queue - j-stage

The Operations Research Society of Japan

NII-Electronic Library Service

The OpeiationsReseaich Society of Japan

Journal Qf the Operations ResearchSociet-y of Japan Vol. 35, No, l, March 1992

INTERPOLATION APPROXIMATIONS FOR THE

MEAN WAITING TIME IN A MULTI-SERVER QUEUE

Toshikazu Kimura

ffokkaddo University

(Received January 10, 1991; Revised May 27, 1991)

Abstrzict [Vhis papeT gives numerical validation of a couple ofinterpelation approximations for the meanwaiting t,ime in a GJIGIs queue, which are. provided by a unified approach similar to that in Kimura (1991).Both approximations are represented as certain eombinations of the mean waiting times for the GIIMIsand GIIDIs queues in which the arrival processes and the rnean service times are the same as in theapproximating GJIG/s queue. To let these approximations be more tractable, we further proyicle simple

lnterpolation approximations for the mean waiting times in GJIM!s and GIfDls queues with low yariable

lnterarrival times. The quality of the approximations is tested by comparing them with exact selutions and

preyious twamoment approximations for a variety of cases. Extensive nurnerical comparisons indicate thatour approximations are mere accurate than the two-moment apprdximations and that the relative percentageerrors are in the order of 5% in moderate traMc and in the order of 1% in heavy trafic.

1. Introduction and Summary

In this paper we provide a couple of approximations for the mean waiting time in a

mu}ti-server queue, which is an improvement of the appraximation proposed in Kimura [14]when interarriyal times are highly variable and/or the traffic is not hearvi}y loaded. As in

[14], we considei the standard GJ!Gls queueing system with s () 2) homogeneous serversin parallel, unlimited waiting room, the first-come first-served discipline and independentsequences of i.i.d. (independent and identicalIy distributed) interarrival times and service

times. Let u and v be generic interarrival- and service-time, respectively; Iet A (B) denotethe interarrival- (service-) time cdf with mean A-i (pt-i); let p == Alsps E [O, 1) be the trafficintensity; and let cZ (c?) be the sqttared coeficient of variation (variance divided by the squareof the mean) of u (v). Assume that the cdf A is not deterministic, i.e., cZ l O. In addition,let EW(GI/G/s) denote the mean waiting time (until beginning service) in this GIIG/s

queue, assuming that the system is in equilibrium state. We approximate EW(Gl/Gls) bycomhining the exact mean waiting times foT the GJ!Mls amd CIIDIs queues both havingthe interarrival-time cdf A.

There are some two-moment approximations for EW(GIIGIs), which are weighted com-

binations of the exact mean waiting times for the MIM!s, M!Dls and DIMIs queues[11, 13, 14, 22]. Although these appicvdmations are tractable and suficiently accuiate fbrmost practical purposes, it has been known that they become less accurate as the variabiiity

parameters cZ and c? (especially c2) get large, i,e., for the cases where detailed informationabout A andlor B is important. Our experlence seems to suggest that EW(GIIGIs) is notsensitive to the shape of B. However, this is not the case with A, The shape of A affects

EW(GIIGIs) quite considerably when the traffic is not heavy and cZ not too sm411. Two-moment approximations are no longer reliable in such circumstances. In this paper we givefull information on A to our approximations by using the GI/Mls and GIIDIs queues as

77



The OpeiationsReseaich Society ofJapan

78 T Kimura

their building blocks, Extensive numerical studies show

accurate than the previous twamoment approximations

The apprQximations we recommend in this paper are

EW(GIIG!s) cr

which will be obtaine

(1.1) have the followmgGIIDIs queues. (iithe appioximation fois exact fbr s= 1.

Our studies indicathe cases that (i) cZ

p2 O･3 for s=2 andp)

(1.1) are in the order of 5%than 101pa. The stud

strongly depend onimportant because algorithmic methods for computmg exact solutions of GIIGIs queues,e,g., [25], beceme infeasible for systems with Iarge s.

In (1,1), the mean waiting times for the building-block systems, i.e., the GflMls andGIIDIs queues, have the same mean service times and tra[ffic intensities as those of theapproximating GJIGIs queue. The exact values of these mean waiting times can be obtainedeither by using the queueing tables of Seelen, Tijms and van Hoorn [24] or by computingtheir analytic andlor algorithmic solutions; see, e,g., [2, 21] for the GIIM!s queue and

[20, 29] for the GIID/s queue with phase-type (abbreviated as Ph) arrival distributions.The algorithms for the PhlM!s and PhlDls queues are computationally feasible for very1arge values of s, e.g., up to s = 250 servers wheit the number of arrival phases is 10; see

[29]. The appreximations in <1.1) are useful not only fbr quick calculation of EW(GIIGIs)but also for obtaining approximations for the distributions of the number of customers and

of the waiting time: Wu and Chan [30] proposed simp}e apprQximations for these queueingcharacteristics in the GIIG!s queue by the use of maximum entropy analysis. Shore [26,27] derived more heuristic apprQximations for the characteristics by viewing the GJIG/s

queue as a system alternating between associated GI/G/1 and GI!Gloo queues. In bothof the approximations, EW(GJIGIs) plays a key role to derive explicit fbrmulas. Ourapprcncimations can be directly applied to these approaches.

This paper is organized as fbllows: In Section 2, fbllowing Kimura [14], we fbcus on

the ratio EW(GllG/s)IEW(GIIG/1) and its reciprocal. We approximate each of thesequantities by a Iinearly weighted sum of the corresponding quantities for the GJ/A41s and

GJIDIs queues. Erom these approximate reiations, we derive a cottple of apprc"cimations

which are consistent with exact properties for particular cases, 'Ib

}et our approximations

be more tractable for systerms with cZ S 1, we further provide simple apprc"dmations forEW(GIIMIs) and EW(GJ/Dls) in Section 3. In each section, the quality of the approxi-mations is tested by comparing them with exact solutions fbr a variety of cases. Finally, inSection 4, we give a collcluding remark on a possible direction for future extensions.

that our approximations are more

in the circumstances above.

(cZ + c?)(EiS"iG+ii}ti'?!,) + Elll2:5iii')/,) )-',

if i s cZ < 2 (i.o

c? EW(GI IM/s) + (1 - c?)EW(GIIDIs), otherwise,

d in a unided way. We immediately see that the approximations in' characteristics: (i) They are, ef course, exact for the GIIMIs and

) They are asymptotically exact as p - 1. (iii) For the MIGIs queue,

r cZ = 1 coincides with the exce]ent appraximation in Kimura [ll] and

te that (1.1) will usually yield satisfactory approximations at least for

g 4 and c? S 4, and (ii) the trafic intensity is not too smal1, e.g.,

O.7 for s = 20. Roughly speaking, the relative percentage errors of

(1%) if the approximate value of EW(GIIGIs) is less (greater) ies also indicate that the accuracy of our approximatiQns does not so

the number s of servers &t least fbr cZ S 4. This property is practically




AppwwimationsforaMbelti-Servereueue 79

2. Approximating EW(GI/G/s)

[[b obtain accurate appreximations for queueing characteristics, we often need the ex-

treme limiting behavior as s - oo or p . O or p -, 1. For EW i EJVV(GllGl$), it is

obvious that EVV - O as s - co or p . O and EW --+ oo as p - 1. A good way to

analyze asymptotic properties of EW in these extreme cases is to normalize EVV so that

nondegenerate Iimits occur.

In this paper, as in Kimura [14], we fbcus on the ratio EW(GIIGIs)IEW(GI/Gll) and

its reciprocal as such normalized quantities. In these quantities, we assume that the GIIGII

queue has the same mean service time and trafic intensity as those of the approximating

GI/Gls queue. We approximate each of these ratios by a lineatly weighted sum of the

corresponding ratios for the GJ!M/s and GJ!Dls queues both having the interarrival-time

cdf A, i.e.,

S:II:.GZG.il-NwS,W,((2ZilflS,i+(i-w)Slll[.GZD.iS,i･ '(2ai)

and

ZW.[C6ZXIBcrvZW.(,SZM.ll),+(i-u):W.(,ZilD.li,i, (2.2)

where w and v denote weighting coeflicients. This approximation is motivated by a similar

idea of Tijms [28, (4.223)] fbr plain performance measuies and the idea of Cosmetatos [3, 5]

for EW(GI!Mls) and EW(M!Gls). For convenience, assume that these coefficients dependonly on the variability parameters cZ and c? and not on s and p. [lb show the dependellcyof the weights on cZ and c? definitely, we write bl i! w(cZ, c?) and v ii! u(cZ, c?) henceforth.

Flrom the consistency with the buiIding-bleck systems in (2,1) and (2.2), we see that these

weighting functions satisfy the fo1}owing conditions: From the censistency with the GJIMIs

queue, bl(cZ, 1) == u(cZ, 1)=l, (2.3)and from the consistency with the GIIDIs queue,

bl(cZ, O) =u(cZ, O)=O. (2.4)

Moreover, we have the next two theorems.

Theorem 2.1 Assume that E[v3] < oo. Then the approximate relations (2.1) and (2.2) are

asyrnptoticalgy correct as p ---, 1 for any fcnite weights.

Theorem 2.2 libr the M/Gls gueue with the

(2,1) and (2.2) are asymptoticagly correct as s

2c2

w(1,c?)=

1+Sc?,

service-time ccif B, the approxirnate relations

- co if

v(1, c?) == c?. (2.5)

Theorem 2.1 ensures that our approximations are structurally accurate in heavy trafic. The

proofs of these theorems are given by virtue of Theorems 2.1 and 2.2 in Kimura [14] and

hence they are omitted.

'Ib

keep the consistency with the building-block systems and the MIG/oo queue, we

will determine the weighting functions in such a way that they satisfy the conditions (2.3),(2,4) and (2.5) at the same time. Howevei, the weights $atisfying these conditions are not




80 T Kimura

lil.q."tehilhde,:e.ryMti,".e:],Cilih[iitil･h,H.e,fae,[it7a]k,in.g,z:/lfiopui;t.opfpt,h.2iw.e:',k2o::inds.ymbymetryofcZ

w(cz,c?) -'" (Ciz++

ic)2,C?,

and

(2,6)

and

From the appTQximate relation (2

EW(GIIG!1)

Ew(GI!Gls)

C! ,z+c?

and

y(cZ,c?) !u c?.

.1) ((2.2)) with the weight

2 2EW(GIIM/s)((Ca+ 1)c

(2.6) ((2,7)), we have

+ cZ(1 - c?)

(2.7)

s

Ew(GIIMII)EW(GI/Dls)

EW(GllDll)

(2.8))･

Ew(GfiG/,) ,, Ew(GiiGi i) (,? SIII[gZM.ili . (i - ,?) Slr:.G,i$ilj )'i

. (, ,)

Obviously, these two approximations contain the mean waiting times for three single server

queues with the same mean service times and trafflc intensities as in the apprQximating

GI/Gls queue. Among these mean waiting times, EW(GJICII) and EW(GJ!Dll) are

diMcult to compute except some special cases, e.g., Poisson arrival case. Hence, to simplifythe approximations (2.8) and (2.9), we replace the mean waiting times for all single server

queues by a simple two-moment approximation; see Remark 2.2. As such an approximation,

we use the approximation ･

EW(GIIGII) or CZ;C?EW(MIM!1)i

(2.10)

see Remark 2.3, Substituting (2.10) into (2.8) and (2.9), we respective]y obtain the approx-lmatlons as

EW(GI!G!s) r c?EW(GI!Mls) + (1 - c?)EW(GllDls), (2.11)and

Ew(Gf/Gls) or (cz + c?)(EillCiG+fiti?l,) + Ei132"((5il j?)1,)]-i

(2･n)The

appraximation (2.11) is a simpie linear interpolation between EW(GIIMIs) and

EW(GIIDIs), and it coincides with Tijms' approximation (4.223) in [28]. The approxi-

mation (2.12) is & certain harmonic mean of EW(GIIMIs) and EW(GIIDIs). We will

show the quality of these appToximations throllgh extensive Rumerical experiments.

Remark 2.1 For the MIG!s queue, it should be noted that (2.ll) coincides with Page's[22] approximation and (2.12) with Kimura's [11] approximation; see (2.17) and (2.18),respectively.

Remark 2.2 Instead of (2.10), it is pessible to use the exact value of EW(GIIMII) in(2.8) &nd (2.9). HoweveT, we can easily see that the resultant formulas are not exact for theGIIA4!s queue. This is why we use the apprQximation (2.10) for EW(GIIMII).




AppreximationsforaMtilti-Sen,ereuette sl

Remark 2.3 It seems to be agood idea to repla £ e the mean waiting times for the si"gle

server queues appeared in (2.8) and (2.9) by a more accurate approximation, e.g., the Kramerand Langenbach-Belz [18] appTcDcimation fbr EVV(GI/Gll), which also has a simple fbrmsimilar to (2.10). However, from numerical comparisons with some other alteinatives fbr

(2,10), we saw that the simple apprQximation (2.10) fits foi our approximations when we use

the coeMcients (2.6) and (2.7); cf. Kimura [14].

Remark 2.4 By using the light traMc limit theorem in Burman and Smith [1], we can

prove for the MIGIs queue that the approximate relations (2,1) and (2.2) are asymptotically

correct as p - O if

.(1, ,?) .. ,2-S1(1-

(S :i l)fi

(S)) {2.13)

and

v(1, c?) == g +- 1 ((, +1 +1)

£i(,) - 1), (2.14)

where

I(s) .. .1[eO

{1-B.(t)}' dt, (2.15)and B. denote the stationaiy-excess cdf associated with the service-time cdf B, i.e.,

B. (t) = ps y[t {1-B(x)} dx, t2 0. (2.16)

Note that the weighting coeMcients depend on s. As in Remark 2.3, it is also a good idea toreplace (2.5) by (2.13) and (2,14). ffowever, it is relatively diMcult to generalize the weightingcoeficients in (2.13) and (2.14) to the GI/G!s case. Taking the light trafic behavior intoapproximations would be an important subject of our future studies; see Kinmia [16].

Table 1: A List of NumeTical Experiments.

Arrival (cZ) Service (c?) s pE,,(O.1)E2(O.5)2(1)10(5)25O.3,O.5,O.7,O.8,O.9,O.95

E2(O.5)HS(1.5)M (1.0)U8(2.s)H8(1.s)H8(4.o)

HS(2.0)HS(3.0)

H8(4.o)

NUMERICAL COMPARISONS

lible 1 gives a combination list of the parameteTs in queueing systems on which we havemade numerical experiments to test the performance of our apprc}ximations. In [fable 1,Hg denotes an H2 distribution with balanced means. The exact values of the mean waiting

times for these systems are given in Seelen et al. [24]. It should be noted that al1 of theexact values are not necessarily available; for example, the exact values for systems with

s = 10 are available only when p ) O.5, Some typical results of these experiments are given




82 T Kimura

in [Ilables 2-5, in which we denote for convenience the approximations (2.11) and (2.12) asC[New-I"

and `CNew-II",

respectively.

The experiments listed in Table 1 have clarified some qualitative properties of our ap-

proximations. First, we will summarize these properties: The approximation New-I is stablyaccurate fbr any combination of the variability par&meters, while the appraximation New-IIbecomes unstable (e.g,, negative) when cZ < 1 and c? > 1. The approximation New-II is lessaccurate than New-I when cZ < 1 and c? m< 1, but perfbrms about the same as New-I when

cZ 2 l. In particular, New-II becomes more accurate than New-I as c2 - 1 from above.

Table 2 compares five approximations with the exact values of the mean queue }ength(excluding customers in seTvice) fbr PhlPhllO queues with low yariable interanrrival times.Apprc"dmations of the mean queue length can be de:ived from those of EW by using Little'sformula. Since the queue length is intuitively easy to capture the level of congestion, we use

the mean queue length rather than the mean waiting time in Tables 2-4, 6 and 7. In Table 2,C`Sim-I"

denotes a simp!ified veTsion of New-I which will be discttssed in Section 3. We add

three close}y-related two-moment approximatioRs of Page [22] and Kimura [11, 14] in thetable. In terms of our notations, Page's approximation is given by

EW(Gl/Gls) cr

while K

cZc?EW(MIM!s) + cZ(1 - c?)EW(MIDIs) + (1 - cZ)c?EW(D/M/s),

imura's [11, 14] approximations are respectively given by

cZ + c?

EW(GJIG!s) .t

(2,17)

2(cZ + cZ - 1)+ 1- c2 s+

1-c2 a

EW(Mll;4/s)EW(MID/s)EW(D/M/s)

(2.18)

and

- ･2(Ca+ Cs- 1)+ 1-cs+koi(1 ' c.)

EVV(GJIGIs) crEW(M/M!s)EW(M/Dls)EW(D!Mls) if cZ f{ 1

(cZ + c? - 1)EW(MIM/s) + (1 - c?)EW(MIDIs)

l-c2

+ fo,,"EW(DIM/s), if cZ>1

(2.19)

where

ki k(p, cz, c?) = exp (- 2( i3; P) (iz-+Cl'?)2) (2 2o)

and

fooi Ei k(p, O, 1). (2.21)Kimura's apprc"dmations (2,18) and (2,19) are denoted in the table by

"Kimura86" and

"Kimura91", respectively. We omit New-II from comparisons, since New-II is evidently less

accurate than the otheTs when cZ < 1. Table 2 shows that New-I is more accurate thanthe two-moment apprc"cimations of Page and Kimura86 when cZ < 1 and c? < 1. Table 2also shows that Kimura91 performs as well as New-I for c? < 1. However, when c9 < 1and c? > 1, Page's approximation has the be$t perfbrmance. This tendency is remarkableespecia]ly when O.5 m< cZ < 1 and c? > 2.5.



The Opeiations Reseaich Society of Japan

Amproximations for a Mtilti-Sen,er eueue 83

Table 2: A Comparisonwith cZ < 1.

of Approximatlonsof the Mean Queue Length for PhlPh/10 Queues

c2 apMethodc2

=O.5 c2 = 1.5 c2 = 2.5 c2 == 4.0 s s s s

O.1O.7Exact O.05 O.19 O.33 O.52New-I O.06 O.19 O.31 O.50

Sim-I O.07 O.20 O.34 O.55Page O,08 O.19 O,30 O.46Kimura86 O.06 O.16 O,25 O.37

Kimura91 O.06 O.18 O.29 O.44O.9Exact 1,44 4.04 6.59 10.40

New-I 1.44 4.10 6,77 10.76Sim-I 1.45 4.12 6.78 10.79

Page 1.55 4.03 6.51 10.24Kimura86 L51 3.92 6.22 9.47

Kimura91 1,46 4.04 6.49 9.94

O.5O.5Exact O.Ol O.Ol O.02 O.03

New-I O.Ol O.02 O.03 o.e4Sim-I O.Ol O.03 O.04 O.07Page O.02 O.02 O.03 O.04

Kimura86 o.oo o.oo O.Ol O.Ol

Kimura91 o.oo O.Ol O.Ol O.OlO.7Exact O.18 O.36 O.50 O.72

New-I O.17 O.38 O.59 O.91

Sim-I O.19 O.43 0.67 1.03Page O.23 O.39 O.55 O.79Kimura86 O.14 O.27 O.38 O.53

Kimura91 O.17 O.32 O.45 O,6109Exact 2,77 5.50 8.10 IL97

New-I 2.74 5,62 8.50 12.83Sim-I 2,76 5,68 8.59 12,96Page 2,89 5.56 8,24 12.25Kimura86 2.73 5.31 7.76 11.22Kimura91 2.79 5.48 8.02 11.59



The Opeiations Reseaich Society ofJapan

84 Z Kimutzl

Table 3: A Comwith cZ ) 1.pamson

of Appraximations of the Mean Queue Length for PhlPhllO Queues

CapMethodc--O.5s c?=:1.5c2=2,5s c?=4.0

1.0O.5Exact O.03 O,04 O,05 O.06New-I O.03 O.04 O.06 O.07New-II O.03 O,04 O,05 O.05

O.7Exact O.41 O.60 O.76 O.99New-I O.40 O.63 O.86 L19New-II O.41 O.61 0.77 O.96

O.9Exact 4.58 7.37 10.02 13.94New-I 4.56 7.47 10.38 14,74New-II 4.59 7.40 10.05 13.72

2.0O.5Exact O.09 O.10 O.10 O.11New-I O.09 O.10 O.11 O.12New-II O.09 O.10 O.10 e.loPage O.06 O.08 O.11 O.14Kimura91 O.06 O.08 O,09 O.11

O.7Exact O.87 1.08 1.25 1.49New-I O.88 LIO l.33 1.66New-II O.89 1.08 1.23 1.39Page e.76 1,12 1.47 2.00Kimura91 O.80 1.02 1.24 1.58

O.9Exact 8.15 11.02 13.74 17.75New-I 8.16 11.09 14.01 18.39New-II 8.19 10.99 13.57 17.07Page 7.90 11.31 14,70 19.81Kimura91 7.96 10.88 13,79 18.17

4.0O.5Exact O.23 O.24 O.24 O.24New-I O.24 O.24 O,24 O.24New-II O.24 O.24 O.24 O.24Page O.12 O.17 O,21 O.28Kimura91 O.13 O.14 O.16 O.18

O.7Exact L96 2.15 2,31 2.56New-I 1.99 2.14 2.29 2.51New-II 2.00 2.13 223 2.34Page 1.48 2.09 2.70 3.61Kimura91 l.57 1.80 2.02 2.36

O.9Exact 15.65 18.49 2126 25,35New-I 15,72 18,46 21.21 25.32New-II 15,75 18.37 20.75 23.96Page 14.59 18.96 23.33 29.89Kimura91 14,76 17.67 20,59 24.97




Amproximations for a Mbelti-Serv er eueue 85

[fable 3 compares four appraximations with the exact values of the mean queue lengthfbr PhlPhl10 queues With highly variab]e interarrival times. In Table 3, we omit Kimura86from comparisons because it becomes unstable when cZ > 1. In addition, the apprc"[imationof Page (Kimura91) for cZ =1 is excluded from the table because it coincides with New-I

(New-II) for the A41Gls queue. Table 3 shows that both of New-I and New-II aJre merea£ curate than the two-moment approximations when cZ > 1. When cZ == 1, the qualityof New-II (== Kimura86 = Kimura91) is quite excellent. Table 3 also shows that New-IIperfbrms as well as New-I except for cZ = c? =: 4, both providing satisfactory accuracy fbrpractical applications.

Table 4: A Comparison of Approximations of the Mean Queue Length ior H2blff2bls Queueswith c2 =: 4, s

MethodCa=1･5 cZ=2.0

ps=2s=5s=10s=2s=5s:10

O.3ExactNew-INew-IIPageKimura91O,15O.18O.20O.20O.16O.02O.03O.02O,03O.02---o.ooo.ooO.17O.21O.30O.26O.18O.03O.03O.03O.03O.03--rmo.ooo.oo

O.5ExactNew-INew-IIPageKimura91O.86O.981.071.07O.91O.31O.38O,32OAIO.35O,08O.10O.080.11O.09O.98X151.481.351,OlO.37O.46O.41O.53O.40O.11O.12O,10O.14O.11

O,7ExactNew-INew-II

PageKimura91

3.593.854.064.053.682.222,502,262.692.391.241.43L171.591.394.004,364.834.794.052.562.862,653.282.681.49L661.392.001.58

O.9Exat;tNew-INew-IIPageKimura9120.9421.3621.6421.6921,0618.4019.0318.4219.5418.8015.8516.6315.4517s2916.4722.9823.5023.9124.2923.0320.3920.9920.3322.1120,6417.7518.3917.0719.8118.I7

Tb see the differences between New-I and New-II more clearly, we compare them with

the two-moment approximations and the exact values of the mean queue length for somell2blH2bls queues with c? = 4 in Table 4. Tab!e 4 indicates that New-II becomes more accurate

tham New-I as cZ - 1. Table 4 also indicates that New-II tends to underestimate the exactvalue as s grows, and hence New-II tends to be less accurate than New-I, especially when

cZ = 2. Although it is relatively difficult to specify the region of cZ where New-II surpassesNew-I, a practica} guideline for this region is that 1 S cZ < 2 if s is not too large, e.g.,

s< 20.




86 T Kimure

Table 5: Recommended Approximations for EW(GI!G!s).

os c2 <1 c2>1 s-

o<c2<o.saNew-IorKimura91New-I

O.5<c2<1-a Page

cZ=1 New-II=Kimura86==Kimura91

1<c2<2a New-IorNew-II New-II

c2>2a- New-I

zab}e 5 summarizes the results of Tables 2-4 and of the other numerical experiments,

which gives a list of recommended approximations for EW(GIIG/s). Of course, this rec-ommendation is valid enly for a subclass of PhlPhls queues, because we have not checked

all of GIIG!s queues due to the lack of authorized numerical resuits. Further numerical

validation is necessary fbr more precise evaluation of our approximations. For almost allcombinations of cZ amd c?, the approximations provided in this paper are better thain thetwo-moment approximations. The only exception is the case that O.5 S cZ < 1 and c? > 1,in which Page's approximation performs better than our apprc"rimations. It is one of thesubjects of our future studies to improve the accuracy for this case, e.g., by refining theweighting coeMcients w and u.

3. Simplified Formulas

[[b evaluate the approximatiens (2.11) and (2.12) actually, it is of course necessary tocompute the mean waiting times in the bnilding-block systems, i.e., the Gl/Mls and GIIDIs

queues. For EW(GI/Mls) with given A, s and p, one can compute its value in a stable way

through an explicit fbrmula [2, pp. 267-273]. Neuts [21] prQvided an eMcient algorithm fbrcomputing EW(PhlMls). However, for EW(GflDls) with a general interarrival-time cdf,

it is not so easy to compute the exact value, A}gorithins fbr computing EVV(GflD/s) are

available only for some special phase-type interarrival-time cdf's; see [20, '29],

In this section, to let our approximations be more tractable fbr systems with cZ S 1, we

approximate EW(GJIMIs) and EVV(GI/Dls) by the mean waiting times for more basicsysterns (e.g., the MIMIs queue and so on) for which it is easy to compute the mean waiting

times or extensive tables have been prepared. As shown in Section 2, the apprQximation

(2.12) becomes unstable when cZ < 1 and c? > 1. Hence, we are concerned only with thesimplification of (2.11).

3.1 ApproximatingEW(GIIM/s)

Although appraximating EW(GI!Mls) is less important than EW(Gl/Dls), it is veryuseful if one can obtain apprQximate values for EW(GJIMIs) only by using basic queueingtables. For this purpose, apprcDrimations of interpo}ating EW(MIM/s) and EW(D/Mls)are appropriate, because exa £ t values for these meall waiting times for given s and p can befbund in some queueing tables [9, 23, 24]. TheTe are some approximations for EW(GI/Mls)as special cases of interpolation approximations for EW(GJIG/s): Page's appr("cimation

(2.17) fbT jE)VV(GIIM/s) cttn be written as a simple linear interpolation

EVV(GflMls) cr cZEVV(MIM!s) + (1 - cZ)EW(D!Mls), (3.1)




AmproximationsforaAdetlti-Serverquette 87

and Kimura's approximation (2.18) for EW(GIIMIs) can be considered as another inter-

polation which is a dual of (3.1) in some sense; cf. (2.19). Cosmetatos [31 has derived more

accurate but complicated formulas which need the exact value of EW(GIIMII). From some

numerical tests, we saw that Page's approxirnation (3.1) has satisfactory accuracy for prac-tical applications, though it is not uniformly more accurate than the others. We are greatlyconcerned with the simplicity ofthe approximation, not with the relative percentage errors inlight traMc if the absolute differences aie smal1. Hence, we adopt (3.1) as our approximation

for EW(GIIMIs).

[[b let (3.1) be more tractable, we will further approximate EW(DIMIs) by using

EW(MIM/s). Kimura [11, 14] suggested approximating EW(DIM/s) by

k

EW(DIM/s) cr {lilL'

ip(s, p)EW(M!Mls), (3.2)

which is obtained by combining the apprQximations of Cosmetatos [4] and Kramer and

Langenbach-Belz [18], where leoi is defined by (2.21) and

di(s, p) == 1-4min {f(s)g(p), O.25(1- 10-6)} (3.3)

with

(, - 1)(vcfi -gg-

2)

f(S)= 16, , (3.4)

1-p g(p)- . (3-5) p

Note that we have slightly modified the original approximation in [4] by inserting the mini-mum with O.25(1-10-6) in (3.3). Without it, the approximation (3.2) becomes negative and

hence meaningless; c £ Tljms [28, (4,228)] and Kimura [14, Equations (32) and (33)]. From

some numerical tests, we saMr that (3.2) performs well unless p is close to zero. Combining

(3.1) and (3.2), we obtain the final form of ouT simplified formula for EW(GIIM/s) as

EVV(GIIM/s) N (cZ + k20i (1 - c2)ip(s, p)] EW(MIMIs). (3.6)


[fable 6 compares the approximations (3.1) and (3.6) with the exact values of the mean

queue length fbr Ph/M/s queues. The exact values for cZ = e.1, O.25 and O.5 are computed

via the analytic solution for the qlM/s queue (m = 10,4 and 2, respectively), while those

for cZ = O.8 are quoted from the tables of Seelen et al. [24]. In Table 6, we refer to (3.6)as

"Sim-Page". Table 6 shows that the simplified approximation (3.6) is stably a £ curate for

various values of c2 E (O, 1) and s, and hence good enough fbr pra£ tical applications. It is

interesting that (3.6) is more accurate tham (3.1), due to underestimation of (3.2).

Remark 3.5 Of course, there is no problem to apply (3.6) to approximating the mean

waiting times for queues with cZ > 1. Fhrom some numerical comparisons, we saw that (3.6)is also accurate for cZ > 1.




88 T KimunT

Table 6: A Comparison of Approximations of the Mean Queue Length for PhlMls Queues.

s pMethodc=O.1a c2=o.2sc=O.5a c=O.8a

10O,7Exact O.126 O.177 O.277 O.415Page O,138 O.201 O.307 O.433Sim-Page O.137 O.200 e.3o6 O.433

O.8Exact O.583 O.740 1.021 1.383Page O.600 O,773 1.061 1.406Sim-Page O.600 O.772 1.061 1.406

O.9Exact 2.773 3.293 4.181 5.276Page 2.791 3.329 4.223 5.301Sim-Page 2.781 3.320 4,220 5.299

50O.8Exact O.052 O.082 O.151 O.260Page O,067 O,114' O.192 O.285Sim-Page O.054 O,103 O.185 O.283

O.9,Exact 1.148 1.460 2,024 2.758Page 1.185 1.533 2.l14 2.810Sim-Page 1.177 1.526 2.llO 2.809

100O.9Exact Oa525 O.715 1.080 1,583Page O,566 O.797 1.187 1.644Sim-Page O.552 O,786 1.175 1.641

O.95Exact 3a997 4.868 6.388 8.303Page 4.056 4.984 6.530 8.386Sim-Page 4.029 4.962 6,515 8.380

3.2 ApproximatingEW(GllDls)

It is giMcult to ohtain an explicit expression for the mean waiting time in the GJID/s

queue with a general mterarrival-time cdf A, except that A is the exponential distribution.

i6nr`thiwZ.SIVb/S

.XIC>LO"q'ueWueesaPwPi2`hmokMill[`l;enE,i'EGmib2Drs/glsbgrfegMs.billleegrJh:reM8hZ"eeWtrgpikgi:'ill;appraximation:

We first approximate EW(GIID!s) by EW(ll.IDIs)'s; we then utilizethe

equivalence

between EW(E.ID/s) and EVV(MIDInzs); and finally we appraximate

EW(MID!s) by EW(M!Mls).

For given cZ (S 1) of the cdf A, tltere exists an integer m () 1) such that

.I1<CZg#Ii (3.7)

SPw(i3Pli.P,rtriil,3tft,,.EWi.((GtY)Pil.,Si by

ii"eariy

interpoiating

Ew(iilaIDIs)

and

EW(GIID!s) !r q.E]W(E.IDIs) + (1 - g.)EW(JEI.+i!Dls),

where the interpolatioll coeMcient g. E (O, 1] is given by

(3.8)

q. = m{(m + 1)cZ - 1}. (3.9)

NII-Electionic



TheOpeiations Reseaich Society of Japan

AppraxitnationsforaMbelti-Sen,eraueue 89

Using the fact that jE]W(E./Pls) =: EVV(MID/ms) which has been pToved by Iversen [10], we can rewrite (3,8) as

EW(Gl/D/s) rt q.EW(MIDIrns) + (1 - q.)EIV(MID!(rn + 1)s). (3.10)

The meaR waiting times for the building-block systems iR (3.10) can be found in some

queueing tables or can be computed by an explicit formula provided by Crommelin [7]. It

is, however, known that the calculation by Crommelin's formula tends to be unstable when

s grows or p tends te unityl see [6, 12], 'Ib

a:void this unstableness, Kimura [15] has recently

suggested approximating EVV(MID/s) by

EW(M/Dls) !r,EW(M!Mls), (3.11)

as a refinement of Cosmetatos' approximation in [41, where r. i r,(p) is

1

r,(p)=i{1+f(s)g(p)h(s,p)} (3.12)

for f(s) in (3.4) and g(p) in (3.5). The bivariate function h(s,p) is introduced to correct

some defects of the original approximation forlarge s and smal1 p, which is given by

h(s,p)=C(s,a(p))n(b(s),p) (3.13)

with

g(,,,) VII-Slll-(Iri]IS, x)o, (3･i4)

op(y,p) ==1-exp (-IP-Yp), y)O. (3.l5)

In (3.13), the functions a(p) and b(s) &re defined by

25.6

a(P)

:{g(p)op(fi,p)}2,

(3.16)

and s-1

b(s) = (3.17) (s + 1)f(s)g(s, a)

'

respectively, where cr and 6 are arbitrary positive constants satisfying the relation

orfi2=25.6. (3.18)

It has been checked from many numerical experiments in [15] that a == 2.2 (and hence fi =: 25,6/2.2) is an optimal value for the best performance of the approximation. It is

shown in [15] that the approximation (3.11) is exa £ t for s == 1 and asymptotically exact as

s . oo or p . O or p . 1, The relative percelltage errors of (3.11) ame less than 1% fbr

most (s, p) combinations. By the use of (3.10), we obtain the apprQximation

EW(Gl/D/s) cr g.r.,EW(MIM/ms) + (1 - q.)r(.+i),EW(M!M/(rn + 1)s), (3.19)

with m satisfying (3.7).




90 T Kimura

Remark 3.6 When the value of ms is smal1, it is suficient to use h(s,p) i 1 in the ap-prorcimation (3.11) with (3.12). However, if ms is large, (3.19) fairly overestimates the truevalue unless the correcting function h(s,p) is used. It should be noted that the value of ms

increases quite rapidly as s grows for small c2; e,g., consider the case with cZ =: O.1.


thble 7 compares the approximation (3,19) (referred as "New")

with the exact values of

the mean queue length for PhlDls queues. All of the exact values are quoted from [24].Table 7 shows that the quality of (3.19) is quite excellent.

Table 7: A Comparison of ApprQximations of the Mean Queue Length for PhlDls Queues.

spMethodcZ=O.1cZ=O.25cZ=O.5cZ=O.8

10O.7ExactNew o.oooo.ooo O.O09O.O08O.067O.066O.198O.202

O.8ExactNew O.O05O.O05O.073O.072O.287O.287O.639O.643

O.9ExactNew O.111O.111O.498O,5021.2991.3062.3802.388

50O.8ExactNew o,oooo.ooo O.OOIO.OOIO.027O.026O.124O.137

O.9Exa £ tNew O.O08O.O07O,130O.128O,555O.5571.2621.298

100O.9ExactNew LL O.036O.034O,260O.256O.732O.771

O.95ExactNew -m O.594O,5971.8831.9043.7533.841

3.3 Accuracy ofthe Simplified Approximation

Substituting (3.6) and (3.19) into (2.11), we obtain the simplified formula forEW(GIIGIs), which is a weighted combination of the mean waiting times for three MIMIcqueues (c = s,ms and (m + 1)s). In Table 2, we have given the simp}ified approximation

(referred as "Sim-I")

for some PhlPhllO queues. Table 2 shows that Sim-I performs as wellas New-I. This indicates that the approximations (3,6) and (3.19) have good quality whenthey anre coiribined. Pbr cZ < 1, New-I in Table 5 can be Teplaced by Sim-I.

4. ConcludingRemark

Almost no exact results have been known fbr the steady-state probabilities in the GJIGIsqueue. Howeyer, some invariance relations among characteristic quantities in general queueshaye been derived by the theory of point processes; see, e.g., Franken et al. [8] and Miyazawa

[19]. Combining these invariance relations with our apprQximations fbr EW, we can derivesimple approximations for the steady-state probabilities in the GI!G/s queue. Extensionsof ouT approximations to this direction are in progress and will be reported elsewhere.




AmproximationsforaMitlti-Servereueue 9I

Acknowledgments

I am gratefu1 to the referees for their helpful suggestions. This research was supported

in part by the Grants in Aid for Scientific Research of the Japanese Ministry of Education,Science and Culture under the Contracts No. 62302059 (1987-1989) and No. 63780017 (1988-1989), and by the Special Grant-in-Aid for PTomotion of Education and Science in Hokkaido

University provided by the Japanese Ministry of Education, Science and Culture (1987-1988).

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The OpeiationsReseaichSociety of Japan

92 T Kimura

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'Ibshikazu KIMURA

Department of Business AdministrationFaculty of EconomicsHokkaido UniversityNishi 7, Kita 9, Kita-kuSapporo 060Japan

queue - j-stage

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