarding agiven unknown parameter can be considered Two differentprior distributions are described as follows However otheralternatives are possible such as those proposed in the studyby Armero and Bayarri [3] Lingappaiah [19] also considereda Bayesian approach to this distribution

221 Beta Prior Therefore the inference process for estimat-ing 120588 starts with (2) and assumes an a priori beta distributionthat is 119901(120588) sim Beta(119886 119887) which has been successfully used ininference in other Markovian queues [12 13] and results inthe following a posteriori distribution

1199011 (120588 | 119883) prop 119871 (120588 x) times 119901 (120588)prop 120588119910 (1 minus 120588)119899

(1 minus 120588119870+1)119899times Γ (119886 + 119887)Γ (119886) Γ (119887)120588(119886minus1) (1 minus 120588)(119887minus1)

prop 120588119910+119886minus1 (1 minus 120588)119899+119887minus1(1 minus 120588119870+1)119899

(5)

Because the a posteriori distribution of 120588 (5) is not aknown distribution it is necessary to use an approximationmethod to generate samples from the distributionThe imple-mentation of the Bayesian estimator for beta prior is shownin Listing 2 Note that this is a bounded one-dimensionalproblem and numerical integrations could be used as wellto find the a posteriori distribution which should be followedby another numerical integration to compute the estimatorAn open research question is whether or not one methodcould be consistently superior to the other depending on thesample sizes in Gibbs sampling and the precision of the twonumerical integrations

A sample is extracted from the a posteriori distribution byusing the function arms (for size 5000) which is available inRrsquos HI package [20] The a posteriori distribution was repre-sented by the logarithm of the probability density function(without the normalization constant) Because a quadraticloss function is considered the Bayes point estimator issimply the average of the sample Examples of a priori betadistributions are shown in Figure 1

Beta(1525)Beta(1010)Beta(2515)

00

10

15

02 04 06 08 1000휌

05p(휌)

Figure 1 Beta prior distributions for traffic intensity 120588

222 Jeffreys Prior Someone might argue that it would bemore natural to use a noninformative Jeffreysrsquo prior distribu-tionwhich is defined in terms of the Fisher information givenby

119868 (120588) = 119864[minus1205972 log119901 (119883 | 120588)1205971205882 ] (6)

Thus the following prior distribution for 120588 can beobtained

119901 (120588) prop [119868 (120588)]12 (7)

Thus from (1) a 119901(120588) can be found as follows Thelogarithm of 119901(119883 | 120588) from (1) is given by

log119901 (119883 | 120588) = 119883 log 120588 + log (1 minus 120588) minus log (1 minus 120588119870+1) for 120588 = 1 (8)

First and second derivatives are given respectively by

120597 log119901 (119883 | 120588)120597120588 = 119883120588 minus 11 minus 120588 + (119870 + 1) 1205881198701 minus 120588119870+1 (9)

4 Mathematical Problems in Engineering

require(HI)EJeRoMM1Kltminusfunction (K samp)

logJefltminusfunction (rho K n sumxi) logpostltminus(sumxi)lowastlog (rho)+nlowastlog(1minusrho)minusnlowastlog(1minusrhoand(K+1))Irholtminus (1rhoand2)lowast(rho(1minusrho)minus(K+1)lowastrhoand(K+1)(1minusrhoand(K+1))) minus

1(1minusrho)and2 minus (K+1)lowast(Klowastrhoand(minusKminus1)+1)(rhoand(minusK)minusrho)and2if (( is nan(Irho)) ampamp (Irhogt0))

logpostlt minuslogpost+05lowastlog(Irho) return(logpost)

sSizeltminus1000nltminuslength(samp)sumxiltminussum(samp)resltminusarms(runif(1) logJef function(x K n sumxi)((x gt 0)lowast(x lt 1)) sSize K n sumxi)return(mean(res))

Listing 3 Bayes estimator for 120588 for Jeffreysrsquo prior

1205972 log119901 (119883 | 120588)1205971205882 = minus1198831205882 + 1

(1 minus 120588)2

+ (119870 + 1) (119870120588minus119870minus1 + 1)(120588minus119870 minus 120588)2

(10)

It follows that

119868 (120588) = 119864(minus1205972 log119901 (119883 | 120588)1205971205882 )

= 119864[1198831205882 minus 1(1 minus 120588)2 minus

(119870 + 1) (119870120588minus119870minus1 + 1)(120588minus119870 minus 120588)2 ]

= 119864 (119883)1205882 minus 1(1 minus 120588)2 minus

(119870 + 1) (119870120588minus119870minus1 + 1)(120588minus119870 minus 120588)2

(11)

The expectation 119864[119883] is given by [14]

119864 [119883] = 1205881 minus 120588 minus (119870 + 1) 120588119870+11 minus 120588119870+1 (12)

Then

119901 (120588) prop [119868 (120588)]12 = [ 11205882 ( 1205881 minus 120588 minus (119870 + 1) 120588119870+11 minus 120588119870+1 )

minus 1(1 minus 120588)2 minus

(119870 + 1) (119870120588minus119870minus1 + 1)(120588minus119870 minus 120588)2 ]

12

(13)

Thus combining the likelihood (2) and Jeffreys priorgiven by (13) it is possible to find the following posteriorprobability distribution for 120588

1199012 (120588 | x) prop 119871 (120588 x) times 119901 (120588) prop 120588119910 times (1 minus 120588)119899(1 minus 120588119870+1)119899

times [ 11205882 ( 1205881 minus 120588 minus (119870 + 1) 120588119870+11 minus 120588119870+1 ) minus 1(1 minus 120588)2

minus (119870 + 1) (119870120588minus119870minus1 + 1)(120588minus119870 minus 120588)2 ]

12

(14)

in which 0 lt 120588 lt 1Note that unlike the case shown earlier (5) the posterior

distribution given by (14) is fixed Indeed it is not possibleto vary Jeffreysrsquo prior which assumes a specific form Theimplementation of the Bayesian estimator for Jeffreysrsquo prioris shown in Listing 3 Because the logarithm of the posteriordistribution is used variable Irho as defined in Listing 3must be such that 0 lt Irho lt infin

23 Bootstrap Correction Among the bias correction meth-ods that are commonly used for estimators the bootstrapmethod is widely used [21] In its nonparametric versionthe method consists of making several (usually 119861 = 1 000)resamplings xlowast (with replacement)The parameter of interestis reestimated by some (usually biased) method Θlowast Thenthe average is calculated Θlowast and the bias is estimated using

bias = Θlowast minus Θ (15)

where Θ is the estimate obtained from the original sampleThen the following corrected version of the estimator isobtained

Θ119861 = 2Θ minus Θlowast (16)

The procedure is illustrated in Figure 2 The R code [17]for the bootstrap corrected estimator is shown in Listing 4The parameter 119870 is the maximum number of users simul-taneously (in service and waiting) in the 1198721198721119870 queueNote that 1000 bootstrap replicates are used and that thecorrection occurs as part of the MLE (see Listing 1)

Mathematical Problems in Engineering 5

x x1 x2 middot middot middot middot middot middotxn xlowast(1)

xlowast(2)

xlowast1(1) xlowast2(1) xlowastn(1)

middot middot middotxlowast1(2) xlowast2(2) xlowastn(2)

xlowast(B) middot middot middotxlowast1(B) xlowast2(B) xlowastn(B)

Θlowast(1)Θ

Θlowast(2)

Θlowast(B)

Θlowast=

sumBi=1 Θ

lowast(i)

B

Figure 2 The bootstrap method

EBoRoMM1Kltminusfunction (K samp) Bltminus1000summltminus0for (i in 1B) resampltminussample (samp replace=T)estmltminusMLERoMM1K (K resamp)summltminussumm+estm

estmStar=summBreturn (2lowastMLERoMM1K(K samp)minusestmStar) Listing 4 Bootstrap corrected estimator

Besides bias correction the bootstrap method has beenused by many researchers in the past with good results inconfidence interval building and hypothesis testing [22] Asan example an empirical bootstrap confidence interval isused in this work as a simple way of interval estimation for 120588If the distribution of 120575 = Θ minus Θ was known then the criticalvalues 1205751205722 and 1205751minus1205722 could be found where 120575120574 is its 120574100thpercentile and then

Pr (1205751205722 le Θ minus Θ le 1205751minus1205722 | Θ) = (1 minus 120572) lArrrArrPr (Θ minus 1205751205722 ge Θ ge Θ minus 1205751minus1205722 | Θ) = (1 minus 120572) (17)

which gives an (1 minus 120572)100 confidence interval of

CIΘ(1minus120572)100 = [Θ minus 1205751minus1205722 Θ minus 1205751205722] (18)

The bootstrap makes it possible to estimate the distribu-tion of 120575 by the distribution of 120575lowast = Θlowast minus Θ where Θlowast isthe estimate obtained from an empirical bootstrap sample asexplained earlier

24 Simulation MM1K of Queues The number of userspresent in an1198721198721119870 queue follows the distribution givenby (1) To efficiently generate randomvariables fromadiscrete

distribution several methods are widely used in the literatureincluding function sample from R [17] The method usedhere is the discrete analog of the inverse transformationmethod in which it is necessary to generate numbers 119877 simUnif(0 1) ie from a uniform distribution between 0 and 1and to know the probabilities of interest 119875119883 = 119895 = 119901119895 forall119895Therefore to simulate a discrete random variable 119883 with theprobability function

119875 119883 = 119895 = 119901119895 119895 = 0 1 sumforall119895

119901119895 = 1 (19)

it is necessary to compute

119883 =

0 if 119877 le 11990101 if 1199010 lt 119877 le 1199010 + 1199011119895 if

119895minus1sum119894=0

119901119894 lt 119877 le 119895sum119894=0

119901119894

(20)

because

119875 119883 = 119895 = 119875119895minus1sum119894=0

119901119894 lt 119877 le 119895sum119894=0

119901119894 = 119901119895 (21)

and119883 follows the required probability distributionFor 1198721198721119870 queues and from (1) the following must

hold119899sum119894=0

119901119894 = 1 minus 120588119899+11 minus 120588119870+1 119899 = 0 1 119870 (22)

Setting 119888 = 1 minus 120588119870+1 it follows that119883minus1sum119894=0

119901119894 lt 119877 le 119883sum119894=0

119901119894 997904rArr

6 Mathematical Problems in Engineering

Table 1 Mean estimates for 120588 and the mean squared error (MSE in parentheses) for 119870 = 5Estimate 120588 119899

10 20 50 100 200

MLE

001 000928 (000084) 000960 (000045) 001000 (000019) 001003 (000010) 001007 (000005)010 009172 (000715) 009614 (000384) 009845 (000160) 009907 (000081) 009964 (000041)020 018449 (001214) 019235 (000644) 019700 (000258) 019833 (000130) 019921 (000065)050 048492 (001946) 049422 (000932) 049824 (000358) 049884 (000180) 049952 (000088)090 087146 (001703) 088854 (000953) 089839 (000467) 090021 (000271) 090044 (000142)099 092503 (001488) 094638 (000734) 096396 (000303) 097225 (000159) 097854 (000083)

Beta

001 009043 (000699) 005378 (000226) 002864 (000052) 001953 (000018) 001487 (000007)010 015663 (000793) 013021 (000396) 011251 (000160) 010617 (000080) 010321 (000041)020 023444 (000989) 021739 (000567) 020690 (000244) 020327 (000126) 020167 (000064)050 050855 (001688) 050669 (000913) 050295 (000357) 050116 (000180) 050067 (000088)090 081770 (001385) 085318 (000654) 088214 (000283) 089387 (000178) 089954 (000114)099 085425 (002303) 089298 (001185) 092818 (000491) 094668 (000248) 096038 (000121)

Jeffreys

001 005000 (000215) 003182 (000084) 001931 (000026) 001476 (000012) 001244 (000005)010 011630 (000497) 010928 (000314) 010422 (000149) 010202 (000078) 010113 (000040)020 019457 (000908) 019512 (000520) 019804 (000229) 019907 (000123) 019963 (000063)050 049320 (002081) 050148 (001058) 050337 (000377) 050230 (000175) 050121 (000084)090 081749 (001400) 085312 (000655) 088216 (000283) 089387 (000178) 089954 (000115)099 085415 (002308) 089294 (001186) 092819 (000491) 094668 (000249) 096037 (000121)

Bootstrap

001 000992 (000096) 001000 (000049) 001018 (000020) 001013 (000010) 001012 (000005)010 009746 (000791) 009970 (000409) 010009 (000164) 009993 (000082) 010007 (000041)020 019446 (001290) 019837 (000665) 019976 (000262) 019977 (000131) 019994 (000065)050 049647 (001868) 050011 (000901) 050065 (000353) 050007 (000178) 050012 (000088)090 089517 (001990) 090321 (001177) 090473 (000574) 090256 (000314) 090074 (000152)099 095119 (001453) 096522 (000742) 097635 (000317) 098100 (000170) 098475 (000090)

rmm1klt minusfunction (ssizerhoK) simulateRltminusrunif(ssize01)cltminus1minusrhoand(K+1)logrholtminuslog(rho)xltminuslog(1minusRlowastc)logrhoreturn(x)gtgt set seed (13579)gt sampltminusrmm1k (ssize=10 rho=020 K=5)gt samp[1] 0 1 0 1 0 0 0 2 1 0

Listing 5 Sample generation for1198721198721119870 queues

1 minus 120588119883119888 lt 119877 le 1 minus 120588119883+1119888 997904rArrlog (1 minus 119877119888)log (120588) minus 1 le 119883 lt log (1 minus 119877119888)

log (120588) (23)

Therefore

119883 = lceil log (1 minus 119877119888)log (120588) minus 1rceil (24)

MtCaRoMM1Kltminusfunction(ssize rho K fEst) repltminus10000sampltminusnumeric(ssize)estltminusnumeric(rep)for (i in 1rep) sampltminusrmm1k(ssize rho K)oldseedltminusGlobalEnv$Randomseedest [i]ltminusfEst (K samp )GlobalEnv$Randomseedltminusoldseed

return(c(mean(est) var(est)))Listing 6 Monte Carlo simulation

where lceil119910rceil is the ceiling function that is its value is the leastinteger that is not inferior to 119910

Listing 5 presents the R implementation [17] used in thisstudy with a sample call

3 Computational Results

To analyze the performance of the estimators 10000 MonteCarlo replications were made using the R code [17] shownin Listing 6 Note that fESt(K samp ) can be any of the

Mathematical Problems in Engineering 7

Table 2 Mean estimates for 120588 and the mean squared error (MSE in parentheses) for 119870 = 20Estimate 120588 119899

10 20 50 100 200

MLE

001 000928 (000084) 000960 (000045) 001000 (000019) 001003 (000010) 001007 (000005)010 009158 (000710) 009609 (000383) 009844 (000159) 009906 (000081) 009964 (000041)020 018345 (001178) 019186 (000631) 019684 (000255) 019827 (000128) 019919 (000064)050 047371 (001424) 048785 (000666) 049543 (000256) 049750 (000127) 049885 (000062)090 089316 (000326) 089758 (000151) 089922 (000057) 089956 (000029) 089979 (000014)099 097283 (000156) 097979 (000073) 098507 (000030) 098728 (000017) 098877 (000009)

Beta

001 009009 (000692) 005374 (000226) 002864 (000052) 001953 (000018) 001487 (000007)010 015477 (000743) 012978 (000388) 011242 (000159) 010616 (000080) 010320 (000041)020 022893 (000850) 021530 (000529) 020632 (000237) 020303 (000124) 020157 (000063)050 047818 (001132) 048902 (000599) 049557 (000245) 049753 (000125) 049885 (000062)090 088374 (000296) 089484 (000142) 089880 (000058) 089938 (000029) 089970 (000014)099 094835 (000246) 096296 (000107) 097492 (000037) 098068 (000017) 098467 (000008)

Jeffreys

001 005049 (000219) 003182 (000084) 001931 (000026) 001476 (000012) 001244 (000005)010 011678 (000489) 010981 (000320) 010425 (000149) 010203 (000078) 010113 (000040)020 019205 (000801) 019570 (000503) 019866 (000233) 019927 (000123) 019970 (000063)050 046191 (001540) 048175 (000769) 049434 (000302) 049828 (000152) 050063 (000074)090 088365 (000296) 089481 (000142) 089880 (000058) 089938 (000029) 089970 (000014)099 094829 (000247) 096295 (000107) 097493 (000037) 098068 (000017) 098467 (000008)

Bootstrap

001 000993 (000096) 001000 (000049) 001018 (000020) 001013 (000010) 001012 (000005)010 009763 (000795) 009976 (000409) 010010 (000164) 009994 (000082) 010007 (000041)020 019462 (001281) 019846 (000661) 019982 (000260) 019982 (000130) 019997 (000065)050 049385 (001404) 049896 (000659) 050019 (000254) 049995 (000127) 050008 (000062)090 090117 (000310) 090065 (000147) 090021 (000056) 090004 (000028) 090002 (000014)099 098215 (000147) 098590 (000076) 098872 (000035) 098963 (000020) 099023 (000012)

implemented estimation functions for 120588 (MLE Bayesian andbootstrap corrected MLE) and that the state of the randomseed GlobalEnv$Randomseed was stored immediatelybefore the function fESt(K samp ) which can be stochas-tic and was reloaded immediately after its call to ensure thatthe samples generated for the estimates were the same for allestimators

31 Simulation Results Samples were generated from (24)and Listing 5 for sizes 119899 isin 10 20 50 100 200 and trafficintensities 120588 isin 001 010 020 050 090 099 In thesescenarios averages of 10000 Monte Carlo replications werecomputed (via code from Listing 6) and point estimates of 120588were computed by

(i) the MLE via numerical maximization of the likeli-hood function (2) and code from Listing 1

(ii) the Bayesian method via an a priori Beta(10 10)distribution and an average of 5000 samples from thea posteriori distribution (5) obtained by the functionarms [23] from Rrsquos HI package [20] via code fromListing 2

(iii) the Bayesian method via Jeffreys prior distributionand an average of 1000 samples from the a posterioridistribution (14) obtained by the function arms [23]from Rrsquos HI package [20] via code from Listing 3

(iv) the bootstrap corrected MLE (16) and code fromListing 4 Additionally mean squared errors (MSEs)were calculated

The results are shown inTables 1 2 and 3 and summarizedin Figures 3 4 and 5 In Figures 3 4 and 5 the averageestimation error and the mean squared error (MSE) definedas MSE(120588) = Var120588(120588) + Bias2(120588 120588) are shown as functions ofboth the traffic intensity 120588 (averaged over all sample sizes)and the sample size 119899 (averaged over all traffic intensities)

For queues with capacity 119870 = 5 (Figure 3) an approx-imately constant average estimation error is observed forthe MLE and the bootstrap corrected MLE except when120588 asymp 10 (Figure 3(a)) The Bayesian estimators (beta priorand Jeffreysrsquo prior) did not show equivalent performancetheir estimates tended to overestimate the true value (positiveerror) when 120588 lt 05 and to underestimate otherwiseRegarding the sample size 119899 all of the estimators showeda monotonic decrease in error (Figure 3(c)) From the MSEside bothBayesian estimators presented themselves generallyas the best alternative because they presented the lowestvalues most of the time Although the bootstrap correctedMLE presented the lowest bias the method achieves thisperformance at the cost of high variability as reflected by itshighest MSEs

When the queue capacity was increased slightly to119870 = 20(Figure 4) a very similar behavior was noted However the

8 Mathematical Problems in Engineering

Table 3 Mean estimates for 120588 and the mean squared error (MSE in parentheses) for 119870 = 80Estimate 120588 119899

10 20 50 100 200

MLE

001 000928 (000084) 000960 (000045) 001000 (000019) 001003 (000010) 001007 (000005)010 009158 (000710) 009609 (000383) 009844 (000159) 009906 (000081) 009964 (000041)020 018345 (001178) 019186 (000631) 019684 (000255) 019827 (000128) 019919 (000064)050 047366 (001422) 048783 (000666) 049543 (000255) 049750 (000127) 049885 (000062)090 089001 (000144) 089551 (000057) 089835 (000020) 089915 (000010) 089960 (000005)099 098737 (000016) 098903 (000008) 098982 (000003) 098995 (000002) 098998 (000001)

Beta

001 009009 (000692) 005374 (000226) 002864 (000052) 001953 (000018) 001487 (000007)010 015477 (000743) 012978 (000388) 011242 (000159) 010616 (000080) 010320 (000041)020 022892 (000850) 021530 (000529) 020632 (000237) 020304 (000124) 020157 (000063)050 047779 (001125) 048892 (000597) 049556 (000245) 049753 (000125) 049885 (000062)090 088221 (000180) 089167 (000065) 089680 (000021) 089837 (000010) 089921 (000005)099 098149 (000017) 098570 (000007) 098856 (000003) 098953 (000002) 098990 (000001)

Jeffreys

001 005225 (000231) 003334 (000089) 001998 (000027) 001506 (000011) 001253 (000005)010 011744 (000482) 011002 (000316) 010425 (000149) 010203 (000078) 010113 (000040)020 019224 (000794) 019572 (000502) 019866 (000233) 019927 (000123) 019970 (000063)050 046147 (001532) 048167 (000767) 049430 (000302) 049826 (000151) 050064 (000074)090 088217 (000180) 089165 (000065) 089680 (000021) 089837 (000010) 089921 (000005)099 098148 (000017) 098570 (000007) 098856 (000003) 098953 (000002) 098990 (000001)

Bootstrap

001 000993 (000096) 001000 (000049) 001018 (000020) 001013 (000010) 001012 (000005)010 009763 (000795) 009976 (000409) 010010 (000164) 009994 (000082) 010007 (000041)020 019462 (001281) 019846 (000661) 019982 (000260) 019982 (000130) 019997 (000065)050 049390 (001405) 049897 (000659) 050019 (000254) 049995 (000127) 050008 (000062)090 089910 (000118) 089987 (000051) 090008 (000019) 090002 (000009) 090003 (000005)099 099000 (000016) 099042 (000009) 099031 (000004) 099012 (000002) 099002 (000001)

difficulty of estimation for traffic intensities120588 asymp 10 seemed todecrease and the highest MSEs occurred when 120588 asymp 05 TheBayesian estimators maintained the performance presentedearlier for 119870 = 5 that is the estimates have positive biasfor 120588 lt 05 and negative bias otherwise The errors of all ofthe estimators converged to zero as the sample size grewThebootstrap corrected MLE presented an average estimationerror near zero for samples 119899 ge 50 However from the pointof view of the MSE the smaller values were again obtainedusing the Bayesian methods (beta and Jeffreysrsquo prior)

Finally for queues with 119870 = 80 (Figure 5) the observedbehavior could be considered for practical purposes as beingequal to that of an infinite Markovian queue in terms of theaverage estimation error and the MSE This behavior wasobserved for infinite Markovian queues (1198721198721 queues)[15] This finding is merely evidence of the correctness of ourimplementations and the quality of the computational resultspresented

Additionally computational experiments were per-formed for the estimators for 119871 (3) and for 119871119902 (4) and fortheir bootstrap corrected versions with 1000 resamplingsand 119870 = 20 Table 4 and Figure 6 show the resultsobtained for 119871 isin 05 1 2 4 8 16 for sample sizes119899 isin 10 20 50 100 and averages of 10000 Monte Carloreplications Similarly for 119871119902 the results are presented inTable 5 and Figure 7 In summary at an extra cost of thebootstrap method and without inflation of the MSEs the

researcher may achieve with samples of size 119899 = 10 estimatesfor 119871 and 119871119902 with the same average error of the MLE forsamples of size 119899 = 100 a reduction that is relevant inpractical terms because it may lead to reduction in time andcost to obtain the estimates Note that the bootstrap methodalways provides smaller errors and MSEs than the MLEmethod for all estimates of 119871 and 119871119902 even when 120588 gt 1 Alsonote the jump up and down in the errors and MSEs when 120588transitions from 120588 lt 1 to 120588 gt 1

Finally to illustrate the ease of use of the bootstrap in theinterval estimation of the traffic intensity 120588 computationalexperiments were performed The length and coverage ofempirical bootstrap intervals computed from (18) and froma normal distribution approximation (ie Θ plusmn 1199111205722120590Θ where119911120574 is the 120574100th percentile of the standard normal distri-bution and the standard deviation 120590Θ was estimated also bybootstrapping) were evaluated for 120588 isin 010 020 050 090for sample sizes 119899 isin 10 20 50 100 200 averages of 10000Monte Carlo replications and 119870 = 20 The satisfactoryperformance of the bootstrapwas demonstrated as presentedin Table 6 with the coverages approaching the nominalconfidence of 95 (that is 1 minus 120572 = 095) as the sample sizesincrease

32 Numerical Example To better illustrate an application ofthe method a numerical application based on the data con-sidered in Table 7 collected in a large supermarket network

Mathematical Problems in Engineering 9

MLEBeta

JeffreysBootstrap

minus0080

minus0060

minus0040

minus0020

0000

0020

0040

Error

010 020 050 090 099001흆

(a) Average error for 120588

MLEBeta

JeffreysBootstrap

0000000010000200003000040000500006000070000800009000100

MSE

010 020 050 090 099001흆

(b) Average MSE for 120588

MLEBeta

JeffreysBootstrap

minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0000

Error

20 50 100 20010n

(c) Average error for 119899

MLEBeta

JeffreysBootstrap

00000

00020

00040

00060

00080

00100

00120

00140

MSE

20 50 100 20010n

(d) Average MSE for 119899

Figure 3 Performance of estimators for 120588 and119870 = 5

Table 4 Mean estimates for 119871 and the mean squared error (MSE in parentheses) for 119870 = 20Estimator 119871 120588 119899

10 20 50 100

MLE

050 033333 045010 (001698) 047586 (000806) 049098 (000303) 049523 (000150)100 050000 090008 (002353) 095255 (000877) 098190 (000286) 099007 (000137)200 066714 180441 (004755) 190696 (001287) 196438 (000283) 198120 (000112)400 080959 372455 (008082) 387956 (001669) 395673 (000268) 397720 (000092)800 094574 788487 (001625) 797002 (000230) 799390 (000057) 799583 (000029)160 123520 162965 (009872) 161779 (003702) 160743 (000747) 160377 (000238)

Bootstrap

050 033333 048573 (001586) 049686 (000777) 050033 (000299) 050009 (000149)100 050000 097570 (001460) 099583 (000661) 100076 (000254) 099982 (000127)200 066714 196026 (001062) 199245 (000416) 200066 (000154) 199975 (000076)400 080959 397317 (000515) 400156 (000204) 400475 (000081) 400113 (000039)800 094574 796978 (000350) 800755 (000138) 800792 (000059) 800277 (000027)160 123520 161315 (003011) 160376 (000699) 160043 (000188) 160037 (000095)

10 Mathematical Problems in Engineering

MLEBeta

JeffreysBootstrap

010 020 050 090 099001흆

minus0030

minus0020

minus0010

0000

0010

0020

0030

0040

Error

(a) Average error for 120588

MLEBeta

JeffreysBootstrap

00000

00010

00020

00030

00040

00050

00060

MSE

010 020 050 090 099001흆

(b) Average MSE for 120588

MLEBeta

JeffreysBootstrap

minus0015

minus0010

minus0005

0000

0005

0010

0015

0020

Error

20 50 100 20010n

(c) Average error for 119899

MLEBeta

JeffreysBootstrap

00000

00010

00020

00030

00040

00050

00060

00070

00080

MSE

20 50 100 20010n

(d) Average MSE for 119899

Figure 4 Performance of estimators for 120588 and119870 = 20

Table 5 Mean estimates for 119871119902 and the mean squared error (MSE in parentheses) for 119870 = 20Estimator 119871119902 120588 119899

10 20 50 100

MLE

050 050000 042637 (001897) 046470 (000776) 048647 (000272) 049256 (000132)100 061818 086547 (002882) 093515 (000918) 097534 (000248) 098726 (000108)200 073449 177618 (005718) 189501 (001417) 196115 (000266) 197942 (000099)400 084574 374340 (006999) 389252 (001340) 396186 (000214) 397930 (000077)800 097126 794718 (000577) 799972 (000141) 800478 (000056) 800250 (000028)160 132722 161536 (003518) 161286 (002392) 160683 (000800) 160254 (000210)

Bootstrap

050 050000 048185 (001434) 049688 (000660) 050057 (000254) 049986 (000127)100 061818 097175 (001146) 099492 (000493) 100110 (000185) 100049 (000092)200 073449 196521 (000787) 199479 (000303) 200236 (000113) 200027 (000056)400 084574 397906 (000409) 400426 (000174) 400494 (000069) 400067 (000033)800 097126 796303 (000394) 800653 (000136) 800732 (000058) 800362 (000028)160 132722 161202 (003015) 160862 (001687) 160226 (000418) 160228 (000229)

Mathematical Problems in Engineering 11

MLEBeta

JeffreysBootstrap

minus0015minus0010minus000500000005001000150020002500300035

Error

010 020 050 090 099001흆

(a) Average error for 120588

MLEBeta

JeffreysBootstrap

00000

00010

00020

00030

00040

00050

00060

MSE

010 020 050 090 099001흆

(b) Average MSE for 120588

MLEBeta

JeffreysBootstrap

minus0015

minus0010

minus0005

0000

0005

0010

0015

0020

0025

Error

20 50 100 20010n

(c) Average error for 119899

MLEBeta

JeffreysBootstrap

00000

00010

00020

00030

00040

00050

00060

00070

MSE

20 50 100 20010n

(d) Average MSE for 119899

Figure 5 Performance of estimators for 120588 and 119870 = 80

Table 6 Average length (L) and coverage (C) of 95 confidence intervals for 120588 and119870 = 20Method 120588 119899 = 10 119899 = 20 119899 = 50 119899 = 100 119899 = 200

L C L C L C L C L C

Empirical Bootstrap

010 0189 0626 0185 0612 0143 0885 0106 0906 0077 0933020 0308 0619 0265 0797 0186 0902 0135 0923 0097 0940050 0402 0836 0294 0892 0191 0924 0137 0938 0097 0943090 0207 0835 0148 0911 0094 0958 0066 0953 0046 0952

Normal Approximation

010 0219 0636 0200 0865 0146 0890 0107 0911 0077 0936020 0335 0868 0274 0835 0188 0909 0136 0934 0098 0942050 0410 0885 0298 0916 0193 0934 0137 0942 0098 0947090 0213 0906 0149 0937 0094 0953 0066 0950 0046 0950

in a region of interest [12] is providedThe goal is to evaluatetraffic intensity which for managerial reasons should notexceed 87 if it does users may leave The data comprise200 random observations of the number of customers in thesystem at random times sufficiently spaced and previouslydefined by the person responsible for collecting the data to

avoid correlationThe observed values (V) and frequency (F)are presented inTable 7 For instance of the 200 observationsat 8 times no customers were found in the system at 21 timesonly 1 customer was found and so on

The estimates are shown in Table 8 they were calculatedusing the MLE (Listing 1) and the Bayesian estimators

12 Mathematical Problems in Engineering

MLEBootstrap

100 200 400 800 160050L

minus0150

minus0100

minus0050

0000

0050

0100

0150

0200

Error

(a) Average error for 119871

MLEBootstrap

00000

00050

00100

00150

00200

00250

00300

00350

00400

MSE

100 200 400 800 160050L

(b) Average MSE for 119871

MLEBootstrap

minus0080minus0070minus0060minus0050minus0040minus0030minus0020minus001000000010

Error

20 50 10010n

(c) Average error for 119899

MLEBootstrap

0000000050001000015000200002500030000350004000045000500

MSE

20 50 10010n

(d) Average MSE for 119899

Figure 6 Performance of estimates for 119871Table 7 Observed values (V) and frequency (F) for a sample of size119899 = 200 with 119870 = 14 [12]V F0 81 212 273 394 295 286 167 158 69 510 211 114 3

(Listings 2 and 3) based on an a prioriBeta(10 10) and 5000samples and Jeffreysrsquo distribution and 1000 samples from

Table 8 Point estimates for the numerical example

Estimate MLE Bayes Jeffreys Bootstrap Corrected120588 08405396 08403625 08403625 08405683

the a posteriori distribution the bootstrap corrected MLE(Listing 4) was based on 1000 resamplings The completescript is shown in Listing 7

According to the results presented in the previous sectionthe Bayesian estimates should be the most reliable It isnotable that the system utilization seemed to be below thetarget (120588 = 08405396 lt 087) Note that the analysis is basedonly on counts of the number of users in the system It is notnecessary to estimate the arrival and service rates separatelyto determine 1205884 Conclusions and Final Observations

The problem of traffic intensity estimation in finite Markovqueues (1198721198721119870 queues) is presented as quite challengingIn fact no estimator is absolutely superior to another in all

Mathematical Problems in Engineering 13

MLEBootstrap

minus0150

minus0100

minus0050

0000

0050

0100

0150

Error

100 200 400 800 160050푳풒

(a) Average error for 119871119902

MLEBootstrap

00000

00050

00100

00150

00200

00250

MSE

100 200 400 800 160050푳풒

(b) Average MSE for 119871119902

MLEBootstrap

minus0120

minus0100

minus0080

minus0060

minus0040

minus0020

0000

0020

Error

20 50 10010n

(c) Average error for 119899

MLEBootstrap

00000

00050

00100

00150

00200

00250

00300

00350

00400

MSE

20 50 10010n

(d) Average MSE for 119899

Figure 7 Performance of estimates for 119871119902

read samplesampltminusc(rep(0 8) rep(1 21) rep(2 27) rep(3 39) rep(4 29) rep(5 28) rep(6 16)rep(7 15) rep(8 6) rep(9 5) rep(10 2) rep(11 1) rep(14 3)) MLE estimateKltminus14hatrhoMLEltminusMLERoMM1K(K samp) Bayesian estimatealtminus10bltminus10setseed(13579)hatrhoBayesltminusEBaRoMM1K(K samp a b) Bayesian Jeffreys estimatesetseed(13579)hatrhoJeltminusEJeRoMM1K(K samp) Bootstrap corrected estimatesetseed(13579)hatrhoBootltminusEBoRoMM1K(K samp)c(hatrhoMLE hatrhoBayes hatrhoJe hatrhoBoot)gt [1] 08405396 08403625 08403625 08405683

Listing 7 Estimations from real data

14 Mathematical Problems in Engineering

parametric space Although the estimates of the MLE andthe bootstrap corrected MLE exhibit less bias the Bayesianestimates (beta and Jeffreysrsquo prior) present the lowest MSEin general Perhaps due to the skewness of the a posteriordistribution the Bayesian estimators do not present low biasIn general for sample size 119899 = 50 and queues with 119870 ge 20the average estimation error is less than 0005 this value wasonly exceeded by the Bayesian estimator for queues with totalcapacity 119870 = 5

In regard to the behavior of the average estimation errorand the average MSE as functions of the traffic intensity 120588the major errors are observed when the sample size is small(119899 le 20) and the traffic intensities are 120588 asymp 10 unlike inthe case of the 1198721198721 queues which exhibit higher biaseswhen 120588 asymp 05 Perhaps due to the truncation of the numberof users to the maximum queue length119870 systems with hightraffic intensities require more computational effort and arethe most difficult to estimate

Finally it is important to note that for queues withcapacity 119870 = 80 the systemrsquos behavior is similar to that of aninfinite Markovian queue (an 1198721198721 queue) as expectedThat is the average estimation error is greater and the MSEis highest when 120588 asymp 05

Future work in this area includes testing other Bayesianpoint estimators (eg the median because of the asym-metry of the a posterior distribution) developing intervalestimators hypothesis testingmethods or even Kernel-basedmethods [24]

Data Availability

The data used to support the findings of this study areincluded within the article

Disclosure

The Brazilian government funding agencies mentioned hadno role in the study

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

Special thanks are due to Gabriel and Carolina for helpingwith the algebra This work was supported by the Brazil-ian agencies CNPq (Conselho Nacional de DesenvolvimentoCientıfico e Tecnologico of the Ministry for Science andTechnology) [Grant nos 3046712014-2 3058412016-5] andFAPEMIG (Fundacao de Amparo a Pesquisa do Estado deMinas Gerais) [Grant nos CEX-PPM-00564-17 APQ-02119-15 and BIP-00106-16]

References

[1] D Ma D Wang Y Bie F Sun and S Jin ldquoA method for queuelength estimation in an urban street network based on roll time

occupancy datardquo Mathematical Problems in Engineering vol2012 p 12 2012 httpdxdoiorg1011552012892575

[2] S Zhao S Liang H Liu and M Ma ldquoCTM based real-timequeue length estimation at signalized intersectionrdquo Mathemat-ical Problems in Engineering vol 2015 Article ID 328712 12pages 2015

[3] C Armero and M J Bayarri ldquoBayesian prediction in 1198721198721queuesrdquo Queueing Systems vol 15 no 1-4 pp 401ndash417 1994

[4] C Armero and M J Bayarri ldquoQueuesrdquo in Proceedings of theInternational Encyclopedia of the Social Behavioral Sciences J DWright Ed pp 784ndash789 Oxford UK 2015

[5] C Armero and D Conesa ldquoInference and prediction in bulkarrival queues and queues with service in stagesrdquo AppliedStochastic Models in Business and Industry vol 14 no 1 pp 35ndash46 1998

[6] C Armero and D Conesa ldquoPrediction in Markovian bulkarrival queuesrdquo Queueing Systems vol 34 no 1-4 pp 327ndash3502000

[7] C Armero and D Conesa ldquoStatistical performance of a mul-ticlass bulk production queueing systemrdquo European Journal ofOperational Research vol 158 no 3 pp 649ndash661 2004

[8] C Armero and D Conesa ldquoBayesian hierarchical models inmanufacturing bulk service queuesrdquo Journal of Statistical Plan-ning and Inference vol 136 no 2 pp 335ndash354 2006

[9] A Choudhury and A C Borthakur ldquoBayesian inference andprediction in the single server Markovian queuerdquo MetrikaInternational Journal for Theoretical and Applied Statistics vol67 no 3 pp 371ndash383 2008

[10] S Chowdhury and S P Mukherjee ldquoEstimation of trafficintensity based on queue length in a single MM1 queuerdquoCommunications in StatisticsmdashTheory and Methods vol 42 no13 pp 2376ndash2390 2013

[11] S Chowdhury and S P Mukherjee ldquoBayes estimation inMM1 queues with bivariate priorrdquo Journal of Statistics andManagement Systems vol 19 no 5 pp 681ndash699 2016

[12] F R Cruz R C Quinino and L L Ho ldquoBayesian estimationof traffic intensity based on queue length in a multi-serverMMS queuerdquo Communications in StatisticsmdashSimulation andComputation vol 46 no 9 pp 7319ndash7331 2017

[13] R C Quinino and F R B Cruz ldquoBayesian sample sizesin an MM1 queueing systemsrdquo The International Journal ofAdvanced Manufacturing Technology vol 88 no 1-4 pp 995ndash1002 2017

[14] D Gross J F Shortle and J M Thompson Fundamentals ofQueueing Theory Wiley-Interscience New York NY USA 4thedition 2009

[15] M A C Almeida and F R B Cruz ldquoA note on Bayesianestimation of traffic intensity in single-server MarkovianqueuesrdquoCommunications in Statistics -SimulationComputationhttpdxdoiorg1010800361091820171353614

[16] R LThomasson and C H Kapadia ldquoOn estimating the param-eter of a truncated geometric distributionrdquo Annals of theInstitute of Statistical Mathematics vol 20 pp 519ndash523 1968

[17] R Core Team R A Language and Environment for StatisticalComputing R Foundation for Statistical Computing ViennaAustria 2017 httpwwwR-projectorg

[18] N Mukhopadhyay Probability and Statistical Inference MarcelDekker New York NY USA 2000

[19] G S Lingappaiah ldquoBayes inference in right truncated geo-metric distributionrdquo Malaysian Mathematical Society BulletinSecond Series vol 15 no 2 pp 61ndash67 1992

4 Mathematical Problems in Engineering

require(HI)EJeRoMM1Kltminusfunction (K samp)

logJefltminusfunction (rho K n sumxi) logpostltminus(sumxi)lowastlog (rho)+nlowastlog(1minusrho)minusnlowastlog(1minusrhoand(K+1))Irholtminus (1rhoand2)lowast(rho(1minusrho)minus(K+1)lowastrhoand(K+1)(1minusrhoand(K+1))) minus

1(1minusrho)and2 minus (K+1)lowast(Klowastrhoand(minusKminus1)+1)(rhoand(minusK)minusrho)and2if (( is nan(Irho)) ampamp (Irhogt0))

logpostlt minuslogpost+05lowastlog(Irho) return(logpost)

sSizeltminus1000nltminuslength(samp)sumxiltminussum(samp)resltminusarms(runif(1) logJef function(x K n sumxi)((x gt 0)lowast(x lt 1)) sSize K n sumxi)return(mean(res))

Listing 3 Bayes estimator for 120588 for Jeffreysrsquo prior

1205972 log119901 (119883 | 120588)1205971205882 = minus1198831205882 + 1

(1 minus 120588)2

+ (119870 + 1) (119870120588minus119870minus1 + 1)(120588minus119870 minus 120588)2

(10)

It follows that

119868 (120588) = 119864(minus1205972 log119901 (119883 | 120588)1205971205882 )

= 119864[1198831205882 minus 1(1 minus 120588)2 minus

(119870 + 1) (119870120588minus119870minus1 + 1)(120588minus119870 minus 120588)2 ]

= 119864 (119883)1205882 minus 1(1 minus 120588)2 minus

(119870 + 1) (119870120588minus119870minus1 + 1)(120588minus119870 minus 120588)2

(11)

The expectation 119864[119883] is given by [14]

119864 [119883] = 1205881 minus 120588 minus (119870 + 1) 120588119870+11 minus 120588119870+1 (12)

Then

119901 (120588) prop [119868 (120588)]12 = [ 11205882 ( 1205881 minus 120588 minus (119870 + 1) 120588119870+11 minus 120588119870+1 )

minus 1(1 minus 120588)2 minus

(119870 + 1) (119870120588minus119870minus1 + 1)(120588minus119870 minus 120588)2 ]

12

(13)

Thus combining the likelihood (2) and Jeffreys priorgiven by (13) it is possible to find the following posteriorprobability distribution for 120588

1199012 (120588 | x) prop 119871 (120588 x) times 119901 (120588) prop 120588119910 times (1 minus 120588)119899(1 minus 120588119870+1)119899

times [ 11205882 ( 1205881 minus 120588 minus (119870 + 1) 120588119870+11 minus 120588119870+1 ) minus 1(1 minus 120588)2

minus (119870 + 1) (119870120588minus119870minus1 + 1)(120588minus119870 minus 120588)2 ]

12

(14)

in which 0 lt 120588 lt 1Note that unlike the case shown earlier (5) the posterior

distribution given by (14) is fixed Indeed it is not possibleto vary Jeffreysrsquo prior which assumes a specific form Theimplementation of the Bayesian estimator for Jeffreysrsquo prioris shown in Listing 3 Because the logarithm of the posteriordistribution is used variable Irho as defined in Listing 3must be such that 0 lt Irho lt infin

23 Bootstrap Correction Among the bias correction meth-ods that are commonly used for estimators the bootstrapmethod is widely used [21] In its nonparametric versionthe method consists of making several (usually 119861 = 1 000)resamplings xlowast (with replacement)The parameter of interestis reestimated by some (usually biased) method Θlowast Thenthe average is calculated Θlowast and the bias is estimated using

bias = Θlowast minus Θ (15)

where Θ is the estimate obtained from the original sampleThen the following corrected version of the estimator isobtained

Θ119861 = 2Θ minus Θlowast (16)

The procedure is illustrated in Figure 2 The R code [17]for the bootstrap corrected estimator is shown in Listing 4The parameter 119870 is the maximum number of users simul-taneously (in service and waiting) in the 1198721198721119870 queueNote that 1000 bootstrap replicates are used and that thecorrection occurs as part of the MLE (see Listing 1)

Mathematical Problems in Engineering 5

x x1 x2 middot middot middot middot middot middotxn xlowast(1)

xlowast(2)

xlowast1(1) xlowast2(1) xlowastn(1)

middot middot middotxlowast1(2) xlowast2(2) xlowastn(2)

xlowast(B) middot middot middotxlowast1(B) xlowast2(B) xlowastn(B)

Θlowast(1)Θ

Θlowast(2)

Θlowast(B)

Θlowast=

sumBi=1 Θ

lowast(i)

B

Figure 2 The bootstrap method

EBoRoMM1Kltminusfunction (K samp) Bltminus1000summltminus0for (i in 1B) resampltminussample (samp replace=T)estmltminusMLERoMM1K (K resamp)summltminussumm+estm

estmStar=summBreturn (2lowastMLERoMM1K(K samp)minusestmStar) Listing 4 Bootstrap corrected estimator

Besides bias correction the bootstrap method has beenused by many researchers in the past with good results inconfidence interval building and hypothesis testing [22] Asan example an empirical bootstrap confidence interval isused in this work as a simple way of interval estimation for 120588If the distribution of 120575 = Θ minus Θ was known then the criticalvalues 1205751205722 and 1205751minus1205722 could be found where 120575120574 is its 120574100thpercentile and then

Pr (1205751205722 le Θ minus Θ le 1205751minus1205722 | Θ) = (1 minus 120572) lArrrArrPr (Θ minus 1205751205722 ge Θ ge Θ minus 1205751minus1205722 | Θ) = (1 minus 120572) (17)

which gives an (1 minus 120572)100 confidence interval of

CIΘ(1minus120572)100 = [Θ minus 1205751minus1205722 Θ minus 1205751205722] (18)

The bootstrap makes it possible to estimate the distribu-tion of 120575 by the distribution of 120575lowast = Θlowast minus Θ where Θlowast isthe estimate obtained from an empirical bootstrap sample asexplained earlier

24 Simulation MM1K of Queues The number of userspresent in an1198721198721119870 queue follows the distribution givenby (1) To efficiently generate randomvariables fromadiscrete

distribution several methods are widely used in the literatureincluding function sample from R [17] The method usedhere is the discrete analog of the inverse transformationmethod in which it is necessary to generate numbers 119877 simUnif(0 1) ie from a uniform distribution between 0 and 1and to know the probabilities of interest 119875119883 = 119895 = 119901119895 forall119895Therefore to simulate a discrete random variable 119883 with theprobability function

119875 119883 = 119895 = 119901119895 119895 = 0 1 sumforall119895

119901119895 = 1 (19)

it is necessary to compute

119883 =

0 if 119877 le 11990101 if 1199010 lt 119877 le 1199010 + 1199011119895 if

119895minus1sum119894=0

119901119894 lt 119877 le 119895sum119894=0

119901119894

(20)

because

119875 119883 = 119895 = 119875119895minus1sum119894=0

119901119894 lt 119877 le 119895sum119894=0

119901119894 = 119901119895 (21)

and119883 follows the required probability distributionFor 1198721198721119870 queues and from (1) the following must

hold119899sum119894=0

119901119894 = 1 minus 120588119899+11 minus 120588119870+1 119899 = 0 1 119870 (22)

Setting 119888 = 1 minus 120588119870+1 it follows that119883minus1sum119894=0

119901119894 lt 119877 le 119883sum119894=0

119901119894 997904rArr

6 Mathematical Problems in Engineering

Table 1 Mean estimates for 120588 and the mean squared error (MSE in parentheses) for 119870 = 5Estimate 120588 119899

10 20 50 100 200

MLE

001 000928 (000084) 000960 (000045) 001000 (000019) 001003 (000010) 001007 (000005)010 009172 (000715) 009614 (000384) 009845 (000160) 009907 (000081) 009964 (000041)020 018449 (001214) 019235 (000644) 019700 (000258) 019833 (000130) 019921 (000065)050 048492 (001946) 049422 (000932) 049824 (000358) 049884 (000180) 049952 (000088)090 087146 (001703) 088854 (000953) 089839 (000467) 090021 (000271) 090044 (000142)099 092503 (001488) 094638 (000734) 096396 (000303) 097225 (000159) 097854 (000083)

Beta

001 009043 (000699) 005378 (000226) 002864 (000052) 001953 (000018) 001487 (000007)010 015663 (000793) 013021 (000396) 011251 (000160) 010617 (000080) 010321 (000041)020 023444 (000989) 021739 (000567) 020690 (000244) 020327 (000126) 020167 (000064)050 050855 (001688) 050669 (000913) 050295 (000357) 050116 (000180) 050067 (000088)090 081770 (001385) 085318 (000654) 088214 (000283) 089387 (000178) 089954 (000114)099 085425 (002303) 089298 (001185) 092818 (000491) 094668 (000248) 096038 (000121)

Jeffreys

001 005000 (000215) 003182 (000084) 001931 (000026) 001476 (000012) 001244 (000005)010 011630 (000497) 010928 (000314) 010422 (000149) 010202 (000078) 010113 (000040)020 019457 (000908) 019512 (000520) 019804 (000229) 019907 (000123) 019963 (000063)050 049320 (002081) 050148 (001058) 050337 (000377) 050230 (000175) 050121 (000084)090 081749 (001400) 085312 (000655) 088216 (000283) 089387 (000178) 089954 (000115)099 085415 (002308) 089294 (001186) 092819 (000491) 094668 (000249) 096037 (000121)

Bootstrap

001 000992 (000096) 001000 (000049) 001018 (000020) 001013 (000010) 001012 (000005)010 009746 (000791) 009970 (000409) 010009 (000164) 009993 (000082) 010007 (000041)020 019446 (001290) 019837 (000665) 019976 (000262) 019977 (000131) 019994 (000065)050 049647 (001868) 050011 (000901) 050065 (000353) 050007 (000178) 050012 (000088)090 089517 (001990) 090321 (001177) 090473 (000574) 090256 (000314) 090074 (000152)099 095119 (001453) 096522 (000742) 097635 (000317) 098100 (000170) 098475 (000090)

rmm1klt minusfunction (ssizerhoK) simulateRltminusrunif(ssize01)cltminus1minusrhoand(K+1)logrholtminuslog(rho)xltminuslog(1minusRlowastc)logrhoreturn(x)gtgt set seed (13579)gt sampltminusrmm1k (ssize=10 rho=020 K=5)gt samp[1] 0 1 0 1 0 0 0 2 1 0

Listing 5 Sample generation for1198721198721119870 queues

1 minus 120588119883119888 lt 119877 le 1 minus 120588119883+1119888 997904rArrlog (1 minus 119877119888)log (120588) minus 1 le 119883 lt log (1 minus 119877119888)

log (120588) (23)

Therefore

119883 = lceil log (1 minus 119877119888)log (120588) minus 1rceil (24)

MtCaRoMM1Kltminusfunction(ssize rho K fEst) repltminus10000sampltminusnumeric(ssize)estltminusnumeric(rep)for (i in 1rep) sampltminusrmm1k(ssize rho K)oldseedltminusGlobalEnv$Randomseedest [i]ltminusfEst (K samp )GlobalEnv$Randomseedltminusoldseed

return(c(mean(est) var(est)))Listing 6 Monte Carlo simulation

where lceil119910rceil is the ceiling function that is its value is the leastinteger that is not inferior to 119910

Listing 5 presents the R implementation [17] used in thisstudy with a sample call

3 Computational Results

To analyze the performance of the estimators 10000 MonteCarlo replications were made using the R code [17] shownin Listing 6 Note that fESt(K samp ) can be any of the

Mathematical Problems in Engineering 7

Table 2 Mean estimates for 120588 and the mean squared error (MSE in parentheses) for 119870 = 20Estimate 120588 119899

10 20 50 100 200

MLE

001 000928 (000084) 000960 (000045) 001000 (000019) 001003 (000010) 001007 (000005)010 009158 (000710) 009609 (000383) 009844 (000159) 009906 (000081) 009964 (000041)020 018345 (001178) 019186 (000631) 019684 (000255) 019827 (000128) 019919 (000064)050 047371 (001424) 048785 (000666) 049543 (000256) 049750 (000127) 049885 (000062)090 089316 (000326) 089758 (000151) 089922 (000057) 089956 (000029) 089979 (000014)099 097283 (000156) 097979 (000073) 098507 (000030) 098728 (000017) 098877 (000009)

Beta

001 009009 (000692) 005374 (000226) 002864 (000052) 001953 (000018) 001487 (000007)010 015477 (000743) 012978 (000388) 011242 (000159) 010616 (000080) 010320 (000041)020 022893 (000850) 021530 (000529) 020632 (000237) 020303 (000124) 020157 (000063)050 047818 (001132) 048902 (000599) 049557 (000245) 049753 (000125) 049885 (000062)090 088374 (000296) 089484 (000142) 089880 (000058) 089938 (000029) 089970 (000014)099 094835 (000246) 096296 (000107) 097492 (000037) 098068 (000017) 098467 (000008)

Jeffreys

001 005049 (000219) 003182 (000084) 001931 (000026) 001476 (000012) 001244 (000005)010 011678 (000489) 010981 (000320) 010425 (000149) 010203 (000078) 010113 (000040)020 019205 (000801) 019570 (000503) 019866 (000233) 019927 (000123) 019970 (000063)050 046191 (001540) 048175 (000769) 049434 (000302) 049828 (000152) 050063 (000074)090 088365 (000296) 089481 (000142) 089880 (000058) 089938 (000029) 089970 (000014)099 094829 (000247) 096295 (000107) 097493 (000037) 098068 (000017) 098467 (000008)

Bootstrap

001 000993 (000096) 001000 (000049) 001018 (000020) 001013 (000010) 001012 (000005)010 009763 (000795) 009976 (000409) 010010 (000164) 009994 (000082) 010007 (000041)020 019462 (001281) 019846 (000661) 019982 (000260) 019982 (000130) 019997 (000065)050 049385 (001404) 049896 (000659) 050019 (000254) 049995 (000127) 050008 (000062)090 090117 (000310) 090065 (000147) 090021 (000056) 090004 (000028) 090002 (000014)099 098215 (000147) 098590 (000076) 098872 (000035) 098963 (000020) 099023 (000012)

implemented estimation functions for 120588 (MLE Bayesian andbootstrap corrected MLE) and that the state of the randomseed GlobalEnv$Randomseed was stored immediatelybefore the function fESt(K samp ) which can be stochas-tic and was reloaded immediately after its call to ensure thatthe samples generated for the estimates were the same for allestimators

31 Simulation Results Samples were generated from (24)and Listing 5 for sizes 119899 isin 10 20 50 100 200 and trafficintensities 120588 isin 001 010 020 050 090 099 In thesescenarios averages of 10000 Monte Carlo replications werecomputed (via code from Listing 6) and point estimates of 120588were computed by

(i) the MLE via numerical maximization of the likeli-hood function (2) and code from Listing 1

(ii) the Bayesian method via an a priori Beta(10 10)distribution and an average of 5000 samples from thea posteriori distribution (5) obtained by the functionarms [23] from Rrsquos HI package [20] via code fromListing 2

(iii) the Bayesian method via Jeffreys prior distributionand an average of 1000 samples from the a posterioridistribution (14) obtained by the function arms [23]from Rrsquos HI package [20] via code from Listing 3

(iv) the bootstrap corrected MLE (16) and code fromListing 4 Additionally mean squared errors (MSEs)were calculated

The results are shown inTables 1 2 and 3 and summarizedin Figures 3 4 and 5 In Figures 3 4 and 5 the averageestimation error and the mean squared error (MSE) definedas MSE(120588) = Var120588(120588) + Bias2(120588 120588) are shown as functions ofboth the traffic intensity 120588 (averaged over all sample sizes)and the sample size 119899 (averaged over all traffic intensities)

For queues with capacity 119870 = 5 (Figure 3) an approx-imately constant average estimation error is observed forthe MLE and the bootstrap corrected MLE except when120588 asymp 10 (Figure 3(a)) The Bayesian estimators (beta priorand Jeffreysrsquo prior) did not show equivalent performancetheir estimates tended to overestimate the true value (positiveerror) when 120588 lt 05 and to underestimate otherwiseRegarding the sample size 119899 all of the estimators showeda monotonic decrease in error (Figure 3(c)) From the MSEside bothBayesian estimators presented themselves generallyas the best alternative because they presented the lowestvalues most of the time Although the bootstrap correctedMLE presented the lowest bias the method achieves thisperformance at the cost of high variability as reflected by itshighest MSEs

When the queue capacity was increased slightly to119870 = 20(Figure 4) a very similar behavior was noted However the

8 Mathematical Problems in Engineering

Table 3 Mean estimates for 120588 and the mean squared error (MSE in parentheses) for 119870 = 80Estimate 120588 119899

10 20 50 100 200

MLE

001 000928 (000084) 000960 (000045) 001000 (000019) 001003 (000010) 001007 (000005)010 009158 (000710) 009609 (000383) 009844 (000159) 009906 (000081) 009964 (000041)020 018345 (001178) 019186 (000631) 019684 (000255) 019827 (000128) 019919 (000064)050 047366 (001422) 048783 (000666) 049543 (000255) 049750 (000127) 049885 (000062)090 089001 (000144) 089551 (000057) 089835 (000020) 089915 (000010) 089960 (000005)099 098737 (000016) 098903 (000008) 098982 (000003) 098995 (000002) 098998 (000001)

Beta

001 009009 (000692) 005374 (000226) 002864 (000052) 001953 (000018) 001487 (000007)010 015477 (000743) 012978 (000388) 011242 (000159) 010616 (000080) 010320 (000041)020 022892 (000850) 021530 (000529) 020632 (000237) 020304 (000124) 020157 (000063)050 047779 (001125) 048892 (000597) 049556 (000245) 049753 (000125) 049885 (000062)090 088221 (000180) 089167 (000065) 089680 (000021) 089837 (000010) 089921 (000005)099 098149 (000017) 098570 (000007) 098856 (000003) 098953 (000002) 098990 (000001)

Jeffreys

001 005225 (000231) 003334 (000089) 001998 (000027) 001506 (000011) 001253 (000005)010 011744 (000482) 011002 (000316) 010425 (000149) 010203 (000078) 010113 (000040)020 019224 (000794) 019572 (000502) 019866 (000233) 019927 (000123) 019970 (000063)050 046147 (001532) 048167 (000767) 049430 (000302) 049826 (000151) 050064 (000074)090 088217 (000180) 089165 (000065) 089680 (000021) 089837 (000010) 089921 (000005)099 098148 (000017) 098570 (000007) 098856 (000003) 098953 (000002) 098990 (000001)

Bootstrap

001 000993 (000096) 001000 (000049) 001018 (000020) 001013 (000010) 001012 (000005)010 009763 (000795) 009976 (000409) 010010 (000164) 009994 (000082) 010007 (000041)020 019462 (001281) 019846 (000661) 019982 (000260) 019982 (000130) 019997 (000065)050 049390 (001405) 049897 (000659) 050019 (000254) 049995 (000127) 050008 (000062)090 089910 (000118) 089987 (000051) 090008 (000019) 090002 (000009) 090003 (000005)099 099000 (000016) 099042 (000009) 099031 (000004) 099012 (000002) 099002 (000001)

difficulty of estimation for traffic intensities120588 asymp 10 seemed todecrease and the highest MSEs occurred when 120588 asymp 05 TheBayesian estimators maintained the performance presentedearlier for 119870 = 5 that is the estimates have positive biasfor 120588 lt 05 and negative bias otherwise The errors of all ofthe estimators converged to zero as the sample size grewThebootstrap corrected MLE presented an average estimationerror near zero for samples 119899 ge 50 However from the pointof view of the MSE the smaller values were again obtainedusing the Bayesian methods (beta and Jeffreysrsquo prior)

Finally for queues with 119870 = 80 (Figure 5) the observedbehavior could be considered for practical purposes as beingequal to that of an infinite Markovian queue in terms of theaverage estimation error and the MSE This behavior wasobserved for infinite Markovian queues (1198721198721 queues)[15] This finding is merely evidence of the correctness of ourimplementations and the quality of the computational resultspresented

Additionally computational experiments were per-formed for the estimators for 119871 (3) and for 119871119902 (4) and fortheir bootstrap corrected versions with 1000 resamplingsand 119870 = 20 Table 4 and Figure 6 show the resultsobtained for 119871 isin 05 1 2 4 8 16 for sample sizes119899 isin 10 20 50 100 and averages of 10000 Monte Carloreplications Similarly for 119871119902 the results are presented inTable 5 and Figure 7 In summary at an extra cost of thebootstrap method and without inflation of the MSEs the

researcher may achieve with samples of size 119899 = 10 estimatesfor 119871 and 119871119902 with the same average error of the MLE forsamples of size 119899 = 100 a reduction that is relevant inpractical terms because it may lead to reduction in time andcost to obtain the estimates Note that the bootstrap methodalways provides smaller errors and MSEs than the MLEmethod for all estimates of 119871 and 119871119902 even when 120588 gt 1 Alsonote the jump up and down in the errors and MSEs when 120588transitions from 120588 lt 1 to 120588 gt 1

Finally to illustrate the ease of use of the bootstrap in theinterval estimation of the traffic intensity 120588 computationalexperiments were performed The length and coverage ofempirical bootstrap intervals computed from (18) and froma normal distribution approximation (ie Θ plusmn 1199111205722120590Θ where119911120574 is the 120574100th percentile of the standard normal distri-bution and the standard deviation 120590Θ was estimated also bybootstrapping) were evaluated for 120588 isin 010 020 050 090for sample sizes 119899 isin 10 20 50 100 200 averages of 10000Monte Carlo replications and 119870 = 20 The satisfactoryperformance of the bootstrapwas demonstrated as presentedin Table 6 with the coverages approaching the nominalconfidence of 95 (that is 1 minus 120572 = 095) as the sample sizesincrease

32 Numerical Example To better illustrate an application ofthe method a numerical application based on the data con-sidered in Table 7 collected in a large supermarket network

Mathematical Problems in Engineering 9

MLEBeta

JeffreysBootstrap

minus0080

minus0060

minus0040

minus0020

0000

0020

0040

Error

010 020 050 090 099001흆

(a) Average error for 120588

MLEBeta

JeffreysBootstrap

0000000010000200003000040000500006000070000800009000100

MSE

010 020 050 090 099001흆

(b) Average MSE for 120588

MLEBeta

JeffreysBootstrap

minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0000

Error

20 50 100 20010n

(c) Average error for 119899

MLEBeta

JeffreysBootstrap

00000

00020

00040

00060

00080

00100

00120

00140

MSE

20 50 100 20010n

(d) Average MSE for 119899

Figure 3 Performance of estimators for 120588 and119870 = 5

Table 4 Mean estimates for 119871 and the mean squared error (MSE in parentheses) for 119870 = 20Estimator 119871 120588 119899

10 20 50 100

MLE

050 033333 045010 (001698) 047586 (000806) 049098 (000303) 049523 (000150)100 050000 090008 (002353) 095255 (000877) 098190 (000286) 099007 (000137)200 066714 180441 (004755) 190696 (001287) 196438 (000283) 198120 (000112)400 080959 372455 (008082) 387956 (001669) 395673 (000268) 397720 (000092)800 094574 788487 (001625) 797002 (000230) 799390 (000057) 799583 (000029)160 123520 162965 (009872) 161779 (003702) 160743 (000747) 160377 (000238)

Bootstrap

050 033333 048573 (001586) 049686 (000777) 050033 (000299) 050009 (000149)100 050000 097570 (001460) 099583 (000661) 100076 (000254) 099982 (000127)200 066714 196026 (001062) 199245 (000416) 200066 (000154) 199975 (000076)400 080959 397317 (000515) 400156 (000204) 400475 (000081) 400113 (000039)800 094574 796978 (000350) 800755 (000138) 800792 (000059) 800277 (000027)160 123520 161315 (003011) 160376 (000699) 160043 (000188) 160037 (000095)

10 Mathematical Problems in Engineering

MLEBeta

JeffreysBootstrap

010 020 050 090 099001흆

minus0030

minus0020

minus0010

0000

0010

0020

0030

0040

Error

(a) Average error for 120588

MLEBeta

JeffreysBootstrap

00000

00010

00020

00030

00040

00050

00060

MSE

010 020 050 090 099001흆

(b) Average MSE for 120588

MLEBeta

JeffreysBootstrap

minus0015

minus0010

minus0005

0000

0005

0010

0015

0020

Error

20 50 100 20010n

(c) Average error for 119899

MLEBeta

JeffreysBootstrap

00000

00010

00020

00030

00040

00050

00060

00070

00080

MSE

20 50 100 20010n

(d) Average MSE for 119899

Figure 4 Performance of estimators for 120588 and119870 = 20

Table 5 Mean estimates for 119871119902 and the mean squared error (MSE in parentheses) for 119870 = 20Estimator 119871119902 120588 119899

10 20 50 100

MLE

050 050000 042637 (001897) 046470 (000776) 048647 (000272) 049256 (000132)100 061818 086547 (002882) 093515 (000918) 097534 (000248) 098726 (000108)200 073449 177618 (005718) 189501 (001417) 196115 (000266) 197942 (000099)400 084574 374340 (006999) 389252 (001340) 396186 (000214) 397930 (000077)800 097126 794718 (000577) 799972 (000141) 800478 (000056) 800250 (000028)160 132722 161536 (003518) 161286 (002392) 160683 (000800) 160254 (000210)

Bootstrap

050 050000 048185 (001434) 049688 (000660) 050057 (000254) 049986 (000127)100 061818 097175 (001146) 099492 (000493) 100110 (000185) 100049 (000092)200 073449 196521 (000787) 199479 (000303) 200236 (000113) 200027 (000056)400 084574 397906 (000409) 400426 (000174) 400494 (000069) 400067 (000033)800 097126 796303 (000394) 800653 (000136) 800732 (000058) 800362 (000028)160 132722 161202 (003015) 160862 (001687) 160226 (000418) 160228 (000229)

Mathematical Problems in Engineering 11

MLEBeta

JeffreysBootstrap

minus0015minus0010minus000500000005001000150020002500300035

Error

010 020 050 090 099001흆

(a) Average error for 120588

MLEBeta

JeffreysBootstrap

00000

00010

00020

00030

00040

00050

00060

MSE

010 020 050 090 099001흆

(b) Average MSE for 120588

MLEBeta

JeffreysBootstrap

minus0015

minus0010

minus0005

0000

0005

0010

0015

0020

0025

Error

20 50 100 20010n

(c) Average error for 119899

MLEBeta

JeffreysBootstrap

00000

00010

00020

00030

00040

00050

00060

00070

MSE

20 50 100 20010n

(d) Average MSE for 119899

Figure 5 Performance of estimators for 120588 and 119870 = 80

Table 6 Average length (L) and coverage (C) of 95 confidence intervals for 120588 and119870 = 20Method 120588 119899 = 10 119899 = 20 119899 = 50 119899 = 100 119899 = 200

L C L C L C L C L C

Empirical Bootstrap

010 0189 0626 0185 0612 0143 0885 0106 0906 0077 0933020 0308 0619 0265 0797 0186 0902 0135 0923 0097 0940050 0402 0836 0294 0892 0191 0924 0137 0938 0097 0943090 0207 0835 0148 0911 0094 0958 0066 0953 0046 0952

Normal Approximation

010 0219 0636 0200 0865 0146 0890 0107 0911 0077 0936020 0335 0868 0274 0835 0188 0909 0136 0934 0098 0942050 0410 0885 0298 0916 0193 0934 0137 0942 0098 0947090 0213 0906 0149 0937 0094 0953 0066 0950 0046 0950

in a region of interest [12] is providedThe goal is to evaluatetraffic intensity which for managerial reasons should notexceed 87 if it does users may leave The data comprise200 random observations of the number of customers in thesystem at random times sufficiently spaced and previouslydefined by the person responsible for collecting the data to

avoid correlationThe observed values (V) and frequency (F)are presented inTable 7 For instance of the 200 observationsat 8 times no customers were found in the system at 21 timesonly 1 customer was found and so on

The estimates are shown in Table 8 they were calculatedusing the MLE (Listing 1) and the Bayesian estimators

12 Mathematical Problems in Engineering

MLEBootstrap

100 200 400 800 160050L

minus0150

minus0100

minus0050

0000

0050

0100

0150

0200

Error

(a) Average error for 119871

MLEBootstrap

00000

00050

00100

00150

00200

00250

00300

00350

00400

MSE

100 200 400 800 160050L

(b) Average MSE for 119871

MLEBootstrap

minus0080minus0070minus0060minus0050minus0040minus0030minus0020minus001000000010

Error

20 50 10010n

(c) Average error for 119899

MLEBootstrap

0000000050001000015000200002500030000350004000045000500

MSE

20 50 10010n

(d) Average MSE for 119899

Figure 6 Performance of estimates for 119871Table 7 Observed values (V) and frequency (F) for a sample of size119899 = 200 with 119870 = 14 [12]V F0 81 212 273 394 295 286 167 158 69 510 211 114 3

(Listings 2 and 3) based on an a prioriBeta(10 10) and 5000samples and Jeffreysrsquo distribution and 1000 samples from

Table 8 Point estimates for the numerical example

Estimate MLE Bayes Jeffreys Bootstrap Corrected120588 08405396 08403625 08403625 08405683

the a posteriori distribution the bootstrap corrected MLE(Listing 4) was based on 1000 resamplings The completescript is shown in Listing 7

According to the results presented in the previous sectionthe Bayesian estimates should be the most reliable It isnotable that the system utilization seemed to be below thetarget (120588 = 08405396 lt 087) Note that the analysis is basedonly on counts of the number of users in the system It is notnecessary to estimate the arrival and service rates separatelyto determine 1205884 Conclusions and Final Observations

The problem of traffic intensity estimation in finite Markovqueues (1198721198721119870 queues) is presented as quite challengingIn fact no estimator is absolutely superior to another in all

Mathematical Problems in Engineering 13

MLEBootstrap

minus0150

minus0100

minus0050

0000

0050

0100

0150

Error

100 200 400 800 160050푳풒

(a) Average error for 119871119902

MLEBootstrap

00000

00050

00100

00150

00200

00250

MSE

100 200 400 800 160050푳풒

(b) Average MSE for 119871119902

MLEBootstrap

minus0120

minus0100

minus0080

minus0060

minus0040

minus0020

0000

0020

Error

20 50 10010n

(c) Average error for 119899

MLEBootstrap

00000

00050

00100

00150

00200

00250

00300

00350

00400

MSE

20 50 10010n

(d) Average MSE for 119899

Figure 7 Performance of estimates for 119871119902

read samplesampltminusc(rep(0 8) rep(1 21) rep(2 27) rep(3 39) rep(4 29) rep(5 28) rep(6 16)rep(7 15) rep(8 6) rep(9 5) rep(10 2) rep(11 1) rep(14 3)) MLE estimateKltminus14hatrhoMLEltminusMLERoMM1K(K samp) Bayesian estimatealtminus10bltminus10setseed(13579)hatrhoBayesltminusEBaRoMM1K(K samp a b) Bayesian Jeffreys estimatesetseed(13579)hatrhoJeltminusEJeRoMM1K(K samp) Bootstrap corrected estimatesetseed(13579)hatrhoBootltminusEBoRoMM1K(K samp)c(hatrhoMLE hatrhoBayes hatrhoJe hatrhoBoot)gt [1] 08405396 08403625 08403625 08405683

Listing 7 Estimations from real data

14 Mathematical Problems in Engineering

parametric space Although the estimates of the MLE andthe bootstrap corrected MLE exhibit less bias the Bayesianestimates (beta and Jeffreysrsquo prior) present the lowest MSEin general Perhaps due to the skewness of the a posteriordistribution the Bayesian estimators do not present low biasIn general for sample size 119899 = 50 and queues with 119870 ge 20the average estimation error is less than 0005 this value wasonly exceeded by the Bayesian estimator for queues with totalcapacity 119870 = 5

In regard to the behavior of the average estimation errorand the average MSE as functions of the traffic intensity 120588the major errors are observed when the sample size is small(119899 le 20) and the traffic intensities are 120588 asymp 10 unlike inthe case of the 1198721198721 queues which exhibit higher biaseswhen 120588 asymp 05 Perhaps due to the truncation of the numberof users to the maximum queue length119870 systems with hightraffic intensities require more computational effort and arethe most difficult to estimate

Finally it is important to note that for queues withcapacity 119870 = 80 the systemrsquos behavior is similar to that of aninfinite Markovian queue (an 1198721198721 queue) as expectedThat is the average estimation error is greater and the MSEis highest when 120588 asymp 05

Future work in this area includes testing other Bayesianpoint estimators (eg the median because of the asym-metry of the a posterior distribution) developing intervalestimators hypothesis testingmethods or even Kernel-basedmethods [24]

Data Availability

The data used to support the findings of this study areincluded within the article

Disclosure

The Brazilian government funding agencies mentioned hadno role in the study

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

Special thanks are due to Gabriel and Carolina for helpingwith the algebra This work was supported by the Brazil-ian agencies CNPq (Conselho Nacional de DesenvolvimentoCientıfico e Tecnologico of the Ministry for Science andTechnology) [Grant nos 3046712014-2 3058412016-5] andFAPEMIG (Fundacao de Amparo a Pesquisa do Estado deMinas Gerais) [Grant nos CEX-PPM-00564-17 APQ-02119-15 and BIP-00106-16]

References

[1] D Ma D Wang Y Bie F Sun and S Jin ldquoA method for queuelength estimation in an urban street network based on roll time

occupancy datardquo Mathematical Problems in Engineering vol2012 p 12 2012 httpdxdoiorg1011552012892575

[2] S Zhao S Liang H Liu and M Ma ldquoCTM based real-timequeue length estimation at signalized intersectionrdquo Mathemat-ical Problems in Engineering vol 2015 Article ID 328712 12pages 2015

[3] C Armero and M J Bayarri ldquoBayesian prediction in 1198721198721queuesrdquo Queueing Systems vol 15 no 1-4 pp 401ndash417 1994

[4] C Armero and M J Bayarri ldquoQueuesrdquo in Proceedings of theInternational Encyclopedia of the Social Behavioral Sciences J DWright Ed pp 784ndash789 Oxford UK 2015

[5] C Armero and D Conesa ldquoInference and prediction in bulkarrival queues and queues with service in stagesrdquo AppliedStochastic Models in Business and Industry vol 14 no 1 pp 35ndash46 1998

[6] C Armero and D Conesa ldquoPrediction in Markovian bulkarrival queuesrdquo Queueing Systems vol 34 no 1-4 pp 327ndash3502000

[7] C Armero and D Conesa ldquoStatistical performance of a mul-ticlass bulk production queueing systemrdquo European Journal ofOperational Research vol 158 no 3 pp 649ndash661 2004

[8] C Armero and D Conesa ldquoBayesian hierarchical models inmanufacturing bulk service queuesrdquo Journal of Statistical Plan-ning and Inference vol 136 no 2 pp 335ndash354 2006

[9] A Choudhury and A C Borthakur ldquoBayesian inference andprediction in the single server Markovian queuerdquo MetrikaInternational Journal for Theoretical and Applied Statistics vol67 no 3 pp 371ndash383 2008

[10] S Chowdhury and S P Mukherjee ldquoEstimation of trafficintensity based on queue length in a single MM1 queuerdquoCommunications in StatisticsmdashTheory and Methods vol 42 no13 pp 2376ndash2390 2013

[11] S Chowdhury and S P Mukherjee ldquoBayes estimation inMM1 queues with bivariate priorrdquo Journal of Statistics andManagement Systems vol 19 no 5 pp 681ndash699 2016

[12] F R Cruz R C Quinino and L L Ho ldquoBayesian estimationof traffic intensity based on queue length in a multi-serverMMS queuerdquo Communications in StatisticsmdashSimulation andComputation vol 46 no 9 pp 7319ndash7331 2017

[13] R C Quinino and F R B Cruz ldquoBayesian sample sizesin an MM1 queueing systemsrdquo The International Journal ofAdvanced Manufacturing Technology vol 88 no 1-4 pp 995ndash1002 2017

[14] D Gross J F Shortle and J M Thompson Fundamentals ofQueueing Theory Wiley-Interscience New York NY USA 4thedition 2009

[15] M A C Almeida and F R B Cruz ldquoA note on Bayesianestimation of traffic intensity in single-server MarkovianqueuesrdquoCommunications in Statistics -SimulationComputationhttpdxdoiorg1010800361091820171353614

[16] R LThomasson and C H Kapadia ldquoOn estimating the param-eter of a truncated geometric distributionrdquo Annals of theInstitute of Statistical Mathematics vol 20 pp 519ndash523 1968

[17] R Core Team R A Language and Environment for StatisticalComputing R Foundation for Statistical Computing ViennaAustria 2017 httpwwwR-projectorg

[18] N Mukhopadhyay Probability and Statistical Inference MarcelDekker New York NY USA 2000

[19] G S Lingappaiah ldquoBayes inference in right truncated geo-metric distributionrdquo Malaysian Mathematical Society BulletinSecond Series vol 15 no 2 pp 61ndash67 1992

Mathematical Problems in Engineering 5

x x1 x2 middot middot middot middot middot middotxn xlowast(1)

xlowast(2)

xlowast1(1) xlowast2(1) xlowastn(1)

middot middot middotxlowast1(2) xlowast2(2) xlowastn(2)

xlowast(B) middot middot middotxlowast1(B) xlowast2(B) xlowastn(B)

Θlowast(1)Θ

Θlowast(2)

Θlowast(B)

Θlowast=

sumBi=1 Θ

lowast(i)

B

Figure 2 The bootstrap method

EBoRoMM1Kltminusfunction (K samp) Bltminus1000summltminus0for (i in 1B) resampltminussample (samp replace=T)estmltminusMLERoMM1K (K resamp)summltminussumm+estm

estmStar=summBreturn (2lowastMLERoMM1K(K samp)minusestmStar) Listing 4 Bootstrap corrected estimator

Besides bias correction the bootstrap method has beenused by many researchers in the past with good results inconfidence interval building and hypothesis testing [22] Asan example an empirical bootstrap confidence interval isused in this work as a simple way of interval estimation for 120588If the distribution of 120575 = Θ minus Θ was known then the criticalvalues 1205751205722 and 1205751minus1205722 could be found where 120575120574 is its 120574100thpercentile and then

Pr (1205751205722 le Θ minus Θ le 1205751minus1205722 | Θ) = (1 minus 120572) lArrrArrPr (Θ minus 1205751205722 ge Θ ge Θ minus 1205751minus1205722 | Θ) = (1 minus 120572) (17)

which gives an (1 minus 120572)100 confidence interval of

CIΘ(1minus120572)100 = [Θ minus 1205751minus1205722 Θ minus 1205751205722] (18)

The bootstrap makes it possible to estimate the distribu-tion of 120575 by the distribution of 120575lowast = Θlowast minus Θ where Θlowast isthe estimate obtained from an empirical bootstrap sample asexplained earlier

24 Simulation MM1K of Queues The number of userspresent in an1198721198721119870 queue follows the distribution givenby (1) To efficiently generate randomvariables fromadiscrete

distribution several methods are widely used in the literatureincluding function sample from R [17] The method usedhere is the discrete analog of the inverse transformationmethod in which it is necessary to generate numbers 119877 simUnif(0 1) ie from a uniform distribution between 0 and 1and to know the probabilities of interest 119875119883 = 119895 = 119901119895 forall119895Therefore to simulate a discrete random variable 119883 with theprobability function

119875 119883 = 119895 = 119901119895 119895 = 0 1 sumforall119895

119901119895 = 1 (19)

it is necessary to compute

119883 =

0 if 119877 le 11990101 if 1199010 lt 119877 le 1199010 + 1199011119895 if

119895minus1sum119894=0

119901119894 lt 119877 le 119895sum119894=0

119901119894

(20)

because

119875 119883 = 119895 = 119875119895minus1sum119894=0

119901119894 lt 119877 le 119895sum119894=0

119901119894 = 119901119895 (21)

and119883 follows the required probability distributionFor 1198721198721119870 queues and from (1) the following must

hold119899sum119894=0

119901119894 = 1 minus 120588119899+11 minus 120588119870+1 119899 = 0 1 119870 (22)

Setting 119888 = 1 minus 120588119870+1 it follows that119883minus1sum119894=0

119901119894 lt 119877 le 119883sum119894=0

119901119894 997904rArr

6 Mathematical Problems in Engineering

Table 1 Mean estimates for 120588 and the mean squared error (MSE in parentheses) for 119870 = 5Estimate 120588 119899

10 20 50 100 200

MLE

001 000928 (000084) 000960 (000045) 001000 (000019) 001003 (000010) 001007 (000005)010 009172 (000715) 009614 (000384) 009845 (000160) 009907 (000081) 009964 (000041)020 018449 (001214) 019235 (000644) 019700 (000258) 019833 (000130) 019921 (000065)050 048492 (001946) 049422 (000932) 049824 (000358) 049884 (000180) 049952 (000088)090 087146 (001703) 088854 (000953) 089839 (000467) 090021 (000271) 090044 (000142)099 092503 (001488) 094638 (000734) 096396 (000303) 097225 (000159) 097854 (000083)

Beta

001 009043 (000699) 005378 (000226) 002864 (000052) 001953 (000018) 001487 (000007)010 015663 (000793) 013021 (000396) 011251 (000160) 010617 (000080) 010321 (000041)020 023444 (000989) 021739 (000567) 020690 (000244) 020327 (000126) 020167 (000064)050 050855 (001688) 050669 (000913) 050295 (000357) 050116 (000180) 050067 (000088)090 081770 (001385) 085318 (000654) 088214 (000283) 089387 (000178) 089954 (000114)099 085425 (002303) 089298 (001185) 092818 (000491) 094668 (000248) 096038 (000121)

Jeffreys

001 005000 (000215) 003182 (000084) 001931 (000026) 001476 (000012) 001244 (000005)010 011630 (000497) 010928 (000314) 010422 (000149) 010202 (000078) 010113 (000040)020 019457 (000908) 019512 (000520) 019804 (000229) 019907 (000123) 019963 (000063)050 049320 (002081) 050148 (001058) 050337 (000377) 050230 (000175) 050121 (000084)090 081749 (001400) 085312 (000655) 088216 (000283) 089387 (000178) 089954 (000115)099 085415 (002308) 089294 (001186) 092819 (000491) 094668 (000249) 096037 (000121)

Bootstrap

001 000992 (000096) 001000 (000049) 001018 (000020) 001013 (000010) 001012 (000005)010 009746 (000791) 009970 (000409) 010009 (000164) 009993 (000082) 010007 (000041)020 019446 (001290) 019837 (000665) 019976 (000262) 019977 (000131) 019994 (000065)050 049647 (001868) 050011 (000901) 050065 (000353) 050007 (000178) 050012 (000088)090 089517 (001990) 090321 (001177) 090473 (000574) 090256 (000314) 090074 (000152)099 095119 (001453) 096522 (000742) 097635 (000317) 098100 (000170) 098475 (000090)

rmm1klt minusfunction (ssizerhoK) simulateRltminusrunif(ssize01)cltminus1minusrhoand(K+1)logrholtminuslog(rho)xltminuslog(1minusRlowastc)logrhoreturn(x)gtgt set seed (13579)gt sampltminusrmm1k (ssize=10 rho=020 K=5)gt samp[1] 0 1 0 1 0 0 0 2 1 0

Listing 5 Sample generation for1198721198721119870 queues

1 minus 120588119883119888 lt 119877 le 1 minus 120588119883+1119888 997904rArrlog (1 minus 119877119888)log (120588) minus 1 le 119883 lt log (1 minus 119877119888)

log (120588) (23)

Therefore

119883 = lceil log (1 minus 119877119888)log (120588) minus 1rceil (24)

MtCaRoMM1Kltminusfunction(ssize rho K fEst) repltminus10000sampltminusnumeric(ssize)estltminusnumeric(rep)for (i in 1rep) sampltminusrmm1k(ssize rho K)oldseedltminusGlobalEnv$Randomseedest [i]ltminusfEst (K samp )GlobalEnv$Randomseedltminusoldseed

return(c(mean(est) var(est)))Listing 6 Monte Carlo simulation

where lceil119910rceil is the ceiling function that is its value is the leastinteger that is not inferior to 119910

Listing 5 presents the R implementation [17] used in thisstudy with a sample call

3 Computational Results

To analyze the performance of the estimators 10000 MonteCarlo replications were made using the R code [17] shownin Listing 6 Note that fESt(K samp ) can be any of the

Mathematical Problems in Engineering 7

Table 2 Mean estimates for 120588 and the mean squared error (MSE in parentheses) for 119870 = 20Estimate 120588 119899

10 20 50 100 200

MLE

001 000928 (000084) 000960 (000045) 001000 (000019) 001003 (000010) 001007 (000005)010 009158 (000710) 009609 (000383) 009844 (000159) 009906 (000081) 009964 (000041)020 018345 (001178) 019186 (000631) 019684 (000255) 019827 (000128) 019919 (000064)050 047371 (001424) 048785 (000666) 049543 (000256) 049750 (000127) 049885 (000062)090 089316 (000326) 089758 (000151) 089922 (000057) 089956 (000029) 089979 (000014)099 097283 (000156) 097979 (000073) 098507 (000030) 098728 (000017) 098877 (000009)

Beta

001 009009 (000692) 005374 (000226) 002864 (000052) 001953 (000018) 001487 (000007)010 015477 (000743) 012978 (000388) 011242 (000159) 010616 (000080) 010320 (000041)020 022893 (000850) 021530 (000529) 020632 (000237) 020303 (000124) 020157 (000063)050 047818 (001132) 048902 (000599) 049557 (000245) 049753 (000125) 049885 (000062)090 088374 (000296) 089484 (000142) 089880 (000058) 089938 (000029) 089970 (000014)099 094835 (000246) 096296 (000107) 097492 (000037) 098068 (000017) 098467 (000008)

Jeffreys

001 005049 (000219) 003182 (000084) 001931 (000026) 001476 (000012) 001244 (000005)010 011678 (000489) 010981 (000320) 010425 (000149) 010203 (000078) 010113 (000040)020 019205 (000801) 019570 (000503) 019866 (000233) 019927 (000123) 019970 (000063)050 046191 (001540) 048175 (000769) 049434 (000302) 049828 (000152) 050063 (000074)090 088365 (000296) 089481 (000142) 089880 (000058) 089938 (000029) 089970 (000014)099 094829 (000247) 096295 (000107) 097493 (000037) 098068 (000017) 098467 (000008)

Bootstrap

001 000993 (000096) 001000 (000049) 001018 (000020) 001013 (000010) 001012 (000005)010 009763 (000795) 009976 (000409) 010010 (000164) 009994 (000082) 010007 (000041)020 019462 (001281) 019846 (000661) 019982 (000260) 019982 (000130) 019997 (000065)050 049385 (001404) 049896 (000659) 050019 (000254) 049995 (000127) 050008 (000062)090 090117 (000310) 090065 (000147) 090021 (000056) 090004 (000028) 090002 (000014)099 098215 (000147) 098590 (000076) 098872 (000035) 098963 (000020) 099023 (000012)

implemented estimation functions for 120588 (MLE Bayesian andbootstrap corrected MLE) and that the state of the randomseed GlobalEnv$Randomseed was stored immediatelybefore the function fESt(K samp ) which can be stochas-tic and was reloaded immediately after its call to ensure thatthe samples generated for the estimates were the same for allestimators

31 Simulation Results Samples were generated from (24)and Listing 5 for sizes 119899 isin 10 20 50 100 200 and trafficintensities 120588 isin 001 010 020 050 090 099 In thesescenarios averages of 10000 Monte Carlo replications werecomputed (via code from Listing 6) and point estimates of 120588were computed by

(i) the MLE via numerical maximization of the likeli-hood function (2) and code from Listing 1

(ii) the Bayesian method via an a priori Beta(10 10)distribution and an average of 5000 samples from thea posteriori distribution (5) obtained by the functionarms [23] from Rrsquos HI package [20] via code fromListing 2

(iii) the Bayesian method via Jeffreys prior distributionand an average of 1000 samples from the a posterioridistribution (14) obtained by the function arms [23]from Rrsquos HI package [20] via code from Listing 3

(iv) the bootstrap corrected MLE (16) and code fromListing 4 Additionally mean squared errors (MSEs)were calculated

The results are shown inTables 1 2 and 3 and summarizedin Figures 3 4 and 5 In Figures 3 4 and 5 the averageestimation error and the mean squared error (MSE) definedas MSE(120588) = Var120588(120588) + Bias2(120588 120588) are shown as functions ofboth the traffic intensity 120588 (averaged over all sample sizes)and the sample size 119899 (averaged over all traffic intensities)

For queues with capacity 119870 = 5 (Figure 3) an approx-imately constant average estimation error is observed forthe MLE and the bootstrap corrected MLE except when120588 asymp 10 (Figure 3(a)) The Bayesian estimators (beta priorand Jeffreysrsquo prior) did not show equivalent performancetheir estimates tended to overestimate the true value (positiveerror) when 120588 lt 05 and to underestimate otherwiseRegarding the sample size 119899 all of the estimators showeda monotonic decrease in error (Figure 3(c)) From the MSEside bothBayesian estimators presented themselves generallyas the best alternative because they presented the lowestvalues most of the time Although the bootstrap correctedMLE presented the lowest bias the method achieves thisperformance at the cost of high variability as reflected by itshighest MSEs

When the queue capacity was increased slightly to119870 = 20(Figure 4) a very similar behavior was noted However the

8 Mathematical Problems in Engineering

Table 3 Mean estimates for 120588 and the mean squared error (MSE in parentheses) for 119870 = 80Estimate 120588 119899

10 20 50 100 200

MLE

001 000928 (000084) 000960 (000045) 001000 (000019) 001003 (000010) 001007 (000005)010 009158 (000710) 009609 (000383) 009844 (000159) 009906 (000081) 009964 (000041)020 018345 (001178) 019186 (000631) 019684 (000255) 019827 (000128) 019919 (000064)050 047366 (001422) 048783 (000666) 049543 (000255) 049750 (000127) 049885 (000062)090 089001 (000144) 089551 (000057) 089835 (000020) 089915 (000010) 089960 (000005)099 098737 (000016) 098903 (000008) 098982 (000003) 098995 (000002) 098998 (000001)

Beta

001 009009 (000692) 005374 (000226) 002864 (000052) 001953 (000018) 001487 (000007)010 015477 (000743) 012978 (000388) 011242 (000159) 010616 (000080) 010320 (000041)020 022892 (000850) 021530 (000529) 020632 (000237) 020304 (000124) 020157 (000063)050 047779 (001125) 048892 (000597) 049556 (000245) 049753 (000125) 049885 (000062)090 088221 (000180) 089167 (000065) 089680 (000021) 089837 (000010) 089921 (000005)099 098149 (000017) 098570 (000007) 098856 (000003) 098953 (000002) 098990 (000001)

Jeffreys

001 005225 (000231) 003334 (000089) 001998 (000027) 001506 (000011) 001253 (000005)010 011744 (000482) 011002 (000316) 010425 (000149) 010203 (000078) 010113 (000040)020 019224 (000794) 019572 (000502) 019866 (000233) 019927 (000123) 019970 (000063)050 046147 (001532) 048167 (000767) 049430 (000302) 049826 (000151) 050064 (000074)090 088217 (000180) 089165 (000065) 089680 (000021) 089837 (000010) 089921 (000005)099 098148 (000017) 098570 (000007) 098856 (000003) 098953 (000002) 098990 (000001)

Bootstrap

001 000993 (000096) 001000 (000049) 001018 (000020) 001013 (000010) 001012 (000005)010 009763 (000795) 009976 (000409) 010010 (000164) 009994 (000082) 010007 (000041)020 019462 (001281) 019846 (000661) 019982 (000260) 019982 (000130) 019997 (000065)050 049390 (001405) 049897 (000659) 050019 (000254) 049995 (000127) 050008 (000062)090 089910 (000118) 089987 (000051) 090008 (000019) 090002 (000009) 090003 (000005)099 099000 (000016) 099042 (000009) 099031 (000004) 099012 (000002) 099002 (000001)

difficulty of estimation for traffic intensities120588 asymp 10 seemed todecrease and the highest MSEs occurred when 120588 asymp 05 TheBayesian estimators maintained the performance presentedearlier for 119870 = 5 that is the estimates have positive biasfor 120588 lt 05 and negative bias otherwise The errors of all ofthe estimators converged to zero as the sample size grewThebootstrap corrected MLE presented an average estimationerror near zero for samples 119899 ge 50 However from the pointof view of the MSE the smaller values were again obtainedusing the Bayesian methods (beta and Jeffreysrsquo prior)

Finally for queues with 119870 = 80 (Figure 5) the observedbehavior could be considered for practical purposes as beingequal to that of an infinite Markovian queue in terms of theaverage estimation error and the MSE This behavior wasobserved for infinite Markovian queues (1198721198721 queues)[15] This finding is merely evidence of the correctness of ourimplementations and the quality of the computational resultspresented

Additionally computational experiments were per-formed for the estimators for 119871 (3) and for 119871119902 (4) and fortheir bootstrap corrected versions with 1000 resamplingsand 119870 = 20 Table 4 and Figure 6 show the resultsobtained for 119871 isin 05 1 2 4 8 16 for sample sizes119899 isin 10 20 50 100 and averages of 10000 Monte Carloreplications Similarly for 119871119902 the results are presented inTable 5 and Figure 7 In summary at an extra cost of thebootstrap method and without inflation of the MSEs the

researcher may achieve with samples of size 119899 = 10 estimatesfor 119871 and 119871119902 with the same average error of the MLE forsamples of size 119899 = 100 a reduction that is relevant inpractical terms because it may lead to reduction in time andcost to obtain the estimates Note that the bootstrap methodalways provides smaller errors and MSEs than the MLEmethod for all estimates of 119871 and 119871119902 even when 120588 gt 1 Alsonote the jump up and down in the errors and MSEs when 120588transitions from 120588 lt 1 to 120588 gt 1

Finally to illustrate the ease of use of the bootstrap in theinterval estimation of the traffic intensity 120588 computationalexperiments were performed The length and coverage ofempirical bootstrap intervals computed from (18) and froma normal distribution approximation (ie Θ plusmn 1199111205722120590Θ where119911120574 is the 120574100th percentile of the standard normal distri-bution and the standard deviation 120590Θ was estimated also bybootstrapping) were evaluated for 120588 isin 010 020 050 090for sample sizes 119899 isin 10 20 50 100 200 averages of 10000Monte Carlo replications and 119870 = 20 The satisfactoryperformance of the bootstrapwas demonstrated as presentedin Table 6 with the coverages approaching the nominalconfidence of 95 (that is 1 minus 120572 = 095) as the sample sizesincrease

32 Numerical Example To better illustrate an application ofthe method a numerical application based on the data con-sidered in Table 7 collected in a large supermarket network

Mathematical Problems in Engineering 9

MLEBeta

JeffreysBootstrap

minus0080

minus0060

minus0040

minus0020

0000

0020

0040

Error

010 020 050 090 099001흆

(a) Average error for 120588

MLEBeta

JeffreysBootstrap

0000000010000200003000040000500006000070000800009000100

MSE

010 020 050 090 099001흆

(b) Average MSE for 120588

MLEBeta

JeffreysBootstrap

minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0000

Error

20 50 100 20010n

(c) Average error for 119899

MLEBeta

JeffreysBootstrap

00000

00020

00040

00060

00080

00100

00120

00140

MSE

20 50 100 20010n

(d) Average MSE for 119899

Figure 3 Performance of estimators for 120588 and119870 = 5

Table 4 Mean estimates for 119871 and the mean squared error (MSE in parentheses) for 119870 = 20Estimator 119871 120588 119899

10 20 50 100

MLE

050 033333 045010 (001698) 047586 (000806) 049098 (000303) 049523 (000150)100 050000 090008 (002353) 095255 (000877) 098190 (000286) 099007 (000137)200 066714 180441 (004755) 190696 (001287) 196438 (000283) 198120 (000112)400 080959 372455 (008082) 387956 (001669) 395673 (000268) 397720 (000092)800 094574 788487 (001625) 797002 (000230) 799390 (000057) 799583 (000029)160 123520 162965 (009872) 161779 (003702) 160743 (000747) 160377 (000238)

Bootstrap

050 033333 048573 (001586) 049686 (000777) 050033 (000299) 050009 (000149)100 050000 097570 (001460) 099583 (000661) 100076 (000254) 099982 (000127)200 066714 196026 (001062) 199245 (000416) 200066 (000154) 199975 (000076)400 080959 397317 (000515) 400156 (000204) 400475 (000081) 400113 (000039)800 094574 796978 (000350) 800755 (000138) 800792 (000059) 800277 (000027)160 123520 161315 (003011) 160376 (000699) 160043 (000188) 160037 (000095)

10 Mathematical Problems in Engineering

MLEBeta

JeffreysBootstrap

010 020 050 090 099001흆

minus0030

minus0020

minus0010

0000

0010

0020

0030

0040

Error

(a) Average error for 120588

MLEBeta

JeffreysBootstrap

00000

00010

00020

00030

00040

00050

00060

MSE

010 020 050 090 099001흆

(b) Average MSE for 120588

MLEBeta

JeffreysBootstrap

minus0015

minus0010

minus0005

0000

0005

0010

0015

0020

Error

20 50 100 20010n

(c) Average error for 119899

MLEBeta

JeffreysBootstrap

00000

00010

00020

00030

00040

00050

00060

00070

00080

MSE

20 50 100 20010n

(d) Average MSE for 119899

Figure 4 Performance of estimators for 120588 and119870 = 20

Table 5 Mean estimates for 119871119902 and the mean squared error (MSE in parentheses) for 119870 = 20Estimator 119871119902 120588 119899

10 20 50 100

MLE

050 050000 042637 (001897) 046470 (000776) 048647 (000272) 049256 (000132)100 061818 086547 (002882) 093515 (000918) 097534 (000248) 098726 (000108)200 073449 177618 (005718) 189501 (001417) 196115 (000266) 197942 (000099)400 084574 374340 (006999) 389252 (001340) 396186 (000214) 397930 (000077)800 097126 794718 (000577) 799972 (000141) 800478 (000056) 800250 (000028)160 132722 161536 (003518) 161286 (002392) 160683 (000800) 160254 (000210)

Bootstrap

050 050000 048185 (001434) 049688 (000660) 050057 (000254) 049986 (000127)100 061818 097175 (001146) 099492 (000493) 100110 (000185) 100049 (000092)200 073449 196521 (000787) 199479 (000303) 200236 (000113) 200027 (000056)400 084574 397906 (000409) 400426 (000174) 400494 (000069) 400067 (000033)800 097126 796303 (000394) 800653 (000136) 800732 (000058) 800362 (000028)160 132722 161202 (003015) 160862 (001687) 160226 (000418) 160228 (000229)

Mathematical Problems in Engineering 11

MLEBeta

JeffreysBootstrap

minus0015minus0010minus000500000005001000150020002500300035

Error

010 020 050 090 099001흆

(a) Average error for 120588

MLEBeta

JeffreysBootstrap

00000

00010

00020

00030

00040

00050

00060

MSE

010 020 050 090 099001흆

(b) Average MSE for 120588

MLEBeta

JeffreysBootstrap

minus0015

minus0010

minus0005

0000

0005

0010

0015

0020

0025

Error

20 50 100 20010n

(c) Average error for 119899

MLEBeta

JeffreysBootstrap

00000

00010

00020

00030

00040

00050

00060

00070

MSE

20 50 100 20010n

(d) Average MSE for 119899

Figure 5 Performance of estimators for 120588 and 119870 = 80

Table 6 Average length (L) and coverage (C) of 95 confidence intervals for 120588 and119870 = 20Method 120588 119899 = 10 119899 = 20 119899 = 50 119899 = 100 119899 = 200

L C L C L C L C L C

Empirical Bootstrap

010 0189 0626 0185 0612 0143 0885 0106 0906 0077 0933020 0308 0619 0265 0797 0186 0902 0135 0923 0097 0940050 0402 0836 0294 0892 0191 0924 0137 0938 0097 0943090 0207 0835 0148 0911 0094 0958 0066 0953 0046 0952

Normal Approximation

010 0219 0636 0200 0865 0146 0890 0107 0911 0077 0936020 0335 0868 0274 0835 0188 0909 0136 0934 0098 0942050 0410 0885 0298 0916 0193 0934 0137 0942 0098 0947090 0213 0906 0149 0937 0094 0953 0066 0950 0046 0950

in a region of interest [12] is providedThe goal is to evaluatetraffic intensity which for managerial reasons should notexceed 87 if it does users may leave The data comprise200 random observations of the number of customers in thesystem at random times sufficiently spaced and previouslydefined by the person responsible for collecting the data to

avoid correlationThe observed values (V) and frequency (F)are presented inTable 7 For instance of the 200 observationsat 8 times no customers were found in the system at 21 timesonly 1 customer was found and so on

The estimates are shown in Table 8 they were calculatedusing the MLE (Listing 1) and the Bayesian estimators

12 Mathematical Problems in Engineering

MLEBootstrap

100 200 400 800 160050L

minus0150

minus0100

minus0050

0000

0050

0100

0150

0200

Error

(a) Average error for 119871

MLEBootstrap

00000

00050

00100

00150

00200

00250

00300

00350

00400

MSE

100 200 400 800 160050L

(b) Average MSE for 119871

MLEBootstrap

minus0080minus0070minus0060minus0050minus0040minus0030minus0020minus001000000010

Error

20 50 10010n

(c) Average error for 119899

MLEBootstrap

0000000050001000015000200002500030000350004000045000500

MSE

20 50 10010n

(d) Average MSE for 119899

Figure 6 Performance of estimates for 119871Table 7 Observed values (V) and frequency (F) for a sample of size119899 = 200 with 119870 = 14 [12]V F0 81 212 273 394 295 286 167 158 69 510 211 114 3

(Listings 2 and 3) based on an a prioriBeta(10 10) and 5000samples and Jeffreysrsquo distribution and 1000 samples from

Table 8 Point estimates for the numerical example

Estimate MLE Bayes Jeffreys Bootstrap Corrected120588 08405396 08403625 08403625 08405683

the a posteriori distribution the bootstrap corrected MLE(Listing 4) was based on 1000 resamplings The completescript is shown in Listing 7

According to the results presented in the previous sectionthe Bayesian estimates should be the most reliable It isnotable that the system utilization seemed to be below thetarget (120588 = 08405396 lt 087) Note that the analysis is basedonly on counts of the number of users in the system It is notnecessary to estimate the arrival and service rates separatelyto determine 1205884 Conclusions and Final Observations

The problem of traffic intensity estimation in finite Markovqueues (1198721198721119870 queues) is presented as quite challengingIn fact no estimator is absolutely superior to another in all

Mathematical Problems in Engineering 13

MLEBootstrap

minus0150

minus0100

minus0050

0000

0050

0100

0150

Error

100 200 400 800 160050푳풒

(a) Average error for 119871119902

MLEBootstrap

00000

00050

00100

00150

00200

00250

MSE

100 200 400 800 160050푳풒

(b) Average MSE for 119871119902

MLEBootstrap

minus0120

minus0100

minus0080

minus0060

minus0040

minus0020

0000

0020

Error

20 50 10010n

(c) Average error for 119899

MLEBootstrap

00000

00050

00100

00150

00200

00250

00300

00350

00400

MSE

20 50 10010n

(d) Average MSE for 119899

Figure 7 Performance of estimates for 119871119902

read samplesampltminusc(rep(0 8) rep(1 21) rep(2 27) rep(3 39) rep(4 29) rep(5 28) rep(6 16)rep(7 15) rep(8 6) rep(9 5) rep(10 2) rep(11 1) rep(14 3)) MLE estimateKltminus14hatrhoMLEltminusMLERoMM1K(K samp) Bayesian estimatealtminus10bltminus10setseed(13579)hatrhoBayesltminusEBaRoMM1K(K samp a b) Bayesian Jeffreys estimatesetseed(13579)hatrhoJeltminusEJeRoMM1K(K samp) Bootstrap corrected estimatesetseed(13579)hatrhoBootltminusEBoRoMM1K(K samp)c(hatrhoMLE hatrhoBayes hatrhoJe hatrhoBoot)gt [1] 08405396 08403625 08403625 08405683

Listing 7 Estimations from real data

14 Mathematical Problems in Engineering

parametric space Although the estimates of the MLE andthe bootstrap corrected MLE exhibit less bias the Bayesianestimates (beta and Jeffreysrsquo prior) present the lowest MSEin general Perhaps due to the skewness of the a posteriordistribution the Bayesian estimators do not present low biasIn general for sample size 119899 = 50 and queues with 119870 ge 20the average estimation error is less than 0005 this value wasonly exceeded by the Bayesian estimator for queues with totalcapacity 119870 = 5

In regard to the behavior of the average estimation errorand the average MSE as functions of the traffic intensity 120588the major errors are observed when the sample size is small(119899 le 20) and the traffic intensities are 120588 asymp 10 unlike inthe case of the 1198721198721 queues which exhibit higher biaseswhen 120588 asymp 05 Perhaps due to the truncation of the numberof users to the maximum queue length119870 systems with hightraffic intensities require more computational effort and arethe most difficult to estimate

Finally it is important to note that for queues withcapacity 119870 = 80 the systemrsquos behavior is similar to that of aninfinite Markovian queue (an 1198721198721 queue) as expectedThat is the average estimation error is greater and the MSEis highest when 120588 asymp 05

Future work in this area includes testing other Bayesianpoint estimators (eg the median because of the asym-metry of the a posterior distribution) developing intervalestimators hypothesis testingmethods or even Kernel-basedmethods [24]

Data Availability

The data used to support the findings of this study areincluded within the article

Disclosure

The Brazilian government funding agencies mentioned hadno role in the study

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

Special thanks are due to Gabriel and Carolina for helpingwith the algebra This work was supported by the Brazil-ian agencies CNPq (Conselho Nacional de DesenvolvimentoCientıfico e Tecnologico of the Ministry for Science andTechnology) [Grant nos 3046712014-2 3058412016-5] andFAPEMIG (Fundacao de Amparo a Pesquisa do Estado deMinas Gerais) [Grant nos CEX-PPM-00564-17 APQ-02119-15 and BIP-00106-16]

References

[1] D Ma D Wang Y Bie F Sun and S Jin ldquoA method for queuelength estimation in an urban street network based on roll time

occupancy datardquo Mathematical Problems in Engineering vol2012 p 12 2012 httpdxdoiorg1011552012892575

[2] S Zhao S Liang H Liu and M Ma ldquoCTM based real-timequeue length estimation at signalized intersectionrdquo Mathemat-ical Problems in Engineering vol 2015 Article ID 328712 12pages 2015

[3] C Armero and M J Bayarri ldquoBayesian prediction in 1198721198721queuesrdquo Queueing Systems vol 15 no 1-4 pp 401ndash417 1994

[4] C Armero and M J Bayarri ldquoQueuesrdquo in Proceedings of theInternational Encyclopedia of the Social Behavioral Sciences J DWright Ed pp 784ndash789 Oxford UK 2015

[5] C Armero and D Conesa ldquoInference and prediction in bulkarrival queues and queues with service in stagesrdquo AppliedStochastic Models in Business and Industry vol 14 no 1 pp 35ndash46 1998

[6] C Armero and D Conesa ldquoPrediction in Markovian bulkarrival queuesrdquo Queueing Systems vol 34 no 1-4 pp 327ndash3502000

[7] C Armero and D Conesa ldquoStatistical performance of a mul-ticlass bulk production queueing systemrdquo European Journal ofOperational Research vol 158 no 3 pp 649ndash661 2004

[8] C Armero and D Conesa ldquoBayesian hierarchical models inmanufacturing bulk service queuesrdquo Journal of Statistical Plan-ning and Inference vol 136 no 2 pp 335ndash354 2006

[9] A Choudhury and A C Borthakur ldquoBayesian inference andprediction in the single server Markovian queuerdquo MetrikaInternational Journal for Theoretical and Applied Statistics vol67 no 3 pp 371ndash383 2008

[10] S Chowdhury and S P Mukherjee ldquoEstimation of trafficintensity based on queue length in a single MM1 queuerdquoCommunications in StatisticsmdashTheory and Methods vol 42 no13 pp 2376ndash2390 2013

[11] S Chowdhury and S P Mukherjee ldquoBayes estimation inMM1 queues with bivariate priorrdquo Journal of Statistics andManagement Systems vol 19 no 5 pp 681ndash699 2016

[12] F R Cruz R C Quinino and L L Ho ldquoBayesian estimationof traffic intensity based on queue length in a multi-serverMMS queuerdquo Communications in StatisticsmdashSimulation andComputation vol 46 no 9 pp 7319ndash7331 2017

[13] R C Quinino and F R B Cruz ldquoBayesian sample sizesin an MM1 queueing systemsrdquo The International Journal ofAdvanced Manufacturing Technology vol 88 no 1-4 pp 995ndash1002 2017

[14] D Gross J F Shortle and J M Thompson Fundamentals ofQueueing Theory Wiley-Interscience New York NY USA 4thedition 2009

[15] M A C Almeida and F R B Cruz ldquoA note on Bayesianestimation of traffic intensity in single-server MarkovianqueuesrdquoCommunications in Statistics -SimulationComputationhttpdxdoiorg1010800361091820171353614

[16] R LThomasson and C H Kapadia ldquoOn estimating the param-eter of a truncated geometric distributionrdquo Annals of theInstitute of Statistical Mathematics vol 20 pp 519ndash523 1968

[17] R Core Team R A Language and Environment for StatisticalComputing R Foundation for Statistical Computing ViennaAustria 2017 httpwwwR-projectorg

[18] N Mukhopadhyay Probability and Statistical Inference MarcelDekker New York NY USA 2000

[19] G S Lingappaiah ldquoBayes inference in right truncated geo-metric distributionrdquo Malaysian Mathematical Society BulletinSecond Series vol 15 no 2 pp 61ndash67 1992

6 Mathematical Problems in Engineering

Table 1 Mean estimates for 120588 and the mean squared error (MSE in parentheses) for 119870 = 5Estimate 120588 119899

10 20 50 100 200

MLE

001 000928 (000084) 000960 (000045) 001000 (000019) 001003 (000010) 001007 (000005)010 009172 (000715) 009614 (000384) 009845 (000160) 009907 (000081) 009964 (000041)020 018449 (001214) 019235 (000644) 019700 (000258) 019833 (000130) 019921 (000065)050 048492 (001946) 049422 (000932) 049824 (000358) 049884 (000180) 049952 (000088)090 087146 (001703) 088854 (000953) 089839 (000467) 090021 (000271) 090044 (000142)099 092503 (001488) 094638 (000734) 096396 (000303) 097225 (000159) 097854 (000083)

Beta

001 009043 (000699) 005378 (000226) 002864 (000052) 001953 (000018) 001487 (000007)010 015663 (000793) 013021 (000396) 011251 (000160) 010617 (000080) 010321 (000041)020 023444 (000989) 021739 (000567) 020690 (000244) 020327 (000126) 020167 (000064)050 050855 (001688) 050669 (000913) 050295 (000357) 050116 (000180) 050067 (000088)090 081770 (001385) 085318 (000654) 088214 (000283) 089387 (000178) 089954 (000114)099 085425 (002303) 089298 (001185) 092818 (000491) 094668 (000248) 096038 (000121)

Jeffreys

001 005000 (000215) 003182 (000084) 001931 (000026) 001476 (000012) 001244 (000005)010 011630 (000497) 010928 (000314) 010422 (000149) 010202 (000078) 010113 (000040)020 019457 (000908) 019512 (000520) 019804 (000229) 019907 (000123) 019963 (000063)050 049320 (002081) 050148 (001058) 050337 (000377) 050230 (000175) 050121 (000084)090 081749 (001400) 085312 (000655) 088216 (000283) 089387 (000178) 089954 (000115)099 085415 (002308) 089294 (001186) 092819 (000491) 094668 (000249) 096037 (000121)

Bootstrap

001 000992 (000096) 001000 (000049) 001018 (000020) 001013 (000010) 001012 (000005)010 009746 (000791) 009970 (000409) 010009 (000164) 009993 (000082) 010007 (000041)020 019446 (001290) 019837 (000665) 019976 (000262) 019977 (000131) 019994 (000065)050 049647 (001868) 050011 (000901) 050065 (000353) 050007 (000178) 050012 (000088)090 089517 (001990) 090321 (001177) 090473 (000574) 090256 (000314) 090074 (000152)099 095119 (001453) 096522 (000742) 097635 (000317) 098100 (000170) 098475 (000090)

rmm1klt minusfunction (ssizerhoK) simulateRltminusrunif(ssize01)cltminus1minusrhoand(K+1)logrholtminuslog(rho)xltminuslog(1minusRlowastc)logrhoreturn(x)gtgt set seed (13579)gt sampltminusrmm1k (ssize=10 rho=020 K=5)gt samp[1] 0 1 0 1 0 0 0 2 1 0

Listing 5 Sample generation for1198721198721119870 queues

1 minus 120588119883119888 lt 119877 le 1 minus 120588119883+1119888 997904rArrlog (1 minus 119877119888)log (120588) minus 1 le 119883 lt log (1 minus 119877119888)

log (120588) (23)

Therefore

119883 = lceil log (1 minus 119877119888)log (120588) minus 1rceil (24)

MtCaRoMM1Kltminusfunction(ssize rho K fEst) repltminus10000sampltminusnumeric(ssize)estltminusnumeric(rep)for (i in 1rep) sampltminusrmm1k(ssize rho K)oldseedltminusGlobalEnv$Randomseedest [i]ltminusfEst (K samp )GlobalEnv$Randomseedltminusoldseed

return(c(mean(est) var(est)))Listing 6 Monte Carlo simulation

where lceil119910rceil is the ceiling function that is its value is the leastinteger that is not inferior to 119910

Listing 5 presents the R implementation [17] used in thisstudy with a sample call

3 Computational Results

To analyze the performance of the estimators 10000 MonteCarlo replications were made using the R code [17] shownin Listing 6 Note that fESt(K samp ) can be any of the

Mathematical Problems in Engineering 7

Table 2 Mean estimates for 120588 and the mean squared error (MSE in parentheses) for 119870 = 20Estimate 120588 119899

10 20 50 100 200

MLE

001 000928 (000084) 000960 (000045) 001000 (000019) 001003 (000010) 001007 (000005)010 009158 (000710) 009609 (000383) 009844 (000159) 009906 (000081) 009964 (000041)020 018345 (001178) 019186 (000631) 019684 (000255) 019827 (000128) 019919 (000064)050 047371 (001424) 048785 (000666) 049543 (000256) 049750 (000127) 049885 (000062)090 089316 (000326) 089758 (000151) 089922 (000057) 089956 (000029) 089979 (000014)099 097283 (000156) 097979 (000073) 098507 (000030) 098728 (000017) 098877 (000009)

Beta

001 009009 (000692) 005374 (000226) 002864 (000052) 001953 (000018) 001487 (000007)010 015477 (000743) 012978 (000388) 011242 (000159) 010616 (000080) 010320 (000041)020 022893 (000850) 021530 (000529) 020632 (000237) 020303 (000124) 020157 (000063)050 047818 (001132) 048902 (000599) 049557 (000245) 049753 (000125) 049885 (000062)090 088374 (000296) 089484 (000142) 089880 (000058) 089938 (000029) 089970 (000014)099 094835 (000246) 096296 (000107) 097492 (000037) 098068 (000017) 098467 (000008)

Jeffreys

001 005049 (000219) 003182 (000084) 001931 (000026) 001476 (000012) 001244 (000005)010 011678 (000489) 010981 (000320) 010425 (000149) 010203 (000078) 010113 (000040)020 019205 (000801) 019570 (000503) 019866 (000233) 019927 (000123) 019970 (000063)050 046191 (001540) 048175 (000769) 049434 (000302) 049828 (000152) 050063 (000074)090 088365 (000296) 089481 (000142) 089880 (000058) 089938 (000029) 089970 (000014)099 094829 (000247) 096295 (000107) 097493 (000037) 098068 (000017) 098467 (000008)

Bootstrap

001 000993 (000096) 001000 (000049) 001018 (000020) 001013 (000010) 001012 (000005)010 009763 (000795) 009976 (000409) 010010 (000164) 009994 (000082) 010007 (000041)020 019462 (001281) 019846 (000661) 019982 (000260) 019982 (000130) 019997 (000065)050 049385 (001404) 049896 (000659) 050019 (000254) 049995 (000127) 050008 (000062)090 090117 (000310) 090065 (000147) 090021 (000056) 090004 (000028) 090002 (000014)099 098215 (000147) 098590 (000076) 098872 (000035) 098963 (000020) 099023 (000012)

implemented estimation functions for 120588 (MLE Bayesian andbootstrap corrected MLE) and that the state of the randomseed GlobalEnv$Randomseed was stored immediatelybefore the function fESt(K samp ) which can be stochas-tic and was reloaded immediately after its call to ensure thatthe samples generated for the estimates were the same for allestimators

31 Simulation Results Samples were generated from (24)and Listing 5 for sizes 119899 isin 10 20 50 100 200 and trafficintensities 120588 isin 001 010 020 050 090 099 In thesescenarios averages of 10000 Monte Carlo replications werecomputed (via code from Listing 6) and point estimates of 120588were computed by

(i) the MLE via numerical maximization of the likeli-hood function (2) and code from Listing 1

(ii) the Bayesian method via an a priori Beta(10 10)distribution and an average of 5000 samples from thea posteriori distribution (5) obtained by the functionarms [23] from Rrsquos HI package [20] via code fromListing 2

(iii) the Bayesian method via Jeffreys prior distributionand an average of 1000 samples from the a posterioridistribution (14) obtained by the function arms [23]from Rrsquos HI package [20] via code from Listing 3

(iv) the bootstrap corrected MLE (16) and code fromListing 4 Additionally mean squared errors (MSEs)were calculated

The results are shown inTables 1 2 and 3 and summarizedin Figures 3 4 and 5 In Figures 3 4 and 5 the averageestimation error and the mean squared error (MSE) definedas MSE(120588) = Var120588(120588) + Bias2(120588 120588) are shown as functions ofboth the traffic intensity 120588 (averaged over all sample sizes)and the sample size 119899 (averaged over all traffic intensities)

For queues with capacity 119870 = 5 (Figure 3) an approx-imately constant average estimation error is observed forthe MLE and the bootstrap corrected MLE except when120588 asymp 10 (Figure 3(a)) The Bayesian estimators (beta priorand Jeffreysrsquo prior) did not show equivalent performancetheir estimates tended to overestimate the true value (positiveerror) when 120588 lt 05 and to underestimate otherwiseRegarding the sample size 119899 all of the estimators showeda monotonic decrease in error (Figure 3(c)) From the MSEside bothBayesian estimators presented themselves generallyas the best alternative because they presented the lowestvalues most of the time Although the bootstrap correctedMLE presented the lowest bias the method achieves thisperformance at the cost of high variability as reflected by itshighest MSEs

When the queue capacity was increased slightly to119870 = 20(Figure 4) a very similar behavior was noted However the

8 Mathematical Problems in Engineering

Table 3 Mean estimates for 120588 and the mean squared error (MSE in parentheses) for 119870 = 80Estimate 120588 119899

10 20 50 100 200

MLE

001 000928 (000084) 000960 (000045) 001000 (000019) 001003 (000010) 001007 (000005)010 009158 (000710) 009609 (000383) 009844 (000159) 009906 (000081) 009964 (000041)020 018345 (001178) 019186 (000631) 019684 (000255) 019827 (000128) 019919 (000064)050 047366 (001422) 048783 (000666) 049543 (000255) 049750 (000127) 049885 (000062)090 089001 (000144) 089551 (000057) 089835 (000020) 089915 (000010) 089960 (000005)099 098737 (000016) 098903 (000008) 098982 (000003) 098995 (000002) 098998 (000001)

Beta

001 009009 (000692) 005374 (000226) 002864 (000052) 001953 (000018) 001487 (000007)010 015477 (000743) 012978 (000388) 011242 (000159) 010616 (000080) 010320 (000041)020 022892 (000850) 021530 (000529) 020632 (000237) 020304 (000124) 020157 (000063)050 047779 (001125) 048892 (000597) 049556 (000245) 049753 (000125) 049885 (000062)090 088221 (000180) 089167 (000065) 089680 (000021) 089837 (000010) 089921 (000005)099 098149 (000017) 098570 (000007) 098856 (000003) 098953 (000002) 098990 (000001)

Jeffreys

001 005225 (000231) 003334 (000089) 001998 (000027) 001506 (000011) 001253 (000005)010 011744 (000482) 011002 (000316) 010425 (000149) 010203 (000078) 010113 (000040)020 019224 (000794) 019572 (000502) 019866 (000233) 019927 (000123) 019970 (000063)050 046147 (001532) 048167 (000767) 049430 (000302) 049826 (000151) 050064 (000074)090 088217 (000180) 089165 (000065) 089680 (000021) 089837 (000010) 089921 (000005)099 098148 (000017) 098570 (000007) 098856 (000003) 098953 (000002) 098990 (000001)

Bootstrap

001 000993 (000096) 001000 (000049) 001018 (000020) 001013 (000010) 001012 (000005)010 009763 (000795) 009976 (000409) 010010 (000164) 009994 (000082) 010007 (000041)020 019462 (001281) 019846 (000661) 019982 (000260) 019982 (000130) 019997 (000065)050 049390 (001405) 049897 (000659) 050019 (000254) 049995 (000127) 050008 (000062)090 089910 (000118) 089987 (000051) 090008 (000019) 090002 (000009) 090003 (000005)099 099000 (000016) 099042 (000009) 099031 (000004) 099012 (000002) 099002 (000001)

difficulty of estimation for traffic intensities120588 asymp 10 seemed todecrease and the highest MSEs occurred when 120588 asymp 05 TheBayesian estimators maintained the performance presentedearlier for 119870 = 5 that is the estimates have positive biasfor 120588 lt 05 and negative bias otherwise The errors of all ofthe estimators converged to zero as the sample size grewThebootstrap corrected MLE presented an average estimationerror near zero for samples 119899 ge 50 However from the pointof view of the MSE the smaller values were again obtainedusing the Bayesian methods (beta and Jeffreysrsquo prior)

Finally for queues with 119870 = 80 (Figure 5) the observedbehavior could be considered for practical purposes as beingequal to that of an infinite Markovian queue in terms of theaverage estimation error and the MSE This behavior wasobserved for infinite Markovian queues (1198721198721 queues)[15] This finding is merely evidence of the correctness of ourimplementations and the quality of the computational resultspresented

Additionally computational experiments were per-formed for the estimators for 119871 (3) and for 119871119902 (4) and fortheir bootstrap corrected versions with 1000 resamplingsand 119870 = 20 Table 4 and Figure 6 show the resultsobtained for 119871 isin 05 1 2 4 8 16 for sample sizes119899 isin 10 20 50 100 and averages of 10000 Monte Carloreplications Similarly for 119871119902 the results are presented inTable 5 and Figure 7 In summary at an extra cost of thebootstrap method and without inflation of the MSEs the

researcher may achieve with samples of size 119899 = 10 estimatesfor 119871 and 119871119902 with the same average error of the MLE forsamples of size 119899 = 100 a reduction that is relevant inpractical terms because it may lead to reduction in time andcost to obtain the estimates Note that the bootstrap methodalways provides smaller errors and MSEs than the MLEmethod for all estimates of 119871 and 119871119902 even when 120588 gt 1 Alsonote the jump up and down in the errors and MSEs when 120588transitions from 120588 lt 1 to 120588 gt 1

Finally to illustrate the ease of use of the bootstrap in theinterval estimation of the traffic intensity 120588 computationalexperiments were performed The length and coverage ofempirical bootstrap intervals computed from (18) and froma normal distribution approximation (ie Θ plusmn 1199111205722120590Θ where119911120574 is the 120574100th percentile of the standard normal distri-bution and the standard deviation 120590Θ was estimated also bybootstrapping) were evaluated for 120588 isin 010 020 050 090for sample sizes 119899 isin 10 20 50 100 200 averages of 10000Monte Carlo replications and 119870 = 20 The satisfactoryperformance of the bootstrapwas demonstrated as presentedin Table 6 with the coverages approaching the nominalconfidence of 95 (that is 1 minus 120572 = 095) as the sample sizesincrease

32 Numerical Example To better illustrate an application ofthe method a numerical application based on the data con-sidered in Table 7 collected in a large supermarket network

Mathematical Problems in Engineering 9

MLEBeta

JeffreysBootstrap

minus0080

minus0060

minus0040

minus0020

0000

0020

0040

Error

010 020 050 090 099001흆

(a) Average error for 120588

MLEBeta

JeffreysBootstrap

0000000010000200003000040000500006000070000800009000100

MSE

010 020 050 090 099001흆

(b) Average MSE for 120588

MLEBeta

JeffreysBootstrap

minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0000

Error

20 50 100 20010n

(c) Average error for 119899

MLEBeta

JeffreysBootstrap

00000

00020

00040

00060

00080

00100

00120

00140

MSE

20 50 100 20010n

(d) Average MSE for 119899

Figure 3 Performance of estimators for 120588 and119870 = 5

Table 4 Mean estimates for 119871 and the mean squared error (MSE in parentheses) for 119870 = 20Estimator 119871 120588 119899

10 20 50 100

MLE

050 033333 045010 (001698) 047586 (000806) 049098 (000303) 049523 (000150)100 050000 090008 (002353) 095255 (000877) 098190 (000286) 099007 (000137)200 066714 180441 (004755) 190696 (001287) 196438 (000283) 198120 (000112)400 080959 372455 (008082) 387956 (001669) 395673 (000268) 397720 (000092)800 094574 788487 (001625) 797002 (000230) 799390 (000057) 799583 (000029)160 123520 162965 (009872) 161779 (003702) 160743 (000747) 160377 (000238)

Bootstrap

050 033333 048573 (001586) 049686 (000777) 050033 (000299) 050009 (000149)100 050000 097570 (001460) 099583 (000661) 100076 (000254) 099982 (000127)200 066714 196026 (001062) 199245 (000416) 200066 (000154) 199975 (000076)400 080959 397317 (000515) 400156 (000204) 400475 (000081) 400113 (000039)800 094574 796978 (000350) 800755 (000138) 800792 (000059) 800277 (000027)160 123520 161315 (003011) 160376 (000699) 160043 (000188) 160037 (000095)

10 Mathematical Problems in Engineering

MLEBeta

JeffreysBootstrap

010 020 050 090 099001흆

minus0030

minus0020

minus0010

0000

0010

0020

0030

0040

Error

(a) Average error for 120588

MLEBeta

JeffreysBootstrap

00000

00010

00020

00030

00040

00050

00060

MSE

010 020 050 090 099001흆

(b) Average MSE for 120588

MLEBeta

JeffreysBootstrap

minus0015

minus0010

minus0005

0000

0005

0010

0015

0020

Error

20 50 100 20010n

(c) Average error for 119899

MLEBeta

JeffreysBootstrap

00000

00010

00020

00030

00040

00050

00060

00070

00080

MSE

20 50 100 20010n

(d) Average MSE for 119899

Figure 4 Performance of estimators for 120588 and119870 = 20

Table 5 Mean estimates for 119871119902 and the mean squared error (MSE in parentheses) for 119870 = 20Estimator 119871119902 120588 119899

10 20 50 100

MLE

050 050000 042637 (001897) 046470 (000776) 048647 (000272) 049256 (000132)100 061818 086547 (002882) 093515 (000918) 097534 (000248) 098726 (000108)200 073449 177618 (005718) 189501 (001417) 196115 (000266) 197942 (000099)400 084574 374340 (006999) 389252 (001340) 396186 (000214) 397930 (000077)800 097126 794718 (000577) 799972 (000141) 800478 (000056) 800250 (000028)160 132722 161536 (003518) 161286 (002392) 160683 (000800) 160254 (000210)

Bootstrap

050 050000 048185 (001434) 049688 (000660) 050057 (000254) 049986 (000127)100 061818 097175 (001146) 099492 (000493) 100110 (000185) 100049 (000092)200 073449 196521 (000787) 199479 (000303) 200236 (000113) 200027 (000056)400 084574 397906 (000409) 400426 (000174) 400494 (000069) 400067 (000033)800 097126 796303 (000394) 800653 (000136) 800732 (000058) 800362 (000028)160 132722 161202 (003015) 160862 (001687) 160226 (000418) 160228 (000229)

Mathematical Problems in Engineering 11

MLEBeta

JeffreysBootstrap

minus0015minus0010minus000500000005001000150020002500300035

Error

010 020 050 090 099001흆

(a) Average error for 120588

MLEBeta

JeffreysBootstrap

00000

00010

00020

00030

00040

00050

00060

MSE

010 020 050 090 099001흆

(b) Average MSE for 120588

MLEBeta

JeffreysBootstrap

minus0015

minus0010

minus0005

0000

0005

0010

0015

0020

0025

Error

20 50 100 20010n

(c) Average error for 119899

MLEBeta

JeffreysBootstrap

00000

00010

00020

00030

00040

00050

00060

00070

MSE

20 50 100 20010n

(d) Average MSE for 119899

Figure 5 Performance of estimators for 120588 and 119870 = 80

Table 6 Average length (L) and coverage (C) of 95 confidence intervals for 120588 and119870 = 20Method 120588 119899 = 10 119899 = 20 119899 = 50 119899 = 100 119899 = 200

L C L C L C L C L C

Empirical Bootstrap

010 0189 0626 0185 0612 0143 0885 0106 0906 0077 0933020 0308 0619 0265 0797 0186 0902 0135 0923 0097 0940050 0402 0836 0294 0892 0191 0924 0137 0938 0097 0943090 0207 0835 0148 0911 0094 0958 0066 0953 0046 0952

Normal Approximation

010 0219 0636 0200 0865 0146 0890 0107 0911 0077 0936020 0335 0868 0274 0835 0188 0909 0136 0934 0098 0942050 0410 0885 0298 0916 0193 0934 0137 0942 0098 0947090 0213 0906 0149 0937 0094 0953 0066 0950 0046 0950

in a region of interest [12] is providedThe goal is to evaluatetraffic intensity which for managerial reasons should notexceed 87 if it does users may leave The data comprise200 random observations of the number of customers in thesystem at random times sufficiently spaced and previouslydefined by the person responsible for collecting the data to

avoid correlationThe observed values (V) and frequency (F)are presented inTable 7 For instance of the 200 observationsat 8 times no customers were found in the system at 21 timesonly 1 customer was found and so on

The estimates are shown in Table 8 they were calculatedusing the MLE (Listing 1) and the Bayesian estimators

12 Mathematical Problems in Engineering

MLEBootstrap

100 200 400 800 160050L

minus0150

minus0100

minus0050

0000

0050

0100

0150

0200

Error

(a) Average error for 119871

MLEBootstrap

00000

00050

00100

00150

00200

00250

00300

00350

00400

MSE

100 200 400 800 160050L

(b) Average MSE for 119871

MLEBootstrap

minus0080minus0070minus0060minus0050minus0040minus0030minus0020minus001000000010

Error

20 50 10010n

(c) Average error for 119899

MLEBootstrap

0000000050001000015000200002500030000350004000045000500

MSE

20 50 10010n

(d) Average MSE for 119899

Figure 6 Performance of estimates for 119871Table 7 Observed values (V) and frequency (F) for a sample of size119899 = 200 with 119870 = 14 [12]V F0 81 212 273 394 295 286 167 158 69 510 211 114 3

(Listings 2 and 3) based on an a prioriBeta(10 10) and 5000samples and Jeffreysrsquo distribution and 1000 samples from

Table 8 Point estimates for the numerical example

Estimate MLE Bayes Jeffreys Bootstrap Corrected120588 08405396 08403625 08403625 08405683

the a posteriori distribution the bootstrap corrected MLE(Listing 4) was based on 1000 resamplings The completescript is shown in Listing 7

According to the results presented in the previous sectionthe Bayesian estimates should be the most reliable It isnotable that the system utilization seemed to be below thetarget (120588 = 08405396 lt 087) Note that the analysis is basedonly on counts of the number of users in the system It is notnecessary to estimate the arrival and service rates separatelyto determine 1205884 Conclusions and Final Observations

The problem of traffic intensity estimation in finite Markovqueues (1198721198721119870 queues) is presented as quite challengingIn fact no estimator is absolutely superior to another in all

Mathematical Problems in Engineering 13

MLEBootstrap

minus0150

minus0100

minus0050

0000

0050

0100

0150

Error

100 200 400 800 160050푳풒

(a) Average error for 119871119902

MLEBootstrap

00000

00050

00100

00150

00200

00250

MSE

100 200 400 800 160050푳풒

(b) Average MSE for 119871119902

MLEBootstrap

minus0120

minus0100

minus0080

minus0060

minus0040

minus0020

0000

0020

Error

20 50 10010n

(c) Average error for 119899

MLEBootstrap

00000

00050

00100

00150

00200

00250

00300

00350

00400

MSE

20 50 10010n

(d) Average MSE for 119899

Figure 7 Performance of estimates for 119871119902

read samplesampltminusc(rep(0 8) rep(1 21) rep(2 27) rep(3 39) rep(4 29) rep(5 28) rep(6 16)rep(7 15) rep(8 6) rep(9 5) rep(10 2) rep(11 1) rep(14 3)) MLE estimateKltminus14hatrhoMLEltminusMLERoMM1K(K samp) Bayesian estimatealtminus10bltminus10setseed(13579)hatrhoBayesltminusEBaRoMM1K(K samp a b) Bayesian Jeffreys estimatesetseed(13579)hatrhoJeltminusEJeRoMM1K(K samp) Bootstrap corrected estimatesetseed(13579)hatrhoBootltminusEBoRoMM1K(K samp)c(hatrhoMLE hatrhoBayes hatrhoJe hatrhoBoot)gt [1] 08405396 08403625 08403625 08405683

Listing 7 Estimations from real data

14 Mathematical Problems in Engineering

parametric space Although the estimates of the MLE andthe bootstrap corrected MLE exhibit less bias the Bayesianestimates (beta and Jeffreysrsquo prior) present the lowest MSEin general Perhaps due to the skewness of the a posteriordistribution the Bayesian estimators do not present low biasIn general for sample size 119899 = 50 and queues with 119870 ge 20the average estimation error is less than 0005 this value wasonly exceeded by the Bayesian estimator for queues with totalcapacity 119870 = 5

In regard to the behavior of the average estimation errorand the average MSE as functions of the traffic intensity 120588the major errors are observed when the sample size is small(119899 le 20) and the traffic intensities are 120588 asymp 10 unlike inthe case of the 1198721198721 queues which exhibit higher biaseswhen 120588 asymp 05 Perhaps due to the truncation of the numberof users to the maximum queue length119870 systems with hightraffic intensities require more computational effort and arethe most difficult to estimate

Finally it is important to note that for queues withcapacity 119870 = 80 the systemrsquos behavior is similar to that of aninfinite Markovian queue (an 1198721198721 queue) as expectedThat is the average estimation error is greater and the MSEis highest when 120588 asymp 05

Future work in this area includes testing other Bayesianpoint estimators (eg the median because of the asym-metry of the a posterior distribution) developing intervalestimators hypothesis testingmethods or even Kernel-basedmethods [24]

Data Availability

The data used to support the findings of this study areincluded within the article

Disclosure

The Brazilian government funding agencies mentioned hadno role in the study

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

Special thanks are due to Gabriel and Carolina for helpingwith the algebra This work was supported by the Brazil-ian agencies CNPq (Conselho Nacional de DesenvolvimentoCientıfico e Tecnologico of the Ministry for Science andTechnology) [Grant nos 3046712014-2 3058412016-5] andFAPEMIG (Fundacao de Amparo a Pesquisa do Estado deMinas Gerais) [Grant nos CEX-PPM-00564-17 APQ-02119-15 and BIP-00106-16]

References

[1] D Ma D Wang Y Bie F Sun and S Jin ldquoA method for queuelength estimation in an urban street network based on roll time

occupancy datardquo Mathematical Problems in Engineering vol2012 p 12 2012 httpdxdoiorg1011552012892575

[2] S Zhao S Liang H Liu and M Ma ldquoCTM based real-timequeue length estimation at signalized intersectionrdquo Mathemat-ical Problems in Engineering vol 2015 Article ID 328712 12pages 2015

[3] C Armero and M J Bayarri ldquoBayesian prediction in 1198721198721queuesrdquo Queueing Systems vol 15 no 1-4 pp 401ndash417 1994

[4] C Armero and M J Bayarri ldquoQueuesrdquo in Proceedings of theInternational Encyclopedia of the Social Behavioral Sciences J DWright Ed pp 784ndash789 Oxford UK 2015

[5] C Armero and D Conesa ldquoInference and prediction in bulkarrival queues and queues with service in stagesrdquo AppliedStochastic Models in Business and Industry vol 14 no 1 pp 35ndash46 1998

[6] C Armero and D Conesa ldquoPrediction in Markovian bulkarrival queuesrdquo Queueing Systems vol 34 no 1-4 pp 327ndash3502000

[7] C Armero and D Conesa ldquoStatistical performance of a mul-ticlass bulk production queueing systemrdquo European Journal ofOperational Research vol 158 no 3 pp 649ndash661 2004

[8] C Armero and D Conesa ldquoBayesian hierarchical models inmanufacturing bulk service queuesrdquo Journal of Statistical Plan-ning and Inference vol 136 no 2 pp 335ndash354 2006

[9] A Choudhury and A C Borthakur ldquoBayesian inference andprediction in the single server Markovian queuerdquo MetrikaInternational Journal for Theoretical and Applied Statistics vol67 no 3 pp 371ndash383 2008

[10] S Chowdhury and S P Mukherjee ldquoEstimation of trafficintensity based on queue length in a single MM1 queuerdquoCommunications in StatisticsmdashTheory and Methods vol 42 no13 pp 2376ndash2390 2013

[11] S Chowdhury and S P Mukherjee ldquoBayes estimation inMM1 queues with bivariate priorrdquo Journal of Statistics andManagement Systems vol 19 no 5 pp 681ndash699 2016

[12] F R Cruz R C Quinino and L L Ho ldquoBayesian estimationof traffic intensity based on queue length in a multi-serverMMS queuerdquo Communications in StatisticsmdashSimulation andComputation vol 46 no 9 pp 7319ndash7331 2017

[13] R C Quinino and F R B Cruz ldquoBayesian sample sizesin an MM1 queueing systemsrdquo The International Journal ofAdvanced Manufacturing Technology vol 88 no 1-4 pp 995ndash1002 2017

[14] D Gross J F Shortle and J M Thompson Fundamentals ofQueueing Theory Wiley-Interscience New York NY USA 4thedition 2009

[15] M A C Almeida and F R B Cruz ldquoA note on Bayesianestimation of traffic intensity in single-server MarkovianqueuesrdquoCommunications in Statistics -SimulationComputationhttpdxdoiorg1010800361091820171353614

[16] R LThomasson and C H Kapadia ldquoOn estimating the param-eter of a truncated geometric distributionrdquo Annals of theInstitute of Statistical Mathematics vol 20 pp 519ndash523 1968

[17] R Core Team R A Language and Environment for StatisticalComputing R Foundation for Statistical Computing ViennaAustria 2017 httpwwwR-projectorg

[18] N Mukhopadhyay Probability and Statistical Inference MarcelDekker New York NY USA 2000

[19] G S Lingappaiah ldquoBayes inference in right truncated geo-metric distributionrdquo Malaysian Mathematical Society BulletinSecond Series vol 15 no 2 pp 61ndash67 1992

Mathematical Problems in Engineering 7

Table 2 Mean estimates for 120588 and the mean squared error (MSE in parentheses) for 119870 = 20Estimate 120588 119899

10 20 50 100 200

MLE

001 000928 (000084) 000960 (000045) 001000 (000019) 001003 (000010) 001007 (000005)010 009158 (000710) 009609 (000383) 009844 (000159) 009906 (000081) 009964 (000041)020 018345 (001178) 019186 (000631) 019684 (000255) 019827 (000128) 019919 (000064)050 047371 (001424) 048785 (000666) 049543 (000256) 049750 (000127) 049885 (000062)090 089316 (000326) 089758 (000151) 089922 (000057) 089956 (000029) 089979 (000014)099 097283 (000156) 097979 (000073) 098507 (000030) 098728 (000017) 098877 (000009)

Beta

001 009009 (000692) 005374 (000226) 002864 (000052) 001953 (000018) 001487 (000007)010 015477 (000743) 012978 (000388) 011242 (000159) 010616 (000080) 010320 (000041)020 022893 (000850) 021530 (000529) 020632 (000237) 020303 (000124) 020157 (000063)050 047818 (001132) 048902 (000599) 049557 (000245) 049753 (000125) 049885 (000062)090 088374 (000296) 089484 (000142) 089880 (000058) 089938 (000029) 089970 (000014)099 094835 (000246) 096296 (000107) 097492 (000037) 098068 (000017) 098467 (000008)

Jeffreys

001 005049 (000219) 003182 (000084) 001931 (000026) 001476 (000012) 001244 (000005)010 011678 (000489) 010981 (000320) 010425 (000149) 010203 (000078) 010113 (000040)020 019205 (000801) 019570 (000503) 019866 (000233) 019927 (000123) 019970 (000063)050 046191 (001540) 048175 (000769) 049434 (000302) 049828 (000152) 050063 (000074)090 088365 (000296) 089481 (000142) 089880 (000058) 089938 (000029) 089970 (000014)099 094829 (000247) 096295 (000107) 097493 (000037) 098068 (000017) 098467 (000008)

Bootstrap

001 000993 (000096) 001000 (000049) 001018 (000020) 001013 (000010) 001012 (000005)010 009763 (000795) 009976 (000409) 010010 (000164) 009994 (000082) 010007 (000041)020 019462 (001281) 019846 (000661) 019982 (000260) 019982 (000130) 019997 (000065)050 049385 (001404) 049896 (000659) 050019 (000254) 049995 (000127) 050008 (000062)090 090117 (000310) 090065 (000147) 090021 (000056) 090004 (000028) 090002 (000014)099 098215 (000147) 098590 (000076) 098872 (000035) 098963 (000020) 099023 (000012)

implemented estimation functions for 120588 (MLE Bayesian andbootstrap corrected MLE) and that the state of the randomseed GlobalEnv$Randomseed was stored immediatelybefore the function fESt(K samp ) which can be stochas-tic and was reloaded immediately after its call to ensure thatthe samples generated for the estimates were the same for allestimators

31 Simulation Results Samples were generated from (24)and Listing 5 for sizes 119899 isin 10 20 50 100 200 and trafficintensities 120588 isin 001 010 020 050 090 099 In thesescenarios averages of 10000 Monte Carlo replications werecomputed (via code from Listing 6) and point estimates of 120588were computed by

(i) the MLE via numerical maximization of the likeli-hood function (2) and code from Listing 1

(ii) the Bayesian method via an a priori Beta(10 10)distribution and an average of 5000 samples from thea posteriori distribution (5) obtained by the functionarms [23] from Rrsquos HI package [20] via code fromListing 2

(iii) the Bayesian method via Jeffreys prior distributionand an average of 1000 samples from the a posterioridistribution (14) obtained by the function arms [23]from Rrsquos HI package [20] via code from Listing 3

(iv) the bootstrap corrected MLE (16) and code fromListing 4 Additionally mean squared errors (MSEs)were calculated

The results are shown inTables 1 2 and 3 and summarizedin Figures 3 4 and 5 In Figures 3 4 and 5 the averageestimation error and the mean squared error (MSE) definedas MSE(120588) = Var120588(120588) + Bias2(120588 120588) are shown as functions ofboth the traffic intensity 120588 (averaged over all sample sizes)and the sample size 119899 (averaged over all traffic intensities)

For queues with capacity 119870 = 5 (Figure 3) an approx-imately constant average estimation error is observed forthe MLE and the bootstrap corrected MLE except when120588 asymp 10 (Figure 3(a)) The Bayesian estimators (beta priorand Jeffreysrsquo prior) did not show equivalent performancetheir estimates tended to overestimate the true value (positiveerror) when 120588 lt 05 and to underestimate otherwiseRegarding the sample size 119899 all of the estimators showeda monotonic decrease in error (Figure 3(c)) From the MSEside bothBayesian estimators presented themselves generallyas the best alternative because they presented the lowestvalues most of the time Although the bootstrap correctedMLE presented the lowest bias the method achieves thisperformance at the cost of high variability as reflected by itshighest MSEs

When the queue capacity was increased slightly to119870 = 20(Figure 4) a very similar behavior was noted However the

8 Mathematical Problems in Engineering

Table 3 Mean estimates for 120588 and the mean squared error (MSE in parentheses) for 119870 = 80Estimate 120588 119899

10 20 50 100 200

MLE

001 000928 (000084) 000960 (000045) 001000 (000019) 001003 (000010) 001007 (000005)010 009158 (000710) 009609 (000383) 009844 (000159) 009906 (000081) 009964 (000041)020 018345 (001178) 019186 (000631) 019684 (000255) 019827 (000128) 019919 (000064)050 047366 (001422) 048783 (000666) 049543 (000255) 049750 (000127) 049885 (000062)090 089001 (000144) 089551 (000057) 089835 (000020) 089915 (000010) 089960 (000005)099 098737 (000016) 098903 (000008) 098982 (000003) 098995 (000002) 098998 (000001)

Beta

001 009009 (000692) 005374 (000226) 002864 (000052) 001953 (000018) 001487 (000007)010 015477 (000743) 012978 (000388) 011242 (000159) 010616 (000080) 010320 (000041)020 022892 (000850) 021530 (000529) 020632 (000237) 020304 (000124) 020157 (000063)050 047779 (001125) 048892 (000597) 049556 (000245) 049753 (000125) 049885 (000062)090 088221 (000180) 089167 (000065) 089680 (000021) 089837 (000010) 089921 (000005)099 098149 (000017) 098570 (000007) 098856 (000003) 098953 (000002) 098990 (000001)

Jeffreys

001 005225 (000231) 003334 (000089) 001998 (000027) 001506 (000011) 001253 (000005)010 011744 (000482) 011002 (000316) 010425 (000149) 010203 (000078) 010113 (000040)020 019224 (000794) 019572 (000502) 019866 (000233) 019927 (000123) 019970 (000063)050 046147 (001532) 048167 (000767) 049430 (000302) 049826 (000151) 050064 (000074)090 088217 (000180) 089165 (000065) 089680 (000021) 089837 (000010) 089921 (000005)099 098148 (000017) 098570 (000007) 098856 (000003) 098953 (000002) 098990 (000001)

Bootstrap

001 000993 (000096) 001000 (000049) 001018 (000020) 001013 (000010) 001012 (000005)010 009763 (000795) 009976 (000409) 010010 (000164) 009994 (000082) 010007 (000041)020 019462 (001281) 019846 (000661) 019982 (000260) 019982 (000130) 019997 (000065)050 049390 (001405) 049897 (000659) 050019 (000254) 049995 (000127) 050008 (000062)090 089910 (000118) 089987 (000051) 090008 (000019) 090002 (000009) 090003 (000005)099 099000 (000016) 099042 (000009) 099031 (000004) 099012 (000002) 099002 (000001)

difficulty of estimation for traffic intensities120588 asymp 10 seemed todecrease and the highest MSEs occurred when 120588 asymp 05 TheBayesian estimators maintained the performance presentedearlier for 119870 = 5 that is the estimates have positive biasfor 120588 lt 05 and negative bias otherwise The errors of all ofthe estimators converged to zero as the sample size grewThebootstrap corrected MLE presented an average estimationerror near zero for samples 119899 ge 50 However from the pointof view of the MSE the smaller values were again obtainedusing the Bayesian methods (beta and Jeffreysrsquo prior)

Finally for queues with 119870 = 80 (Figure 5) the observedbehavior could be considered for practical purposes as beingequal to that of an infinite Markovian queue in terms of theaverage estimation error and the MSE This behavior wasobserved for infinite Markovian queues (1198721198721 queues)[15] This finding is merely evidence of the correctness of ourimplementations and the quality of the computational resultspresented

Additionally computational experiments were per-formed for the estimators for 119871 (3) and for 119871119902 (4) and fortheir bootstrap corrected versions with 1000 resamplingsand 119870 = 20 Table 4 and Figure 6 show the resultsobtained for 119871 isin 05 1 2 4 8 16 for sample sizes119899 isin 10 20 50 100 and averages of 10000 Monte Carloreplications Similarly for 119871119902 the results are presented inTable 5 and Figure 7 In summary at an extra cost of thebootstrap method and without inflation of the MSEs the

researcher may achieve with samples of size 119899 = 10 estimatesfor 119871 and 119871119902 with the same average error of the MLE forsamples of size 119899 = 100 a reduction that is relevant inpractical terms because it may lead to reduction in time andcost to obtain the estimates Note that the bootstrap methodalways provides smaller errors and MSEs than the MLEmethod for all estimates of 119871 and 119871119902 even when 120588 gt 1 Alsonote the jump up and down in the errors and MSEs when 120588transitions from 120588 lt 1 to 120588 gt 1

Finally to illustrate the ease of use of the bootstrap in theinterval estimation of the traffic intensity 120588 computationalexperiments were performed The length and coverage ofempirical bootstrap intervals computed from (18) and froma normal distribution approximation (ie Θ plusmn 1199111205722120590Θ where119911120574 is the 120574100th percentile of the standard normal distri-bution and the standard deviation 120590Θ was estimated also bybootstrapping) were evaluated for 120588 isin 010 020 050 090for sample sizes 119899 isin 10 20 50 100 200 averages of 10000Monte Carlo replications and 119870 = 20 The satisfactoryperformance of the bootstrapwas demonstrated as presentedin Table 6 with the coverages approaching the nominalconfidence of 95 (that is 1 minus 120572 = 095) as the sample sizesincrease

32 Numerical Example To better illustrate an application ofthe method a numerical application based on the data con-sidered in Table 7 collected in a large supermarket network

Mathematical Problems in Engineering 9

MLEBeta

JeffreysBootstrap

minus0080

minus0060

minus0040

minus0020

0000

0020

0040

Error

010 020 050 090 099001흆

(a) Average error for 120588

MLEBeta

JeffreysBootstrap

0000000010000200003000040000500006000070000800009000100

MSE

010 020 050 090 099001흆

(b) Average MSE for 120588

MLEBeta

JeffreysBootstrap

minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0000

Error

20 50 100 20010n

(c) Average error for 119899

MLEBeta

JeffreysBootstrap

00000

00020

00040

00060

00080

00100

00120

00140

MSE

20 50 100 20010n

(d) Average MSE for 119899

Figure 3 Performance of estimators for 120588 and119870 = 5

Table 4 Mean estimates for 119871 and the mean squared error (MSE in parentheses) for 119870 = 20Estimator 119871 120588 119899

10 20 50 100

MLE

050 033333 045010 (001698) 047586 (000806) 049098 (000303) 049523 (000150)100 050000 090008 (002353) 095255 (000877) 098190 (000286) 099007 (000137)200 066714 180441 (004755) 190696 (001287) 196438 (000283) 198120 (000112)400 080959 372455 (008082) 387956 (001669) 395673 (000268) 397720 (000092)800 094574 788487 (001625) 797002 (000230) 799390 (000057) 799583 (000029)160 123520 162965 (009872) 161779 (003702) 160743 (000747) 160377 (000238)

Bootstrap

050 033333 048573 (001586) 049686 (000777) 050033 (000299) 050009 (000149)100 050000 097570 (001460) 099583 (000661) 100076 (000254) 099982 (000127)200 066714 196026 (001062) 199245 (000416) 200066 (000154) 199975 (000076)400 080959 397317 (000515) 400156 (000204) 400475 (000081) 400113 (000039)800 094574 796978 (000350) 800755 (000138) 800792 (000059) 800277 (000027)160 123520 161315 (003011) 160376 (000699) 160043 (000188) 160037 (000095)

10 Mathematical Problems in Engineering

MLEBeta

JeffreysBootstrap

010 020 050 090 099001흆

minus0030

minus0020

minus0010

0000

0010

0020

0030

0040

Error

(a) Average error for 120588

MLEBeta

JeffreysBootstrap

00000

00010

00020

00030

00040

00050

00060

MSE

010 020 050 090 099001흆

(b) Average MSE for 120588

MLEBeta

JeffreysBootstrap

minus0015

minus0010

minus0005

0000

0005

0010

0015

0020

Error

20 50 100 20010n

(c) Average error for 119899

MLEBeta

JeffreysBootstrap

00000

00010

00020

00030

00040

00050

00060

00070

00080

MSE

20 50 100 20010n

(d) Average MSE for 119899

Figure 4 Performance of estimators for 120588 and119870 = 20

Table 5 Mean estimates for 119871119902 and the mean squared error (MSE in parentheses) for 119870 = 20Estimator 119871119902 120588 119899

10 20 50 100

MLE

050 050000 042637 (001897) 046470 (000776) 048647 (000272) 049256 (000132)100 061818 086547 (002882) 093515 (000918) 097534 (000248) 098726 (000108)200 073449 177618 (005718) 189501 (001417) 196115 (000266) 197942 (000099)400 084574 374340 (006999) 389252 (001340) 396186 (000214) 397930 (000077)800 097126 794718 (000577) 799972 (000141) 800478 (000056) 800250 (000028)160 132722 161536 (003518) 161286 (002392) 160683 (000800) 160254 (000210)

Bootstrap

050 050000 048185 (001434) 049688 (000660) 050057 (000254) 049986 (000127)100 061818 097175 (001146) 099492 (000493) 100110 (000185) 100049 (000092)200 073449 196521 (000787) 199479 (000303) 200236 (000113) 200027 (000056)400 084574 397906 (000409) 400426 (000174) 400494 (000069) 400067 (000033)800 097126 796303 (000394) 800653 (000136) 800732 (000058) 800362 (000028)160 132722 161202 (003015) 160862 (001687) 160226 (000418) 160228 (000229)

Mathematical Problems in Engineering 11

MLEBeta

JeffreysBootstrap

minus0015minus0010minus000500000005001000150020002500300035

Error

010 020 050 090 099001흆

(a) Average error for 120588

MLEBeta

JeffreysBootstrap

00000

00010

00020

00030

00040

00050

00060

MSE

010 020 050 090 099001흆

(b) Average MSE for 120588

MLEBeta

JeffreysBootstrap

minus0015

minus0010

minus0005

0000

0005

0010

0015

0020

0025

Error

20 50 100 20010n

(c) Average error for 119899

MLEBeta

JeffreysBootstrap

00000

00010

00020

00030

00040

00050

00060

00070

MSE

20 50 100 20010n

(d) Average MSE for 119899

Figure 5 Performance of estimators for 120588 and 119870 = 80

Table 6 Average length (L) and coverage (C) of 95 confidence intervals for 120588 and119870 = 20Method 120588 119899 = 10 119899 = 20 119899 = 50 119899 = 100 119899 = 200

L C L C L C L C L C

Empirical Bootstrap

010 0189 0626 0185 0612 0143 0885 0106 0906 0077 0933020 0308 0619 0265 0797 0186 0902 0135 0923 0097 0940050 0402 0836 0294 0892 0191 0924 0137 0938 0097 0943090 0207 0835 0148 0911 0094 0958 0066 0953 0046 0952

Normal Approximation

010 0219 0636 0200 0865 0146 0890 0107 0911 0077 0936020 0335 0868 0274 0835 0188 0909 0136 0934 0098 0942050 0410 0885 0298 0916 0193 0934 0137 0942 0098 0947090 0213 0906 0149 0937 0094 0953 0066 0950 0046 0950

in a region of interest [12] is providedThe goal is to evaluatetraffic intensity which for managerial reasons should notexceed 87 if it does users may leave The data comprise200 random observations of the number of customers in thesystem at random times sufficiently spaced and previouslydefined by the person responsible for collecting the data to

avoid correlationThe observed values (V) and frequency (F)are presented inTable 7 For instance of the 200 observationsat 8 times no customers were found in the system at 21 timesonly 1 customer was found and so on

The estimates are shown in Table 8 they were calculatedusing the MLE (Listing 1) and the Bayesian estimators

12 Mathematical Problems in Engineering

MLEBootstrap

100 200 400 800 160050L

minus0150

minus0100

minus0050

0000

0050

0100

0150

0200

Error

(a) Average error for 119871

MLEBootstrap

00000

00050

00100

00150

00200

00250

00300

00350

00400

MSE

100 200 400 800 160050L

(b) Average MSE for 119871

MLEBootstrap

minus0080minus0070minus0060minus0050minus0040minus0030minus0020minus001000000010

Error

20 50 10010n

(c) Average error for 119899

MLEBootstrap

0000000050001000015000200002500030000350004000045000500

MSE

20 50 10010n

(d) Average MSE for 119899

Figure 6 Performance of estimates for 119871Table 7 Observed values (V) and frequency (F) for a sample of size119899 = 200 with 119870 = 14 [12]V F0 81 212 273 394 295 286 167 158 69 510 211 114 3

(Listings 2 and 3) based on an a prioriBeta(10 10) and 5000samples and Jeffreysrsquo distribution and 1000 samples from

Table 8 Point estimates for the numerical example

Estimate MLE Bayes Jeffreys Bootstrap Corrected120588 08405396 08403625 08403625 08405683

the a posteriori distribution the bootstrap corrected MLE(Listing 4) was based on 1000 resamplings The completescript is shown in Listing 7

According to the results presented in the previous sectionthe Bayesian estimates should be the most reliable It isnotable that the system utilization seemed to be below thetarget (120588 = 08405396 lt 087) Note that the analysis is basedonly on counts of the number of users in the system It is notnecessary to estimate the arrival and service rates separatelyto determine 1205884 Conclusions and Final Observations

The problem of traffic intensity estimation in finite Markovqueues (1198721198721119870 queues) is presented as quite challengingIn fact no estimator is absolutely superior to another in all

Mathematical Problems in Engineering 13

MLEBootstrap

minus0150

minus0100

minus0050

0000

0050

0100

0150

Error

100 200 400 800 160050푳풒

(a) Average error for 119871119902

MLEBootstrap

00000

00050

00100

00150

00200

00250

MSE

100 200 400 800 160050푳풒

(b) Average MSE for 119871119902

MLEBootstrap

minus0120

minus0100

minus0080

minus0060

minus0040

minus0020

0000

0020

Error

20 50 10010n

(c) Average error for 119899

MLEBootstrap

00000

00050

00100

00150

00200

00250

00300

00350

00400

MSE

20 50 10010n

(d) Average MSE for 119899

Figure 7 Performance of estimates for 119871119902

read samplesampltminusc(rep(0 8) rep(1 21) rep(2 27) rep(3 39) rep(4 29) rep(5 28) rep(6 16)rep(7 15) rep(8 6) rep(9 5) rep(10 2) rep(11 1) rep(14 3)) MLE estimateKltminus14hatrhoMLEltminusMLERoMM1K(K samp) Bayesian estimatealtminus10bltminus10setseed(13579)hatrhoBayesltminusEBaRoMM1K(K samp a b) Bayesian Jeffreys estimatesetseed(13579)hatrhoJeltminusEJeRoMM1K(K samp) Bootstrap corrected estimatesetseed(13579)hatrhoBootltminusEBoRoMM1K(K samp)c(hatrhoMLE hatrhoBayes hatrhoJe hatrhoBoot)gt [1] 08405396 08403625 08403625 08405683

Listing 7 Estimations from real data

14 Mathematical Problems in Engineering

parametric space Although the estimates of the MLE andthe bootstrap corrected MLE exhibit less bias the Bayesianestimates (beta and Jeffreysrsquo prior) present the lowest MSEin general Perhaps due to the skewness of the a posteriordistribution the Bayesian estimators do not present low biasIn general for sample size 119899 = 50 and queues with 119870 ge 20the average estimation error is less than 0005 this value wasonly exceeded by the Bayesian estimator for queues with totalcapacity 119870 = 5

In regard to the behavior of the average estimation errorand the average MSE as functions of the traffic intensity 120588the major errors are observed when the sample size is small(119899 le 20) and the traffic intensities are 120588 asymp 10 unlike inthe case of the 1198721198721 queues which exhibit higher biaseswhen 120588 asymp 05 Perhaps due to the truncation of the numberof users to the maximum queue length119870 systems with hightraffic intensities require more computational effort and arethe most difficult to estimate

Finally it is important to note that for queues withcapacity 119870 = 80 the systemrsquos behavior is similar to that of aninfinite Markovian queue (an 1198721198721 queue) as expectedThat is the average estimation error is greater and the MSEis highest when 120588 asymp 05

Future work in this area includes testing other Bayesianpoint estimators (eg the median because of the asym-metry of the a posterior distribution) developing intervalestimators hypothesis testingmethods or even Kernel-basedmethods [24]

Data Availability

The data used to support the findings of this study areincluded within the article

Disclosure

The Brazilian government funding agencies mentioned hadno role in the study

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

Special thanks are due to Gabriel and Carolina for helpingwith the algebra This work was supported by the Brazil-ian agencies CNPq (Conselho Nacional de DesenvolvimentoCientıfico e Tecnologico of the Ministry for Science andTechnology) [Grant nos 3046712014-2 3058412016-5] andFAPEMIG (Fundacao de Amparo a Pesquisa do Estado deMinas Gerais) [Grant nos CEX-PPM-00564-17 APQ-02119-15 and BIP-00106-16]

References

[1] D Ma D Wang Y Bie F Sun and S Jin ldquoA method for queuelength estimation in an urban street network based on roll time

occupancy datardquo Mathematical Problems in Engineering vol2012 p 12 2012 httpdxdoiorg1011552012892575

[2] S Zhao S Liang H Liu and M Ma ldquoCTM based real-timequeue length estimation at signalized intersectionrdquo Mathemat-ical Problems in Engineering vol 2015 Article ID 328712 12pages 2015

[3] C Armero and M J Bayarri ldquoBayesian prediction in 1198721198721queuesrdquo Queueing Systems vol 15 no 1-4 pp 401ndash417 1994

[4] C Armero and M J Bayarri ldquoQueuesrdquo in Proceedings of theInternational Encyclopedia of the Social Behavioral Sciences J DWright Ed pp 784ndash789 Oxford UK 2015

[5] C Armero and D Conesa ldquoInference and prediction in bulkarrival queues and queues with service in stagesrdquo AppliedStochastic Models in Business and Industry vol 14 no 1 pp 35ndash46 1998

[6] C Armero and D Conesa ldquoPrediction in Markovian bulkarrival queuesrdquo Queueing Systems vol 34 no 1-4 pp 327ndash3502000

[7] C Armero and D Conesa ldquoStatistical performance of a mul-ticlass bulk production queueing systemrdquo European Journal ofOperational Research vol 158 no 3 pp 649ndash661 2004

[8] C Armero and D Conesa ldquoBayesian hierarchical models inmanufacturing bulk service queuesrdquo Journal of Statistical Plan-ning and Inference vol 136 no 2 pp 335ndash354 2006

[9] A Choudhury and A C Borthakur ldquoBayesian inference andprediction in the single server Markovian queuerdquo MetrikaInternational Journal for Theoretical and Applied Statistics vol67 no 3 pp 371ndash383 2008

[10] S Chowdhury and S P Mukherjee ldquoEstimation of trafficintensity based on queue length in a single MM1 queuerdquoCommunications in StatisticsmdashTheory and Methods vol 42 no13 pp 2376ndash2390 2013

[11] S Chowdhury and S P Mukherjee ldquoBayes estimation inMM1 queues with bivariate priorrdquo Journal of Statistics andManagement Systems vol 19 no 5 pp 681ndash699 2016

[12] F R Cruz R C Quinino and L L Ho ldquoBayesian estimationof traffic intensity based on queue length in a multi-serverMMS queuerdquo Communications in StatisticsmdashSimulation andComputation vol 46 no 9 pp 7319ndash7331 2017

[13] R C Quinino and F R B Cruz ldquoBayesian sample sizesin an MM1 queueing systemsrdquo The International Journal ofAdvanced Manufacturing Technology vol 88 no 1-4 pp 995ndash1002 2017

[14] D Gross J F Shortle and J M Thompson Fundamentals ofQueueing Theory Wiley-Interscience New York NY USA 4thedition 2009

[15] M A C Almeida and F R B Cruz ldquoA note on Bayesianestimation of traffic intensity in single-server MarkovianqueuesrdquoCommunications in Statistics -SimulationComputationhttpdxdoiorg1010800361091820171353614

[16] R LThomasson and C H Kapadia ldquoOn estimating the param-eter of a truncated geometric distributionrdquo Annals of theInstitute of Statistical Mathematics vol 20 pp 519ndash523 1968

[17] R Core Team R A Language and Environment for StatisticalComputing R Foundation for Statistical Computing ViennaAustria 2017 httpwwwR-projectorg

[18] N Mukhopadhyay Probability and Statistical Inference MarcelDekker New York NY USA 2000

[19] G S Lingappaiah ldquoBayes inference in right truncated geo-metric distributionrdquo Malaysian Mathematical Society BulletinSecond Series vol 15 no 2 pp 61ndash67 1992

8 Mathematical Problems in Engineering

Table 3 Mean estimates for 120588 and the mean squared error (MSE in parentheses) for 119870 = 80Estimate 120588 119899

10 20 50 100 200

MLE

001 000928 (000084) 000960 (000045) 001000 (000019) 001003 (000010) 001007 (000005)010 009158 (000710) 009609 (000383) 009844 (000159) 009906 (000081) 009964 (000041)020 018345 (001178) 019186 (000631) 019684 (000255) 019827 (000128) 019919 (000064)050 047366 (001422) 048783 (000666) 049543 (000255) 049750 (000127) 049885 (000062)090 089001 (000144) 089551 (000057) 089835 (000020) 089915 (000010) 089960 (000005)099 098737 (000016) 098903 (000008) 098982 (000003) 098995 (000002) 098998 (000001)

Beta

001 009009 (000692) 005374 (000226) 002864 (000052) 001953 (000018) 001487 (000007)010 015477 (000743) 012978 (000388) 011242 (000159) 010616 (000080) 010320 (000041)020 022892 (000850) 021530 (000529) 020632 (000237) 020304 (000124) 020157 (000063)050 047779 (001125) 048892 (000597) 049556 (000245) 049753 (000125) 049885 (000062)090 088221 (000180) 089167 (000065) 089680 (000021) 089837 (000010) 089921 (000005)099 098149 (000017) 098570 (000007) 098856 (000003) 098953 (000002) 098990 (000001)

Jeffreys

001 005225 (000231) 003334 (000089) 001998 (000027) 001506 (000011) 001253 (000005)010 011744 (000482) 011002 (000316) 010425 (000149) 010203 (000078) 010113 (000040)020 019224 (000794) 019572 (000502) 019866 (000233) 019927 (000123) 019970 (000063)050 046147 (001532) 048167 (000767) 049430 (000302) 049826 (000151) 050064 (000074)090 088217 (000180) 089165 (000065) 089680 (000021) 089837 (000010) 089921 (000005)099 098148 (000017) 098570 (000007) 098856 (000003) 098953 (000002) 098990 (000001)

Bootstrap

001 000993 (000096) 001000 (000049) 001018 (000020) 001013 (000010) 001012 (000005)010 009763 (000795) 009976 (000409) 010010 (000164) 009994 (000082) 010007 (000041)020 019462 (001281) 019846 (000661) 019982 (000260) 019982 (000130) 019997 (000065)050 049390 (001405) 049897 (000659) 050019 (000254) 049995 (000127) 050008 (000062)090 089910 (000118) 089987 (000051) 090008 (000019) 090002 (000009) 090003 (000005)099 099000 (000016) 099042 (000009) 099031 (000004) 099012 (000002) 099002 (000001)

difficulty of estimation for traffic intensities120588 asymp 10 seemed todecrease and the highest MSEs occurred when 120588 asymp 05 TheBayesian estimators maintained the performance presentedearlier for 119870 = 5 that is the estimates have positive biasfor 120588 lt 05 and negative bias otherwise The errors of all ofthe estimators converged to zero as the sample size grewThebootstrap corrected MLE presented an average estimationerror near zero for samples 119899 ge 50 However from the pointof view of the MSE the smaller values were again obtainedusing the Bayesian methods (beta and Jeffreysrsquo prior)

Finally for queues with 119870 = 80 (Figure 5) the observedbehavior could be considered for practical purposes as beingequal to that of an infinite Markovian queue in terms of theaverage estimation error and the MSE This behavior wasobserved for infinite Markovian queues (1198721198721 queues)[15] This finding is merely evidence of the correctness of ourimplementations and the quality of the computational resultspresented

Additionally computational experiments were per-formed for the estimators for 119871 (3) and for 119871119902 (4) and fortheir bootstrap corrected versions with 1000 resamplingsand 119870 = 20 Table 4 and Figure 6 show the resultsobtained for 119871 isin 05 1 2 4 8 16 for sample sizes119899 isin 10 20 50 100 and averages of 10000 Monte Carloreplications Similarly for 119871119902 the results are presented inTable 5 and Figure 7 In summary at an extra cost of thebootstrap method and without inflation of the MSEs the

researcher may achieve with samples of size 119899 = 10 estimatesfor 119871 and 119871119902 with the same average error of the MLE forsamples of size 119899 = 100 a reduction that is relevant inpractical terms because it may lead to reduction in time andcost to obtain the estimates Note that the bootstrap methodalways provides smaller errors and MSEs than the MLEmethod for all estimates of 119871 and 119871119902 even when 120588 gt 1 Alsonote the jump up and down in the errors and MSEs when 120588transitions from 120588 lt 1 to 120588 gt 1

Finally to illustrate the ease of use of the bootstrap in theinterval estimation of the traffic intensity 120588 computationalexperiments were performed The length and coverage ofempirical bootstrap intervals computed from (18) and froma normal distribution approximation (ie Θ plusmn 1199111205722120590Θ where119911120574 is the 120574100th percentile of the standard normal distri-bution and the standard deviation 120590Θ was estimated also bybootstrapping) were evaluated for 120588 isin 010 020 050 090for sample sizes 119899 isin 10 20 50 100 200 averages of 10000Monte Carlo replications and 119870 = 20 The satisfactoryperformance of the bootstrapwas demonstrated as presentedin Table 6 with the coverages approaching the nominalconfidence of 95 (that is 1 minus 120572 = 095) as the sample sizesincrease

32 Numerical Example To better illustrate an application ofthe method a numerical application based on the data con-sidered in Table 7 collected in a large supermarket network

Mathematical Problems in Engineering 9

MLEBeta

JeffreysBootstrap

minus0080

minus0060

minus0040

minus0020

0000

0020

0040

Error

010 020 050 090 099001흆

(a) Average error for 120588

MLEBeta

JeffreysBootstrap

0000000010000200003000040000500006000070000800009000100

MSE

010 020 050 090 099001흆

(b) Average MSE for 120588

MLEBeta

JeffreysBootstrap

minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0000

Error

20 50 100 20010n

(c) Average error for 119899

MLEBeta

JeffreysBootstrap

00000

00020

00040

00060

00080

00100

00120

00140

MSE

20 50 100 20010n

(d) Average MSE for 119899

Figure 3 Performance of estimators for 120588 and119870 = 5

Table 4 Mean estimates for 119871 and the mean squared error (MSE in parentheses) for 119870 = 20Estimator 119871 120588 119899

10 20 50 100

MLE

050 033333 045010 (001698) 047586 (000806) 049098 (000303) 049523 (000150)100 050000 090008 (002353) 095255 (000877) 098190 (000286) 099007 (000137)200 066714 180441 (004755) 190696 (001287) 196438 (000283) 198120 (000112)400 080959 372455 (008082) 387956 (001669) 395673 (000268) 397720 (000092)800 094574 788487 (001625) 797002 (000230) 799390 (000057) 799583 (000029)160 123520 162965 (009872) 161779 (003702) 160743 (000747) 160377 (000238)

Bootstrap

050 033333 048573 (001586) 049686 (000777) 050033 (000299) 050009 (000149)100 050000 097570 (001460) 099583 (000661) 100076 (000254) 099982 (000127)200 066714 196026 (001062) 199245 (000416) 200066 (000154) 199975 (000076)400 080959 397317 (000515) 400156 (000204) 400475 (000081) 400113 (000039)800 094574 796978 (000350) 800755 (000138) 800792 (000059) 800277 (000027)160 123520 161315 (003011) 160376 (000699) 160043 (000188) 160037 (000095)

10 Mathematical Problems in Engineering

MLEBeta

JeffreysBootstrap

010 020 050 090 099001흆

minus0030

minus0020

minus0010

0000

0010

0020

0030

0040

Error

(a) Average error for 120588

MLEBeta

JeffreysBootstrap

00000

00010

00020

00030

00040

00050

00060

MSE

010 020 050 090 099001흆

(b) Average MSE for 120588

MLEBeta

JeffreysBootstrap

minus0015

minus0010

minus0005

0000

0005

0010

0015

0020

Error

20 50 100 20010n

(c) Average error for 119899

MLEBeta

JeffreysBootstrap

00000

00010

00020

00030

00040

00050

00060

00070

00080

MSE

20 50 100 20010n

(d) Average MSE for 119899

Figure 4 Performance of estimators for 120588 and119870 = 20

Table 5 Mean estimates for 119871119902 and the mean squared error (MSE in parentheses) for 119870 = 20Estimator 119871119902 120588 119899

10 20 50 100

MLE

050 050000 042637 (001897) 046470 (000776) 048647 (000272) 049256 (000132)100 061818 086547 (002882) 093515 (000918) 097534 (000248) 098726 (000108)200 073449 177618 (005718) 189501 (001417) 196115 (000266) 197942 (000099)400 084574 374340 (006999) 389252 (001340) 396186 (000214) 397930 (000077)800 097126 794718 (000577) 799972 (000141) 800478 (000056) 800250 (000028)160 132722 161536 (003518) 161286 (002392) 160683 (000800) 160254 (000210)

Bootstrap

050 050000 048185 (001434) 049688 (000660) 050057 (000254) 049986 (000127)100 061818 097175 (001146) 099492 (000493) 100110 (000185) 100049 (000092)200 073449 196521 (000787) 199479 (000303) 200236 (000113) 200027 (000056)400 084574 397906 (000409) 400426 (000174) 400494 (000069) 400067 (000033)800 097126 796303 (000394) 800653 (000136) 800732 (000058) 800362 (000028)160 132722 161202 (003015) 160862 (001687) 160226 (000418) 160228 (000229)

Mathematical Problems in Engineering 11

MLEBeta

JeffreysBootstrap

minus0015minus0010minus000500000005001000150020002500300035

Error

010 020 050 090 099001흆

(a) Average error for 120588

MLEBeta

JeffreysBootstrap

00000

00010

00020

00030

00040

00050

00060

MSE

010 020 050 090 099001흆

(b) Average MSE for 120588

MLEBeta

JeffreysBootstrap

minus0015

minus0010

minus0005

0000

0005

0010

0015

0020

0025

Error

20 50 100 20010n

(c) Average error for 119899

MLEBeta

JeffreysBootstrap

00000

00010

00020

00030

00040

00050

00060

00070

MSE

20 50 100 20010n

(d) Average MSE for 119899

Figure 5 Performance of estimators for 120588 and 119870 = 80

Table 6 Average length (L) and coverage (C) of 95 confidence intervals for 120588 and119870 = 20Method 120588 119899 = 10 119899 = 20 119899 = 50 119899 = 100 119899 = 200

L C L C L C L C L C

Empirical Bootstrap

010 0189 0626 0185 0612 0143 0885 0106 0906 0077 0933020 0308 0619 0265 0797 0186 0902 0135 0923 0097 0940050 0402 0836 0294 0892 0191 0924 0137 0938 0097 0943090 0207 0835 0148 0911 0094 0958 0066 0953 0046 0952

Normal Approximation

010 0219 0636 0200 0865 0146 0890 0107 0911 0077 0936020 0335 0868 0274 0835 0188 0909 0136 0934 0098 0942050 0410 0885 0298 0916 0193 0934 0137 0942 0098 0947090 0213 0906 0149 0937 0094 0953 0066 0950 0046 0950

in a region of interest [12] is providedThe goal is to evaluatetraffic intensity which for managerial reasons should notexceed 87 if it does users may leave The data comprise200 random observations of the number of customers in thesystem at random times sufficiently spaced and previouslydefined by the person responsible for collecting the data to

avoid correlationThe observed values (V) and frequency (F)are presented inTable 7 For instance of the 200 observationsat 8 times no customers were found in the system at 21 timesonly 1 customer was found and so on

The estimates are shown in Table 8 they were calculatedusing the MLE (Listing 1) and the Bayesian estimators

12 Mathematical Problems in Engineering

MLEBootstrap

100 200 400 800 160050L

minus0150

minus0100

minus0050

0000

0050

0100

0150

0200

Error

(a) Average error for 119871

MLEBootstrap

00000

00050

00100

00150

00200

00250

00300

00350

00400

MSE

100 200 400 800 160050L

(b) Average MSE for 119871

MLEBootstrap

minus0080minus0070minus0060minus0050minus0040minus0030minus0020minus001000000010

Error

20 50 10010n

(c) Average error for 119899

MLEBootstrap

0000000050001000015000200002500030000350004000045000500

MSE

20 50 10010n

(d) Average MSE for 119899

Figure 6 Performance of estimates for 119871Table 7 Observed values (V) and frequency (F) for a sample of size119899 = 200 with 119870 = 14 [12]V F0 81 212 273 394 295 286 167 158 69 510 211 114 3

(Listings 2 and 3) based on an a prioriBeta(10 10) and 5000samples and Jeffreysrsquo distribution and 1000 samples from

Table 8 Point estimates for the numerical example

Estimate MLE Bayes Jeffreys Bootstrap Corrected120588 08405396 08403625 08403625 08405683

the a posteriori distribution the bootstrap corrected MLE(Listing 4) was based on 1000 resamplings The completescript is shown in Listing 7

According to the results presented in the previous sectionthe Bayesian estimates should be the most reliable It isnotable that the system utilization seemed to be below thetarget (120588 = 08405396 lt 087) Note that the analysis is basedonly on counts of the number of users in the system It is notnecessary to estimate the arrival and service rates separatelyto determine 1205884 Conclusions and Final Observations

The problem of traffic intensity estimation in finite Markovqueues (1198721198721119870 queues) is presented as quite challengingIn fact no estimator is absolutely superior to another in all

Mathematical Problems in Engineering 13

MLEBootstrap

minus0150

minus0100

minus0050

0000

0050

0100

0150

Error

100 200 400 800 160050푳풒

(a) Average error for 119871119902

MLEBootstrap

00000

00050

00100

00150

00200

00250

MSE

100 200 400 800 160050푳풒

(b) Average MSE for 119871119902

MLEBootstrap

minus0120

minus0100

minus0080

minus0060

minus0040

minus0020

0000

0020

Error

20 50 10010n

(c) Average error for 119899

MLEBootstrap

00000

00050

00100

00150

00200

00250

00300

00350

00400

MSE

20 50 10010n

(d) Average MSE for 119899

Figure 7 Performance of estimates for 119871119902

read samplesampltminusc(rep(0 8) rep(1 21) rep(2 27) rep(3 39) rep(4 29) rep(5 28) rep(6 16)rep(7 15) rep(8 6) rep(9 5) rep(10 2) rep(11 1) rep(14 3)) MLE estimateKltminus14hatrhoMLEltminusMLERoMM1K(K samp) Bayesian estimatealtminus10bltminus10setseed(13579)hatrhoBayesltminusEBaRoMM1K(K samp a b) Bayesian Jeffreys estimatesetseed(13579)hatrhoJeltminusEJeRoMM1K(K samp) Bootstrap corrected estimatesetseed(13579)hatrhoBootltminusEBoRoMM1K(K samp)c(hatrhoMLE hatrhoBayes hatrhoJe hatrhoBoot)gt [1] 08405396 08403625 08403625 08405683

Listing 7 Estimations from real data

14 Mathematical Problems in Engineering

parametric space Although the estimates of the MLE andthe bootstrap corrected MLE exhibit less bias the Bayesianestimates (beta and Jeffreysrsquo prior) present the lowest MSEin general Perhaps due to the skewness of the a posteriordistribution the Bayesian estimators do not present low biasIn general for sample size 119899 = 50 and queues with 119870 ge 20the average estimation error is less than 0005 this value wasonly exceeded by the Bayesian estimator for queues with totalcapacity 119870 = 5

In regard to the behavior of the average estimation errorand the average MSE as functions of the traffic intensity 120588the major errors are observed when the sample size is small(119899 le 20) and the traffic intensities are 120588 asymp 10 unlike inthe case of the 1198721198721 queues which exhibit higher biaseswhen 120588 asymp 05 Perhaps due to the truncation of the numberof users to the maximum queue length119870 systems with hightraffic intensities require more computational effort and arethe most difficult to estimate

Finally it is important to note that for queues withcapacity 119870 = 80 the systemrsquos behavior is similar to that of aninfinite Markovian queue (an 1198721198721 queue) as expectedThat is the average estimation error is greater and the MSEis highest when 120588 asymp 05

Future work in this area includes testing other Bayesianpoint estimators (eg the median because of the asym-metry of the a posterior distribution) developing intervalestimators hypothesis testingmethods or even Kernel-basedmethods [24]

Data Availability

The data used to support the findings of this study areincluded within the article

Disclosure

The Brazilian government funding agencies mentioned hadno role in the study

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

Special thanks are due to Gabriel and Carolina for helpingwith the algebra This work was supported by the Brazil-ian agencies CNPq (Conselho Nacional de DesenvolvimentoCientıfico e Tecnologico of the Ministry for Science andTechnology) [Grant nos 3046712014-2 3058412016-5] andFAPEMIG (Fundacao de Amparo a Pesquisa do Estado deMinas Gerais) [Grant nos CEX-PPM-00564-17 APQ-02119-15 and BIP-00106-16]

References

[1] D Ma D Wang Y Bie F Sun and S Jin ldquoA method for queuelength estimation in an urban street network based on roll time

occupancy datardquo Mathematical Problems in Engineering vol2012 p 12 2012 httpdxdoiorg1011552012892575

[2] S Zhao S Liang H Liu and M Ma ldquoCTM based real-timequeue length estimation at signalized intersectionrdquo Mathemat-ical Problems in Engineering vol 2015 Article ID 328712 12pages 2015

[3] C Armero and M J Bayarri ldquoBayesian prediction in 1198721198721queuesrdquo Queueing Systems vol 15 no 1-4 pp 401ndash417 1994

[4] C Armero and M J Bayarri ldquoQueuesrdquo in Proceedings of theInternational Encyclopedia of the Social Behavioral Sciences J DWright Ed pp 784ndash789 Oxford UK 2015

[5] C Armero and D Conesa ldquoInference and prediction in bulkarrival queues and queues with service in stagesrdquo AppliedStochastic Models in Business and Industry vol 14 no 1 pp 35ndash46 1998

[6] C Armero and D Conesa ldquoPrediction in Markovian bulkarrival queuesrdquo Queueing Systems vol 34 no 1-4 pp 327ndash3502000

[7] C Armero and D Conesa ldquoStatistical performance of a mul-ticlass bulk production queueing systemrdquo European Journal ofOperational Research vol 158 no 3 pp 649ndash661 2004

[8] C Armero and D Conesa ldquoBayesian hierarchical models inmanufacturing bulk service queuesrdquo Journal of Statistical Plan-ning and Inference vol 136 no 2 pp 335ndash354 2006

[9] A Choudhury and A C Borthakur ldquoBayesian inference andprediction in the single server Markovian queuerdquo MetrikaInternational Journal for Theoretical and Applied Statistics vol67 no 3 pp 371ndash383 2008

[10] S Chowdhury and S P Mukherjee ldquoEstimation of trafficintensity based on queue length in a single MM1 queuerdquoCommunications in StatisticsmdashTheory and Methods vol 42 no13 pp 2376ndash2390 2013

[11] S Chowdhury and S P Mukherjee ldquoBayes estimation inMM1 queues with bivariate priorrdquo Journal of Statistics andManagement Systems vol 19 no 5 pp 681ndash699 2016

[12] F R Cruz R C Quinino and L L Ho ldquoBayesian estimationof traffic intensity based on queue length in a multi-serverMMS queuerdquo Communications in StatisticsmdashSimulation andComputation vol 46 no 9 pp 7319ndash7331 2017

[13] R C Quinino and F R B Cruz ldquoBayesian sample sizesin an MM1 queueing systemsrdquo The International Journal ofAdvanced Manufacturing Technology vol 88 no 1-4 pp 995ndash1002 2017

[14] D Gross J F Shortle and J M Thompson Fundamentals ofQueueing Theory Wiley-Interscience New York NY USA 4thedition 2009

[15] M A C Almeida and F R B Cruz ldquoA note on Bayesianestimation of traffic intensity in single-server MarkovianqueuesrdquoCommunications in Statistics -SimulationComputationhttpdxdoiorg1010800361091820171353614

[16] R LThomasson and C H Kapadia ldquoOn estimating the param-eter of a truncated geometric distributionrdquo Annals of theInstitute of Statistical Mathematics vol 20 pp 519ndash523 1968

[17] R Core Team R A Language and Environment for StatisticalComputing R Foundation for Statistical Computing ViennaAustria 2017 httpwwwR-projectorg

[18] N Mukhopadhyay Probability and Statistical Inference MarcelDekker New York NY USA 2000

[19] G S Lingappaiah ldquoBayes inference in right truncated geo-metric distributionrdquo Malaysian Mathematical Society BulletinSecond Series vol 15 no 2 pp 61ndash67 1992

Mathematical Problems in Engineering 9

MLEBeta

JeffreysBootstrap

minus0080

minus0060

minus0040

minus0020

0000

0020

0040

Error

010 020 050 090 099001흆

(a) Average error for 120588

MLEBeta

JeffreysBootstrap

0000000010000200003000040000500006000070000800009000100

MSE

010 020 050 090 099001흆

(b) Average MSE for 120588

MLEBeta

JeffreysBootstrap

minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0000

Error

20 50 100 20010n

(c) Average error for 119899

MLEBeta

JeffreysBootstrap

00000

00020

00040

00060

00080

00100

00120

00140

MSE

20 50 100 20010n

(d) Average MSE for 119899

Figure 3 Performance of estimators for 120588 and119870 = 5

Table 4 Mean estimates for 119871 and the mean squared error (MSE in parentheses) for 119870 = 20Estimator 119871 120588 119899

10 20 50 100

MLE

050 033333 045010 (001698) 047586 (000806) 049098 (000303) 049523 (000150)100 050000 090008 (002353) 095255 (000877) 098190 (000286) 099007 (000137)200 066714 180441 (004755) 190696 (001287) 196438 (000283) 198120 (000112)400 080959 372455 (008082) 387956 (001669) 395673 (000268) 397720 (000092)800 094574 788487 (001625) 797002 (000230) 799390 (000057) 799583 (000029)160 123520 162965 (009872) 161779 (003702) 160743 (000747) 160377 (000238)

Bootstrap

050 033333 048573 (001586) 049686 (000777) 050033 (000299) 050009 (000149)100 050000 097570 (001460) 099583 (000661) 100076 (000254) 099982 (000127)200 066714 196026 (001062) 199245 (000416) 200066 (000154) 199975 (000076)400 080959 397317 (000515) 400156 (000204) 400475 (000081) 400113 (000039)800 094574 796978 (000350) 800755 (000138) 800792 (000059) 800277 (000027)160 123520 161315 (003011) 160376 (000699) 160043 (000188) 160037 (000095)

10 Mathematical Problems in Engineering

MLEBeta

JeffreysBootstrap

010 020 050 090 099001흆

minus0030

minus0020

minus0010

0000

0010

0020

0030

0040

Error

(a) Average error for 120588

MLEBeta

JeffreysBootstrap

00000

00010

00020

00030

00040

00050

00060

MSE

010 020 050 090 099001흆

(b) Average MSE for 120588

MLEBeta

JeffreysBootstrap

minus0015

minus0010

minus0005

0000

0005

0010

0015

0020

Error

20 50 100 20010n

(c) Average error for 119899

MLEBeta

JeffreysBootstrap

00000

00010

00020

00030

00040

00050

00060

00070

00080

MSE

20 50 100 20010n

(d) Average MSE for 119899

Figure 4 Performance of estimators for 120588 and119870 = 20

Table 5 Mean estimates for 119871119902 and the mean squared error (MSE in parentheses) for 119870 = 20Estimator 119871119902 120588 119899

10 20 50 100

MLE

050 050000 042637 (001897) 046470 (000776) 048647 (000272) 049256 (000132)100 061818 086547 (002882) 093515 (000918) 097534 (000248) 098726 (000108)200 073449 177618 (005718) 189501 (001417) 196115 (000266) 197942 (000099)400 084574 374340 (006999) 389252 (001340) 396186 (000214) 397930 (000077)800 097126 794718 (000577) 799972 (000141) 800478 (000056) 800250 (000028)160 132722 161536 (003518) 161286 (002392) 160683 (000800) 160254 (000210)

Bootstrap

050 050000 048185 (001434) 049688 (000660) 050057 (000254) 049986 (000127)100 061818 097175 (001146) 099492 (000493) 100110 (000185) 100049 (000092)200 073449 196521 (000787) 199479 (000303) 200236 (000113) 200027 (000056)400 084574 397906 (000409) 400426 (000174) 400494 (000069) 400067 (000033)800 097126 796303 (000394) 800653 (000136) 800732 (000058) 800362 (000028)160 132722 161202 (003015) 160862 (001687) 160226 (000418) 160228 (000229)

Mathematical Problems in Engineering 11

MLEBeta

JeffreysBootstrap

minus0015minus0010minus000500000005001000150020002500300035

Error

010 020 050 090 099001흆

(a) Average error for 120588

MLEBeta

JeffreysBootstrap

00000

00010

00020

00030

00040

00050

00060

MSE

010 020 050 090 099001흆

(b) Average MSE for 120588

MLEBeta

JeffreysBootstrap

minus0015

minus0010

minus0005

0000

0005

0010

0015

0020

0025

Error

20 50 100 20010n

(c) Average error for 119899

MLEBeta

JeffreysBootstrap

00000

00010

00020

00030

00040

00050

00060

00070

MSE

20 50 100 20010n

(d) Average MSE for 119899

Figure 5 Performance of estimators for 120588 and 119870 = 80

Table 6 Average length (L) and coverage (C) of 95 confidence intervals for 120588 and119870 = 20Method 120588 119899 = 10 119899 = 20 119899 = 50 119899 = 100 119899 = 200

L C L C L C L C L C

Empirical Bootstrap

010 0189 0626 0185 0612 0143 0885 0106 0906 0077 0933020 0308 0619 0265 0797 0186 0902 0135 0923 0097 0940050 0402 0836 0294 0892 0191 0924 0137 0938 0097 0943090 0207 0835 0148 0911 0094 0958 0066 0953 0046 0952

Normal Approximation

010 0219 0636 0200 0865 0146 0890 0107 0911 0077 0936020 0335 0868 0274 0835 0188 0909 0136 0934 0098 0942050 0410 0885 0298 0916 0193 0934 0137 0942 0098 0947090 0213 0906 0149 0937 0094 0953 0066 0950 0046 0950

in a region of interest [12] is providedThe goal is to evaluatetraffic intensity which for managerial reasons should notexceed 87 if it does users may leave The data comprise200 random observations of the number of customers in thesystem at random times sufficiently spaced and previouslydefined by the person responsible for collecting the data to

avoid correlationThe observed values (V) and frequency (F)are presented inTable 7 For instance of the 200 observationsat 8 times no customers were found in the system at 21 timesonly 1 customer was found and so on

The estimates are shown in Table 8 they were calculatedusing the MLE (Listing 1) and the Bayesian estimators

12 Mathematical Problems in Engineering

MLEBootstrap

100 200 400 800 160050L

minus0150

minus0100

minus0050

0000

0050

0100

0150

0200

Error

(a) Average error for 119871

MLEBootstrap

00000

00050

00100

00150

00200

00250

00300

00350

00400

MSE

100 200 400 800 160050L

(b) Average MSE for 119871

MLEBootstrap

minus0080minus0070minus0060minus0050minus0040minus0030minus0020minus001000000010

Error

20 50 10010n

(c) Average error for 119899

MLEBootstrap

0000000050001000015000200002500030000350004000045000500

MSE

20 50 10010n

(d) Average MSE for 119899

Figure 6 Performance of estimates for 119871Table 7 Observed values (V) and frequency (F) for a sample of size119899 = 200 with 119870 = 14 [12]V F0 81 212 273 394 295 286 167 158 69 510 211 114 3

(Listings 2 and 3) based on an a prioriBeta(10 10) and 5000samples and Jeffreysrsquo distribution and 1000 samples from

Table 8 Point estimates for the numerical example

Estimate MLE Bayes Jeffreys Bootstrap Corrected120588 08405396 08403625 08403625 08405683

the a posteriori distribution the bootstrap corrected MLE(Listing 4) was based on 1000 resamplings The completescript is shown in Listing 7

According to the results presented in the previous sectionthe Bayesian estimates should be the most reliable It isnotable that the system utilization seemed to be below thetarget (120588 = 08405396 lt 087) Note that the analysis is basedonly on counts of the number of users in the system It is notnecessary to estimate the arrival and service rates separatelyto determine 1205884 Conclusions and Final Observations

The problem of traffic intensity estimation in finite Markovqueues (1198721198721119870 queues) is presented as quite challengingIn fact no estimator is absolutely superior to another in all

Mathematical Problems in Engineering 13

MLEBootstrap

minus0150

minus0100

minus0050

0000

0050

0100

0150

Error

100 200 400 800 160050푳풒

(a) Average error for 119871119902

MLEBootstrap

00000

00050

00100

00150

00200

00250

MSE

100 200 400 800 160050푳풒

(b) Average MSE for 119871119902

MLEBootstrap

minus0120

minus0100

minus0080

minus0060

minus0040

minus0020

0000

0020

Error

20 50 10010n

(c) Average error for 119899

MLEBootstrap

00000

00050

00100

00150

00200

00250

00300

00350

00400

MSE

20 50 10010n

(d) Average MSE for 119899

Figure 7 Performance of estimates for 119871119902

read samplesampltminusc(rep(0 8) rep(1 21) rep(2 27) rep(3 39) rep(4 29) rep(5 28) rep(6 16)rep(7 15) rep(8 6) rep(9 5) rep(10 2) rep(11 1) rep(14 3)) MLE estimateKltminus14hatrhoMLEltminusMLERoMM1K(K samp) Bayesian estimatealtminus10bltminus10setseed(13579)hatrhoBayesltminusEBaRoMM1K(K samp a b) Bayesian Jeffreys estimatesetseed(13579)hatrhoJeltminusEJeRoMM1K(K samp) Bootstrap corrected estimatesetseed(13579)hatrhoBootltminusEBoRoMM1K(K samp)c(hatrhoMLE hatrhoBayes hatrhoJe hatrhoBoot)gt [1] 08405396 08403625 08403625 08405683

Listing 7 Estimations from real data

14 Mathematical Problems in Engineering

parametric space Although the estimates of the MLE andthe bootstrap corrected MLE exhibit less bias the Bayesianestimates (beta and Jeffreysrsquo prior) present the lowest MSEin general Perhaps due to the skewness of the a posteriordistribution the Bayesian estimators do not present low biasIn general for sample size 119899 = 50 and queues with 119870 ge 20the average estimation error is less than 0005 this value wasonly exceeded by the Bayesian estimator for queues with totalcapacity 119870 = 5

In regard to the behavior of the average estimation errorand the average MSE as functions of the traffic intensity 120588the major errors are observed when the sample size is small(119899 le 20) and the traffic intensities are 120588 asymp 10 unlike inthe case of the 1198721198721 queues which exhibit higher biaseswhen 120588 asymp 05 Perhaps due to the truncation of the numberof users to the maximum queue length119870 systems with hightraffic intensities require more computational effort and arethe most difficult to estimate

Finally it is important to note that for queues withcapacity 119870 = 80 the systemrsquos behavior is similar to that of aninfinite Markovian queue (an 1198721198721 queue) as expectedThat is the average estimation error is greater and the MSEis highest when 120588 asymp 05

Future work in this area includes testing other Bayesianpoint estimators (eg the median because of the asym-metry of the a posterior distribution) developing intervalestimators hypothesis testingmethods or even Kernel-basedmethods [24]

Data Availability

The data used to support the findings of this study areincluded within the article

Disclosure

The Brazilian government funding agencies mentioned hadno role in the study

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

Special thanks are due to Gabriel and Carolina for helpingwith the algebra This work was supported by the Brazil-ian agencies CNPq (Conselho Nacional de DesenvolvimentoCientıfico e Tecnologico of the Ministry for Science andTechnology) [Grant nos 3046712014-2 3058412016-5] andFAPEMIG (Fundacao de Amparo a Pesquisa do Estado deMinas Gerais) [Grant nos CEX-PPM-00564-17 APQ-02119-15 and BIP-00106-16]

References

[1] D Ma D Wang Y Bie F Sun and S Jin ldquoA method for queuelength estimation in an urban street network based on roll time

occupancy datardquo Mathematical Problems in Engineering vol2012 p 12 2012 httpdxdoiorg1011552012892575

[2] S Zhao S Liang H Liu and M Ma ldquoCTM based real-timequeue length estimation at signalized intersectionrdquo Mathemat-ical Problems in Engineering vol 2015 Article ID 328712 12pages 2015

[3] C Armero and M J Bayarri ldquoBayesian prediction in 1198721198721queuesrdquo Queueing Systems vol 15 no 1-4 pp 401ndash417 1994

[4] C Armero and M J Bayarri ldquoQueuesrdquo in Proceedings of theInternational Encyclopedia of the Social Behavioral Sciences J DWright Ed pp 784ndash789 Oxford UK 2015

[5] C Armero and D Conesa ldquoInference and prediction in bulkarrival queues and queues with service in stagesrdquo AppliedStochastic Models in Business and Industry vol 14 no 1 pp 35ndash46 1998

[6] C Armero and D Conesa ldquoPrediction in Markovian bulkarrival queuesrdquo Queueing Systems vol 34 no 1-4 pp 327ndash3502000

[7] C Armero and D Conesa ldquoStatistical performance of a mul-ticlass bulk production queueing systemrdquo European Journal ofOperational Research vol 158 no 3 pp 649ndash661 2004

[8] C Armero and D Conesa ldquoBayesian hierarchical models inmanufacturing bulk service queuesrdquo Journal of Statistical Plan-ning and Inference vol 136 no 2 pp 335ndash354 2006

[9] A Choudhury and A C Borthakur ldquoBayesian inference andprediction in the single server Markovian queuerdquo MetrikaInternational Journal for Theoretical and Applied Statistics vol67 no 3 pp 371ndash383 2008

[10] S Chowdhury and S P Mukherjee ldquoEstimation of trafficintensity based on queue length in a single MM1 queuerdquoCommunications in StatisticsmdashTheory and Methods vol 42 no13 pp 2376ndash2390 2013

[11] S Chowdhury and S P Mukherjee ldquoBayes estimation inMM1 queues with bivariate priorrdquo Journal of Statistics andManagement Systems vol 19 no 5 pp 681ndash699 2016

[12] F R Cruz R C Quinino and L L Ho ldquoBayesian estimationof traffic intensity based on queue length in a multi-serverMMS queuerdquo Communications in StatisticsmdashSimulation andComputation vol 46 no 9 pp 7319ndash7331 2017

[13] R C Quinino and F R B Cruz ldquoBayesian sample sizesin an MM1 queueing systemsrdquo The International Journal ofAdvanced Manufacturing Technology vol 88 no 1-4 pp 995ndash1002 2017

[14] D Gross J F Shortle and J M Thompson Fundamentals ofQueueing Theory Wiley-Interscience New York NY USA 4thedition 2009

[15] M A C Almeida and F R B Cruz ldquoA note on Bayesianestimation of traffic intensity in single-server MarkovianqueuesrdquoCommunications in Statistics -SimulationComputationhttpdxdoiorg1010800361091820171353614

[16] R LThomasson and C H Kapadia ldquoOn estimating the param-eter of a truncated geometric distributionrdquo Annals of theInstitute of Statistical Mathematics vol 20 pp 519ndash523 1968

[17] R Core Team R A Language and Environment for StatisticalComputing R Foundation for Statistical Computing ViennaAustria 2017 httpwwwR-projectorg

[18] N Mukhopadhyay Probability and Statistical Inference MarcelDekker New York NY USA 2000

[19] G S Lingappaiah ldquoBayes inference in right truncated geo-metric distributionrdquo Malaysian Mathematical Society BulletinSecond Series vol 15 no 2 pp 61ndash67 1992

10 Mathematical Problems in Engineering

MLEBeta

JeffreysBootstrap

010 020 050 090 099001흆

minus0030

minus0020

minus0010

0000

0010

0020

0030

0040

Error

(a) Average error for 120588

MLEBeta

JeffreysBootstrap

00000

00010

00020

00030

00040

00050

00060

MSE

010 020 050 090 099001흆

(b) Average MSE for 120588

MLEBeta

JeffreysBootstrap

minus0015

minus0010

minus0005

0000

0005

0010

0015

0020

Error

20 50 100 20010n

(c) Average error for 119899

MLEBeta

JeffreysBootstrap

00000

00010

00020

00030

00040

00050

00060

00070

00080

MSE

20 50 100 20010n

(d) Average MSE for 119899

Figure 4 Performance of estimators for 120588 and119870 = 20

Table 5 Mean estimates for 119871119902 and the mean squared error (MSE in parentheses) for 119870 = 20Estimator 119871119902 120588 119899

10 20 50 100

MLE

050 050000 042637 (001897) 046470 (000776) 048647 (000272) 049256 (000132)100 061818 086547 (002882) 093515 (000918) 097534 (000248) 098726 (000108)200 073449 177618 (005718) 189501 (001417) 196115 (000266) 197942 (000099)400 084574 374340 (006999) 389252 (001340) 396186 (000214) 397930 (000077)800 097126 794718 (000577) 799972 (000141) 800478 (000056) 800250 (000028)160 132722 161536 (003518) 161286 (002392) 160683 (000800) 160254 (000210)

Bootstrap

050 050000 048185 (001434) 049688 (000660) 050057 (000254) 049986 (000127)100 061818 097175 (001146) 099492 (000493) 100110 (000185) 100049 (000092)200 073449 196521 (000787) 199479 (000303) 200236 (000113) 200027 (000056)400 084574 397906 (000409) 400426 (000174) 400494 (000069) 400067 (000033)800 097126 796303 (000394) 800653 (000136) 800732 (000058) 800362 (000028)160 132722 161202 (003015) 160862 (001687) 160226 (000418) 160228 (000229)

Mathematical Problems in Engineering 11

MLEBeta

JeffreysBootstrap

minus0015minus0010minus000500000005001000150020002500300035

Error

010 020 050 090 099001흆

(a) Average error for 120588

MLEBeta

JeffreysBootstrap

00000

00010

00020

00030

00040

00050

00060

MSE

010 020 050 090 099001흆

(b) Average MSE for 120588

MLEBeta

JeffreysBootstrap

minus0015

minus0010

minus0005

0000

0005

0010

0015

0020

0025

Error

20 50 100 20010n

(c) Average error for 119899

MLEBeta

JeffreysBootstrap

00000

00010

00020

00030

00040

00050

00060

00070

MSE

20 50 100 20010n

(d) Average MSE for 119899

Figure 5 Performance of estimators for 120588 and 119870 = 80

Table 6 Average length (L) and coverage (C) of 95 confidence intervals for 120588 and119870 = 20Method 120588 119899 = 10 119899 = 20 119899 = 50 119899 = 100 119899 = 200

L C L C L C L C L C

Empirical Bootstrap

010 0189 0626 0185 0612 0143 0885 0106 0906 0077 0933020 0308 0619 0265 0797 0186 0902 0135 0923 0097 0940050 0402 0836 0294 0892 0191 0924 0137 0938 0097 0943090 0207 0835 0148 0911 0094 0958 0066 0953 0046 0952

Normal Approximation

010 0219 0636 0200 0865 0146 0890 0107 0911 0077 0936020 0335 0868 0274 0835 0188 0909 0136 0934 0098 0942050 0410 0885 0298 0916 0193 0934 0137 0942 0098 0947090 0213 0906 0149 0937 0094 0953 0066 0950 0046 0950

in a region of interest [12] is providedThe goal is to evaluatetraffic intensity which for managerial reasons should notexceed 87 if it does users may leave The data comprise200 random observations of the number of customers in thesystem at random times sufficiently spaced and previouslydefined by the person responsible for collecting the data to

avoid correlationThe observed values (V) and frequency (F)are presented inTable 7 For instance of the 200 observationsat 8 times no customers were found in the system at 21 timesonly 1 customer was found and so on

The estimates are shown in Table 8 they were calculatedusing the MLE (Listing 1) and the Bayesian estimators

12 Mathematical Problems in Engineering

MLEBootstrap

100 200 400 800 160050L

minus0150

minus0100

minus0050

0000

0050

0100

0150

0200

Error

(a) Average error for 119871

MLEBootstrap

00000

00050

00100

00150

00200

00250

00300

00350

00400

MSE

100 200 400 800 160050L

(b) Average MSE for 119871

MLEBootstrap

minus0080minus0070minus0060minus0050minus0040minus0030minus0020minus001000000010

Error

20 50 10010n

(c) Average error for 119899

MLEBootstrap

0000000050001000015000200002500030000350004000045000500

MSE

20 50 10010n

(d) Average MSE for 119899

Figure 6 Performance of estimates for 119871Table 7 Observed values (V) and frequency (F) for a sample of size119899 = 200 with 119870 = 14 [12]V F0 81 212 273 394 295 286 167 158 69 510 211 114 3

(Listings 2 and 3) based on an a prioriBeta(10 10) and 5000samples and Jeffreysrsquo distribution and 1000 samples from

Table 8 Point estimates for the numerical example

Estimate MLE Bayes Jeffreys Bootstrap Corrected120588 08405396 08403625 08403625 08405683

the a posteriori distribution the bootstrap corrected MLE(Listing 4) was based on 1000 resamplings The completescript is shown in Listing 7

According to the results presented in the previous sectionthe Bayesian estimates should be the most reliable It isnotable that the system utilization seemed to be below thetarget (120588 = 08405396 lt 087) Note that the analysis is basedonly on counts of the number of users in the system It is notnecessary to estimate the arrival and service rates separatelyto determine 1205884 Conclusions and Final Observations

The problem of traffic intensity estimation in finite Markovqueues (1198721198721119870 queues) is presented as quite challengingIn fact no estimator is absolutely superior to another in all

Mathematical Problems in Engineering 13

MLEBootstrap

minus0150

minus0100

minus0050

0000

0050

0100

0150

Error

100 200 400 800 160050푳풒

(a) Average error for 119871119902

MLEBootstrap

00000

00050

00100

00150

00200

00250

MSE

100 200 400 800 160050푳풒

(b) Average MSE for 119871119902

MLEBootstrap

minus0120

minus0100

minus0080

minus0060

minus0040

minus0020

0000

0020

Error

20 50 10010n

(c) Average error for 119899

MLEBootstrap

00000

00050

00100

00150

00200

00250

00300

00350

00400

MSE

20 50 10010n

(d) Average MSE for 119899

Figure 7 Performance of estimates for 119871119902

read samplesampltminusc(rep(0 8) rep(1 21) rep(2 27) rep(3 39) rep(4 29) rep(5 28) rep(6 16)rep(7 15) rep(8 6) rep(9 5) rep(10 2) rep(11 1) rep(14 3)) MLE estimateKltminus14hatrhoMLEltminusMLERoMM1K(K samp) Bayesian estimatealtminus10bltminus10setseed(13579)hatrhoBayesltminusEBaRoMM1K(K samp a b) Bayesian Jeffreys estimatesetseed(13579)hatrhoJeltminusEJeRoMM1K(K samp) Bootstrap corrected estimatesetseed(13579)hatrhoBootltminusEBoRoMM1K(K samp)c(hatrhoMLE hatrhoBayes hatrhoJe hatrhoBoot)gt [1] 08405396 08403625 08403625 08405683

Listing 7 Estimations from real data

14 Mathematical Problems in Engineering

parametric space Although the estimates of the MLE andthe bootstrap corrected MLE exhibit less bias the Bayesianestimates (beta and Jeffreysrsquo prior) present the lowest MSEin general Perhaps due to the skewness of the a posteriordistribution the Bayesian estimators do not present low biasIn general for sample size 119899 = 50 and queues with 119870 ge 20the average estimation error is less than 0005 this value wasonly exceeded by the Bayesian estimator for queues with totalcapacity 119870 = 5

In regard to the behavior of the average estimation errorand the average MSE as functions of the traffic intensity 120588the major errors are observed when the sample size is small(119899 le 20) and the traffic intensities are 120588 asymp 10 unlike inthe case of the 1198721198721 queues which exhibit higher biaseswhen 120588 asymp 05 Perhaps due to the truncation of the numberof users to the maximum queue length119870 systems with hightraffic intensities require more computational effort and arethe most difficult to estimate

Finally it is important to note that for queues withcapacity 119870 = 80 the systemrsquos behavior is similar to that of aninfinite Markovian queue (an 1198721198721 queue) as expectedThat is the average estimation error is greater and the MSEis highest when 120588 asymp 05

Future work in this area includes testing other Bayesianpoint estimators (eg the median because of the asym-metry of the a posterior distribution) developing intervalestimators hypothesis testingmethods or even Kernel-basedmethods [24]

Data Availability

The data used to support the findings of this study areincluded within the article

Disclosure

The Brazilian government funding agencies mentioned hadno role in the study

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

Special thanks are due to Gabriel and Carolina for helpingwith the algebra This work was supported by the Brazil-ian agencies CNPq (Conselho Nacional de DesenvolvimentoCientıfico e Tecnologico of the Ministry for Science andTechnology) [Grant nos 3046712014-2 3058412016-5] andFAPEMIG (Fundacao de Amparo a Pesquisa do Estado deMinas Gerais) [Grant nos CEX-PPM-00564-17 APQ-02119-15 and BIP-00106-16]

References

[1] D Ma D Wang Y Bie F Sun and S Jin ldquoA method for queuelength estimation in an urban street network based on roll time

occupancy datardquo Mathematical Problems in Engineering vol2012 p 12 2012 httpdxdoiorg1011552012892575

[2] S Zhao S Liang H Liu and M Ma ldquoCTM based real-timequeue length estimation at signalized intersectionrdquo Mathemat-ical Problems in Engineering vol 2015 Article ID 328712 12pages 2015

[3] C Armero and M J Bayarri ldquoBayesian prediction in 1198721198721queuesrdquo Queueing Systems vol 15 no 1-4 pp 401ndash417 1994

[4] C Armero and M J Bayarri ldquoQueuesrdquo in Proceedings of theInternational Encyclopedia of the Social Behavioral Sciences J DWright Ed pp 784ndash789 Oxford UK 2015

[5] C Armero and D Conesa ldquoInference and prediction in bulkarrival queues and queues with service in stagesrdquo AppliedStochastic Models in Business and Industry vol 14 no 1 pp 35ndash46 1998

[6] C Armero and D Conesa ldquoPrediction in Markovian bulkarrival queuesrdquo Queueing Systems vol 34 no 1-4 pp 327ndash3502000

[7] C Armero and D Conesa ldquoStatistical performance of a mul-ticlass bulk production queueing systemrdquo European Journal ofOperational Research vol 158 no 3 pp 649ndash661 2004

[8] C Armero and D Conesa ldquoBayesian hierarchical models inmanufacturing bulk service queuesrdquo Journal of Statistical Plan-ning and Inference vol 136 no 2 pp 335ndash354 2006

[9] A Choudhury and A C Borthakur ldquoBayesian inference andprediction in the single server Markovian queuerdquo MetrikaInternational Journal for Theoretical and Applied Statistics vol67 no 3 pp 371ndash383 2008

[10] S Chowdhury and S P Mukherjee ldquoEstimation of trafficintensity based on queue length in a single MM1 queuerdquoCommunications in StatisticsmdashTheory and Methods vol 42 no13 pp 2376ndash2390 2013

[11] S Chowdhury and S P Mukherjee ldquoBayes estimation inMM1 queues with bivariate priorrdquo Journal of Statistics andManagement Systems vol 19 no 5 pp 681ndash699 2016

[12] F R Cruz R C Quinino and L L Ho ldquoBayesian estimationof traffic intensity based on queue length in a multi-serverMMS queuerdquo Communications in StatisticsmdashSimulation andComputation vol 46 no 9 pp 7319ndash7331 2017

[13] R C Quinino and F R B Cruz ldquoBayesian sample sizesin an MM1 queueing systemsrdquo The International Journal ofAdvanced Manufacturing Technology vol 88 no 1-4 pp 995ndash1002 2017

[14] D Gross J F Shortle and J M Thompson Fundamentals ofQueueing Theory Wiley-Interscience New York NY USA 4thedition 2009

[15] M A C Almeida and F R B Cruz ldquoA note on Bayesianestimation of traffic intensity in single-server MarkovianqueuesrdquoCommunications in Statistics -SimulationComputationhttpdxdoiorg1010800361091820171353614

[16] R LThomasson and C H Kapadia ldquoOn estimating the param-eter of a truncated geometric distributionrdquo Annals of theInstitute of Statistical Mathematics vol 20 pp 519ndash523 1968

[17] R Core Team R A Language and Environment for StatisticalComputing R Foundation for Statistical Computing ViennaAustria 2017 httpwwwR-projectorg

[18] N Mukhopadhyay Probability and Statistical Inference MarcelDekker New York NY USA 2000

[19] G S Lingappaiah ldquoBayes inference in right truncated geo-metric distributionrdquo Malaysian Mathematical Society BulletinSecond Series vol 15 no 2 pp 61ndash67 1992

Mathematical Problems in Engineering 11

MLEBeta

JeffreysBootstrap

minus0015minus0010minus000500000005001000150020002500300035

Error

010 020 050 090 099001흆

(a) Average error for 120588

MLEBeta

JeffreysBootstrap

00000

00010

00020

00030

00040

00050

00060

MSE

010 020 050 090 099001흆

(b) Average MSE for 120588

MLEBeta

JeffreysBootstrap

minus0015

minus0010

minus0005

0000

0005

0010

0015

0020

0025

Error

20 50 100 20010n

(c) Average error for 119899

MLEBeta

JeffreysBootstrap

00000

00010

00020

00030

00040

00050

00060

00070

MSE

20 50 100 20010n

(d) Average MSE for 119899

Figure 5 Performance of estimators for 120588 and 119870 = 80

Table 6 Average length (L) and coverage (C) of 95 confidence intervals for 120588 and119870 = 20Method 120588 119899 = 10 119899 = 20 119899 = 50 119899 = 100 119899 = 200

L C L C L C L C L C

Empirical Bootstrap

010 0189 0626 0185 0612 0143 0885 0106 0906 0077 0933020 0308 0619 0265 0797 0186 0902 0135 0923 0097 0940050 0402 0836 0294 0892 0191 0924 0137 0938 0097 0943090 0207 0835 0148 0911 0094 0958 0066 0953 0046 0952

Normal Approximation

010 0219 0636 0200 0865 0146 0890 0107 0911 0077 0936020 0335 0868 0274 0835 0188 0909 0136 0934 0098 0942050 0410 0885 0298 0916 0193 0934 0137 0942 0098 0947090 0213 0906 0149 0937 0094 0953 0066 0950 0046 0950

in a region of interest [12] is providedThe goal is to evaluatetraffic intensity which for managerial reasons should notexceed 87 if it does users may leave The data comprise200 random observations of the number of customers in thesystem at random times sufficiently spaced and previouslydefined by the person responsible for collecting the data to

avoid correlationThe observed values (V) and frequency (F)are presented inTable 7 For instance of the 200 observationsat 8 times no customers were found in the system at 21 timesonly 1 customer was found and so on

The estimates are shown in Table 8 they were calculatedusing the MLE (Listing 1) and the Bayesian estimators

12 Mathematical Problems in Engineering

MLEBootstrap

100 200 400 800 160050L

minus0150

minus0100

minus0050

0000

0050

0100

0150

0200

Error

(a) Average error for 119871

MLEBootstrap

00000

00050

00100

00150

00200

00250

00300

00350

00400

MSE

100 200 400 800 160050L

(b) Average MSE for 119871

MLEBootstrap

minus0080minus0070minus0060minus0050minus0040minus0030minus0020minus001000000010

Error

20 50 10010n

(c) Average error for 119899

MLEBootstrap

0000000050001000015000200002500030000350004000045000500

MSE

20 50 10010n

(d) Average MSE for 119899

Figure 6 Performance of estimates for 119871Table 7 Observed values (V) and frequency (F) for a sample of size119899 = 200 with 119870 = 14 [12]V F0 81 212 273 394 295 286 167 158 69 510 211 114 3

(Listings 2 and 3) based on an a prioriBeta(10 10) and 5000samples and Jeffreysrsquo distribution and 1000 samples from

Table 8 Point estimates for the numerical example

Estimate MLE Bayes Jeffreys Bootstrap Corrected120588 08405396 08403625 08403625 08405683

the a posteriori distribution the bootstrap corrected MLE(Listing 4) was based on 1000 resamplings The completescript is shown in Listing 7

According to the results presented in the previous sectionthe Bayesian estimates should be the most reliable It isnotable that the system utilization seemed to be below thetarget (120588 = 08405396 lt 087) Note that the analysis is basedonly on counts of the number of users in the system It is notnecessary to estimate the arrival and service rates separatelyto determine 1205884 Conclusions and Final Observations

The problem of traffic intensity estimation in finite Markovqueues (1198721198721119870 queues) is presented as quite challengingIn fact no estimator is absolutely superior to another in all

Mathematical Problems in Engineering 13

MLEBootstrap

minus0150

minus0100

minus0050

0000

0050

0100

0150

Error

100 200 400 800 160050푳풒

(a) Average error for 119871119902

MLEBootstrap

00000

00050

00100

00150

00200

00250

MSE

100 200 400 800 160050푳풒

(b) Average MSE for 119871119902

MLEBootstrap

minus0120

minus0100

minus0080

minus0060

minus0040

minus0020

0000

0020

Error

20 50 10010n

(c) Average error for 119899

MLEBootstrap

00000

00050

00100

00150

00200

00250

00300

00350

00400

MSE

20 50 10010n

(d) Average MSE for 119899

Figure 7 Performance of estimates for 119871119902

read samplesampltminusc(rep(0 8) rep(1 21) rep(2 27) rep(3 39) rep(4 29) rep(5 28) rep(6 16)rep(7 15) rep(8 6) rep(9 5) rep(10 2) rep(11 1) rep(14 3)) MLE estimateKltminus14hatrhoMLEltminusMLERoMM1K(K samp) Bayesian estimatealtminus10bltminus10setseed(13579)hatrhoBayesltminusEBaRoMM1K(K samp a b) Bayesian Jeffreys estimatesetseed(13579)hatrhoJeltminusEJeRoMM1K(K samp) Bootstrap corrected estimatesetseed(13579)hatrhoBootltminusEBoRoMM1K(K samp)c(hatrhoMLE hatrhoBayes hatrhoJe hatrhoBoot)gt [1] 08405396 08403625 08403625 08405683

Listing 7 Estimations from real data

14 Mathematical Problems in Engineering

parametric space Although the estimates of the MLE andthe bootstrap corrected MLE exhibit less bias the Bayesianestimates (beta and Jeffreysrsquo prior) present the lowest MSEin general Perhaps due to the skewness of the a posteriordistribution the Bayesian estimators do not present low biasIn general for sample size 119899 = 50 and queues with 119870 ge 20the average estimation error is less than 0005 this value wasonly exceeded by the Bayesian estimator for queues with totalcapacity 119870 = 5

In regard to the behavior of the average estimation errorand the average MSE as functions of the traffic intensity 120588the major errors are observed when the sample size is small(119899 le 20) and the traffic intensities are 120588 asymp 10 unlike inthe case of the 1198721198721 queues which exhibit higher biaseswhen 120588 asymp 05 Perhaps due to the truncation of the numberof users to the maximum queue length119870 systems with hightraffic intensities require more computational effort and arethe most difficult to estimate

Finally it is important to note that for queues withcapacity 119870 = 80 the systemrsquos behavior is similar to that of aninfinite Markovian queue (an 1198721198721 queue) as expectedThat is the average estimation error is greater and the MSEis highest when 120588 asymp 05

Future work in this area includes testing other Bayesianpoint estimators (eg the median because of the asym-metry of the a posterior distribution) developing intervalestimators hypothesis testingmethods or even Kernel-basedmethods [24]

Data Availability

The data used to support the findings of this study areincluded within the article

Disclosure

The Brazilian government funding agencies mentioned hadno role in the study

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

Special thanks are due to Gabriel and Carolina for helpingwith the algebra This work was supported by the Brazil-ian agencies CNPq (Conselho Nacional de DesenvolvimentoCientıfico e Tecnologico of the Ministry for Science andTechnology) [Grant nos 3046712014-2 3058412016-5] andFAPEMIG (Fundacao de Amparo a Pesquisa do Estado deMinas Gerais) [Grant nos CEX-PPM-00564-17 APQ-02119-15 and BIP-00106-16]

References

[1] D Ma D Wang Y Bie F Sun and S Jin ldquoA method for queuelength estimation in an urban street network based on roll time

occupancy datardquo Mathematical Problems in Engineering vol2012 p 12 2012 httpdxdoiorg1011552012892575

[2] S Zhao S Liang H Liu and M Ma ldquoCTM based real-timequeue length estimation at signalized intersectionrdquo Mathemat-ical Problems in Engineering vol 2015 Article ID 328712 12pages 2015

[3] C Armero and M J Bayarri ldquoBayesian prediction in 1198721198721queuesrdquo Queueing Systems vol 15 no 1-4 pp 401ndash417 1994

[4] C Armero and M J Bayarri ldquoQueuesrdquo in Proceedings of theInternational Encyclopedia of the Social Behavioral Sciences J DWright Ed pp 784ndash789 Oxford UK 2015

[5] C Armero and D Conesa ldquoInference and prediction in bulkarrival queues and queues with service in stagesrdquo AppliedStochastic Models in Business and Industry vol 14 no 1 pp 35ndash46 1998

[6] C Armero and D Conesa ldquoPrediction in Markovian bulkarrival queuesrdquo Queueing Systems vol 34 no 1-4 pp 327ndash3502000

[7] C Armero and D Conesa ldquoStatistical performance of a mul-ticlass bulk production queueing systemrdquo European Journal ofOperational Research vol 158 no 3 pp 649ndash661 2004

[8] C Armero and D Conesa ldquoBayesian hierarchical models inmanufacturing bulk service queuesrdquo Journal of Statistical Plan-ning and Inference vol 136 no 2 pp 335ndash354 2006

[9] A Choudhury and A C Borthakur ldquoBayesian inference andprediction in the single server Markovian queuerdquo MetrikaInternational Journal for Theoretical and Applied Statistics vol67 no 3 pp 371ndash383 2008

[10] S Chowdhury and S P Mukherjee ldquoEstimation of trafficintensity based on queue length in a single MM1 queuerdquoCommunications in StatisticsmdashTheory and Methods vol 42 no13 pp 2376ndash2390 2013

[11] S Chowdhury and S P Mukherjee ldquoBayes estimation inMM1 queues with bivariate priorrdquo Journal of Statistics andManagement Systems vol 19 no 5 pp 681ndash699 2016

[12] F R Cruz R C Quinino and L L Ho ldquoBayesian estimationof traffic intensity based on queue length in a multi-serverMMS queuerdquo Communications in StatisticsmdashSimulation andComputation vol 46 no 9 pp 7319ndash7331 2017

[13] R C Quinino and F R B Cruz ldquoBayesian sample sizesin an MM1 queueing systemsrdquo The International Journal ofAdvanced Manufacturing Technology vol 88 no 1-4 pp 995ndash1002 2017

[14] D Gross J F Shortle and J M Thompson Fundamentals ofQueueing Theory Wiley-Interscience New York NY USA 4thedition 2009

[15] M A C Almeida and F R B Cruz ldquoA note on Bayesianestimation of traffic intensity in single-server MarkovianqueuesrdquoCommunications in Statistics -SimulationComputationhttpdxdoiorg1010800361091820171353614

[16] R LThomasson and C H Kapadia ldquoOn estimating the param-eter of a truncated geometric distributionrdquo Annals of theInstitute of Statistical Mathematics vol 20 pp 519ndash523 1968

[17] R Core Team R A Language and Environment for StatisticalComputing R Foundation for Statistical Computing ViennaAustria 2017 httpwwwR-projectorg

[18] N Mukhopadhyay Probability and Statistical Inference MarcelDekker New York NY USA 2000

[19] G S Lingappaiah ldquoBayes inference in right truncated geo-metric distributionrdquo Malaysian Mathematical Society BulletinSecond Series vol 15 no 2 pp 61ndash67 1992

12 Mathematical Problems in Engineering

MLEBootstrap

100 200 400 800 160050L

minus0150

minus0100

minus0050

0000

0050

0100

0150

0200

Error

(a) Average error for 119871

MLEBootstrap

00000

00050

00100

00150

00200

00250

00300

00350

00400

MSE

100 200 400 800 160050L

(b) Average MSE for 119871

MLEBootstrap

minus0080minus0070minus0060minus0050minus0040minus0030minus0020minus001000000010

Error

20 50 10010n

(c) Average error for 119899

MLEBootstrap

0000000050001000015000200002500030000350004000045000500

MSE

20 50 10010n

(d) Average MSE for 119899

Figure 6 Performance of estimates for 119871Table 7 Observed values (V) and frequency (F) for a sample of size119899 = 200 with 119870 = 14 [12]V F0 81 212 273 394 295 286 167 158 69 510 211 114 3

(Listings 2 and 3) based on an a prioriBeta(10 10) and 5000samples and Jeffreysrsquo distribution and 1000 samples from

Table 8 Point estimates for the numerical example

Estimate MLE Bayes Jeffreys Bootstrap Corrected120588 08405396 08403625 08403625 08405683

the a posteriori distribution the bootstrap corrected MLE(Listing 4) was based on 1000 resamplings The completescript is shown in Listing 7

According to the results presented in the previous sectionthe Bayesian estimates should be the most reliable It isnotable that the system utilization seemed to be below thetarget (120588 = 08405396 lt 087) Note that the analysis is basedonly on counts of the number of users in the system It is notnecessary to estimate the arrival and service rates separatelyto determine 1205884 Conclusions and Final Observations

The problem of traffic intensity estimation in finite Markovqueues (1198721198721119870 queues) is presented as quite challengingIn fact no estimator is absolutely superior to another in all

Mathematical Problems in Engineering 13

MLEBootstrap

minus0150

minus0100

minus0050

0000

0050

0100

0150

Error

100 200 400 800 160050푳풒

(a) Average error for 119871119902

MLEBootstrap

00000

00050

00100

00150

00200

00250

MSE

100 200 400 800 160050푳풒

(b) Average MSE for 119871119902

MLEBootstrap

minus0120

minus0100

minus0080

minus0060

minus0040

minus0020

0000

0020

Error

20 50 10010n

(c) Average error for 119899

MLEBootstrap

00000

00050

00100

00150

00200

00250

00300

00350

00400

MSE

20 50 10010n

(d) Average MSE for 119899

Figure 7 Performance of estimates for 119871119902

read samplesampltminusc(rep(0 8) rep(1 21) rep(2 27) rep(3 39) rep(4 29) rep(5 28) rep(6 16)rep(7 15) rep(8 6) rep(9 5) rep(10 2) rep(11 1) rep(14 3)) MLE estimateKltminus14hatrhoMLEltminusMLERoMM1K(K samp) Bayesian estimatealtminus10bltminus10setseed(13579)hatrhoBayesltminusEBaRoMM1K(K samp a b) Bayesian Jeffreys estimatesetseed(13579)hatrhoJeltminusEJeRoMM1K(K samp) Bootstrap corrected estimatesetseed(13579)hatrhoBootltminusEBoRoMM1K(K samp)c(hatrhoMLE hatrhoBayes hatrhoJe hatrhoBoot)gt [1] 08405396 08403625 08403625 08405683

Listing 7 Estimations from real data

14 Mathematical Problems in Engineering

parametric space Although the estimates of the MLE andthe bootstrap corrected MLE exhibit less bias the Bayesianestimates (beta and Jeffreysrsquo prior) present the lowest MSEin general Perhaps due to the skewness of the a posteriordistribution the Bayesian estimators do not present low biasIn general for sample size 119899 = 50 and queues with 119870 ge 20the average estimation error is less than 0005 this value wasonly exceeded by the Bayesian estimator for queues with totalcapacity 119870 = 5

In regard to the behavior of the average estimation errorand the average MSE as functions of the traffic intensity 120588the major errors are observed when the sample size is small(119899 le 20) and the traffic intensities are 120588 asymp 10 unlike inthe case of the 1198721198721 queues which exhibit higher biaseswhen 120588 asymp 05 Perhaps due to the truncation of the numberof users to the maximum queue length119870 systems with hightraffic intensities require more computational effort and arethe most difficult to estimate

Finally it is important to note that for queues withcapacity 119870 = 80 the systemrsquos behavior is similar to that of aninfinite Markovian queue (an 1198721198721 queue) as expectedThat is the average estimation error is greater and the MSEis highest when 120588 asymp 05

Future work in this area includes testing other Bayesianpoint estimators (eg the median because of the asym-metry of the a posterior distribution) developing intervalestimators hypothesis testingmethods or even Kernel-basedmethods [24]

Data Availability

The data used to support the findings of this study areincluded within the article

Disclosure

The Brazilian government funding agencies mentioned hadno role in the study

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

Special thanks are due to Gabriel and Carolina for helpingwith the algebra This work was supported by the Brazil-ian agencies CNPq (Conselho Nacional de DesenvolvimentoCientıfico e Tecnologico of the Ministry for Science andTechnology) [Grant nos 3046712014-2 3058412016-5] andFAPEMIG (Fundacao de Amparo a Pesquisa do Estado deMinas Gerais) [Grant nos CEX-PPM-00564-17 APQ-02119-15 and BIP-00106-16]

References

[1] D Ma D Wang Y Bie F Sun and S Jin ldquoA method for queuelength estimation in an urban street network based on roll time

occupancy datardquo Mathematical Problems in Engineering vol2012 p 12 2012 httpdxdoiorg1011552012892575

[2] S Zhao S Liang H Liu and M Ma ldquoCTM based real-timequeue length estimation at signalized intersectionrdquo Mathemat-ical Problems in Engineering vol 2015 Article ID 328712 12pages 2015

[3] C Armero and M J Bayarri ldquoBayesian prediction in 1198721198721queuesrdquo Queueing Systems vol 15 no 1-4 pp 401ndash417 1994

[4] C Armero and M J Bayarri ldquoQueuesrdquo in Proceedings of theInternational Encyclopedia of the Social Behavioral Sciences J DWright Ed pp 784ndash789 Oxford UK 2015

[5] C Armero and D Conesa ldquoInference and prediction in bulkarrival queues and queues with service in stagesrdquo AppliedStochastic Models in Business and Industry vol 14 no 1 pp 35ndash46 1998

[6] C Armero and D Conesa ldquoPrediction in Markovian bulkarrival queuesrdquo Queueing Systems vol 34 no 1-4 pp 327ndash3502000

[7] C Armero and D Conesa ldquoStatistical performance of a mul-ticlass bulk production queueing systemrdquo European Journal ofOperational Research vol 158 no 3 pp 649ndash661 2004

[8] C Armero and D Conesa ldquoBayesian hierarchical models inmanufacturing bulk service queuesrdquo Journal of Statistical Plan-ning and Inference vol 136 no 2 pp 335ndash354 2006

[9] A Choudhury and A C Borthakur ldquoBayesian inference andprediction in the single server Markovian queuerdquo MetrikaInternational Journal for Theoretical and Applied Statistics vol67 no 3 pp 371ndash383 2008

[10] S Chowdhury and S P Mukherjee ldquoEstimation of trafficintensity based on queue length in a single MM1 queuerdquoCommunications in StatisticsmdashTheory and Methods vol 42 no13 pp 2376ndash2390 2013

[11] S Chowdhury and S P Mukherjee ldquoBayes estimation inMM1 queues with bivariate priorrdquo Journal of Statistics andManagement Systems vol 19 no 5 pp 681ndash699 2016

[12] F R Cruz R C Quinino and L L Ho ldquoBayesian estimationof traffic intensity based on queue length in a multi-serverMMS queuerdquo Communications in StatisticsmdashSimulation andComputation vol 46 no 9 pp 7319ndash7331 2017

[13] R C Quinino and F R B Cruz ldquoBayesian sample sizesin an MM1 queueing systemsrdquo The International Journal ofAdvanced Manufacturing Technology vol 88 no 1-4 pp 995ndash1002 2017

[14] D Gross J F Shortle and J M Thompson Fundamentals ofQueueing Theory Wiley-Interscience New York NY USA 4thedition 2009

[15] M A C Almeida and F R B Cruz ldquoA note on Bayesianestimation of traffic intensity in single-server MarkovianqueuesrdquoCommunications in Statistics -SimulationComputationhttpdxdoiorg1010800361091820171353614

[16] R LThomasson and C H Kapadia ldquoOn estimating the param-eter of a truncated geometric distributionrdquo Annals of theInstitute of Statistical Mathematics vol 20 pp 519ndash523 1968

[17] R Core Team R A Language and Environment for StatisticalComputing R Foundation for Statistical Computing ViennaAustria 2017 httpwwwR-projectorg

[18] N Mukhopadhyay Probability and Statistical Inference MarcelDekker New York NY USA 2000

[19] G S Lingappaiah ldquoBayes inference in right truncated geo-metric distributionrdquo Malaysian Mathematical Society BulletinSecond Series vol 15 no 2 pp 61ndash67 1992

Mathematical Problems in Engineering 13

MLEBootstrap

minus0150

minus0100

minus0050

0000

0050

0100

0150

Error

100 200 400 800 160050푳풒

(a) Average error for 119871119902

MLEBootstrap

00000

00050

00100

00150

00200

00250

MSE

100 200 400 800 160050푳풒

(b) Average MSE for 119871119902

MLEBootstrap

minus0120

minus0100

minus0080

minus0060

minus0040

minus0020

0000

0020

Error

20 50 10010n

(c) Average error for 119899

MLEBootstrap

00000

00050

00100

00150

00200

00250

00300

00350

00400

MSE

20 50 10010n

(d) Average MSE for 119899

Figure 7 Performance of estimates for 119871119902

read samplesampltminusc(rep(0 8) rep(1 21) rep(2 27) rep(3 39) rep(4 29) rep(5 28) rep(6 16)rep(7 15) rep(8 6) rep(9 5) rep(10 2) rep(11 1) rep(14 3)) MLE estimateKltminus14hatrhoMLEltminusMLERoMM1K(K samp) Bayesian estimatealtminus10bltminus10setseed(13579)hatrhoBayesltminusEBaRoMM1K(K samp a b) Bayesian Jeffreys estimatesetseed(13579)hatrhoJeltminusEJeRoMM1K(K samp) Bootstrap corrected estimatesetseed(13579)hatrhoBootltminusEBoRoMM1K(K samp)c(hatrhoMLE hatrhoBayes hatrhoJe hatrhoBoot)gt [1] 08405396 08403625 08403625 08405683

Listing 7 Estimations from real data

14 Mathematical Problems in Engineering

parametric space Although the estimates of the MLE andthe bootstrap corrected MLE exhibit less bias the Bayesianestimates (beta and Jeffreysrsquo prior) present the lowest MSEin general Perhaps due to the skewness of the a posteriordistribution the Bayesian estimators do not present low biasIn general for sample size 119899 = 50 and queues with 119870 ge 20the average estimation error is less than 0005 this value wasonly exceeded by the Bayesian estimator for queues with totalcapacity 119870 = 5

In regard to the behavior of the average estimation errorand the average MSE as functions of the traffic intensity 120588the major errors are observed when the sample size is small(119899 le 20) and the traffic intensities are 120588 asymp 10 unlike inthe case of the 1198721198721 queues which exhibit higher biaseswhen 120588 asymp 05 Perhaps due to the truncation of the numberof users to the maximum queue length119870 systems with hightraffic intensities require more computational effort and arethe most difficult to estimate

Finally it is important to note that for queues withcapacity 119870 = 80 the systemrsquos behavior is similar to that of aninfinite Markovian queue (an 1198721198721 queue) as expectedThat is the average estimation error is greater and the MSEis highest when 120588 asymp 05

Future work in this area includes testing other Bayesianpoint estimators (eg the median because of the asym-metry of the a posterior distribution) developing intervalestimators hypothesis testingmethods or even Kernel-basedmethods [24]

Data Availability

The data used to support the findings of this study areincluded within the article

Disclosure

The Brazilian government funding agencies mentioned hadno role in the study

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

Special thanks are due to Gabriel and Carolina for helpingwith the algebra This work was supported by the Brazil-ian agencies CNPq (Conselho Nacional de DesenvolvimentoCientıfico e Tecnologico of the Ministry for Science andTechnology) [Grant nos 3046712014-2 3058412016-5] andFAPEMIG (Fundacao de Amparo a Pesquisa do Estado deMinas Gerais) [Grant nos CEX-PPM-00564-17 APQ-02119-15 and BIP-00106-16]

References

[1] D Ma D Wang Y Bie F Sun and S Jin ldquoA method for queuelength estimation in an urban street network based on roll time

occupancy datardquo Mathematical Problems in Engineering vol2012 p 12 2012 httpdxdoiorg1011552012892575

[2] S Zhao S Liang H Liu and M Ma ldquoCTM based real-timequeue length estimation at signalized intersectionrdquo Mathemat-ical Problems in Engineering vol 2015 Article ID 328712 12pages 2015

[3] C Armero and M J Bayarri ldquoBayesian prediction in 1198721198721queuesrdquo Queueing Systems vol 15 no 1-4 pp 401ndash417 1994

[4] C Armero and M J Bayarri ldquoQueuesrdquo in Proceedings of theInternational Encyclopedia of the Social Behavioral Sciences J DWright Ed pp 784ndash789 Oxford UK 2015

[5] C Armero and D Conesa ldquoInference and prediction in bulkarrival queues and queues with service in stagesrdquo AppliedStochastic Models in Business and Industry vol 14 no 1 pp 35ndash46 1998

[6] C Armero and D Conesa ldquoPrediction in Markovian bulkarrival queuesrdquo Queueing Systems vol 34 no 1-4 pp 327ndash3502000

[7] C Armero and D Conesa ldquoStatistical performance of a mul-ticlass bulk production queueing systemrdquo European Journal ofOperational Research vol 158 no 3 pp 649ndash661 2004

[8] C Armero and D Conesa ldquoBayesian hierarchical models inmanufacturing bulk service queuesrdquo Journal of Statistical Plan-ning and Inference vol 136 no 2 pp 335ndash354 2006

[9] A Choudhury and A C Borthakur ldquoBayesian inference andprediction in the single server Markovian queuerdquo MetrikaInternational Journal for Theoretical and Applied Statistics vol67 no 3 pp 371ndash383 2008

[10] S Chowdhury and S P Mukherjee ldquoEstimation of trafficintensity based on queue length in a single MM1 queuerdquoCommunications in StatisticsmdashTheory and Methods vol 42 no13 pp 2376ndash2390 2013

[11] S Chowdhury and S P Mukherjee ldquoBayes estimation inMM1 queues with bivariate priorrdquo Journal of Statistics andManagement Systems vol 19 no 5 pp 681ndash699 2016

[12] F R Cruz R C Quinino and L L Ho ldquoBayesian estimationof traffic intensity based on queue length in a multi-serverMMS queuerdquo Communications in StatisticsmdashSimulation andComputation vol 46 no 9 pp 7319ndash7331 2017

[13] R C Quinino and F R B Cruz ldquoBayesian sample sizesin an MM1 queueing systemsrdquo The International Journal ofAdvanced Manufacturing Technology vol 88 no 1-4 pp 995ndash1002 2017

[14] D Gross J F Shortle and J M Thompson Fundamentals ofQueueing Theory Wiley-Interscience New York NY USA 4thedition 2009

[15] M A C Almeida and F R B Cruz ldquoA note on Bayesianestimation of traffic intensity in single-server MarkovianqueuesrdquoCommunications in Statistics -SimulationComputationhttpdxdoiorg1010800361091820171353614

[16] R LThomasson and C H Kapadia ldquoOn estimating the param-eter of a truncated geometric distributionrdquo Annals of theInstitute of Statistical Mathematics vol 20 pp 519ndash523 1968

[17] R Core Team R A Language and Environment for StatisticalComputing R Foundation for Statistical Computing ViennaAustria 2017 httpwwwR-projectorg

[18] N Mukhopadhyay Probability and Statistical Inference MarcelDekker New York NY USA 2000

[19] G S Lingappaiah ldquoBayes inference in right truncated geo-metric distributionrdquo Malaysian Mathematical Society BulletinSecond Series vol 15 no 2 pp 61ndash67 1992

14 Mathematical Problems in Engineering

parametric space Although the estimates of the MLE andthe bootstrap corrected MLE exhibit less bias the Bayesianestimates (beta and Jeffreysrsquo prior) present the lowest MSEin general Perhaps due to the skewness of the a posteriordistribution the Bayesian estimators do not present low biasIn general for sample size 119899 = 50 and queues with 119870 ge 20the average estimation error is less than 0005 this value wasonly exceeded by the Bayesian estimator for queues with totalcapacity 119870 = 5

In regard to the behavior of the average estimation errorand the average MSE as functions of the traffic intensity 120588the major errors are observed when the sample size is small(119899 le 20) and the traffic intensities are 120588 asymp 10 unlike inthe case of the 1198721198721 queues which exhibit higher biaseswhen 120588 asymp 05 Perhaps due to the truncation of the numberof users to the maximum queue length119870 systems with hightraffic intensities require more computational effort and arethe most difficult to estimate

Finally it is important to note that for queues withcapacity 119870 = 80 the systemrsquos behavior is similar to that of aninfinite Markovian queue (an 1198721198721 queue) as expectedThat is the average estimation error is greater and the MSEis highest when 120588 asymp 05

Future work in this area includes testing other Bayesianpoint estimators (eg the median because of the asym-metry of the a posterior distribution) developing intervalestimators hypothesis testingmethods or even Kernel-basedmethods [24]

Data Availability

The data used to support the findings of this study areincluded within the article

Disclosure

The Brazilian government funding agencies mentioned hadno role in the study

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

Special thanks are due to Gabriel and Carolina for helpingwith the algebra This work was supported by the Brazil-ian agencies CNPq (Conselho Nacional de DesenvolvimentoCientıfico e Tecnologico of the Ministry for Science andTechnology) [Grant nos 3046712014-2 3058412016-5] andFAPEMIG (Fundacao de Amparo a Pesquisa do Estado deMinas Gerais) [Grant nos CEX-PPM-00564-17 APQ-02119-15 and BIP-00106-16]

References

[1] D Ma D Wang Y Bie F Sun and S Jin ldquoA method for queuelength estimation in an urban street network based on roll time

occupancy datardquo Mathematical Problems in Engineering vol2012 p 12 2012 httpdxdoiorg1011552012892575

[2] S Zhao S Liang H Liu and M Ma ldquoCTM based real-timequeue length estimation at signalized intersectionrdquo Mathemat-ical Problems in Engineering vol 2015 Article ID 328712 12pages 2015

[3] C Armero and M J Bayarri ldquoBayesian prediction in 1198721198721queuesrdquo Queueing Systems vol 15 no 1-4 pp 401ndash417 1994

[4] C Armero and M J Bayarri ldquoQueuesrdquo in Proceedings of theInternational Encyclopedia of the Social Behavioral Sciences J DWright Ed pp 784ndash789 Oxford UK 2015

[5] C Armero and D Conesa ldquoInference and prediction in bulkarrival queues and queues with service in stagesrdquo AppliedStochastic Models in Business and Industry vol 14 no 1 pp 35ndash46 1998

[6] C Armero and D Conesa ldquoPrediction in Markovian bulkarrival queuesrdquo Queueing Systems vol 34 no 1-4 pp 327ndash3502000

[7] C Armero and D Conesa ldquoStatistical performance of a mul-ticlass bulk production queueing systemrdquo European Journal ofOperational Research vol 158 no 3 pp 649ndash661 2004

[8] C Armero and D Conesa ldquoBayesian hierarchical models inmanufacturing bulk service queuesrdquo Journal of Statistical Plan-ning and Inference vol 136 no 2 pp 335ndash354 2006

[9] A Choudhury and A C Borthakur ldquoBayesian inference andprediction in the single server Markovian queuerdquo MetrikaInternational Journal for Theoretical and Applied Statistics vol67 no 3 pp 371ndash383 2008

[10] S Chowdhury and S P Mukherjee ldquoEstimation of trafficintensity based on queue length in a single MM1 queuerdquoCommunications in StatisticsmdashTheory and Methods vol 42 no13 pp 2376ndash2390 2013

[11] S Chowdhury and S P Mukherjee ldquoBayes estimation inMM1 queues with bivariate priorrdquo Journal of Statistics andManagement Systems vol 19 no 5 pp 681ndash699 2016

[12] F R Cruz R C Quinino and L L Ho ldquoBayesian estimationof traffic intensity based on queue length in a multi-serverMMS queuerdquo Communications in StatisticsmdashSimulation andComputation vol 46 no 9 pp 7319ndash7331 2017

[13] R C Quinino and F R B Cruz ldquoBayesian sample sizesin an MM1 queueing systemsrdquo The International Journal ofAdvanced Manufacturing Technology vol 88 no 1-4 pp 995ndash1002 2017

[14] D Gross J F Shortle and J M Thompson Fundamentals ofQueueing Theory Wiley-Interscience New York NY USA 4thedition 2009

[15] M A C Almeida and F R B Cruz ldquoA note on Bayesianestimation of traffic intensity in single-server MarkovianqueuesrdquoCommunications in Statistics -SimulationComputationhttpdxdoiorg1010800361091820171353614

[16] R LThomasson and C H Kapadia ldquoOn estimating the param-eter of a truncated geometric distributionrdquo Annals of theInstitute of Statistical Mathematics vol 20 pp 519ndash523 1968

[17] R Core Team R A Language and Environment for StatisticalComputing R Foundation for Statistical Computing ViennaAustria 2017 httpwwwR-projectorg

[18] N Mukhopadhyay Probability and Statistical Inference MarcelDekker New York NY USA 2000

[19] G S Lingappaiah ldquoBayes inference in right truncated geo-metric distributionrdquo Malaysian Mathematical Society BulletinSecond Series vol 15 no 2 pp 61ndash67 1992

Mathematical Problems in Engineering 15

[20] G Petris and L Tardella ldquoHI Simulation from distribu-tions supported by nested hyperplanes 2013 (original Ccode for ARMS by Wally R Gilks R package version 04)rdquohttpCRANR-projectorgpackage=HI

[21] B Efron and R J Tibshirani An Introduction to the BootstrapMonographs on Statistics and Applied Probability Chapmanand Hall New York NY USA 1993

[22] B Efron and R Tibshirani ldquoBootstrap methods for standarderrors confidence intervals and other measures of statisticalaccuracyrdquo Statistical Science vol 1 no 1 pp 54ndash75 1986

[23] W R Gilks N G Best and K K C Tan ldquoAdaptive rejectionmetropolis sampling within gibbs samplingrdquo Journal of theRoyal Statistical Society vol 44 Series C no 4 pp 455ndash4721995

[24] G M Gontijo G S Atuncar F R B Cruz and L KerbacheldquoPerformance evaluation and dimensioning of 119866119868XMcNsystems through kernel estimationrdquo Mathematical Problemsin Engineering vol 2011 Article ID 348262 20 pages 2011httpdxdoiorg1011552011348262

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