research article the assessment and foundation of bell...

Research ArticleThe Assessment and Foundation of Bell-ShapedTestability Growth Effort Functions Dependent SystemTestability Growth Models Based on NHPP

Tian-Mei Li1 Cong-Qi Xu2 Jing Qiu3 Guan-Jun Liu3 and Qi Zhang1

1Department of Automation Xirsquoan Institute of High-Tech Xirsquoan Shaanxi 710025 China2Institute of Construction Engineering Research General Logistics Department of PLA Xirsquoan Shaanxi 710032 China3Laboratory of Science and Technology on Integrated Logistics Support National University of Defense TechnologyChangsha Hunan 410073 China

Correspondence should be addressed to Tian-Mei Li tmlixjtu163com

Received 29 October 2014 Accepted 8 December 2014

Academic Editor Gang Li

Copyright copy 2015 Tian-Mei Li et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper investigates a type of STGM (system testability growth model) based on the nonhomogeneous Poisson process whichincorporates TGEF (testability growth effort function) First we analyze the process of TGT (testability growth test) for equipmentwhich shows that the TGT can be divided into two committed steps make the unit under test be in broken condition to identifyTDL (testability design limitation) and remove the TDL We consider that the amount of TGF (testability growth effort) spent onidentifying TDL is a crucial issue which decides the shape of testability growth curve and that the TGF increases firstly and thendecreases at different rates in the whole life cycle Furthermore we incorporate five TGEFs an Exponential curve a Rayleigh curvea logistic curve a delayed S-shape curve or an inflected S-shaped curve which are collectively referred to as Bell-shaped TGEFs intoSTGM Results from applications to a real data set of a stable tracking platform are analyzed and evaluated in testability predictioncapability and show that the Bell-shaped function can be expressed as a TGF curve and that the logistic TGEF dependent STGMgives better predictions based on the real data set

1 Introduction

Generally system testability is defined as the probability offault detection or isolation for a specific period of time in aspecified environment which is quantified by various testa-bility indexes such as FDR FIR and FAR [1ndash3] Over thelast several decades many testability test technologies espe-cially testability demonstration test for equipment have beenresearched [4ndash15] However some research work indicatesthat the result of testability demonstration test departuresfrom the actual value of testability greatly and makes theresult of testability demonstration test unauthentic [4] Theroot cause for this is that the vacancy of TGT in trackingandmeasuring the growth of testability as equipment is beingdeveloped

In general any fault diagnosis system similar to reliabilityfor large-scale and complicated equipment like a missilemay be premature when the fault diagnosis system has been

developed in which many TDLs hide such as nonexpectantfailure test vacancy ambiguity group fuzzy point improperresistance tolerance or threshold And then the designersanalyze and judge the root cause of TDL and further trackto the design of equipment testability like UUT redesign oftest equipment or interface equipment circuit of BIT faultdiagnosis software and test program of ATEThus almost allthe fault diagnosis systems for large-scale and complicatedequipment need a certain extent of time period to developTGT to identify and remove the TDLs to attain the expectedvalue under contract

The TGT is an important and expensive part of equip-ment testability development A TDL in UUT leads to anoutput that differs from specifications and requirements Theaim of TGT is to identify and remove the TDL and furtherto increase the fault detectionisolation probability that adesigned testability will work as intended in the hands of

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 613170 17 pageshttpdxdoiorg1011552015613170

2 Mathematical Problems in Engineering

the designers The testability growth test phase aims atidentifying and removing the TDLs Typically TGT is moreconstructive and has high confidence level for system testa-bility rather than the testability demonstration test at theacceptance stage only

A common approach for measuring system testability isby using an analytic model whose parameters are generallyestimated from available data on fault detectionisolation Inthis paper the analytic model is referred to by us as STGMA STGM provides a mathematical relationship between thenumber of TDLs removed and the test time during thewhole life cycle and generally is used as a tool to estimateand predict the progress of system testability At the sametime the STGM can be used for planning and controlling alltest resources during development and can assure us aboutthe testability of equipment There is an extensive body ofliterature emphasizing the importance of TGT but no depthof research is done on STGM [3 8]

In the context of TGT the basic objective is to identify theTDLs and remove them one by one Therefore the processof TGT contains not only a TDL identification process butalso a TDL removal process and that the key issues arethe TGE which are consumed to identify the TDL and theeffectiveness of TDL removal respectively In general theTGE can be represented as man-hour TGT cost the times offault injection and so forth The functions that describe howa TGE is distributed over the TGT phase are referred to by usas TGEF The shape of the observed testability growth curvedepends strongly on the time distribution of the TGE Due tothe randomness of TGT and consider the designerrsquos capabilityand familiarity to UUT synchronously the other key issue(ie the effectiveness of TDL removal) can be referred to byus as a constant TDL identification rate and a constant TDLremoval rate respectively

There is an extensive body of literature [16ndash40] onsoftware reliability growth model describing the relationshipbetween the test time and the amount of test-effort expendedduring that time in which the test effort was often describedby the traditional Exponential [19 21] Rayleigh [16 19 21]logistic [29 30 37 38] delayed-shaped [19 39] or inflectedS-shaped [18 39] curves Thus to address the issue of TGEFthis paper will use the above five test effort curves which arecollectively referred to as Bell-shaped TGEFs to describe therelationship between the test time and the amount of TGEexpended during that time by analyzing the consumptionrule of TGE Sometimes the TGE can be represented asthe number of faults injected or occurred naturally insteadof man-hours or TGT cost Similarly to address the issueof effectiveness of TDL removal we assume that the TDLidentification rate and the TDL removal rate are constant

In this paper the process of TGT is decomposed intofault injection or occurrence TDL identification and TDLremoval processes which are all based on NHPP We showhow to integrate time dependent TGEF constant TDL iden-tification rate and constant TDL removal rate into STGM tofind the framework of STGMWe further pay main attentionon the consumption pattern of TGE based on fault injectioncollection of fault occurring naturally and then discuss howto integrate five Bell-shaped TGEFs into STGM A method

to estimate the model parameter is provided Experimentalresults from a stable tracking platform are analyzed and thefive STGMs based on the above five Bell-shaped TGEFs arecompared with each other to show which STGM can givebetter prediction

The remainder of the paper is organized as followsIn Section 2 we found a framework of STGM consideringTGEF The fault occurrence TDL identification and TDLremoval process based on NHPP are also described andanalyzed in this section In Section 3 we pay main atten-tion to analyzing the consumption rule of TGEF which isrepresented by fault injection or collection of fault occurrednaturally at designampdevelopment stage and trial amp in-servicestage respectively At the same time we show how to incor-porate the five Bell-shaped TGEFs into STGM Parametersof the proposed STGMs as estimated by the method of LSEare discussed in Section 4 Finally Section 5 discusses thegoodness of the above five STGMs on prediction with theapplication of these models to a real data set

2 A General Framework of STGMBased on NHPP

In general the precondition for testability growth test is thatthe UUT is in failed condition That is to say there is oneor more faults occurred in the UUT Under the precondi-tion TGT is the process of running a testability diagnosticprogram and capturing the fault detection and isolation datato identify the TDLs and remove them In other words theTGT aims at identifying the TDLs and removing them oneby one under the precondition of fault occurred Thereforethe process of removing TDLs can roughly be divided intothree steps fault occurrence TDL identification and TDLremoval At the same time each of the above three steps canbe described effectively as a counting process based onNHPP

In practice the most important factor which affects theSTGMrsquos evaluation and predication accuracy is TGE TGEis the cumulative testability growth effort consumed in TGTwhich can be measured by the number of fault injection testcost staff and so onThe consumedTGE indicates howTDLsare identified effectively in the UUT and can be modeled bydifferent curves considering the consumption pattern inTGTIn this section a framework for foundation of STGM withTGEF is proposed

21 Fault Occurrence TDL Identification and Removal TDLidentification may be realized under the condition thatthe UUT must be broken Once the UUT is in brokencondition which usually can be realized by fault injection orcollection information of fault occurred naturally the designof testability begins to run to identify the TDL and analyzethe root cause of the TDL When the specific root cause isrecognized the designers can remove the TDL accordingly

We can use the number of residual TDLs as a measure ofsystem testability The great the number of TDLs the lowerof testability level would be So the process of TGT is theprocess to identify and remove the TDLs existing in the UUTIn other words the process of TGT is to identify and removethe TDL one by one So the process of TGT can be described

Mathematical Problems in Engineering 3

0

0

0

0

0

t1

t2

t3

t4

ti

tM

middot middot middotmiddot middot middot





f1 f2 f3 f4 fi fM

The valid detection and The valid detection and The remaining invalid

(d) The process of removing TDL exist in the UUT

(e) The process of fault detection and isolation after TGT

The event of fault occurred or injected

The event of detecting and isolating a fault successfully

The event of identifying a TDL

The event of removing a TDL successfully

The event of removing a TDL unsuccessfullyThe event of detecting and isolating a fault unsuccessfully

TDL1 TDL2 TDLj

(b) The process of fault detection and isolation designed in the UUT

(c) The process of identifying TDL exist in the UUT

A valid removal to TDL1 A valid removal to TDL2

isolation to f1 after TGT isolation to f4 after TGT detection and isolation to fiafter TGT

An identification to TDL1 An identification to TDL2 An identification to TDLj

AnAn invalid removal to TDLj

t

t

t

t

t

An invalid detectionand isolation to f1


An invalid detectionand isolation to fi

A valid detectionand isolation to f2


A valid detectionand isolation to fM

(a) The process of fault occurred or injected in the UUT

Figure 1 An apt sketch map of TGT

by Poisson process considering the randomness of faultinjection fault detection and isolation TDL identificationand removal etc Here we give an apt shown in Figure 1 todescribe the process of TGT

The fives processes given by Figure 1 about TGT and faultdetection and isolation are (a) the process of fault occurrednaturally or injected of theUUT (c) the process of identifyingthat TDL exists in the UUT (d) the process of removing TDLexist in the UUT (b) and (e) depict the fault detection andisolation before and after the TGT respectively

In (a) 1198911 1198912 119891

119894 119891

119872 is the failure mode set which

is obtained by FMEA and 119872 is the total number of failures

in the UUT With the progress of TGT the time-dependentfailure phenomena which can be realized by fault injection orcollection of fault information occurred naturally at randomtime serial 119905

1 1199052 119905

119894 119905

119872

In (b) When some failures occurred at random timeserial 119905

1 1199052 119905

119894 119905

119872 the system testability is activated

to detect and isolate the failures accordingly If a failure can-not be detected and isolated successfully a TDL is identifiedPractically some failures may be easy to detect and isolateand some others are not For example 119891

1and 119891

4cannot be

detected and isolated correctly so two TDLs are identifiedand are noted as TDL

1and TDL

2 respectively


With the progress of TDL identification increasing TDLsare identified gradually thus we have the process (c) whichdescribes the counting process of TDL identified

In (d) the designers analyze the root cause of theTDL andtry their best to modify the design of testability and removethe TDL accordingly But not all the TDLs identified canbe removed successfully because of designerrsquos familiarity toUUT For example the TDL

119895presented in process (c) still

cannot be removed successfully after an infinite amount oftest time and test effort consumed All the TDLs which canbe identified but cannot be removed successfully form theprocess (d) In TGT we should pay more attention to theexpected average number of TDLs identified and removedsuccessfully

By comparison with process (b) process (e) gives thegrowth effect of fault detection and isolation after TGTapparently

22 NHPP in TGT There is an extensive body of literature[12 32] on NHPP used for description of counting processeffectively such as the probabilistic failure process in softwareand equipment In this paper the NHPP is introduced todescribe the TDL detection and removal process 119873(119905) isa nonnegative integer and a time-dependent nondecreasingfunction which describes the cumulative number of TDLsidentified and removed up to time 119905 If 119904 lt 119905 119873(119904) minus 119873(119905) isthe counts of the TDL identified and removed in the interval(119904 119905)

The following characters of 119873(119905) can be derived fromthe assumption that ldquothe probability that two or more faultsoccurred synchronously is wee and can be neglected in testa-bility engineeringrdquo [3]

(1) 119873(0) = 0(2) The process has independent increment(3) lim

ℎrarr0119875[119873(119905 + ℎ) minus 119873(119905) = 1] = 120582(119905)ℎ + 119900(ℎ)

(4) limℎrarr0

119875[119873(119905 + ℎ) minus 119873(119905) ge 2] = 119900(ℎ)

Thus the process of 119873(119905) 119905 ge 0 follows a Poissonprocess with a parameter 120582(119905) where 120582(119905) is the TDL intensityof 119898119903(119905) (ie number of TDLs identified and removed cor-

rectly per unit time) The counting process with a constant120582(119905) has a smooth increment which is named homogeneousPoisson processOtherwise the counting processwith a time-dependent 120582(119905) has a fluctuant increment which is namednonhomogeneous Poisson process HPP can be considered asa special case of NHPP and NHPP is an extend case of HPP

Make 119898119903(119905) = 119864[119873(119905)] then 119898

119903(119905) is the expected mean

number of TDLs identified and removed successfully in time(0 119905] so119898

119903(119905) can be expressed as

119898119903(119905) = int

119905

0

120582(119906)d119906 (1)

So

120582(119905) = 1198981015840

119903(119905) = limΔ119905rarr0

119864[119873(119905 + Δ119905) minus 119873(119905)]

Δ119905 (2)

To identify the inherentTDLs in the UUT theprocess is expressed by

To remove the TDLsidentified by the tester the

To evaluate the value ofsystem testability after TGT

noted as Te

b2byprocess is expressed

To estimate the value ofsystem testability before

TGT noted as T(t0)

W(t) and b1

Figure 2 The basic process of TGT

To any 119905 ge 0 119904 ge 0 119873(119905 + 119904) minus 119873(119905) follows the Poissondistribution with parameters119898

119903(119905 + 119904) minus 119898

119903(119905) so we have

119875[119873(119905 + 119904) minus 119873(119905) = 119899] = expminus[119898119903(119905 + 119904) minus 119898

119903(119905)]

sdot[119898119903(119905 + 119904) minus 119898

119903(119905)]119899

119899

(3)

We can use either the number of TDLs identified andremoved or the number of remaining TDLs as a measureof testability quality Here we use the number of TDLsidentified and removed successfully (ie119898

119903(119905)) as a measure

of system testability quality Parallel to the analysis of Fig-ure 1 119898

119903(119905) is critical for both the mean and variance of

testability estimation and prediction Hence calculation of119898119903(119905) is our main focus in the following parts of this paper

23 A General Framework of STGM Considering TGEF ASTGM provides a mathematical relationship between thenumber of TDLs identified and removed successfully (119898

119903(119905))

from the UUT and the test time STGM can be used as atool to estimate and predict the progress of system testabilityA TDL inherent in the UUT leads to an output that thefault occurred or injected cannot be detected and isolatedcorrectly The TGT aims at identifying these TDLs andremoving them At any time during the TGT phase thebasic process of TGT includes four steps which are shown inFigure 2

In Figure 2119879119890is the target testability level which has been

fixed at the beginning of TGT But it is frequently realized thatthis target may not be achievable for a number of reasonslike inadequacy of TGE or inefficiency of the test team SoTGT is a repetitive work of ldquotest-identification-correction-testrdquo generally until the conditional expression 119879(119905

0) lt 119879119890is

satisfied 119879(1199050) is the estimation value of testability at time 119905

0

Firstly we need to estimate 119879(1199050) at time 119905

0 From the

above analysis the main cause leading to 119879(1199050) lt 119879

119890is the

TDL existing in the UUT In this paper we assume that theremoval of a TDL from the UUT includes two phases In thefirst phase the TDL identification team primarily consisting


of test personnel identifies a TDL by making the UUT bein failed condition Concretely the identification team caneither collect the fault occurred naturally or injected to verifythe level of testability During theTGTphasemuch testabilitygrowth effort is consumedThe consumedTGE indicates howthe TDLs are identified effectively in the UUT and can bemodeled by different curves Actually the system testabilityis highly related to the amount of testability growth effortexpenditures spent on identifyingTDLs In this step theworkin the process of TDL identification is calculated by 119882(119905)such as man-hour TGT cost and the times of fault injection119882(119905) signifies the cumulative TGE consumed in identifyingthe TDLs up to the given time 119905Then another team primarilyconsisting of designers analyzes the reason for the TDLand modifies the design of testability to remove the TDL byredesigning of UUT or ATE and the corresponding interfaceoptimizing the circuit of BIT debugging the diagnosticsoftware and rewriting the diagnostic program of ATE etcAll the work about removing the TDLs will be done in thethird step of Figure 2 by the designers of system testabilitywhich is described by a constant 119887

2 Finally all the teams

evaluate the value of system testability after TGTnoted as119879(119905)and give a test conclusion that if 119879(119905) ge 119879

119890 the TGT can be

stopped else the TGT will be continued until the conditionexpression 119879(119905) ge 119879

119890is satisfied Apparently testability can

be enhanced if TGT is done for a prolonged period that is119879119890ge 119879(119905) gt 119879(119905

0)

The STGM developed below is based on the followingassumptions

(1) The TDL identification and its removal processes allfollow NHPP

(2) The failure of fault detectionisolation at randomtimes is caused by the TDLs which are inherent in theUUT

(3) All TDLs are mutually independent and have thesame contribution to the failure of fault detectioniso-lation occurred in the system

(4) There is a one-to-one correspondence between afailure and the corresponding test

(5) Two phases can be observed within the TGT processTDL identification andTDL removalThere is no timelag between the TDL identification and its removalWhenever a TDL is identified the goal of TDLremoval is to analyze the root cause of the TDL andto remove it At the same time no new TDLs areintroduced

(6) The mean number of TDL identified in the timeinterval [119905 119905 + Δ119905] by the current TGE is proportionalto themeannumber of TDLs unidentified in theUUTThe mean number of TDL identified and removed inthe time interval [119905 119905 + Δ119905] by the current testabilitygrowth effort is proportional to the mean number ofTDLs uncorrected in the UUT The proportions areconstants and expressed by 119887

1and 1198872 respectively

(7) The consumption of TGE is modeled by119882(119905)

Let 119898119894(119905) be the MVF of the expected number of TDL

identified and let119898119903(119905) be theMVFof the expected number of

TDLs identified and removed in time (0 119905] Then accordingto the above assumptions we describe the STGM based onTGEF as follows

d119898119894(119905)

d119905times

1

119908(119905)= 1198871[119886 minus 119898

119894(119905)]

d119898119903(119905)

d119905times

1

119908(119905)= 1198872[119898119894(119905) minus 119898

119903(119905)]

(4)

Note that here we assume 1198871

= 1198872 Solving (4) under the

boundary condition119898119894(0) = 119898

119903(0) = 0 we have

119898119894(119905) = 119886 times 1 minus exp[minus119887

1119882lowast(119905)] (5)

119898119903(119905) = 119886times1 minus

1198871exp[minus119887

2119882lowast(119905)]minus119887

2exp[minus119887

1119882lowast(119905)]

1198871minus 1198872

(6)

119886remaining = 119886 minus 119898119903(infin)

= 1198861198871exp[minus119887

2119882lowast(infin)] minus 119887

2exp[minus119887

1119882lowast(infin)]

1198871minus 1198872

(7)

Due to the space limitations here we only propose thetestability growth model of FDR in general FDR is definedas the capability to detect fault occurred in the UUT andFDR is also used to describe the average detection probabilityto the failure mode set which is gained from FMEA Themathematical model of FDR can be formulated as [1ndash3]

FDR =119873119863

119872 (8)

where 119872 is the total number of failures 119873119863is the number

of failures which can be detected accurately by the systemtestability design

According to the above assumption (4) we can have thatthe number of failures which can be detected accurately bythe system testability design is

119873119863= 119872 minus 119886(119905) = 119872 minus [119886 minus 119898

119903(119905)] (9)

Substituting it into (8) we obtain

119902(119905) =119872 minus [119886 minus 119898

119903(119905)]

119872 (10)

Substitute119898119903(119905) (in (6)) into (10) the 119902(119905) can be rewritten as

119902(119905) = (119872 minus 119886((1198871exp[minus119887

2119882lowast(119905)] minus 119887

2exp[minus119887

1119882lowast(119905)])

times(1198871minus 1198872)minus1

))119872minus1

(11)

Depending on how elaborate a model one wishes toobtain is one can use 119908(119905) to yield more complex or lesscomplex analytic solutions for 119902(119905) Different 119908(119905) reflectvarious assumption patterns of TGE in TGT


Equation (11) is themodeling framework which considersTGEF into the NHPP STGM At the same time (11) canprovide designers or testers with an estimation of the timeneeded to reach a given level of FDR And that it may also beused to determine the appropriate ending time for TGT andcan provide useful information for making decision on turnphase

3 STGM with Bell-Shaped TGEF

Actually the system testability is highly related to the amountof testability growth effort expenditures spent on identifyingthe TDLs In this section we develop some STGMs in whichfive Bell-shaped TGEFs an Exponential TGEF a RayleighTGEF a logistic TGEF a delayed S-shaped TGEF and aninflected S-shaped TGEF are taken into account The formsof TGEF are all discussed in the literature about softwarereliability growthmodel [16ndash39]Here we choose appropriateforms of TGEF in view of the TGTrsquos specialty

31 Bell-Shaped TGEF In TGT one of the key factors isthe TGE which can be represented as the number of man-hours the times of fault injection and the cost of TGTand so forth The functions that describe how an effort isdistributed over the testability growth phase are referred toby us as TGEF TGEF describes the relationship between thetest time and the amount of TGE expended in the test timeActually fault injection is one of the best effective measuresto make the UUT be in broken condition and to identify theinherent TDLs in theUUT furtherThere is an extensive bodyof literature on fault injection used for identifying TDL intestability test [1ndash3 41ndash43] Therefore we will use the timesof fault injection as a measurement of TGE which has anadvantage that it is very intuitive and that testability growtheffort can be quantified exactly at the same time

In general the whole life-cycle of equipment can bedivided into three stages demonstration stage design ampdevelopment stage and trial amp in-service stage But theempirical analysis shows that the effective and credible stagesto conduct TGT are design amp development stage and trial ampin-service stage considering the cost effectiveness in whichfault injection and collection information for naturallyoccurred faults are done respectively [3 6] At the same timeTGE are calculated by times of fault injection and the numberof naturally occurred faults at design amp development stageand trial amp in-service stage respectively

Actually the more faults we inject or collect the moreconfidence we obtain in the estimation of system testabilityUnfortunately TGT with redundant fault injections or alarge number of failure data may lead to excessive cost andtoo much time consumption It is impracticable with theconstraint of test cycle and total cost

To give an accurate and reasonable description of TGEwe need to know not only how many faults need to inject atdesign amp development stage but also how many failure dataneed to collect at trial amp in-service stage That is to say weneed to analyze the TGErsquos consumption pattern in TGT bycombining Figure 3 which gives an abridged general view oftestability growth process in the whole life-cycle

QE desired value of FDR

Testability growth testing stage0

Design amp development stage Trial amp in-service stage

True value of FDR

t

q(te) = QE

q(t)

(nm+1 fm+1)qk

q(0)ts te

(nm fm)

(n1 f1)

(n2 f2)(nk fk) (nmminus1 fmminus1)

at time t (0 le t le Te)

Figure 3 The testability growth chart in the whole life-cycle

In general the TGT time can be represented by the serialnumber of testability growth test stage instead of the actualexecution time The assumptions which are shown in Fig-ure 3 and will be used for description of TGEF are as follows

(1) TheTGT stage is expressed as the time span [0 119905119890] and

is composed by the design amp development stage timespan [0 119905

119904] and the trail amp in-service stage time span

[119905119904 119905119890] respectively

(2) The times of TGT based on fault injection at design ampdevelopment stage are119898 and the maximum numberof fault injection at the design amp development stage ismarked as 119873DD At the design amp development stagewe can observe 119899 data pairs in the form (119899

119896 119891119896) 1 le

119896 le 119898 119899119896is the number of fault injection at the 119896th

TGT stage and 119891119896is the number of TDLs identified

successfully at the 119896th TGT stage(3) After a period of time especially the 119898th TGT at

design amp development stage the UUT is put intothe trail amp in-service stage at which the number ofTGT based on fault occurred naturally is 119898

119878and the

maximum number of failures collecting is marked as119873119878 In a similar way the same data pairs in the form

(119899119895 119891119895) 119898 lt 119895 le 119898

119878are observed by collecting the

failure data occurred naturally 119899119895is the number of

failures by collecting the naturally occurred fault and119891119895is the number of TDLs identified successfully at the

119895th TGT stage(4) With the progress of TGT 119902

119894 1 le 119894 le 119898

119878is the

true value of FDR at the 119894th TGT stage and that 1199021lt

sdot sdot sdot 119902119894sdot sdot sdot lt 119902

119898119878 The TGT can be stopped only when

the 119876119864is achieved in which 119876

119864is the FDRrsquos desired

value of TGT shown in Figure 3

Based on the above assumptions we will analyze theTGE consumption pattern from two perspectives one is faultinjection based TGE consumption at design amp developmentstage the other is naturally occurred fault based TGE con-sumption at trail amp in-service stage

At the beginning of the design amp development stage FDRlevel is relatively lower hence a certain number of TDLs will


01 02 03 04 05 06 07 08 090

20

40

60

80

100

120

FDR level

Num

ber o

f fau

lt in

ject

ions

TDL = 3

TDL = 5

TDL = 10

Figure 4The number of faults needs to be injected with the changeof FDR level and number of TDLs respectively

be identified as long as we inject less fault In order to identifya certain number of TDLs at the design amp development stagethe number of faults need to inject grow up with the level ofFDR which can be formulated as [1ndash3]

119902119896=

119899119896minus 119891119896

119899119896

1 le 119896 le 119898 (12)

For example with reference to (12) the number of faultswhich need to be injected with the change of FDR leveland number of TDLs respectively are shown in Figure 4As seen from Figure 4 we find that at the same numberof TDLs the FDR level needs to achieve increase as timegoes on and the number of faults needs to inject increasesaccordingly At the same level of FDR the greater number ofTDLs needs to identify the greater number of faults needs toinject Consequently TGEF is a time-dependent increasingfunction at design amp development stage

Collection of failure which occurred naturally is also aneffective measure to identify TDLWhen the UUT is put intothe trail amp in-service stage Actually the number of faultsat trail amp in-service stage is ldquosmall samplerdquo under severalconstraints such as high reliability requirements or specifiedtest cycle Considering the growth of reliability the expectednumber of faults at regular intervals will decrease graduallyThat is to say at trail amp in-service stage TGEF is a time-dependent decreasing function which has the similar varia-tion tendency shown in Figure 5

In conclusion at the whole TGT phase TGEF increasesfirstly and then decreases at different rate Based on theincrease-decrease characteristic we will use a kind of Bell-shaped function which increases firstly and then decreases tofit the practical growth rate of TGEF Bell-shaped functionjust as its name implies draws a time-dependent curve like abell which has a similar shape to the curve shown in Figure 6

It should be noted that Figure 6 is just a schematic usedto depict the Bell-shaped function the concrete functionalexpression depends on the actual TGE data set Becauseactual testability growth effort data represent various expen-diture patterns sometimes the testability growth effort

Num

ber o

f fai

lure

s occ

urre

d na

tura

lly

Test timets te t

Figure 5 The diagrammatic drawing of TGEF at trail amp in-servicestage

0Test time

ts te

Curr

ent t

eata

bilit

y gr

owth

effor

t

t

cons

umpt

ion

at ti

me t

w(t)

Figure 6 A schematic of Bell-shaped current TGE consumption

expenditures are difficult to describe only by a Bell-shapedcurve In this paper the Exponential TGEF the RayleighTGEF the logistic TGEF the delayed S-shaped TGEF andthe inflected S-shaped TGEF have been used to explain thetestability growth effort which can be derived from theassumption that ldquothe efforts that govern the pace of identi-fying the TDL for the UUT increase firstly and then decreasegraduallyrdquo

311 The Exponential TGEF Yamada et al [21] found thatthe TEF in software reliability growth could be described bya Weibull-type distribution with two cases the Exponentialcurve and the Rayleigh curve


The Exponential TGEF over time period (0 119905] can beexpressed as

119882119864(119905) = 119873

1[1 minus exp(minus120573

1119905)] (13)

The current TGE expenditure rate at time 119905 is

119908119864(119905) = 119873

11205731exp(minus120573

1119905) (14)

where119882119864(119905) = int

119905

0119908119864(120591)d120591

The Exponential curve is used for process that declinesmonotonically to an asymptote

Substitute the119882lowast119864(119905) in (13) into (7) we obtain

119886remaining = 1198861198871exp[minus119887

2119882lowast

119864(infin)] minus 119887

2exp[minus119887

1119882lowast

119864(infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198731] minus 1198872exp[minus119887

11198731]

1198871minus 1198872

(15)

312 The Rayleigh TGEF [16 19 21 37] Another alternativeof the Weibull-type distribution is the Rayleigh curve Ithas been empirically observed that software developmentprojects follow a life-cycle pattern described by the Rayleighcurve

The Rayleigh TGEF over time period (0 119905] can beexpressed as

119882119877(119905) = 119873

2[1 minus exp(

minus1205732

21199052)] (16)


119908119877(119905) = 119873

21205732119905 exp(minus

1205732

21199052) (17)

where119882119877(119905) = int

119905

0119908119877(120591)d120591

Therefore 119908119877(119905) is a smooth Bell-shaped curve and

reaches its maximum value at time

119905max =1

1205732

(18)

The Rayleigh curve first increases to a peak and thendecreases at a decelerating rate



2119882lowast

119877(infin)] minus 119887

2exp[minus119887

1119882lowast

119877(infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198732] minus 1198872exp[minus119887

11198732]

1198871minus 1198872

(19)

313 The Logistic TGEF Logistic TEF was originally pro-posed by Parr [16] about software reliability growth model Itexhibits similar behavior to the Rayleigh curve except duringthe early part of the project Huang et al [28 30 37] proposed

that a logistic testing effort function can be used instead ofthe Weibull-type curve to describe the test effort patternsduring the software development process In some two dozenprojects studied in the Yourdon 1978ndash1980 project survey theLogistic TEF appeared to be fairly accurate in describing theexpended test efforts [44]

The logistic TGEF over time period (0 119905] can be expressedas

119882119871(119905) =

1198733

1 + 119860 exp(minus1205733119905) (20)


119908119871(119905) =

11987331198601205733exp(minus120573

3119905)

[1 + 119860 exp(minus1205733119905)]2 (21)

where119882119871(119905) = int

119905

0119908119871(120591)d120591

Therefore 119908119871(119905) is a smooth Bell-shaped function and


119905max =1

1205733

ln119860 (22)


119886remaining

= 1198861198871exp[minus119887

2119882lowast

119877(infin)] minus 119887

2exp[minus119887

1119882lowast

119877(infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198733(1 + 119860)] minus 119887

2exp[minus119887

11198733(1 + 119860)]

1198871minus 1198872

(23)

Ohba [17 18] found that the test effort function in soft-ware reliability growth could be described by S-shaped dis-tribution with two cases the delayed S-shaped curve and theinflected S-shaped curve

314 The Delayed S-Shaped TGEF [17 40] The delayed S-shaped TGEF over time period (0 119905] can be expressed as

119882DS(119905) = 1198734[1 minus (1 + 120573

4119905) exp(minus120573

4119905)] (24)


119908DS(119905) = 11987341205732

4119905 exp (minus120573

4119905) (25)

where119882DS(119905) = int119905

0119908DS(120591)d120591

Therefore 119908DS(119905) is a smooth Bell-shaped function andreaches its maximum value at time

119905max =1

1205734

(26)

Substitute the119882lowastDS(119905) in (24) into (7) we obtain


2119882lowast

DS (infin)] minus 1198872exp[minus119887

1119882lowast

DS (infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198734] minus 1198872exp[minus119887

11198734]

1198871minus 1198872

(27)


0 10 20 30 40 50 600

5

10

15

20

25

30

Time (weeks)

Curr

ent T

GE

N1 = 240 1205731 = 01111

Figure 7 An Exponential curve with fixed parameters

315 The Inflected S-Shaped TGEF [18 40] An inflected S-shaped TGEF over time period (0 119905] can be expressed as

119882IS(119905) = 1198735

1 minus exp(minus1205735119905)

1 + 120593 exp(minus1205735119905) (28)


119908IS(119905) =11987351205735(1 + 120593) exp(minus120573

5119905)

[1 + 120593 exp(minus1205735119905)]2

(29)

where119882IS(119905) = int119905

0119908IS(120591)d120591

Therefore 119908IS(119905) is a smooth Bell-shaped function andreaches its maximum value at time

119905max =1

1205735

ln120593 (30)

Substitute the119882lowastIS(119905) in (28) into (7) we obtain


2119882lowast

IS(infin)] minus 1198872exp[minus119887

1119882lowast

IS(infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198735] minus 1198872exp[minus119887

11198735]

1198871minus 1198872

(31)

With reference to the above five Bell-shaped TGEFs theinstantaneous TGEs which are shown in Figures 7 8 9 10and 11 decrease ultimately during the testability growth life-cycle because the cumulative TGEs approach a finite limit Asshown from (15) (19) (23) (27) and (31) we find that not allthe inherent TDLs in the UUT can be fully removed evenafter a long TGT period because the total amount of TGE tobe consumed during the TGT phase is limited to 119882max Thisassumption is reasonable because no system testability designcompany will spend infinite resources on TGT consideringthe restricted cost amp development cycle

32 Bell-Shaped TGEF Dependent STGM In this section wewill use the above five TGEFs in the foundation of STGM

0

5

10

15

20

25

Time (weeks)

Curr

ent T

GE

0 2 4 6 8 10 12 14 16 18 20

N2 = 120 1205732 = 01111

Figure 8 A Rayleigh curve with fixed parameters

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

Time (weeks)

Curr

ent T

GE

N3 = 240 A = 2303 1205733 = 01111

Figure 9 A logistic curve with fixed parameters

One of these can be substituted into (10) to have five kinds ofSTGM which are formulated as follows

321 EX-STGM Substitute (13) into (11) the EX-STGM canbe formulated as

119902(119905) = (119872 minus 119886 times ((1198871exp[minus119887

21198731[1 minus exp (minus120573

1119905)]]

minus1198872exp[minus119887


1119905)]])

times (1198871minus 1198872)minus1

))119872minus1

(32)

322 RA-STGM Substitute (16) into (11) the RA-STGM canbe formulated as


21198732[1 minus exp(minus

1205732

21199052)]]



1205732

21199052)]])


))119872minus1

(33)


0 10 20 30 40 50 60 70 800

21

4

6

8

10

Time (weeks)

Curr

ent T

GE

N4 = 240 1205734 = 01111

9

7

5

3

Figure 10 A delayed S-shaped curve with fixed parameters

0 10 20 30 40 50 60 70 80Time (weeks)

Curr

ent T

GE

N5 = 240 120593 = 1984 1205735 = 01111

005

115

225

335

445

5

Figure 11 An inflected S-shaped curve with fixed parameters

323 LO-STGM Substitute (20) into (11) the LO-STGM canbe formulated as

119902(119905) = (119872minus119886((1198871exp[minus119887

21198733[

1

1+119860 exp (minus120572119905)minus

1

1+119860]]

minus 1198872exp[minus119887

11198733

times[1

1+119860 exp (minus120572119905)minus

1

1+119860]])


))119872minus1

(34)

324 DS-STGM Substitute (24) into (11) the DS-STGM canbe formulated as

119902(119905) = (119872minus119886((1198871exp[minus119887

21198734[1 minus (1+120573

1119905) exp (minus120573

1119905)]]


11198734[1 minus (1+120573

1119905) exp (minus120573

1119905)]])


))119872minus1

(35)

325 IS-STGM Substitute (28) into (11) the IS-STGM canbe formulated as

119902(119905) = (119872 minus 119886((1198871exp[minus119887

21198735[

1 + exp (minus1205732119905)

1 + 120593 exp (minus1205732119905)]]


11198735[

1 + exp (minus1205732119905)

1 + 120593 exp (minus1205732119905)]])


))119872minus1

(36)

4 Estimation of STGM Parameters

Fitting a proposedmodel to actual TGTdata involves estimat-ing the model parameters from the real TGT data set Twopopular estimation techniques are MLE and LSE [45] TheMLE estimates parameters by solving a set of simultaneousequations However the equation set may be very complexand usually must be solved numerically The LSE minimizesthe sumof squares of the deviations betweenwhat we actuallyobserve and what we expect In order to avoid the solutionof complex simultaneous equations of MLE in this sectionwe employ the method of LSE to estimate the parameters ofthe above five TGEFs Using the estimated TGEFs the otherparameters 119886 119887

1 1198872in (5) (6) can also be estimated by LSE

Due to the limitations of paper size reference to the abovefive TGEFs only the parameters 119873

1 1205731of the Exponential

TGEF in (13) are estimated by themethod of LSE At the sametime the parameters 119886 119887

1given in (5) and 119887

2given in (6) can

all be estimated by LSE For themethod of LSE the evaluationformula 119878

119864(1198731 1205731) 119878119872119888(119886 1198871) and 119878

119872119903(119898119888 1198872) are as follows

Minimize 119878119864(1198731 1205731) =

119899

sum119896=1

[119882119864119896

minus119882119864(119905119896)]2

Minimize 119878119872119888(119886 1198871) =

119899

sum119896=1

[119872119862119896

minus119872119862(119905119896)]2

Minimize 119878119872119903(119898119888 1198872) =

119899

sum119896=1

[119898119903119896minus 119898119903(119905119896)]2

(37)

Differentiating 119878119864 119878119872119894 119878119872119888

with respect to (1198731 1205731)

(119886 1198871) and (119898

119894 1198872) respectively setting the partial derivatives

to zero and rearranging these terms we can solve this type ofnonlinear least square problems For a simple illustration weonly consider the generalized Exponential TGEF

Take the partial derivatives of 119878119864with respect to 119873

1 1205731

we get

120597119878119864

1205971198731

=

119899

sum119896=1

minus 2119882119864119896

minus 1198731[1 minus exp(minus120573

1119896)]

times [1 minus exp(minus1205731119896)] = 0

(38)


Table 1 Data set for a stable tracking platform

Test time(weeks)

CumulativeTGE

(fault times)

CumulativeTDLs identified

CumulativeTDLs removed

Test time(weeks)

CumulativeTGE

(fault times)



1 4 3 2 13 223 74 482 9 6 4 14 241 76 493 16 11 7 15 256 77 504 25 17 11 16 268 79 515 38 25 16 17 278 80 526 55 33 21 18 286 80 527 74 41 27 19 294 82 538 95 50 33 20 301 83 549 119 57 37 21 306 84 5410 145 63 41 22 310 85 5511 173 68 44 23 312 85 5512 203 72 46 24 313 86 55

Thus the least squares estimator 1198731is given by solving

the above equation yielding

1198731=

sum119899

119896=2119882119896[1 minus exp (minus120573

1119896)]

sum119899

119896=2[1 minus exp(minus120573

1119896)]2 (39)

Next we have

120597119878119864

1205971205731

=

119899

sum119896=1

minus 21198731119896119882119896minus 1198731[1 minus exp(minus120573

1119896)]

times exp(minus1205731119896) = 0

(40)

The parameter 1205731can also be obtained by substituting the

least squares estimator1198731into the above equation

Similarly the parameters1198732 1205732of the Rayleigh TGEF in

(16) 1198733 119860 120573

3of the logistic TGEF in (20) 119873

4 1205734of the

delayed S-shaped TGEF in (24) 1198735 1205735of the inflected S-

shaped TGEF in (28) can also be estimated by LSE

5 Data Analysis and STGM Comparisons

51 Data Description To validate the proposed STGMswith the above five Bell-shaped TGEFs TGT on a stabletracking platform have been performed The TGT data setemployed (listed in Table 1) was from the testability labo-ratory of National University of Defense Technology for astable tracking platform which can isolate the movement ofmoving vehicle such as car ship and aircraft FMEA of thestable tracking platform had been done and had gained thatthe stable tracking platform consisted of approximately 350functional circuit level failures Over the course of 12 weeks atthe design amp development stage 72 TDLs were identified byinjecting 203 functional circuits level failures Failures wereinjected by 1553B fault injection equipment ARINC 429 faultinjection equipment RS232422 fault injection equipmentCAN bus fault injection equipment and the like Further

testability designers analyzed the root cause of the TDLs andhad tried their best to modify the design of testability andremoved 46 TDLs successfully On the other hand over thecourse of 12weeks at the trialamp in-service stage 110 functionalcircuit level failures had occurred naturally 14 TDLs wereidentified in which 7 TDLs were removed successfully

52 Criteria for Model Comparison A STGM can generallybe analyzed according to its estimation capability fitnesscapability and predictive capability That is to say a STGMcan be analyzed according to its ability to reproduce theidentified TDL and behavior of the UUT and to predict thefuture behavior of the UUT from the observed TDL data Inthis paper the STGMs are compared with each other basedon the following three criteria

521 The Accuracy of Estimation Criterion [39] For practicalpurposes we will use AE to calculate the accuracy of esti-mation AE is defined as

AE =10038161003816100381610038161003816100381610038161003816

119898119894minus 119886

119898119894

10038161003816100381610038161003816100381610038161003816 (41)

119898119894is the actual cumulative number of identifiedTDLafter

the TGT and 119886 is the estimated number of initial TDLs119898119894is

obtained from system testability TDL tracking after TGT

522 The Goodness-of-Fit Criteria To quantitatively com-pare long-term predictions we use MSE because it providesa well-understood measure of the difference between actualand predicted values The MSE is defined as [30 37 45]

MSE =sum119899

119894=1[119898119903(119905119894) minus 119898119903119894]2

119899 (42)

A smaller MSE indicates a smaller fitting error and betterperformance


523 The Predictive Validity Criterion The capability of themodel to predict TDL identification and removal behaviorfrompresentamppast TDLbehavior is called predictive validityThis approach which was proposed byMusa et al [45] can berepresented by computing RE for a data set

RE =119898119903(119905119886) minus 119890

119890 (43)

Assuming that we have identified and removed 119890TDLs bythe end of TGT time 119905

119890 we employ the TDL data up to time 119905

119886

(119905119886le 119905119890) to estimate the parameters of119898

119903(119905) Substituting the

estimates of these parameters in the MVF yields the estimateof the number of TDLs 119898

119903(119905119890) by time 119905

119890 The estimate

is compared with the actual number 119890 The procedure isrepeated for various values of 119905

119886 We can check the predictive

validity by plotting the relative error for different values of119905119886 Numbers closer to zero imply more accurate prediction

Positive values of error indicate overestimation and negativevalues indicate underestimation

53 STGM Performance Analysis In this section we presentour evaluation of the performance of the proposed STGMswhen applied to DS listed in Table 1

All the parameters of the Exponential TGEF the RayleighTGEF the logistic TGEF the delayed S-shaped TGEF and theinflected S-shaped TGEF are also estimated by LSE Firstlythe two unknown parameters 119873


TGEF are solved by LSE giving the estimated values 1198731=

25927 (fault times) and 1205731= 00060week Correspondingly

the estimated parameters of the Rayleigh TGEF are 1198732

=

32957 (fault times) and 1205732

= 00123week In similarlythe estimated parameters of the logistic TGEF the delayed S-shaped TGEF the inflected S-shaped TGEF are119873

3= 31118

(fault times) 119860 = 3936 1205733= 03524week 119873

4= 40606

(fault times) 1205734= 01313week 119873

5= 31444 (fault times)

120593 = 2709 and1205735= 03232week In order to clearly show the

comparative effectiveness of the fitness for the observed TGEdata the comparisons between the observed five current TGEdata and the estimated five current TGE data are illustratedgraphically in Figures 12 and 13 respectively Similarly thecomparisons between the observed five cumulative TGE dataand the estimated five cumulative TGE data are illustratedgraphically in Figures 14 and 15 respectively

In order to check the performance of the five aboveTGEFs andmake comparisons with each other here we selectsome comparison criterions for the estimation of TGEFs[30 37 45]

PE119894= Actural(observed)

119894minus Predicted(estimated)

119894

Bias = 1

119899

119899

sum119894=1

PE119894

Variation = radicsum119899

119894=1(PE119894minus Bias)2

119899 minus 1

(44)

The PE Bias and Variation for the above five TGEFs arelisted in Table 2 From Table 2 we see that the inflected S-shaped TGEF has lower values of PE Bias andVariation than

Table 2 Comparison results for different TGEFs applied to DS

TGEF Bias Variation PEend of TGT

Exponential TGEF 426 2555 3471Rayleigh TGEF 127 707 703Logistic TGEF 070 378 minus440Delayed S-shaped TGEF 273 1386 2091Inflected S-shaped TGEF 018 228 minus229

0 5 10 15 200

5

10

15

20

25

30

35

Time (weeks)Cu

rren

t TG

E (fa

ult t

imes

)ExponentialRayleigh

LogisticActural

Figure 12 Observed and three estimated current TGEFs for DS

0 5 10 15 200

5

10

15

20

25

30

35

Time (weeks)

Curr

ent T

GE

(faul

t tim

es)

Delay S-shapedInflected S-shapedActural

Figure 13 Observed and two estimated current TGEFs for DS

the other four TGEFs On average the inflected S-shapedTGEF yields a better fit for this data set which can also bedrown from Figures 12 to 15 approximately

Table 3 lists the estimated values of parameters of differ-ent STGMs We also give the values of AE RE and MSE inTable 3 It is observed that the STGMwith inflected S-shapedTGEF (ie IS-STGM) has the smallest value of MSE whencompared with other STGMs

From Table 3 we see that the LO-STGM has lower valuesof AE and RE than the other four Bell-shaped TGEF depen-dent STGMs and that the RA-STGM has lower value of MSEthan the other four Bell-shapedTGEFdependent STGMsOnaverage the logistic TGEF dependent STGMyields a better fitfor this data set


Table 3 Estimated parameters values and model comparisons for DS

STGMs 119886 1198871

1198872

AE () MSE MREEX-STGM 11439 00046 00090 331 mdashlowast 03156RA-STGM 9560 00068 00076 1116 8391 01623LO-STGM 8735 00096 00070 157 8551 01063DS-STGM 10130 00059 00080 1779 9009 02213IS-STGM 9066 00081 00071 542 8613 01226lowastThe symbol ldquomdashrdquo indicates that the value is too large to characterize the MSE

0 5 10 15 20Time (weeks)

Cum

ulat

ive T

GE

(faul

t tim

es)

ExponentialRayleigh

LogisticActural

0

50

150

200

250

300

350

100

Figure 14 Observed and three estimated cumulative TGEFs for DS

0 5 10 15 20Time (weeks)

Cum

ulat

ive T

GE

(faul

t tim

es)


0

50

100

150

200

250

300

350

Figure 15 Observed and two estimated cumulative TGEFs for DS

0 5 10 15 20 25

002040608

11214

Time (weeks)

Rela

tive e

rror

minus04

minus02

Figure 16 RE curve of EX-STGM compared with actual DS

0 5 10 15 20 25

002040608

Time (weeks)

Rela

tive e

rror

minus1

minus08

minus06

minus04

minus02

Figure 17 RE curve of RA-STGM compared with actual DS

0 5 10 15 20 25

010203040506070

Time (weeks)

Rela

tive e

rror

minus10

Figure 18 RE curve of LO-STGM compared with actual DS

Finally Figures 16 17 18 19 and 20 depict the RE curvefor different selected STGMs

Substitute the estimated value of 119886 1198871 1198872listed in

Table 3 into (32) (33) (34) (35) (36) we can find fiveTGEF dependent STGMs of FDR Figures 21 22 23 24 and25 depict the growth curve of FDR at the whole TGT stage

Finally the performance of STGM strongly depends onthe kind of data set If a system testability designer plansto employ STGM for estimation of testability growth ofUUT during system development processes the testabilitydesigners need to select several representative models andapply them at the same time Although models sometimesgive good results there is no single model that can be trustedto give accurate results in all circumstances


0 5 10 15 20 25

002040608

Time (weeks)

Rela

tive e

rror

minus1

minus08

minus06

minus04

minus02

Figure 19 RE curve of DS-STGM compared with actual DS

0 5 10 15 20 25

0

1

2

3

4

5

Time (weeks)

Rela

tive e

rror

minus1

Figure 20 RE curve of IS-STGM compared with actual DS

0 5 10 15 20065

07

075

08

085

09

Time (weeks)

FDR

Figure 21 FDR curve based on EX-STGM

0 5 10 15 20072074076078

08082084086088

09092

Time (weeks)

FDR

Figure 22 FDR curve based on RA-STGM

0 5 10 15 20074076078

08082084086088

09092094

Time (weeks)

FDR

Figure 23 FDR curve based on LO-STGM

0 5 10 15 2007

075

08

085

09

095

Time (weeks)

FDR

Figure 24 FDR curve based on DS-STGM

0 5 10 15 20074076078

08082084086088

09092

Time (weeks)

FDR

Figure 25 FDR curve based on IS-STGM

From our results we can draw the following conclusions

(1) The Bell-shaped TGEF may be a good approach to pro-viding amore accurate description of resource consump-tion during TGT phase Particularly the inflected S-shapedTGEFhas the smallest value of Bias Variationand PE compared with the other four TGEFs whenapplied to DS for a stable tracking platform

(2) By incorporating the Bell-shaped TGEF into the struc-ture of STGM the STGMs with different TGEFs arevery powerful and flexible for various test environ-ments The RA-STGM has the smallest value of MSEcompared with the other four STGMs when appliedtoDS for a stable tracking platform Similarly the LO-STGM has the smallest value of AE and RE comparedwith the other four STGMs


Acronyms

UUT Unit under testATE Automatic test equipmentBIT Built-in testFMEA Failure mode and effect analysisFDR Fault detection rateFIR Fault isolation rateFAR False alarm rateHPP Homogeneous Poisson processNHPP Nonhomogeneous Poisson processSTGM System testability growth modelTGT Testability growth testTDL Testability design limitationTGE Testability growth effortTGEF Testability growth effort functionMVF Mean value functionEX-STGM A STGM with an Exponential TGEFRA-STGM A STGM with a Rayleigh TGEFLO-STGM A STGM with a logistic TGEFDS-STGM A STGM with a delayed S-shaped TGEFIS-STGM A STGM with an inflected S-shaped TGEFLSE Least square estimationMLE Maximum likelihood estimationMSE Mean square of fitting errorRE Relative errorPE Prediction errorMRE Magnitude of relative error

Notations

119873(119905) Counting process for the total numberof TDLs identified and removed in [0 119905)

119872 Total number of failures in the UUTwhich can be acquired by FMEA

119886 Number of initial TDLs present in UUTbefore TGT

119886(119905) Time-dependent TDLs contentfunction which will decrease with theidentification and removal of TDLs

119886remaining Number of TDLs remaining in UUTwhen the TGT has been stopped

119898(119905) MVF of TDL identified and removedin time (0 119905]

119898119894(119905) Cumulative number of TDL identified

up to 119905119898119903(119905) Cumulative number of TDL removed

up to 119905120582(119905) TDL intensity for119898

119903(119905)

1198871 Constant TDL identification rate

which is the TDL identificationintensity for119898

119894(119905)

1198872 Constant TDL removal rate which is

the TDL removal intensity for119898119903(119905)

119882(119905) Cumulative TGE consumption up totime 119905 which can be measured by thenumber of fault injected or occurrednaturally test cost man-hour and so on

119908(119905) Current TGE consumption at time 119905

119902(119905) Time-dependent FDR function inthe whole life cycle

119882lowast(119905) 119882(119905) minus 119882(0)

119882max(119905) Total amount of TGEeventually consumed

119882119864(119905) The Exponential TGEF

1198731 Total amount of TGE eventually

consumed in an Exponential TGEF1205731 TGE consumption rate of

an Exponential TGEF119882lowast

119864(119905) 119882

119864(119905) minus 119882

119864(0)

119882119877(119905) The Rayleigh TGEF

1198732 Total TGE eventually consumed of

a Rayleigh TGEF1205732 TGE consumption rate of

a Rayleigh TGEF119882lowast

119877(119905) 119882

119877(119905) minus 119882

119877(0)

119882119871(119905) The logistic TGEF


a logistic TGEF119860 Constant parameter in the logistic TGEF1205733 TGE consumption rate of

the logistic TGEF119882lowast

119871(119905) 119882

119871(119905) minus 119882

119871(0)

119882DS(119905) The delayed S-shaped TGEF1198734 Total TGE eventually consumed of

a delayed S-shaped TGEF1205734 TGE consumption rate of the delayed

S-shaped TGEF119882lowast

DS(119905) 119882DS(119905) minus 119882DS(0)119882IS(119905) The inflected S-shaped TGEF1198735 Total TGE eventually consumed of

an inflected S-shaped TGEF120593 Constant parameter in the inflected

S-shaped TGEF1205735 TGE consumption rate of the inflected


DS(119905) 119882IS(119905) minus 119882IS(0)119898119903(119905119894) Expected number of TDLs by time 119905

119894

estimated by a model119898119903119894 Actual number of TDLs by time 119905

119894

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to acknowledge the support and con-structive comments from the editor and the anonymousreviewers Particularly we appreciate Dr Hong-Dong Fan forhis constructive and insightful suggestions for improving thedetails of this paper The research reported here is partiallysupported by the NSFC under Grants 61304103

References

[1] Department of Defense ldquoMaintainability verificationdemon-strationevaluationrdquo Tech Rep MIL-STD-471A 1973


[2] GJB2547-1995 ldquoEquipment testability descriptionrdquo 1995[3] Z Tian and J Y Shi Design Analysis and Demonstration of

System Testability BeiHang University Press Beijing China2003

[4] T F Pliska J Angus and F L Jew ADA081128 BITExternalTest Figures of Merit and Demonstration Techniques Rome AirDevelopment Center Griffiss AFB New York NY USA 1979

[5] K Jerome ldquoTestability demonstrationrdquo in Proceedings of theIEEE International Automatic TestingConference (AUTESTCONrsquo82) 1982

[6] Z Tian ldquoStudy of testability demonstration methodsrdquo ActaAeronautica et Astronautica Sinica vol 16 no 1 pp 65ndash70 1995

[7] J-Y Shi and R Kang ldquoStudy on the plan of testability demon-stration based on the general adequacy criterionrdquo Acta Aero-nautica et Astronautica Sinica vol 26 no 6 pp 691ndash695 2005

[8] P Xu S L Liu and Y Li ldquoResearch on concept and model oftestability testrdquo Computer Measurement amp Control vol 14 no9 pp 1149ndash1152 2006

[9] T Li J Qiu and G Liu ldquoNew methodology for determiningtestability integrated test scheme with test data in the develop-ment stagesrdquo Journal of Mechanical Engineering vol 45 no 8pp 52ndash57 2009

[10] T Li J Qiu and G Liu ldquoResearch on testability field statisticsverification based on bayes inference theory of dynamic pop-ulationrdquo Acta Aeronautica et Astronautica Sinica vol 31 no 2pp 335ndash341 2010

[11] Y Zhang J Qiu and G Liu ldquoEnvironmental stress-fault greyrelational analysis for helicopter gyroscoperdquoThe Journal of GreySystem vol 24 no 1 pp 29ndash38 2012

[12] Y Zhang J Qiu G Liu and P Yang ldquoA fault sample simulationapproach for virtual testability demonstration testrdquo ChineseJournal of Aeronautics vol 25 no 4 pp 598ndash604 2012

[13] Z-Q Wang C-H Hu W Wang and X-S Si ldquoAn additivewiener process-based prognostic model for hybrid deteriorat-ing systemsrdquo IEEE Transactions on Reliability vol 63 no 1 pp208ndash222 2014

[14] Z Q Wang C H Hu W B Wang Z J Zhou and X S Si ldquoAcase study of remaining storage life prediction using stochasticfiltering with the influence of condition monitoringrdquo ReliabilityEngineering amp System Safety vol 132 pp 186ndash195 2014

[15] Z QWangW BWang CHHu X S Si and J Li ldquoA real-timeprognostic method for the drift errors in the inertial navigationsystem by a nonlinear random-coefficient regression modelrdquoActa Astronautica vol 103 no 10 pp 45ndash54 2014

[16] F N Parr ldquoAn alternative to the Rayleigh curve model forsoftware development effortrdquo IEEE Transactions on SoftwareEngineering vol SE-6 no 3 pp 291ndash296 1980

[17] M Ohba ldquoSoftware reliability analysis modelsrdquo IBM Journal ofResearch and Development vol 28 no 4 pp 428ndash443 1984

[18] M Ohba ldquoInflection S-shaped software reliability growthmodelsrdquo in Stochas Models in Reliability Theory pp 144ndash162Springer Berlin Germany 1984

[19] S YamadaMOhba and SOsaki ldquoS-shaped software reliabilitygrowth models and their applicationsrdquo IEEE Transactions onReliability vol 33 no 4 pp 289ndash292 1984

[20] Y K Malaiya N Karunanithi and P Verma ldquoPredictabilitymeasures for software reliability modelsrdquo in Proceedings ofthe 14th IEEE annual International Computer Software andApplications Conference Chicago Ill USA October 1990

[21] S Yamada J Hishitani and S Osaki ldquoSoftware-reliabilitygrowth with aWeibull test-effort A model amp applicationrdquo IEEETransactions on Reliability vol 42 no 1 pp 100ndash105 1993

[22] P K Kapur and S Younes ldquoSoftware reliability growth modelwith error dependencyrdquoMicroelectronics Reliability vol 35 no2 pp 273ndash278 1995

[23] K Kanoun M Kaamche and J-C Laprie ldquoQualitative andquantitative reliability assessmenrdquo IEEE Software vol 14 no 2pp 77ndash86 1997

[24] A L Goel and K-Z Yang ldquoSoftware reliability and readinessassessment based on the non-homogeneous Poisson processrdquoAdvances in Computers vol 45 pp 197ndash267 1997

[25] K Pillai and V S S Nair ldquoA model for software developmenteffort and cost estimationrdquo IEEE Transactions on SoftwareEngineering vol 23 no 8 pp 485ndash497 1997

[26] S S Gokhale and K S Trivedi ldquoLog-logistic software reliabilitygrowth modelrdquo in Proceedings of the 3rd IEEE InternationalHigh-Assurance Systems Engineering Symposium pp 34ndash41Washington DC USA 1998

[27] H Pham L Nordmann and X Zhang ldquoA general imperfect-software-debugging model with s-shaped fault-detection raterdquoIEEE Transactions on Reliability vol 48 no 2 pp 169ndash175 1999

[28] C-YHuang S-Y Kuo andMR Lyu ldquoEffort-index-based soft-ware reliability growth models and performance assessmentrdquoin Proceedinsg of the IEEE 24th Annual International ComputerSoftware andApplications Conference (COMPSAC rsquo00) pp 454ndash459 Taipei Taiwan October 2000

[29] S-Y Kuo C-Y Huang and M R Lyu ldquoFramework for mod-eling software reliability using various testing-efforts and fault-detection ratesrdquo IEEE Transactions on Reliability vol 50 no 3pp 310ndash320 2001

[30] C-Y Huang and S-Y Kuo ldquoAnalysis of incorporating logistictesting-effort function into software reliability modelingrdquo IEEETransactions on Reliability vol 51 no 3 pp 261ndash270 2002

[31] P K Kapur and A K Bardhan ldquoTesting effort control throughsoftware reliability growth modellingrdquo International Journal ofModelling and Simulation vol 22 no 2 pp 90ndash96 2002

[32] C-Y Huang M R Lyu and S-Y Kuo ldquoA unified scheme ofsome nonhomogenous poisson process models for softwarereliability estimationrdquo IEEE Transactions on Software Engineer-ing vol 29 no 3 pp 261ndash269 2003

[33] PKKapurDNGoswami andAGupta ldquoA software reliabilitygrowth model with testing effort dependent learning functionfor distributed systemsrdquo International Journal of ReliabilityQuality and Safety Engineering vol 11 no 4 pp 365ndash377 2004

[34] C-Y Huang and M R Lyu ldquoOptimal release time for softwaresystems considering cost testing-effort and test efficiencyrdquoIEEE Transactions on Reliability vol 54 no 4 pp 583ndash5912005

[35] H-W Liu X-Z Yang F Qu and J Dong ldquoSoftware reliabilitygrowth model with bell-shaped fault detection rate functionrdquoChinese Journal of Computers vol 28 no 5 pp 908ndash912 2005

[36] M U Bokhari and N Ahmad ldquoAnalysis of a software reliabilitygrowth models the case of log-logistic test-effort functionrdquoin Proceedings of the 17th IASTED International Conference onModelling and Simulation pp 540ndash545Montreal CanadaMay2006

[37] C-Y Huang S-Y Kuo and M R Lyu ldquoAn assessment oftesting-effort dependent software reliability growth modelsrdquoIEEE Transactions on Reliability vol 56 no 2 pp 198ndash211 2007

[38] C-T Lin and C-Y Huang ldquoEnhancing and measuring thepredictive capabilities of testing-effort dependent software reli-ability modelsrdquoThe Journal of Systems and Software vol 81 no6 pp 1025ndash1038 2008


[39] N Ahmad M G M Khan and L S Rafi ldquoA study of testing-effort dependent inflection S-shaped software reliability growthmodels with imperfect debuggingrdquo International Journal ofQuality and Reliability Management vol 27 no 1 pp 89ndash1102010

[40] Q Li H Li M Lu and X Wang ldquoSoftware reliability growthmodel with S-shaped testing effort functionrdquo Journal of BeijingUniversity of Aeronautics andAstronautics vol 37 no 2 pp 149ndash160 2011

[41] T M Li C H Hu and X Zhou ldquoResearch on testability fieldstatistics test based on Bayes inference theory of dynamic pop-ulationrdquo Acta Aeronautica et Astronautica Sinica vol 32 no 12pp 2277ndash2286 2011

[42] M-C Hsueh T K Tsai and R K Iyer ldquoFault injection tech-niques and toolsrdquo Computer vol 30 no 4 pp 75ndash82 1997

[43] L Antoni R Leveugle and B Feher ldquoUsing run-time reconfig-uration for fault injection applicationsrdquo IEEE Transactions onInstrumentation andMeasurement vol 52 no 5 pp 1468ndash14732003

[44] T DeMarco Controlling Software Projects Management Mea-surement and Estimation Prentice Hall Englewod Cliffs NJUSA 1982

[45] J D Musa A Iannino and K Okumoto Software ReliabilityMeasurement Prediction and Application McGraw Hill NewYork NY USA 1988

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


the designers The testability growth test phase aims atidentifying and removing the TDLs Typically TGT is moreconstructive and has high confidence level for system testa-bility rather than the testability demonstration test at theacceptance stage only

A common approach for measuring system testability isby using an analytic model whose parameters are generallyestimated from available data on fault detectionisolation Inthis paper the analytic model is referred to by us as STGMA STGM provides a mathematical relationship between thenumber of TDLs removed and the test time during thewhole life cycle and generally is used as a tool to estimateand predict the progress of system testability At the sametime the STGM can be used for planning and controlling alltest resources during development and can assure us aboutthe testability of equipment There is an extensive body ofliterature emphasizing the importance of TGT but no depthof research is done on STGM [3 8]

In the context of TGT the basic objective is to identify theTDLs and remove them one by one Therefore the processof TGT contains not only a TDL identification process butalso a TDL removal process and that the key issues arethe TGE which are consumed to identify the TDL and theeffectiveness of TDL removal respectively In general theTGE can be represented as man-hour TGT cost the times offault injection and so forth The functions that describe howa TGE is distributed over the TGT phase are referred to by usas TGEF The shape of the observed testability growth curvedepends strongly on the time distribution of the TGE Due tothe randomness of TGT and consider the designerrsquos capabilityand familiarity to UUT synchronously the other key issue(ie the effectiveness of TDL removal) can be referred to byus as a constant TDL identification rate and a constant TDLremoval rate respectively

There is an extensive body of literature [16ndash40] onsoftware reliability growth model describing the relationshipbetween the test time and the amount of test-effort expendedduring that time in which the test effort was often describedby the traditional Exponential [19 21] Rayleigh [16 19 21]logistic [29 30 37 38] delayed-shaped [19 39] or inflectedS-shaped [18 39] curves Thus to address the issue of TGEFthis paper will use the above five test effort curves which arecollectively referred to as Bell-shaped TGEFs to describe therelationship between the test time and the amount of TGEexpended during that time by analyzing the consumptionrule of TGE Sometimes the TGE can be represented asthe number of faults injected or occurred naturally insteadof man-hours or TGT cost Similarly to address the issueof effectiveness of TDL removal we assume that the TDLidentification rate and the TDL removal rate are constant

In this paper the process of TGT is decomposed intofault injection or occurrence TDL identification and TDLremoval processes which are all based on NHPP We showhow to integrate time dependent TGEF constant TDL iden-tification rate and constant TDL removal rate into STGM tofind the framework of STGMWe further pay main attentionon the consumption pattern of TGE based on fault injectioncollection of fault occurring naturally and then discuss howto integrate five Bell-shaped TGEFs into STGM A method

to estimate the model parameter is provided Experimentalresults from a stable tracking platform are analyzed and thefive STGMs based on the above five Bell-shaped TGEFs arecompared with each other to show which STGM can givebetter prediction

The remainder of the paper is organized as followsIn Section 2 we found a framework of STGM consideringTGEF The fault occurrence TDL identification and TDLremoval process based on NHPP are also described andanalyzed in this section In Section 3 we pay main atten-tion to analyzing the consumption rule of TGEF which isrepresented by fault injection or collection of fault occurrednaturally at designampdevelopment stage and trial amp in-servicestage respectively At the same time we show how to incor-porate the five Bell-shaped TGEFs into STGM Parametersof the proposed STGMs as estimated by the method of LSEare discussed in Section 4 Finally Section 5 discusses thegoodness of the above five STGMs on prediction with theapplication of these models to a real data set

2 A General Framework of STGMBased on NHPP

In general the precondition for testability growth test is thatthe UUT is in failed condition That is to say there is oneor more faults occurred in the UUT Under the precondi-tion TGT is the process of running a testability diagnosticprogram and capturing the fault detection and isolation datato identify the TDLs and remove them In other words theTGT aims at identifying the TDLs and removing them oneby one under the precondition of fault occurred Thereforethe process of removing TDLs can roughly be divided intothree steps fault occurrence TDL identification and TDLremoval At the same time each of the above three steps canbe described effectively as a counting process based onNHPP

In practice the most important factor which affects theSTGMrsquos evaluation and predication accuracy is TGE TGEis the cumulative testability growth effort consumed in TGTwhich can be measured by the number of fault injection testcost staff and so onThe consumedTGE indicates howTDLsare identified effectively in the UUT and can be modeled bydifferent curves considering the consumption pattern inTGTIn this section a framework for foundation of STGM withTGEF is proposed

21 Fault Occurrence TDL Identification and Removal TDLidentification may be realized under the condition thatthe UUT must be broken Once the UUT is in brokencondition which usually can be realized by fault injection orcollection information of fault occurred naturally the designof testability begins to run to identify the TDL and analyzethe root cause of the TDL When the specific root cause isrecognized the designers can remove the TDL accordingly

We can use the number of residual TDLs as a measure ofsystem testability The great the number of TDLs the lowerof testability level would be So the process of TGT is theprocess to identify and remove the TDLs existing in the UUTIn other words the process of TGT is to identify and removethe TDL one by one So the process of TGT can be described


0

0

0

0

0

t1

t2

t3

t4

ti

tM






f1 f2 f3 f4 fi fM









TDL1 TDL2 TDLj







t

t

t

t

t











In (a) 1198911 1198912 119891

119894 119891




1 1199052 119905

119894 119905

119872


1 1199052 119905

119894 119905



1and 119891

4cannot be


1and TDL

2 respectively










ℎrarr0119875[119873(119905 + ℎ) minus 119873(119905) = 1] = 120582(119905)ℎ + 119900(ℎ)

(4) limℎrarr0

119875[119873(119905 + ℎ) minus 119873(119905) ge 2] = 119900(ℎ)



Make 119898119903(119905) = 119864[119873(119905)] then 119898




119898119903(119905) = int

119905

0

120582(119906)d119906 (1)

So

120582(119905) = 1198981015840

119903(119905) = limΔ119905rarr0

119864[119873(119905 + Δ119905) minus 119873(119905)]

Δ119905 (2)




noted as Te



TGT noted as T(t0)

W(t) and b1



119903(119905 + 119904) minus 119898

119903(119905) so we have


119903(119905)]

sdot[119898119903(119905 + 119904) minus 119898

119903(119905)]119899

119899

(3)


119903(119905)) as a measure





119903(119905))




0) lt 119879119890is


0


0 From the


119890is the










0)













d119898119894(119905)

d119905times

1

119908(119905)= 1198871[119886 minus 119898

119894(119905)]

d119898119903(119905)

d119905times

1

119908(119905)= 1198872[119898119894(119905) minus 119898

119903(119905)]

(4)




119903(0) = 0 we have


1119882lowast(119905)] (5)

119898119903(119905) = 119886times1 minus

1198871exp[minus119887

2119882lowast(119905)]minus119887

2exp[minus119887

1119882lowast(119905)]

1198871minus 1198872

(6)


= 1198861198871exp[minus119887


2exp[minus119887


1198871minus 1198872

(7)


FDR =119873119863

119872 (8)





119903(119905)] (9)


119902(119905) =119872 minus [119886 minus 119898

119903(119905)]

119872 (10)


119902(119905) = (119872 minus 119886((1198871exp[minus119887

2119882lowast(119905)] minus 119887

2exp[minus119887

1119882lowast(119905)])


))119872minus1

(11)













True value of FDR

t

q(te) = QE

q(t)

(nm+1 fm+1)qk

q(0)ts te

(nm fm)

(n1 f1)








[119905119904 119905119890] respectively


119896 119891119896) 1 le





119878and the


(119899119895 119891119895) 119898 lt 119895 le 119898





119894 1 le 119894 le 119898

119878is the










01 02 03 04 05 06 07 08 090

20

40

60

80

100

120

FDR level

Num

ber o

f fau

lt in

ject

ions

TDL = 3

TDL = 5

TDL = 10



119902119896=

119899119896minus 119891119896

119899119896

1 le 119896 le 119898 (12)





Num

ber o

f fai

lure

s occ

urre

d na

tura

lly

Test timets te t


0Test time

ts te

Curr

ent t

eata

bilit

y gr

owth

effor

t

t

cons

umpt

ion

at ti

me t

w(t)






119882119864(119905) = 119873


1119905)] (13)


119908119864(119905) = 119873

11205731exp(minus120573

1119905) (14)

where119882119864(119905) = int

119905

0119908119864(120591)d120591




2119882lowast

119864(infin)] minus 119887

2exp[minus119887

1119882lowast

119864(infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198731] minus 1198872exp[minus119887

11198731]

1198871minus 1198872

(15)



119882119877(119905) = 119873

2[1 minus exp(

minus1205732

21199052)] (16)


119908119877(119905) = 119873

21205732119905 exp(minus

1205732

21199052) (17)

where119882119877(119905) = int

119905

0119908119877(120591)d120591



119905max =1

1205732

(18)




2119882lowast

119877(infin)] minus 119887

2exp[minus119887

1119882lowast

119877(infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198732] minus 1198872exp[minus119887

11198732]

1198871minus 1198872

(19)




119882119871(119905) =

1198733

1 + 119860 exp(minus1205733119905) (20)


119908119871(119905) =

11987331198601205733exp(minus120573

3119905)

[1 + 119860 exp(minus1205733119905)]2 (21)

where119882119871(119905) = int

119905

0119908119871(120591)d120591



119905max =1

1205733

ln119860 (22)


119886remaining

= 1198861198871exp[minus119887

2119882lowast

119877(infin)] minus 119887

2exp[minus119887

1119882lowast

119877(infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198733(1 + 119860)] minus 119887

2exp[minus119887

11198733(1 + 119860)]

1198871minus 1198872

(23)



119882DS(119905) = 1198734[1 minus (1 + 120573

4119905) exp(minus120573

4119905)] (24)


119908DS(119905) = 11987341205732

4119905 exp (minus120573

4119905) (25)

where119882DS(119905) = int119905

0119908DS(120591)d120591


119905max =1

1205734

(26)



2119882lowast


1119882lowast

DS (infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198734] minus 1198872exp[minus119887

11198734]

1198871minus 1198872

(27)


0 10 20 30 40 50 600

5

10

15

20

25

30

Time (weeks)

Curr

ent T

GE

N1 = 240 1205731 = 01111



119882IS(119905) = 1198735


1 + 120593 exp(minus1205735119905) (28)


119908IS(119905) =11987351205735(1 + 120593) exp(minus120573

5119905)

[1 + 120593 exp(minus1205735119905)]2

(29)

where119882IS(119905) = int119905

0119908IS(120591)d120591


119905max =1

1205735

ln120593 (30)



2119882lowast


1119882lowast

IS(infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198735] minus 1198872exp[minus119887

11198735]

1198871minus 1198872

(31)



0

5

10

15

20

25

Time (weeks)

Curr

ent T

GE

0 2 4 6 8 10 12 14 16 18 20

N2 = 120 1205732 = 01111


0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

Time (weeks)

Curr

ent T

GE

N3 = 240 A = 2303 1205733 = 01111





21198731[1 minus exp (minus120573

1119905)]]



1119905)]])


))119872minus1

(32)




1205732

21199052)]]



1205732

21199052)]])


))119872minus1

(33)


0 10 20 30 40 50 60 70 800

21

4

6

8

10

Time (weeks)

Curr

ent T

GE

N4 = 240 1205734 = 01111

9

7

5

3


0 10 20 30 40 50 60 70 80Time (weeks)

Curr

ent T

GE

N5 = 240 120593 = 1984 1205735 = 01111

005

115

225

335

445

5



119902(119905) = (119872minus119886((1198871exp[minus119887

21198733[

1

1+119860 exp (minus120572119905)minus

1

1+119860]]


11198733

times[1

1+119860 exp (minus120572119905)minus

1

1+119860]])


))119872minus1

(34)


119902(119905) = (119872minus119886((1198871exp[minus119887

21198734[1 minus (1+120573

1119905) exp (minus120573

1119905)]]


11198734[1 minus (1+120573

1119905) exp (minus120573

1119905)]])


))119872minus1

(35)


119902(119905) = (119872 minus 119886((1198871exp[minus119887

21198735[

1 + exp (minus1205732119905)

1 + 120593 exp (minus1205732119905)]]


11198735[

1 + exp (minus1205732119905)

1 + 120593 exp (minus1205732119905)]])


))119872minus1

(36)








2given in (6) can


119864(1198731 1205731) 119878119872119888(119886 1198871) and 119878

119872119903(119898119888 1198872) are as follows

Minimize 119878119864(1198731 1205731) =

119899

sum119896=1

[119882119864119896

minus119882119864(119905119896)]2

Minimize 119878119872119888(119886 1198871) =

119899

sum119896=1

[119872119862119896

minus119872119862(119905119896)]2

Minimize 119878119872119903(119898119888 1198872) =

119899

sum119896=1

[119898119903119896minus 119898119903(119905119896)]2

(37)


with respect to (1198731 1205731)

(119886 1198871) and (119898




1 1205731

we get

120597119878119864

1205971198731

=

119899

sum119896=1

minus 2119882119864119896


1119896)]


(38)



Test time(weeks)

CumulativeTGE

(fault times)



Test time(weeks)

CumulativeTGE

(fault times)



1 4 3 2 13 223 74 482 9 6 4 14 241 76 493 16 11 7 15 256 77 504 25 17 11 16 268 79 515 38 25 16 17 278 80 526 55 33 21 18 286 80 527 74 41 27 19 294 82 538 95 50 33 20 301 83 549 119 57 37 21 306 84 5410 145 63 41 22 310 85 5511 173 68 44 23 312 85 5512 203 72 46 24 313 86 55



1198731=

sum119899

119896=2119882119896[1 minus exp (minus120573

1119896)]

sum119899


1119896)]2 (39)

Next we have

120597119878119864

1205971205731

=

119899

sum119896=1


1119896)]

times exp(minus1205731119896) = 0

(40)




(16) 1198733 119860 120573


4 1205734of the








AE =10038161003816100381610038161003816100381610038161003816

119898119894minus 119886

119898119894

10038161003816100381610038161003816100381610038161003816 (41)





MSE =sum119899

119894=1[119898119903(119905119894) minus 119898119903119894]2

119899 (42)




RE =119898119903(119905119886) minus 119890

119890 (43)



119886




119903(119905119890) by time 119905

119890 The estimate











=



3= 31118

(fault times) 119860 = 3936 1205733= 03524week 119873

4= 40606

(fault times) 1205734= 01313week 119873







119894

Bias = 1

119899

119899

sum119894=1

PE119894


119894=1(PE119894minus Bias)2

119899 minus 1

(44)





0 5 10 15 200

5

10

15

20

25

30

35

Time (weeks)Cu

rren

t TG

E (fa

ult t

imes


LogisticActural


0 5 10 15 200

5

10

15

20

25

30

35

Time (weeks)

Curr

ent T

GE

(faul

t tim

es)








STGMs 119886 1198871

1198872


0 5 10 15 20Time (weeks)

Cum

ulat

ive T

GE

(faul

t tim

es)

ExponentialRayleigh

LogisticActural

0

50

150

200

250

300

350

100


0 5 10 15 20Time (weeks)

Cum

ulat

ive T

GE

(faul

t tim

es)


0

50

100

150

200

250

300

350


0 5 10 15 20 25

002040608

11214

Time (weeks)

Rela

tive e

rror

minus04

minus02


0 5 10 15 20 25

002040608

Time (weeks)

Rela

tive e

rror

minus1

minus08

minus06

minus04

minus02


0 5 10 15 20 25

010203040506070

Time (weeks)

Rela

tive e

rror

minus10







0 5 10 15 20 25

002040608

Time (weeks)

Rela

tive e

rror

minus1

minus08

minus06

minus04

minus02


0 5 10 15 20 25

0

1

2

3

4

5

Time (weeks)

Rela

tive e

rror

minus1


0 5 10 15 20065

07

075

08

085

09

Time (weeks)

FDR


0 5 10 15 20072074076078

08082084086088

09092

Time (weeks)

FDR


0 5 10 15 20074076078

08082084086088

09092094

Time (weeks)

FDR


0 5 10 15 2007

075

08

085

09

095

Time (weeks)

FDR


0 5 10 15 20074076078

08082084086088

09092

Time (weeks)

FDR






Acronyms


Notations










119903(119905)



119894(119905)






119882lowast(119905) 119882(119905) minus 119882(0)






119864(119905) 119882

119864(119905) minus 119882

119864(0)





119877(119905) 119882

119877(119905) minus 119882

119877(0)





119871(119905) 119882

119871(119905) minus 119882

119871(0)









119894


119894



Acknowledgments


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

0

0

0

0

t1

t2

t3

t4

ti

tM






f1 f2 f3 f4 fi fM









TDL1 TDL2 TDLj







t

t

t

t

t











In (a) 1198911 1198912 119891

119894 119891




1 1199052 119905

119894 119905

119872


1 1199052 119905

119894 119905



1and 119891

4cannot be


1and TDL

2 respectively










ℎrarr0119875[119873(119905 + ℎ) minus 119873(119905) = 1] = 120582(119905)ℎ + 119900(ℎ)

(4) limℎrarr0

119875[119873(119905 + ℎ) minus 119873(119905) ge 2] = 119900(ℎ)



Make 119898119903(119905) = 119864[119873(119905)] then 119898




119898119903(119905) = int

119905

0

120582(119906)d119906 (1)

So

120582(119905) = 1198981015840

119903(119905) = limΔ119905rarr0

119864[119873(119905 + Δ119905) minus 119873(119905)]

Δ119905 (2)




noted as Te



TGT noted as T(t0)

W(t) and b1



119903(119905 + 119904) minus 119898

119903(119905) so we have


119903(119905)]

sdot[119898119903(119905 + 119904) minus 119898

119903(119905)]119899

119899

(3)


119903(119905)) as a measure





119903(119905))




0) lt 119879119890is


0


0 From the


119890is the










0)













d119898119894(119905)

d119905times

1

119908(119905)= 1198871[119886 minus 119898

119894(119905)]

d119898119903(119905)

d119905times

1

119908(119905)= 1198872[119898119894(119905) minus 119898

119903(119905)]

(4)




119903(0) = 0 we have


1119882lowast(119905)] (5)

119898119903(119905) = 119886times1 minus

1198871exp[minus119887

2119882lowast(119905)]minus119887

2exp[minus119887

1119882lowast(119905)]

1198871minus 1198872

(6)


= 1198861198871exp[minus119887


2exp[minus119887


1198871minus 1198872

(7)


FDR =119873119863

119872 (8)





119903(119905)] (9)


119902(119905) =119872 minus [119886 minus 119898

119903(119905)]

119872 (10)


119902(119905) = (119872 minus 119886((1198871exp[minus119887

2119882lowast(119905)] minus 119887

2exp[minus119887

1119882lowast(119905)])


))119872minus1

(11)













True value of FDR

t

q(te) = QE

q(t)

(nm+1 fm+1)qk

q(0)ts te

(nm fm)

(n1 f1)








[119905119904 119905119890] respectively


119896 119891119896) 1 le





119878and the


(119899119895 119891119895) 119898 lt 119895 le 119898





119894 1 le 119894 le 119898

119878is the










01 02 03 04 05 06 07 08 090

20

40

60

80

100

120

FDR level

Num

ber o

f fau

lt in

ject

ions

TDL = 3

TDL = 5

TDL = 10



119902119896=

119899119896minus 119891119896

119899119896

1 le 119896 le 119898 (12)





Num

ber o

f fai

lure

s occ

urre

d na

tura

lly

Test timets te t


0Test time

ts te

Curr

ent t

eata

bilit

y gr

owth

effor

t

t

cons

umpt

ion

at ti

me t

w(t)






119882119864(119905) = 119873


1119905)] (13)


119908119864(119905) = 119873

11205731exp(minus120573

1119905) (14)

where119882119864(119905) = int

119905

0119908119864(120591)d120591




2119882lowast

119864(infin)] minus 119887

2exp[minus119887

1119882lowast

119864(infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198731] minus 1198872exp[minus119887

11198731]

1198871minus 1198872

(15)



119882119877(119905) = 119873

2[1 minus exp(

minus1205732

21199052)] (16)


119908119877(119905) = 119873

21205732119905 exp(minus

1205732

21199052) (17)

where119882119877(119905) = int

119905

0119908119877(120591)d120591



119905max =1

1205732

(18)




2119882lowast

119877(infin)] minus 119887

2exp[minus119887

1119882lowast

119877(infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198732] minus 1198872exp[minus119887

11198732]

1198871minus 1198872

(19)




119882119871(119905) =

1198733

1 + 119860 exp(minus1205733119905) (20)


119908119871(119905) =

11987331198601205733exp(minus120573

3119905)

[1 + 119860 exp(minus1205733119905)]2 (21)

where119882119871(119905) = int

119905

0119908119871(120591)d120591



119905max =1

1205733

ln119860 (22)


119886remaining

= 1198861198871exp[minus119887

2119882lowast

119877(infin)] minus 119887

2exp[minus119887

1119882lowast

119877(infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198733(1 + 119860)] minus 119887

2exp[minus119887

11198733(1 + 119860)]

1198871minus 1198872

(23)



119882DS(119905) = 1198734[1 minus (1 + 120573

4119905) exp(minus120573

4119905)] (24)


119908DS(119905) = 11987341205732

4119905 exp (minus120573

4119905) (25)

where119882DS(119905) = int119905

0119908DS(120591)d120591


119905max =1

1205734

(26)



2119882lowast


1119882lowast

DS (infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198734] minus 1198872exp[minus119887

11198734]

1198871minus 1198872

(27)


0 10 20 30 40 50 600

5

10

15

20

25

30

Time (weeks)

Curr

ent T

GE

N1 = 240 1205731 = 01111



119882IS(119905) = 1198735


1 + 120593 exp(minus1205735119905) (28)


119908IS(119905) =11987351205735(1 + 120593) exp(minus120573

5119905)

[1 + 120593 exp(minus1205735119905)]2

(29)

where119882IS(119905) = int119905

0119908IS(120591)d120591


119905max =1

1205735

ln120593 (30)



2119882lowast


1119882lowast

IS(infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198735] minus 1198872exp[minus119887

11198735]

1198871minus 1198872

(31)



0

5

10

15

20

25

Time (weeks)

Curr

ent T

GE

0 2 4 6 8 10 12 14 16 18 20

N2 = 120 1205732 = 01111


0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

Time (weeks)

Curr

ent T

GE

N3 = 240 A = 2303 1205733 = 01111





21198731[1 minus exp (minus120573

1119905)]]



1119905)]])


))119872minus1

(32)




1205732

21199052)]]



1205732

21199052)]])


))119872minus1

(33)


0 10 20 30 40 50 60 70 800

21

4

6

8

10

Time (weeks)

Curr

ent T

GE

N4 = 240 1205734 = 01111

9

7

5

3


0 10 20 30 40 50 60 70 80Time (weeks)

Curr

ent T

GE

N5 = 240 120593 = 1984 1205735 = 01111

005

115

225

335

445

5



119902(119905) = (119872minus119886((1198871exp[minus119887

21198733[

1

1+119860 exp (minus120572119905)minus

1

1+119860]]


11198733

times[1

1+119860 exp (minus120572119905)minus

1

1+119860]])


))119872minus1

(34)


119902(119905) = (119872minus119886((1198871exp[minus119887

21198734[1 minus (1+120573

1119905) exp (minus120573

1119905)]]


11198734[1 minus (1+120573

1119905) exp (minus120573

1119905)]])


))119872minus1

(35)


119902(119905) = (119872 minus 119886((1198871exp[minus119887

21198735[

1 + exp (minus1205732119905)

1 + 120593 exp (minus1205732119905)]]


11198735[

1 + exp (minus1205732119905)

1 + 120593 exp (minus1205732119905)]])


))119872minus1

(36)








2given in (6) can


119864(1198731 1205731) 119878119872119888(119886 1198871) and 119878

119872119903(119898119888 1198872) are as follows

Minimize 119878119864(1198731 1205731) =

119899

sum119896=1

[119882119864119896

minus119882119864(119905119896)]2

Minimize 119878119872119888(119886 1198871) =

119899

sum119896=1

[119872119862119896

minus119872119862(119905119896)]2

Minimize 119878119872119903(119898119888 1198872) =

119899

sum119896=1

[119898119903119896minus 119898119903(119905119896)]2

(37)


with respect to (1198731 1205731)

(119886 1198871) and (119898




1 1205731

we get

120597119878119864

1205971198731

=

119899

sum119896=1

minus 2119882119864119896


1119896)]


(38)



Test time(weeks)

CumulativeTGE

(fault times)



Test time(weeks)

CumulativeTGE

(fault times)



1 4 3 2 13 223 74 482 9 6 4 14 241 76 493 16 11 7 15 256 77 504 25 17 11 16 268 79 515 38 25 16 17 278 80 526 55 33 21 18 286 80 527 74 41 27 19 294 82 538 95 50 33 20 301 83 549 119 57 37 21 306 84 5410 145 63 41 22 310 85 5511 173 68 44 23 312 85 5512 203 72 46 24 313 86 55



1198731=

sum119899

119896=2119882119896[1 minus exp (minus120573

1119896)]

sum119899


1119896)]2 (39)

Next we have

120597119878119864

1205971205731

=

119899

sum119896=1


1119896)]

times exp(minus1205731119896) = 0

(40)




(16) 1198733 119860 120573


4 1205734of the








AE =10038161003816100381610038161003816100381610038161003816

119898119894minus 119886

119898119894

10038161003816100381610038161003816100381610038161003816 (41)





MSE =sum119899

119894=1[119898119903(119905119894) minus 119898119903119894]2

119899 (42)




RE =119898119903(119905119886) minus 119890

119890 (43)



119886




119903(119905119890) by time 119905

119890 The estimate











=



3= 31118

(fault times) 119860 = 3936 1205733= 03524week 119873

4= 40606

(fault times) 1205734= 01313week 119873







119894

Bias = 1

119899

119899

sum119894=1

PE119894


119894=1(PE119894minus Bias)2

119899 minus 1

(44)





0 5 10 15 200

5

10

15

20

25

30

35

Time (weeks)Cu

rren

t TG

E (fa

ult t

imes


LogisticActural


0 5 10 15 200

5

10

15

20

25

30

35

Time (weeks)

Curr

ent T

GE

(faul

t tim

es)








STGMs 119886 1198871

1198872


0 5 10 15 20Time (weeks)

Cum

ulat

ive T

GE

(faul

t tim

es)

ExponentialRayleigh

LogisticActural

0

50

150

200

250

300

350

100


0 5 10 15 20Time (weeks)

Cum

ulat

ive T

GE

(faul

t tim

es)


0

50

100

150

200

250

300

350


0 5 10 15 20 25

002040608

11214

Time (weeks)

Rela

tive e

rror

minus04

minus02


0 5 10 15 20 25

002040608

Time (weeks)

Rela

tive e

rror

minus1

minus08

minus06

minus04

minus02


0 5 10 15 20 25

010203040506070

Time (weeks)

Rela

tive e

rror

minus10







0 5 10 15 20 25

002040608

Time (weeks)

Rela

tive e

rror

minus1

minus08

minus06

minus04

minus02


0 5 10 15 20 25

0

1

2

3

4

5

Time (weeks)

Rela

tive e

rror

minus1


0 5 10 15 20065

07

075

08

085

09

Time (weeks)

FDR


0 5 10 15 20072074076078

08082084086088

09092

Time (weeks)

FDR


0 5 10 15 20074076078

08082084086088

09092094

Time (weeks)

FDR


0 5 10 15 2007

075

08

085

09

095

Time (weeks)

FDR


0 5 10 15 20074076078

08082084086088

09092

Time (weeks)

FDR






Acronyms


Notations










119903(119905)



119894(119905)






119882lowast(119905) 119882(119905) minus 119882(0)






119864(119905) 119882

119864(119905) minus 119882

119864(0)





119877(119905) 119882

119877(119905) minus 119882

119877(0)





119871(119905) 119882

119871(119905) minus 119882

119871(0)









119894


119894



Acknowledgments


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















ℎrarr0119875[119873(119905 + ℎ) minus 119873(119905) = 1] = 120582(119905)ℎ + 119900(ℎ)

(4) limℎrarr0

119875[119873(119905 + ℎ) minus 119873(119905) ge 2] = 119900(ℎ)



Make 119898119903(119905) = 119864[119873(119905)] then 119898




119898119903(119905) = int

119905

0

120582(119906)d119906 (1)

So

120582(119905) = 1198981015840

119903(119905) = limΔ119905rarr0

119864[119873(119905 + Δ119905) minus 119873(119905)]

Δ119905 (2)




noted as Te



TGT noted as T(t0)

W(t) and b1



119903(119905 + 119904) minus 119898

119903(119905) so we have


119903(119905)]

sdot[119898119903(119905 + 119904) minus 119898

119903(119905)]119899

119899

(3)


119903(119905)) as a measure





119903(119905))




0) lt 119879119890is


0


0 From the


119890is the










0)













d119898119894(119905)

d119905times

1

119908(119905)= 1198871[119886 minus 119898

119894(119905)]

d119898119903(119905)

d119905times

1

119908(119905)= 1198872[119898119894(119905) minus 119898

119903(119905)]

(4)




119903(0) = 0 we have


1119882lowast(119905)] (5)

119898119903(119905) = 119886times1 minus

1198871exp[minus119887

2119882lowast(119905)]minus119887

2exp[minus119887

1119882lowast(119905)]

1198871minus 1198872

(6)


= 1198861198871exp[minus119887


2exp[minus119887


1198871minus 1198872

(7)


FDR =119873119863

119872 (8)





119903(119905)] (9)


119902(119905) =119872 minus [119886 minus 119898

119903(119905)]

119872 (10)


119902(119905) = (119872 minus 119886((1198871exp[minus119887

2119882lowast(119905)] minus 119887

2exp[minus119887

1119882lowast(119905)])


))119872minus1

(11)













True value of FDR

t

q(te) = QE

q(t)

(nm+1 fm+1)qk

q(0)ts te

(nm fm)

(n1 f1)








[119905119904 119905119890] respectively


119896 119891119896) 1 le





119878and the


(119899119895 119891119895) 119898 lt 119895 le 119898





119894 1 le 119894 le 119898

119878is the










01 02 03 04 05 06 07 08 090

20

40

60

80

100

120

FDR level

Num

ber o

f fau

lt in

ject

ions

TDL = 3

TDL = 5

TDL = 10



119902119896=

119899119896minus 119891119896

119899119896

1 le 119896 le 119898 (12)





Num

ber o

f fai

lure

s occ

urre

d na

tura

lly

Test timets te t


0Test time

ts te

Curr

ent t

eata

bilit

y gr

owth

effor

t

t

cons

umpt

ion

at ti

me t

w(t)






119882119864(119905) = 119873


1119905)] (13)


119908119864(119905) = 119873

11205731exp(minus120573

1119905) (14)

where119882119864(119905) = int

119905

0119908119864(120591)d120591




2119882lowast

119864(infin)] minus 119887

2exp[minus119887

1119882lowast

119864(infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198731] minus 1198872exp[minus119887

11198731]

1198871minus 1198872

(15)



119882119877(119905) = 119873

2[1 minus exp(

minus1205732

21199052)] (16)


119908119877(119905) = 119873

21205732119905 exp(minus

1205732

21199052) (17)

where119882119877(119905) = int

119905

0119908119877(120591)d120591



119905max =1

1205732

(18)




2119882lowast

119877(infin)] minus 119887

2exp[minus119887

1119882lowast

119877(infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198732] minus 1198872exp[minus119887

11198732]

1198871minus 1198872

(19)




119882119871(119905) =

1198733

1 + 119860 exp(minus1205733119905) (20)


119908119871(119905) =

11987331198601205733exp(minus120573

3119905)

[1 + 119860 exp(minus1205733119905)]2 (21)

where119882119871(119905) = int

119905

0119908119871(120591)d120591



119905max =1

1205733

ln119860 (22)


119886remaining

= 1198861198871exp[minus119887

2119882lowast

119877(infin)] minus 119887

2exp[minus119887

1119882lowast

119877(infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198733(1 + 119860)] minus 119887

2exp[minus119887

11198733(1 + 119860)]

1198871minus 1198872

(23)



119882DS(119905) = 1198734[1 minus (1 + 120573

4119905) exp(minus120573

4119905)] (24)


119908DS(119905) = 11987341205732

4119905 exp (minus120573

4119905) (25)

where119882DS(119905) = int119905

0119908DS(120591)d120591


119905max =1

1205734

(26)



2119882lowast


1119882lowast

DS (infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198734] minus 1198872exp[minus119887

11198734]

1198871minus 1198872

(27)


0 10 20 30 40 50 600

5

10

15

20

25

30

Time (weeks)

Curr

ent T

GE

N1 = 240 1205731 = 01111



119882IS(119905) = 1198735


1 + 120593 exp(minus1205735119905) (28)


119908IS(119905) =11987351205735(1 + 120593) exp(minus120573

5119905)

[1 + 120593 exp(minus1205735119905)]2

(29)

where119882IS(119905) = int119905

0119908IS(120591)d120591


119905max =1

1205735

ln120593 (30)



2119882lowast


1119882lowast

IS(infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198735] minus 1198872exp[minus119887

11198735]

1198871minus 1198872

(31)



0

5

10

15

20

25

Time (weeks)

Curr

ent T

GE

0 2 4 6 8 10 12 14 16 18 20

N2 = 120 1205732 = 01111


0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

Time (weeks)

Curr

ent T

GE

N3 = 240 A = 2303 1205733 = 01111





21198731[1 minus exp (minus120573

1119905)]]



1119905)]])


))119872minus1

(32)




1205732

21199052)]]



1205732

21199052)]])


))119872minus1

(33)


0 10 20 30 40 50 60 70 800

21

4

6

8

10

Time (weeks)

Curr

ent T

GE

N4 = 240 1205734 = 01111

9

7

5

3


0 10 20 30 40 50 60 70 80Time (weeks)

Curr

ent T

GE

N5 = 240 120593 = 1984 1205735 = 01111

005

115

225

335

445

5



119902(119905) = (119872minus119886((1198871exp[minus119887

21198733[

1

1+119860 exp (minus120572119905)minus

1

1+119860]]


11198733

times[1

1+119860 exp (minus120572119905)minus

1

1+119860]])


))119872minus1

(34)


119902(119905) = (119872minus119886((1198871exp[minus119887

21198734[1 minus (1+120573

1119905) exp (minus120573

1119905)]]


11198734[1 minus (1+120573

1119905) exp (minus120573

1119905)]])


))119872minus1

(35)


119902(119905) = (119872 minus 119886((1198871exp[minus119887

21198735[

1 + exp (minus1205732119905)

1 + 120593 exp (minus1205732119905)]]


11198735[

1 + exp (minus1205732119905)

1 + 120593 exp (minus1205732119905)]])


))119872minus1

(36)








2given in (6) can


119864(1198731 1205731) 119878119872119888(119886 1198871) and 119878

119872119903(119898119888 1198872) are as follows

Minimize 119878119864(1198731 1205731) =

119899

sum119896=1

[119882119864119896

minus119882119864(119905119896)]2

Minimize 119878119872119888(119886 1198871) =

119899

sum119896=1

[119872119862119896

minus119872119862(119905119896)]2

Minimize 119878119872119903(119898119888 1198872) =

119899

sum119896=1

[119898119903119896minus 119898119903(119905119896)]2

(37)


with respect to (1198731 1205731)

(119886 1198871) and (119898




1 1205731

we get

120597119878119864

1205971198731

=

119899

sum119896=1

minus 2119882119864119896


1119896)]


(38)



Test time(weeks)

CumulativeTGE

(fault times)



Test time(weeks)

CumulativeTGE

(fault times)



1 4 3 2 13 223 74 482 9 6 4 14 241 76 493 16 11 7 15 256 77 504 25 17 11 16 268 79 515 38 25 16 17 278 80 526 55 33 21 18 286 80 527 74 41 27 19 294 82 538 95 50 33 20 301 83 549 119 57 37 21 306 84 5410 145 63 41 22 310 85 5511 173 68 44 23 312 85 5512 203 72 46 24 313 86 55



1198731=

sum119899

119896=2119882119896[1 minus exp (minus120573

1119896)]

sum119899


1119896)]2 (39)

Next we have

120597119878119864

1205971205731

=

119899

sum119896=1


1119896)]

times exp(minus1205731119896) = 0

(40)




(16) 1198733 119860 120573


4 1205734of the








AE =10038161003816100381610038161003816100381610038161003816

119898119894minus 119886

119898119894

10038161003816100381610038161003816100381610038161003816 (41)





MSE =sum119899

119894=1[119898119903(119905119894) minus 119898119903119894]2

119899 (42)




RE =119898119903(119905119886) minus 119890

119890 (43)



119886




119903(119905119890) by time 119905

119890 The estimate











=



3= 31118

(fault times) 119860 = 3936 1205733= 03524week 119873

4= 40606

(fault times) 1205734= 01313week 119873







119894

Bias = 1

119899

119899

sum119894=1

PE119894


119894=1(PE119894minus Bias)2

119899 minus 1

(44)





0 5 10 15 200

5

10

15

20

25

30

35

Time (weeks)Cu

rren

t TG

E (fa

ult t

imes


LogisticActural


0 5 10 15 200

5

10

15

20

25

30

35

Time (weeks)

Curr

ent T

GE

(faul

t tim

es)








STGMs 119886 1198871

1198872


0 5 10 15 20Time (weeks)

Cum

ulat

ive T

GE

(faul

t tim

es)

ExponentialRayleigh

LogisticActural

0

50

150

200

250

300

350

100


0 5 10 15 20Time (weeks)

Cum

ulat

ive T

GE

(faul

t tim

es)


0

50

100

150

200

250

300

350


0 5 10 15 20 25

002040608

11214

Time (weeks)

Rela

tive e

rror

minus04

minus02


0 5 10 15 20 25

002040608

Time (weeks)

Rela

tive e

rror

minus1

minus08

minus06

minus04

minus02


0 5 10 15 20 25

010203040506070

Time (weeks)

Rela

tive e

rror

minus10







0 5 10 15 20 25

002040608

Time (weeks)

Rela

tive e

rror

minus1

minus08

minus06

minus04

minus02


0 5 10 15 20 25

0

1

2

3

4

5

Time (weeks)

Rela

tive e

rror

minus1


0 5 10 15 20065

07

075

08

085

09

Time (weeks)

FDR


0 5 10 15 20072074076078

08082084086088

09092

Time (weeks)

FDR


0 5 10 15 20074076078

08082084086088

09092094

Time (weeks)

FDR


0 5 10 15 2007

075

08

085

09

095

Time (weeks)

FDR


0 5 10 15 20074076078

08082084086088

09092

Time (weeks)

FDR






Acronyms


Notations










119903(119905)



119894(119905)






119882lowast(119905) 119882(119905) minus 119882(0)






119864(119905) 119882

119864(119905) minus 119882

119864(0)





119877(119905) 119882

119877(119905) minus 119882

119877(0)





119871(119905) 119882

119871(119905) minus 119882

119871(0)









119894


119894



Acknowledgments


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















0)













d119898119894(119905)

d119905times

1

119908(119905)= 1198871[119886 minus 119898

119894(119905)]

d119898119903(119905)

d119905times

1

119908(119905)= 1198872[119898119894(119905) minus 119898

119903(119905)]

(4)




119903(0) = 0 we have


1119882lowast(119905)] (5)

119898119903(119905) = 119886times1 minus

1198871exp[minus119887

2119882lowast(119905)]minus119887

2exp[minus119887

1119882lowast(119905)]

1198871minus 1198872

(6)


= 1198861198871exp[minus119887


2exp[minus119887


1198871minus 1198872

(7)


FDR =119873119863

119872 (8)





119903(119905)] (9)


119902(119905) =119872 minus [119886 minus 119898

119903(119905)]

119872 (10)


119902(119905) = (119872 minus 119886((1198871exp[minus119887

2119882lowast(119905)] minus 119887

2exp[minus119887

1119882lowast(119905)])


))119872minus1

(11)













True value of FDR

t

q(te) = QE

q(t)

(nm+1 fm+1)qk

q(0)ts te

(nm fm)

(n1 f1)








[119905119904 119905119890] respectively


119896 119891119896) 1 le





119878and the


(119899119895 119891119895) 119898 lt 119895 le 119898





119894 1 le 119894 le 119898

119878is the










01 02 03 04 05 06 07 08 090

20

40

60

80

100

120

FDR level

Num

ber o

f fau

lt in

ject

ions

TDL = 3

TDL = 5

TDL = 10



119902119896=

119899119896minus 119891119896

119899119896

1 le 119896 le 119898 (12)





Num

ber o

f fai

lure

s occ

urre

d na

tura

lly

Test timets te t


0Test time

ts te

Curr

ent t

eata

bilit

y gr

owth

effor

t

t

cons

umpt

ion

at ti

me t

w(t)






119882119864(119905) = 119873


1119905)] (13)


119908119864(119905) = 119873

11205731exp(minus120573

1119905) (14)

where119882119864(119905) = int

119905

0119908119864(120591)d120591




2119882lowast

119864(infin)] minus 119887

2exp[minus119887

1119882lowast

119864(infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198731] minus 1198872exp[minus119887

11198731]

1198871minus 1198872

(15)



119882119877(119905) = 119873

2[1 minus exp(

minus1205732

21199052)] (16)


119908119877(119905) = 119873

21205732119905 exp(minus

1205732

21199052) (17)

where119882119877(119905) = int

119905

0119908119877(120591)d120591



119905max =1

1205732

(18)




2119882lowast

119877(infin)] minus 119887

2exp[minus119887

1119882lowast

119877(infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198732] minus 1198872exp[minus119887

11198732]

1198871minus 1198872

(19)




119882119871(119905) =

1198733

1 + 119860 exp(minus1205733119905) (20)


119908119871(119905) =

11987331198601205733exp(minus120573

3119905)

[1 + 119860 exp(minus1205733119905)]2 (21)

where119882119871(119905) = int

119905

0119908119871(120591)d120591



119905max =1

1205733

ln119860 (22)


119886remaining

= 1198861198871exp[minus119887

2119882lowast

119877(infin)] minus 119887

2exp[minus119887

1119882lowast

119877(infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198733(1 + 119860)] minus 119887

2exp[minus119887

11198733(1 + 119860)]

1198871minus 1198872

(23)



119882DS(119905) = 1198734[1 minus (1 + 120573

4119905) exp(minus120573

4119905)] (24)


119908DS(119905) = 11987341205732

4119905 exp (minus120573

4119905) (25)

where119882DS(119905) = int119905

0119908DS(120591)d120591


119905max =1

1205734

(26)



2119882lowast


1119882lowast

DS (infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198734] minus 1198872exp[minus119887

11198734]

1198871minus 1198872

(27)


0 10 20 30 40 50 600

5

10

15

20

25

30

Time (weeks)

Curr

ent T

GE

N1 = 240 1205731 = 01111



119882IS(119905) = 1198735


1 + 120593 exp(minus1205735119905) (28)


119908IS(119905) =11987351205735(1 + 120593) exp(minus120573

5119905)

[1 + 120593 exp(minus1205735119905)]2

(29)

where119882IS(119905) = int119905

0119908IS(120591)d120591


119905max =1

1205735

ln120593 (30)



2119882lowast


1119882lowast

IS(infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198735] minus 1198872exp[minus119887

11198735]

1198871minus 1198872

(31)



0

5

10

15

20

25

Time (weeks)

Curr

ent T

GE

0 2 4 6 8 10 12 14 16 18 20

N2 = 120 1205732 = 01111


0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

Time (weeks)

Curr

ent T

GE

N3 = 240 A = 2303 1205733 = 01111





21198731[1 minus exp (minus120573

1119905)]]



1119905)]])


))119872minus1

(32)




1205732

21199052)]]



1205732

21199052)]])


))119872minus1

(33)


0 10 20 30 40 50 60 70 800

21

4

6

8

10

Time (weeks)

Curr

ent T

GE

N4 = 240 1205734 = 01111

9

7

5

3


0 10 20 30 40 50 60 70 80Time (weeks)

Curr

ent T

GE

N5 = 240 120593 = 1984 1205735 = 01111

005

115

225

335

445

5



119902(119905) = (119872minus119886((1198871exp[minus119887

21198733[

1

1+119860 exp (minus120572119905)minus

1

1+119860]]


11198733

times[1

1+119860 exp (minus120572119905)minus

1

1+119860]])


))119872minus1

(34)


119902(119905) = (119872minus119886((1198871exp[minus119887

21198734[1 minus (1+120573

1119905) exp (minus120573

1119905)]]


11198734[1 minus (1+120573

1119905) exp (minus120573

1119905)]])


))119872minus1

(35)


119902(119905) = (119872 minus 119886((1198871exp[minus119887

21198735[

1 + exp (minus1205732119905)

1 + 120593 exp (minus1205732119905)]]


11198735[

1 + exp (minus1205732119905)

1 + 120593 exp (minus1205732119905)]])


))119872minus1

(36)








2given in (6) can


119864(1198731 1205731) 119878119872119888(119886 1198871) and 119878

119872119903(119898119888 1198872) are as follows

Minimize 119878119864(1198731 1205731) =

119899

sum119896=1

[119882119864119896

minus119882119864(119905119896)]2

Minimize 119878119872119888(119886 1198871) =

119899

sum119896=1

[119872119862119896

minus119872119862(119905119896)]2

Minimize 119878119872119903(119898119888 1198872) =

119899

sum119896=1

[119898119903119896minus 119898119903(119905119896)]2

(37)


with respect to (1198731 1205731)

(119886 1198871) and (119898




1 1205731

we get

120597119878119864

1205971198731

=

119899

sum119896=1

minus 2119882119864119896


1119896)]


(38)



Test time(weeks)

CumulativeTGE

(fault times)



Test time(weeks)

CumulativeTGE

(fault times)



1 4 3 2 13 223 74 482 9 6 4 14 241 76 493 16 11 7 15 256 77 504 25 17 11 16 268 79 515 38 25 16 17 278 80 526 55 33 21 18 286 80 527 74 41 27 19 294 82 538 95 50 33 20 301 83 549 119 57 37 21 306 84 5410 145 63 41 22 310 85 5511 173 68 44 23 312 85 5512 203 72 46 24 313 86 55



1198731=

sum119899

119896=2119882119896[1 minus exp (minus120573

1119896)]

sum119899


1119896)]2 (39)

Next we have

120597119878119864

1205971205731

=

119899

sum119896=1


1119896)]

times exp(minus1205731119896) = 0

(40)




(16) 1198733 119860 120573


4 1205734of the








AE =10038161003816100381610038161003816100381610038161003816

119898119894minus 119886

119898119894

10038161003816100381610038161003816100381610038161003816 (41)





MSE =sum119899

119894=1[119898119903(119905119894) minus 119898119903119894]2

119899 (42)




RE =119898119903(119905119886) minus 119890

119890 (43)



119886




119903(119905119890) by time 119905

119890 The estimate











=



3= 31118

(fault times) 119860 = 3936 1205733= 03524week 119873

4= 40606

(fault times) 1205734= 01313week 119873







119894

Bias = 1

119899

119899

sum119894=1

PE119894


119894=1(PE119894minus Bias)2

119899 minus 1

(44)





0 5 10 15 200

5

10

15

20

25

30

35

Time (weeks)Cu

rren

t TG

E (fa

ult t

imes


LogisticActural


0 5 10 15 200

5

10

15

20

25

30

35

Time (weeks)

Curr

ent T

GE

(faul

t tim

es)








STGMs 119886 1198871

1198872


0 5 10 15 20Time (weeks)

Cum

ulat

ive T

GE

(faul

t tim

es)

ExponentialRayleigh

LogisticActural

0

50

150

200

250

300

350

100


0 5 10 15 20Time (weeks)

Cum

ulat

ive T

GE

(faul

t tim

es)


0

50

100

150

200

250

300

350


0 5 10 15 20 25

002040608

11214

Time (weeks)

Rela

tive e

rror

minus04

minus02


0 5 10 15 20 25

002040608

Time (weeks)

Rela

tive e

rror

minus1

minus08

minus06

minus04

minus02


0 5 10 15 20 25

010203040506070

Time (weeks)

Rela

tive e

rror

minus10







0 5 10 15 20 25

002040608

Time (weeks)

Rela

tive e

rror

minus1

minus08

minus06

minus04

minus02


0 5 10 15 20 25

0

1

2

3

4

5

Time (weeks)

Rela

tive e

rror

minus1


0 5 10 15 20065

07

075

08

085

09

Time (weeks)

FDR


0 5 10 15 20072074076078

08082084086088

09092

Time (weeks)

FDR


0 5 10 15 20074076078

08082084086088

09092094

Time (weeks)

FDR


0 5 10 15 2007

075

08

085

09

095

Time (weeks)

FDR


0 5 10 15 20074076078

08082084086088

09092

Time (weeks)

FDR






Acronyms


Notations










119903(119905)



119894(119905)






119882lowast(119905) 119882(119905) minus 119882(0)






119864(119905) 119882

119864(119905) minus 119882

119864(0)





119877(119905) 119882

119877(119905) minus 119882

119877(0)





119871(119905) 119882

119871(119905) minus 119882

119871(0)









119894


119894



Acknowledgments


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















True value of FDR

t

q(te) = QE

q(t)

(nm+1 fm+1)qk

q(0)ts te

(nm fm)

(n1 f1)








[119905119904 119905119890] respectively


119896 119891119896) 1 le





119878and the


(119899119895 119891119895) 119898 lt 119895 le 119898





119894 1 le 119894 le 119898

119878is the










01 02 03 04 05 06 07 08 090

20

40

60

80

100

120

FDR level

Num

ber o

f fau

lt in

ject

ions

TDL = 3

TDL = 5

TDL = 10



119902119896=

119899119896minus 119891119896

119899119896

1 le 119896 le 119898 (12)





Num

ber o

f fai

lure

s occ

urre

d na

tura

lly

Test timets te t


0Test time

ts te

Curr

ent t

eata

bilit

y gr

owth

effor

t

t

cons

umpt

ion

at ti

me t

w(t)






119882119864(119905) = 119873


1119905)] (13)


119908119864(119905) = 119873

11205731exp(minus120573

1119905) (14)

where119882119864(119905) = int

119905

0119908119864(120591)d120591




2119882lowast

119864(infin)] minus 119887

2exp[minus119887

1119882lowast

119864(infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198731] minus 1198872exp[minus119887

11198731]

1198871minus 1198872

(15)



119882119877(119905) = 119873

2[1 minus exp(

minus1205732

21199052)] (16)


119908119877(119905) = 119873

21205732119905 exp(minus

1205732

21199052) (17)

where119882119877(119905) = int

119905

0119908119877(120591)d120591



119905max =1

1205732

(18)




2119882lowast

119877(infin)] minus 119887

2exp[minus119887

1119882lowast

119877(infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198732] minus 1198872exp[minus119887

11198732]

1198871minus 1198872

(19)




119882119871(119905) =

1198733

1 + 119860 exp(minus1205733119905) (20)


119908119871(119905) =

11987331198601205733exp(minus120573

3119905)

[1 + 119860 exp(minus1205733119905)]2 (21)

where119882119871(119905) = int

119905

0119908119871(120591)d120591



119905max =1

1205733

ln119860 (22)


119886remaining

= 1198861198871exp[minus119887

2119882lowast

119877(infin)] minus 119887

2exp[minus119887

1119882lowast

119877(infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198733(1 + 119860)] minus 119887

2exp[minus119887

11198733(1 + 119860)]

1198871minus 1198872

(23)



119882DS(119905) = 1198734[1 minus (1 + 120573

4119905) exp(minus120573

4119905)] (24)


119908DS(119905) = 11987341205732

4119905 exp (minus120573

4119905) (25)

where119882DS(119905) = int119905

0119908DS(120591)d120591


119905max =1

1205734

(26)



2119882lowast


1119882lowast

DS (infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198734] minus 1198872exp[minus119887

11198734]

1198871minus 1198872

(27)


0 10 20 30 40 50 600

5

10

15

20

25

30

Time (weeks)

Curr

ent T

GE

N1 = 240 1205731 = 01111



119882IS(119905) = 1198735


1 + 120593 exp(minus1205735119905) (28)


119908IS(119905) =11987351205735(1 + 120593) exp(minus120573

5119905)

[1 + 120593 exp(minus1205735119905)]2

(29)

where119882IS(119905) = int119905

0119908IS(120591)d120591


119905max =1

1205735

ln120593 (30)



2119882lowast


1119882lowast

IS(infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198735] minus 1198872exp[minus119887

11198735]

1198871minus 1198872

(31)



0

5

10

15

20

25

Time (weeks)

Curr

ent T

GE

0 2 4 6 8 10 12 14 16 18 20

N2 = 120 1205732 = 01111


0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

Time (weeks)

Curr

ent T

GE

N3 = 240 A = 2303 1205733 = 01111





21198731[1 minus exp (minus120573

1119905)]]



1119905)]])


))119872minus1

(32)




1205732

21199052)]]



1205732

21199052)]])


))119872minus1

(33)


0 10 20 30 40 50 60 70 800

21

4

6

8

10

Time (weeks)

Curr

ent T

GE

N4 = 240 1205734 = 01111

9

7

5

3


0 10 20 30 40 50 60 70 80Time (weeks)

Curr

ent T

GE

N5 = 240 120593 = 1984 1205735 = 01111

005

115

225

335

445

5



119902(119905) = (119872minus119886((1198871exp[minus119887

21198733[

1

1+119860 exp (minus120572119905)minus

1

1+119860]]


11198733

times[1

1+119860 exp (minus120572119905)minus

1

1+119860]])


))119872minus1

(34)


119902(119905) = (119872minus119886((1198871exp[minus119887

21198734[1 minus (1+120573

1119905) exp (minus120573

1119905)]]


11198734[1 minus (1+120573

1119905) exp (minus120573

1119905)]])


))119872minus1

(35)


119902(119905) = (119872 minus 119886((1198871exp[minus119887

21198735[

1 + exp (minus1205732119905)

1 + 120593 exp (minus1205732119905)]]


11198735[

1 + exp (minus1205732119905)

1 + 120593 exp (minus1205732119905)]])


))119872minus1

(36)








2given in (6) can


119864(1198731 1205731) 119878119872119888(119886 1198871) and 119878

119872119903(119898119888 1198872) are as follows

Minimize 119878119864(1198731 1205731) =

119899

sum119896=1

[119882119864119896

minus119882119864(119905119896)]2

Minimize 119878119872119888(119886 1198871) =

119899

sum119896=1

[119872119862119896

minus119872119862(119905119896)]2

Minimize 119878119872119903(119898119888 1198872) =

119899

sum119896=1

[119898119903119896minus 119898119903(119905119896)]2

(37)


with respect to (1198731 1205731)

(119886 1198871) and (119898




1 1205731

we get

120597119878119864

1205971198731

=

119899

sum119896=1

minus 2119882119864119896


1119896)]


(38)



Test time(weeks)

CumulativeTGE

(fault times)



Test time(weeks)

CumulativeTGE

(fault times)



1 4 3 2 13 223 74 482 9 6 4 14 241 76 493 16 11 7 15 256 77 504 25 17 11 16 268 79 515 38 25 16 17 278 80 526 55 33 21 18 286 80 527 74 41 27 19 294 82 538 95 50 33 20 301 83 549 119 57 37 21 306 84 5410 145 63 41 22 310 85 5511 173 68 44 23 312 85 5512 203 72 46 24 313 86 55



1198731=

sum119899

119896=2119882119896[1 minus exp (minus120573

1119896)]

sum119899


1119896)]2 (39)

Next we have

120597119878119864

1205971205731

=

119899

sum119896=1


1119896)]

times exp(minus1205731119896) = 0

(40)




(16) 1198733 119860 120573


4 1205734of the








AE =10038161003816100381610038161003816100381610038161003816

119898119894minus 119886

119898119894

10038161003816100381610038161003816100381610038161003816 (41)





MSE =sum119899

119894=1[119898119903(119905119894) minus 119898119903119894]2

119899 (42)




RE =119898119903(119905119886) minus 119890

119890 (43)



119886




119903(119905119890) by time 119905

119890 The estimate











=



3= 31118

(fault times) 119860 = 3936 1205733= 03524week 119873

4= 40606

(fault times) 1205734= 01313week 119873







119894

Bias = 1

119899

119899

sum119894=1

PE119894


119894=1(PE119894minus Bias)2

119899 minus 1

(44)





0 5 10 15 200

5

10

15

20

25

30

35

Time (weeks)Cu

rren

t TG

E (fa

ult t

imes


LogisticActural


0 5 10 15 200

5

10

15

20

25

30

35

Time (weeks)

Curr

ent T

GE

(faul

t tim

es)








STGMs 119886 1198871

1198872


0 5 10 15 20Time (weeks)

Cum

ulat

ive T

GE

(faul

t tim

es)

ExponentialRayleigh

LogisticActural

0

50

150

200

250

300

350

100


0 5 10 15 20Time (weeks)

Cum

ulat

ive T

GE

(faul

t tim

es)


0

50

100

150

200

250

300

350


0 5 10 15 20 25

002040608

11214

Time (weeks)

Rela

tive e

rror

minus04

minus02


0 5 10 15 20 25

002040608

Time (weeks)

Rela

tive e

rror

minus1

minus08

minus06

minus04

minus02


0 5 10 15 20 25

010203040506070

Time (weeks)

Rela

tive e

rror

minus10







0 5 10 15 20 25

002040608

Time (weeks)

Rela

tive e

rror

minus1

minus08

minus06

minus04

minus02


0 5 10 15 20 25

0

1

2

3

4

5

Time (weeks)

Rela

tive e

rror

minus1


0 5 10 15 20065

07

075

08

085

09

Time (weeks)

FDR


0 5 10 15 20072074076078

08082084086088

09092

Time (weeks)

FDR


0 5 10 15 20074076078

08082084086088

09092094

Time (weeks)

FDR


0 5 10 15 2007

075

08

085

09

095

Time (weeks)

FDR


0 5 10 15 20074076078

08082084086088

09092

Time (weeks)

FDR






Acronyms


Notations










119903(119905)



119894(119905)






119882lowast(119905) 119882(119905) minus 119882(0)






119864(119905) 119882

119864(119905) minus 119882

119864(0)





119877(119905) 119882

119877(119905) minus 119882

119877(0)





119871(119905) 119882

119871(119905) minus 119882

119871(0)









119894


119894



Acknowledgments


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










01 02 03 04 05 06 07 08 090

20

40

60

80

100

120

FDR level

Num

ber o

f fau

lt in

ject

ions

TDL = 3

TDL = 5

TDL = 10



119902119896=

119899119896minus 119891119896

119899119896

1 le 119896 le 119898 (12)





Num

ber o

f fai

lure

s occ

urre

d na

tura

lly

Test timets te t


0Test time

ts te

Curr

ent t

eata

bilit

y gr

owth

effor

t

t

cons

umpt

ion

at ti

me t

w(t)






119882119864(119905) = 119873


1119905)] (13)


119908119864(119905) = 119873

11205731exp(minus120573

1119905) (14)

where119882119864(119905) = int

119905

0119908119864(120591)d120591




2119882lowast

119864(infin)] minus 119887

2exp[minus119887

1119882lowast

119864(infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198731] minus 1198872exp[minus119887

11198731]

1198871minus 1198872

(15)



119882119877(119905) = 119873

2[1 minus exp(

minus1205732

21199052)] (16)


119908119877(119905) = 119873

21205732119905 exp(minus

1205732

21199052) (17)

where119882119877(119905) = int

119905

0119908119877(120591)d120591



119905max =1

1205732

(18)




2119882lowast

119877(infin)] minus 119887

2exp[minus119887

1119882lowast

119877(infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198732] minus 1198872exp[minus119887

11198732]

1198871minus 1198872

(19)




119882119871(119905) =

1198733

1 + 119860 exp(minus1205733119905) (20)


119908119871(119905) =

11987331198601205733exp(minus120573

3119905)

[1 + 119860 exp(minus1205733119905)]2 (21)

where119882119871(119905) = int

119905

0119908119871(120591)d120591



119905max =1

1205733

ln119860 (22)


119886remaining

= 1198861198871exp[minus119887

2119882lowast

119877(infin)] minus 119887

2exp[minus119887

1119882lowast

119877(infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198733(1 + 119860)] minus 119887

2exp[minus119887

11198733(1 + 119860)]

1198871minus 1198872

(23)



119882DS(119905) = 1198734[1 minus (1 + 120573

4119905) exp(minus120573

4119905)] (24)


119908DS(119905) = 11987341205732

4119905 exp (minus120573

4119905) (25)

where119882DS(119905) = int119905

0119908DS(120591)d120591


119905max =1

1205734

(26)



2119882lowast


1119882lowast

DS (infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198734] minus 1198872exp[minus119887

11198734]

1198871minus 1198872

(27)


0 10 20 30 40 50 600

5

10

15

20

25

30

Time (weeks)

Curr

ent T

GE

N1 = 240 1205731 = 01111



119882IS(119905) = 1198735


1 + 120593 exp(minus1205735119905) (28)


119908IS(119905) =11987351205735(1 + 120593) exp(minus120573

5119905)

[1 + 120593 exp(minus1205735119905)]2

(29)

where119882IS(119905) = int119905

0119908IS(120591)d120591


119905max =1

1205735

ln120593 (30)



2119882lowast


1119882lowast

IS(infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198735] minus 1198872exp[minus119887

11198735]

1198871minus 1198872

(31)



0

5

10

15

20

25

Time (weeks)

Curr

ent T

GE

0 2 4 6 8 10 12 14 16 18 20

N2 = 120 1205732 = 01111


0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

Time (weeks)

Curr

ent T

GE

N3 = 240 A = 2303 1205733 = 01111





21198731[1 minus exp (minus120573

1119905)]]



1119905)]])


))119872minus1

(32)




1205732

21199052)]]



1205732

21199052)]])


))119872minus1

(33)


0 10 20 30 40 50 60 70 800

21

4

6

8

10

Time (weeks)

Curr

ent T

GE

N4 = 240 1205734 = 01111

9

7

5

3


0 10 20 30 40 50 60 70 80Time (weeks)

Curr

ent T

GE

N5 = 240 120593 = 1984 1205735 = 01111

005

115

225

335

445

5



119902(119905) = (119872minus119886((1198871exp[minus119887

21198733[

1

1+119860 exp (minus120572119905)minus

1

1+119860]]


11198733

times[1

1+119860 exp (minus120572119905)minus

1

1+119860]])


))119872minus1

(34)


119902(119905) = (119872minus119886((1198871exp[minus119887

21198734[1 minus (1+120573

1119905) exp (minus120573

1119905)]]


11198734[1 minus (1+120573

1119905) exp (minus120573

1119905)]])


))119872minus1

(35)


119902(119905) = (119872 minus 119886((1198871exp[minus119887

21198735[

1 + exp (minus1205732119905)

1 + 120593 exp (minus1205732119905)]]


11198735[

1 + exp (minus1205732119905)

1 + 120593 exp (minus1205732119905)]])


))119872minus1

(36)








2given in (6) can


119864(1198731 1205731) 119878119872119888(119886 1198871) and 119878

119872119903(119898119888 1198872) are as follows

Minimize 119878119864(1198731 1205731) =

119899

sum119896=1

[119882119864119896

minus119882119864(119905119896)]2

Minimize 119878119872119888(119886 1198871) =

119899

sum119896=1

[119872119862119896

minus119872119862(119905119896)]2

Minimize 119878119872119903(119898119888 1198872) =

119899

sum119896=1

[119898119903119896minus 119898119903(119905119896)]2

(37)


with respect to (1198731 1205731)

(119886 1198871) and (119898




1 1205731

we get

120597119878119864

1205971198731

=

119899

sum119896=1

minus 2119882119864119896


1119896)]


(38)



Test time(weeks)

CumulativeTGE

(fault times)



Test time(weeks)

CumulativeTGE

(fault times)



1 4 3 2 13 223 74 482 9 6 4 14 241 76 493 16 11 7 15 256 77 504 25 17 11 16 268 79 515 38 25 16 17 278 80 526 55 33 21 18 286 80 527 74 41 27 19 294 82 538 95 50 33 20 301 83 549 119 57 37 21 306 84 5410 145 63 41 22 310 85 5511 173 68 44 23 312 85 5512 203 72 46 24 313 86 55



1198731=

sum119899

119896=2119882119896[1 minus exp (minus120573

1119896)]

sum119899


1119896)]2 (39)

Next we have

120597119878119864

1205971205731

=

119899

sum119896=1


1119896)]

times exp(minus1205731119896) = 0

(40)




(16) 1198733 119860 120573


4 1205734of the








AE =10038161003816100381610038161003816100381610038161003816

119898119894minus 119886

119898119894

10038161003816100381610038161003816100381610038161003816 (41)





MSE =sum119899

119894=1[119898119903(119905119894) minus 119898119903119894]2

119899 (42)




RE =119898119903(119905119886) minus 119890

119890 (43)



119886




119903(119905119890) by time 119905

119890 The estimate











=



3= 31118

(fault times) 119860 = 3936 1205733= 03524week 119873

4= 40606

(fault times) 1205734= 01313week 119873







119894

Bias = 1

119899

119899

sum119894=1

PE119894


119894=1(PE119894minus Bias)2

119899 minus 1

(44)





0 5 10 15 200

5

10

15

20

25

30

35

Time (weeks)Cu

rren

t TG

E (fa

ult t

imes


LogisticActural


0 5 10 15 200

5

10

15

20

25

30

35

Time (weeks)

Curr

ent T

GE

(faul

t tim

es)








STGMs 119886 1198871

1198872


0 5 10 15 20Time (weeks)

Cum

ulat

ive T

GE

(faul

t tim

es)

ExponentialRayleigh

LogisticActural

0

50

150

200

250

300

350

100


0 5 10 15 20Time (weeks)

Cum

ulat

ive T

GE

(faul

t tim

es)


0

50

100

150

200

250

300

350


0 5 10 15 20 25

002040608

11214

Time (weeks)

Rela

tive e

rror

minus04

minus02


0 5 10 15 20 25

002040608

Time (weeks)

Rela

tive e

rror

minus1

minus08

minus06

minus04

minus02


0 5 10 15 20 25

010203040506070

Time (weeks)

Rela

tive e

rror

minus10







0 5 10 15 20 25

002040608

Time (weeks)

Rela

tive e

rror

minus1

minus08

minus06

minus04

minus02


0 5 10 15 20 25

0

1

2

3

4

5

Time (weeks)

Rela

tive e

rror

minus1


0 5 10 15 20065

07

075

08

085

09

Time (weeks)

FDR


0 5 10 15 20072074076078

08082084086088

09092

Time (weeks)

FDR


0 5 10 15 20074076078

08082084086088

09092094

Time (weeks)

FDR


0 5 10 15 2007

075

08

085

09

095

Time (weeks)

FDR


0 5 10 15 20074076078

08082084086088

09092

Time (weeks)

FDR






Acronyms


Notations










119903(119905)



119894(119905)






119882lowast(119905) 119882(119905) minus 119882(0)






119864(119905) 119882

119864(119905) minus 119882

119864(0)





119877(119905) 119882

119877(119905) minus 119882

119877(0)





119871(119905) 119882

119871(119905) minus 119882

119871(0)









119894


119894



Acknowledgments


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119882119864(119905) = 119873


1119905)] (13)


119908119864(119905) = 119873

11205731exp(minus120573

1119905) (14)

where119882119864(119905) = int

119905

0119908119864(120591)d120591




2119882lowast

119864(infin)] minus 119887

2exp[minus119887

1119882lowast

119864(infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198731] minus 1198872exp[minus119887

11198731]

1198871minus 1198872

(15)



119882119877(119905) = 119873

2[1 minus exp(

minus1205732

21199052)] (16)


119908119877(119905) = 119873

21205732119905 exp(minus

1205732

21199052) (17)

where119882119877(119905) = int

119905

0119908119877(120591)d120591



119905max =1

1205732

(18)




2119882lowast

119877(infin)] minus 119887

2exp[minus119887

1119882lowast

119877(infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198732] minus 1198872exp[minus119887

11198732]

1198871minus 1198872

(19)




119882119871(119905) =

1198733

1 + 119860 exp(minus1205733119905) (20)


119908119871(119905) =

11987331198601205733exp(minus120573

3119905)

[1 + 119860 exp(minus1205733119905)]2 (21)

where119882119871(119905) = int

119905

0119908119871(120591)d120591



119905max =1

1205733

ln119860 (22)


119886remaining

= 1198861198871exp[minus119887

2119882lowast

119877(infin)] minus 119887

2exp[minus119887

1119882lowast

119877(infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198733(1 + 119860)] minus 119887

2exp[minus119887

11198733(1 + 119860)]

1198871minus 1198872

(23)



119882DS(119905) = 1198734[1 minus (1 + 120573

4119905) exp(minus120573

4119905)] (24)


119908DS(119905) = 11987341205732

4119905 exp (minus120573

4119905) (25)

where119882DS(119905) = int119905

0119908DS(120591)d120591


119905max =1

1205734

(26)



2119882lowast


1119882lowast

DS (infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198734] minus 1198872exp[minus119887

11198734]

1198871minus 1198872

(27)


0 10 20 30 40 50 600

5

10

15

20

25

30

Time (weeks)

Curr

ent T

GE

N1 = 240 1205731 = 01111



119882IS(119905) = 1198735


1 + 120593 exp(minus1205735119905) (28)


119908IS(119905) =11987351205735(1 + 120593) exp(minus120573

5119905)

[1 + 120593 exp(minus1205735119905)]2

(29)

where119882IS(119905) = int119905

0119908IS(120591)d120591


119905max =1

1205735

ln120593 (30)



2119882lowast


1119882lowast

IS(infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198735] minus 1198872exp[minus119887

11198735]

1198871minus 1198872

(31)



0

5

10

15

20

25

Time (weeks)

Curr

ent T

GE

0 2 4 6 8 10 12 14 16 18 20

N2 = 120 1205732 = 01111


0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

Time (weeks)

Curr

ent T

GE

N3 = 240 A = 2303 1205733 = 01111





21198731[1 minus exp (minus120573

1119905)]]



1119905)]])


))119872minus1

(32)




1205732

21199052)]]



1205732

21199052)]])


))119872minus1

(33)


0 10 20 30 40 50 60 70 800

21

4

6

8

10

Time (weeks)

Curr

ent T

GE

N4 = 240 1205734 = 01111

9

7

5

3


0 10 20 30 40 50 60 70 80Time (weeks)

Curr

ent T

GE

N5 = 240 120593 = 1984 1205735 = 01111

005

115

225

335

445

5



119902(119905) = (119872minus119886((1198871exp[minus119887

21198733[

1

1+119860 exp (minus120572119905)minus

1

1+119860]]


11198733

times[1

1+119860 exp (minus120572119905)minus

1

1+119860]])


))119872minus1

(34)


119902(119905) = (119872minus119886((1198871exp[minus119887

21198734[1 minus (1+120573

1119905) exp (minus120573

1119905)]]


11198734[1 minus (1+120573

1119905) exp (minus120573

1119905)]])


))119872minus1

(35)


119902(119905) = (119872 minus 119886((1198871exp[minus119887

21198735[

1 + exp (minus1205732119905)

1 + 120593 exp (minus1205732119905)]]


11198735[

1 + exp (minus1205732119905)

1 + 120593 exp (minus1205732119905)]])


))119872minus1

(36)








2given in (6) can


119864(1198731 1205731) 119878119872119888(119886 1198871) and 119878

119872119903(119898119888 1198872) are as follows

Minimize 119878119864(1198731 1205731) =

119899

sum119896=1

[119882119864119896

minus119882119864(119905119896)]2

Minimize 119878119872119888(119886 1198871) =

119899

sum119896=1

[119872119862119896

minus119872119862(119905119896)]2

Minimize 119878119872119903(119898119888 1198872) =

119899

sum119896=1

[119898119903119896minus 119898119903(119905119896)]2

(37)


with respect to (1198731 1205731)

(119886 1198871) and (119898




1 1205731

we get

120597119878119864

1205971198731

=

119899

sum119896=1

minus 2119882119864119896


1119896)]


(38)



Test time(weeks)

CumulativeTGE

(fault times)



Test time(weeks)

CumulativeTGE

(fault times)



1 4 3 2 13 223 74 482 9 6 4 14 241 76 493 16 11 7 15 256 77 504 25 17 11 16 268 79 515 38 25 16 17 278 80 526 55 33 21 18 286 80 527 74 41 27 19 294 82 538 95 50 33 20 301 83 549 119 57 37 21 306 84 5410 145 63 41 22 310 85 5511 173 68 44 23 312 85 5512 203 72 46 24 313 86 55



1198731=

sum119899

119896=2119882119896[1 minus exp (minus120573

1119896)]

sum119899


1119896)]2 (39)

Next we have

120597119878119864

1205971205731

=

119899

sum119896=1


1119896)]

times exp(minus1205731119896) = 0

(40)




(16) 1198733 119860 120573


4 1205734of the








AE =10038161003816100381610038161003816100381610038161003816

119898119894minus 119886

119898119894

10038161003816100381610038161003816100381610038161003816 (41)





MSE =sum119899

119894=1[119898119903(119905119894) minus 119898119903119894]2

119899 (42)




RE =119898119903(119905119886) minus 119890

119890 (43)



119886




119903(119905119890) by time 119905

119890 The estimate











=



3= 31118

(fault times) 119860 = 3936 1205733= 03524week 119873

4= 40606

(fault times) 1205734= 01313week 119873







119894

Bias = 1

119899

119899

sum119894=1

PE119894


119894=1(PE119894minus Bias)2

119899 minus 1

(44)





0 5 10 15 200

5

10

15

20

25

30

35

Time (weeks)Cu

rren

t TG

E (fa

ult t

imes


LogisticActural


0 5 10 15 200

5

10

15

20

25

30

35

Time (weeks)

Curr

ent T

GE

(faul

t tim

es)








STGMs 119886 1198871

1198872


0 5 10 15 20Time (weeks)

Cum

ulat

ive T

GE

(faul

t tim

es)

ExponentialRayleigh

LogisticActural

0

50

150

200

250

300

350

100


0 5 10 15 20Time (weeks)

Cum

ulat

ive T

GE

(faul

t tim

es)


0

50

100

150

200

250

300

350


0 5 10 15 20 25

002040608

11214

Time (weeks)

Rela

tive e

rror

minus04

minus02


0 5 10 15 20 25

002040608

Time (weeks)

Rela

tive e

rror

minus1

minus08

minus06

minus04

minus02


0 5 10 15 20 25

010203040506070

Time (weeks)

Rela

tive e

rror

minus10







0 5 10 15 20 25

002040608

Time (weeks)

Rela

tive e

rror

minus1

minus08

minus06

minus04

minus02


0 5 10 15 20 25

0

1

2

3

4

5

Time (weeks)

Rela

tive e

rror

minus1


0 5 10 15 20065

07

075

08

085

09

Time (weeks)

FDR


0 5 10 15 20072074076078

08082084086088

09092

Time (weeks)

FDR


0 5 10 15 20074076078

08082084086088

09092094

Time (weeks)

FDR


0 5 10 15 2007

075

08

085

09

095

Time (weeks)

FDR


0 5 10 15 20074076078

08082084086088

09092

Time (weeks)

FDR






Acronyms


Notations










119903(119905)



119894(119905)






119882lowast(119905) 119882(119905) minus 119882(0)






119864(119905) 119882

119864(119905) minus 119882

119864(0)





119877(119905) 119882

119877(119905) minus 119882

119877(0)





119871(119905) 119882

119871(119905) minus 119882

119871(0)









119894


119894



Acknowledgments


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 50 600

5

10

15

20

25

30

Time (weeks)

Curr

ent T

GE

N1 = 240 1205731 = 01111



119882IS(119905) = 1198735


1 + 120593 exp(minus1205735119905) (28)


119908IS(119905) =11987351205735(1 + 120593) exp(minus120573

5119905)

[1 + 120593 exp(minus1205735119905)]2

(29)

where119882IS(119905) = int119905

0119908IS(120591)d120591


119905max =1

1205735

ln120593 (30)



2119882lowast


1119882lowast

IS(infin)]

1198871minus 1198872

asymp 1198861198871exp[minus119887

21198735] minus 1198872exp[minus119887

11198735]

1198871minus 1198872

(31)



0

5

10

15

20

25

Time (weeks)

Curr

ent T

GE

0 2 4 6 8 10 12 14 16 18 20

N2 = 120 1205732 = 01111


0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

Time (weeks)

Curr

ent T

GE

N3 = 240 A = 2303 1205733 = 01111





21198731[1 minus exp (minus120573

1119905)]]



1119905)]])


))119872minus1

(32)




1205732

21199052)]]



1205732

21199052)]])


))119872minus1

(33)


0 10 20 30 40 50 60 70 800

21

4

6

8

10

Time (weeks)

Curr

ent T

GE

N4 = 240 1205734 = 01111

9

7

5

3


0 10 20 30 40 50 60 70 80Time (weeks)

Curr

ent T

GE

N5 = 240 120593 = 1984 1205735 = 01111

005

115

225

335

445

5



119902(119905) = (119872minus119886((1198871exp[minus119887

21198733[

1

1+119860 exp (minus120572119905)minus

1

1+119860]]


11198733

times[1

1+119860 exp (minus120572119905)minus

1

1+119860]])


))119872minus1

(34)


119902(119905) = (119872minus119886((1198871exp[minus119887

21198734[1 minus (1+120573

1119905) exp (minus120573

1119905)]]


11198734[1 minus (1+120573

1119905) exp (minus120573

1119905)]])


))119872minus1

(35)


119902(119905) = (119872 minus 119886((1198871exp[minus119887

21198735[

1 + exp (minus1205732119905)

1 + 120593 exp (minus1205732119905)]]


11198735[

1 + exp (minus1205732119905)

1 + 120593 exp (minus1205732119905)]])


))119872minus1

(36)








2given in (6) can


119864(1198731 1205731) 119878119872119888(119886 1198871) and 119878

119872119903(119898119888 1198872) are as follows

Minimize 119878119864(1198731 1205731) =

119899

sum119896=1

[119882119864119896

minus119882119864(119905119896)]2

Minimize 119878119872119888(119886 1198871) =

119899

sum119896=1

[119872119862119896

minus119872119862(119905119896)]2

Minimize 119878119872119903(119898119888 1198872) =

119899

sum119896=1

[119898119903119896minus 119898119903(119905119896)]2

(37)


with respect to (1198731 1205731)

(119886 1198871) and (119898




1 1205731

we get

120597119878119864

1205971198731

=

119899

sum119896=1

minus 2119882119864119896


1119896)]


(38)



Test time(weeks)

CumulativeTGE

(fault times)



Test time(weeks)

CumulativeTGE

(fault times)



1 4 3 2 13 223 74 482 9 6 4 14 241 76 493 16 11 7 15 256 77 504 25 17 11 16 268 79 515 38 25 16 17 278 80 526 55 33 21 18 286 80 527 74 41 27 19 294 82 538 95 50 33 20 301 83 549 119 57 37 21 306 84 5410 145 63 41 22 310 85 5511 173 68 44 23 312 85 5512 203 72 46 24 313 86 55



1198731=

sum119899

119896=2119882119896[1 minus exp (minus120573

1119896)]

sum119899


1119896)]2 (39)

Next we have

120597119878119864

1205971205731

=

119899

sum119896=1


1119896)]

times exp(minus1205731119896) = 0

(40)




(16) 1198733 119860 120573


4 1205734of the








AE =10038161003816100381610038161003816100381610038161003816

119898119894minus 119886

119898119894

10038161003816100381610038161003816100381610038161003816 (41)





MSE =sum119899

119894=1[119898119903(119905119894) minus 119898119903119894]2

119899 (42)




RE =119898119903(119905119886) minus 119890

119890 (43)



119886




119903(119905119890) by time 119905

119890 The estimate











=



3= 31118

(fault times) 119860 = 3936 1205733= 03524week 119873

4= 40606

(fault times) 1205734= 01313week 119873







119894

Bias = 1

119899

119899

sum119894=1

PE119894


119894=1(PE119894minus Bias)2

119899 minus 1

(44)





0 5 10 15 200

5

10

15

20

25

30

35

Time (weeks)Cu

rren

t TG

E (fa

ult t

imes


LogisticActural


0 5 10 15 200

5

10

15

20

25

30

35

Time (weeks)

Curr

ent T

GE

(faul

t tim

es)








STGMs 119886 1198871

1198872


0 5 10 15 20Time (weeks)

Cum

ulat

ive T

GE

(faul

t tim

es)

ExponentialRayleigh

LogisticActural

0

50

150

200

250

300

350

100


0 5 10 15 20Time (weeks)

Cum

ulat

ive T

GE

(faul

t tim

es)


0

50

100

150

200

250

300

350


0 5 10 15 20 25

002040608

11214

Time (weeks)

Rela

tive e

rror

minus04

minus02


0 5 10 15 20 25

002040608

Time (weeks)

Rela

tive e

rror

minus1

minus08

minus06

minus04

minus02


0 5 10 15 20 25

010203040506070

Time (weeks)

Rela

tive e

rror

minus10







0 5 10 15 20 25

002040608

Time (weeks)

Rela

tive e

rror

minus1

minus08

minus06

minus04

minus02


0 5 10 15 20 25

0

1

2

3

4

5

Time (weeks)

Rela

tive e

rror

minus1


0 5 10 15 20065

07

075

08

085

09

Time (weeks)

FDR


0 5 10 15 20072074076078

08082084086088

09092

Time (weeks)

FDR


0 5 10 15 20074076078

08082084086088

09092094

Time (weeks)

FDR


0 5 10 15 2007

075

08

085

09

095

Time (weeks)

FDR


0 5 10 15 20074076078

08082084086088

09092

Time (weeks)

FDR






Acronyms


Notations










119903(119905)



119894(119905)






119882lowast(119905) 119882(119905) minus 119882(0)






119864(119905) 119882

119864(119905) minus 119882

119864(0)





119877(119905) 119882

119877(119905) minus 119882

119877(0)





119871(119905) 119882

119871(119905) minus 119882

119871(0)









119894


119894



Acknowledgments


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 50 60 70 800

21

4

6

8

10

Time (weeks)

Curr

ent T

GE

N4 = 240 1205734 = 01111

9

7

5

3


0 10 20 30 40 50 60 70 80Time (weeks)

Curr

ent T

GE

N5 = 240 120593 = 1984 1205735 = 01111

005

115

225

335

445

5



119902(119905) = (119872minus119886((1198871exp[minus119887

21198733[

1

1+119860 exp (minus120572119905)minus

1

1+119860]]


11198733

times[1

1+119860 exp (minus120572119905)minus

1

1+119860]])


))119872minus1

(34)


119902(119905) = (119872minus119886((1198871exp[minus119887

21198734[1 minus (1+120573

1119905) exp (minus120573

1119905)]]


11198734[1 minus (1+120573

1119905) exp (minus120573

1119905)]])


))119872minus1

(35)


119902(119905) = (119872 minus 119886((1198871exp[minus119887

21198735[

1 + exp (minus1205732119905)

1 + 120593 exp (minus1205732119905)]]


11198735[

1 + exp (minus1205732119905)

1 + 120593 exp (minus1205732119905)]])


))119872minus1

(36)








2given in (6) can


119864(1198731 1205731) 119878119872119888(119886 1198871) and 119878

119872119903(119898119888 1198872) are as follows

Minimize 119878119864(1198731 1205731) =

119899

sum119896=1

[119882119864119896

minus119882119864(119905119896)]2

Minimize 119878119872119888(119886 1198871) =

119899

sum119896=1

[119872119862119896

minus119872119862(119905119896)]2

Minimize 119878119872119903(119898119888 1198872) =

119899

sum119896=1

[119898119903119896minus 119898119903(119905119896)]2

(37)


with respect to (1198731 1205731)

(119886 1198871) and (119898




1 1205731

we get

120597119878119864

1205971198731

=

119899

sum119896=1

minus 2119882119864119896


1119896)]


(38)



Test time(weeks)

CumulativeTGE

(fault times)



Test time(weeks)

CumulativeTGE

(fault times)



1 4 3 2 13 223 74 482 9 6 4 14 241 76 493 16 11 7 15 256 77 504 25 17 11 16 268 79 515 38 25 16 17 278 80 526 55 33 21 18 286 80 527 74 41 27 19 294 82 538 95 50 33 20 301 83 549 119 57 37 21 306 84 5410 145 63 41 22 310 85 5511 173 68 44 23 312 85 5512 203 72 46 24 313 86 55



1198731=

sum119899

119896=2119882119896[1 minus exp (minus120573

1119896)]

sum119899


1119896)]2 (39)

Next we have

120597119878119864

1205971205731

=

119899

sum119896=1


1119896)]

times exp(minus1205731119896) = 0

(40)




(16) 1198733 119860 120573


4 1205734of the








AE =10038161003816100381610038161003816100381610038161003816

119898119894minus 119886

119898119894

10038161003816100381610038161003816100381610038161003816 (41)





MSE =sum119899

119894=1[119898119903(119905119894) minus 119898119903119894]2

119899 (42)




RE =119898119903(119905119886) minus 119890

119890 (43)



119886




119903(119905119890) by time 119905

119890 The estimate











=



3= 31118

(fault times) 119860 = 3936 1205733= 03524week 119873

4= 40606

(fault times) 1205734= 01313week 119873







119894

Bias = 1

119899

119899

sum119894=1

PE119894


119894=1(PE119894minus Bias)2

119899 minus 1

(44)





0 5 10 15 200

5

10

15

20

25

30

35

Time (weeks)Cu

rren

t TG

E (fa

ult t

imes


LogisticActural


0 5 10 15 200

5

10

15

20

25

30

35

Time (weeks)

Curr

ent T

GE

(faul

t tim

es)








STGMs 119886 1198871

1198872


0 5 10 15 20Time (weeks)

Cum

ulat

ive T

GE

(faul

t tim

es)

ExponentialRayleigh

LogisticActural

0

50

150

200

250

300

350

100


0 5 10 15 20Time (weeks)

Cum

ulat

ive T

GE

(faul

t tim

es)


0

50

100

150

200

250

300

350


0 5 10 15 20 25

002040608

11214

Time (weeks)

Rela

tive e

rror

minus04

minus02


0 5 10 15 20 25

002040608

Time (weeks)

Rela

tive e

rror

minus1

minus08

minus06

minus04

minus02


0 5 10 15 20 25

010203040506070

Time (weeks)

Rela

tive e

rror

minus10







0 5 10 15 20 25

002040608

Time (weeks)

Rela

tive e

rror

minus1

minus08

minus06

minus04

minus02


0 5 10 15 20 25

0

1

2

3

4

5

Time (weeks)

Rela

tive e

rror

minus1


0 5 10 15 20065

07

075

08

085

09

Time (weeks)

FDR


0 5 10 15 20072074076078

08082084086088

09092

Time (weeks)

FDR


0 5 10 15 20074076078

08082084086088

09092094

Time (weeks)

FDR


0 5 10 15 2007

075

08

085

09

095

Time (weeks)

FDR


0 5 10 15 20074076078

08082084086088

09092

Time (weeks)

FDR






Acronyms


Notations










119903(119905)



119894(119905)






119882lowast(119905) 119882(119905) minus 119882(0)






119864(119905) 119882

119864(119905) minus 119882

119864(0)





119877(119905) 119882

119877(119905) minus 119882

119877(0)





119871(119905) 119882

119871(119905) minus 119882

119871(0)









119894


119894



Acknowledgments


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Test time(weeks)

CumulativeTGE

(fault times)



Test time(weeks)

CumulativeTGE

(fault times)



1 4 3 2 13 223 74 482 9 6 4 14 241 76 493 16 11 7 15 256 77 504 25 17 11 16 268 79 515 38 25 16 17 278 80 526 55 33 21 18 286 80 527 74 41 27 19 294 82 538 95 50 33 20 301 83 549 119 57 37 21 306 84 5410 145 63 41 22 310 85 5511 173 68 44 23 312 85 5512 203 72 46 24 313 86 55



1198731=

sum119899

119896=2119882119896[1 minus exp (minus120573

1119896)]

sum119899


1119896)]2 (39)

Next we have

120597119878119864

1205971205731

=

119899

sum119896=1


1119896)]

times exp(minus1205731119896) = 0

(40)




(16) 1198733 119860 120573


4 1205734of the








AE =10038161003816100381610038161003816100381610038161003816

119898119894minus 119886

119898119894

10038161003816100381610038161003816100381610038161003816 (41)





MSE =sum119899

119894=1[119898119903(119905119894) minus 119898119903119894]2

119899 (42)




RE =119898119903(119905119886) minus 119890

119890 (43)



119886




119903(119905119890) by time 119905

119890 The estimate











=



3= 31118

(fault times) 119860 = 3936 1205733= 03524week 119873

4= 40606

(fault times) 1205734= 01313week 119873







119894

Bias = 1

119899

119899

sum119894=1

PE119894


119894=1(PE119894minus Bias)2

119899 minus 1

(44)





0 5 10 15 200

5

10

15

20

25

30

35

Time (weeks)Cu

rren

t TG

E (fa

ult t

imes


LogisticActural


0 5 10 15 200

5

10

15

20

25

30

35

Time (weeks)

Curr

ent T

GE

(faul

t tim

es)








STGMs 119886 1198871

1198872


0 5 10 15 20Time (weeks)

Cum

ulat

ive T

GE

(faul

t tim

es)

ExponentialRayleigh

LogisticActural

0

50

150

200

250

300

350

100


0 5 10 15 20Time (weeks)

Cum

ulat

ive T

GE

(faul

t tim

es)


0

50

100

150

200

250

300

350


0 5 10 15 20 25

002040608

11214

Time (weeks)

Rela

tive e

rror

minus04

minus02


0 5 10 15 20 25

002040608

Time (weeks)

Rela

tive e

rror

minus1

minus08

minus06

minus04

minus02


0 5 10 15 20 25

010203040506070

Time (weeks)

Rela

tive e

rror

minus10







0 5 10 15 20 25

002040608

Time (weeks)

Rela

tive e

rror

minus1

minus08

minus06

minus04

minus02


0 5 10 15 20 25

0

1

2

3

4

5

Time (weeks)

Rela

tive e

rror

minus1


0 5 10 15 20065

07

075

08

085

09

Time (weeks)

FDR


0 5 10 15 20072074076078

08082084086088

09092

Time (weeks)

FDR


0 5 10 15 20074076078

08082084086088

09092094

Time (weeks)

FDR


0 5 10 15 2007

075

08

085

09

095

Time (weeks)

FDR


0 5 10 15 20074076078

08082084086088

09092

Time (weeks)

FDR






Acronyms


Notations










119903(119905)



119894(119905)






119882lowast(119905) 119882(119905) minus 119882(0)






119864(119905) 119882

119864(119905) minus 119882

119864(0)





119877(119905) 119882

119877(119905) minus 119882

119877(0)





119871(119905) 119882

119871(119905) minus 119882

119871(0)









119894


119894



Acknowledgments


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











RE =119898119903(119905119886) minus 119890

119890 (43)



119886




119903(119905119890) by time 119905

119890 The estimate











=



3= 31118

(fault times) 119860 = 3936 1205733= 03524week 119873

4= 40606

(fault times) 1205734= 01313week 119873







119894

Bias = 1

119899

119899

sum119894=1

PE119894


119894=1(PE119894minus Bias)2

119899 minus 1

(44)





0 5 10 15 200

5

10

15

20

25

30

35

Time (weeks)Cu

rren

t TG

E (fa

ult t

imes


LogisticActural


0 5 10 15 200

5

10

15

20

25

30

35

Time (weeks)

Curr

ent T

GE

(faul

t tim

es)








STGMs 119886 1198871

1198872


0 5 10 15 20Time (weeks)

Cum

ulat

ive T

GE

(faul

t tim

es)

ExponentialRayleigh

LogisticActural

0

50

150

200

250

300

350

100


0 5 10 15 20Time (weeks)

Cum

ulat

ive T

GE

(faul

t tim

es)


0

50

100

150

200

250

300

350


0 5 10 15 20 25

002040608

11214

Time (weeks)

Rela

tive e

rror

minus04

minus02


0 5 10 15 20 25

002040608

Time (weeks)

Rela

tive e

rror

minus1

minus08

minus06

minus04

minus02


0 5 10 15 20 25

010203040506070

Time (weeks)

Rela

tive e

rror

minus10







0 5 10 15 20 25

002040608

Time (weeks)

Rela

tive e

rror

minus1

minus08

minus06

minus04

minus02


0 5 10 15 20 25

0

1

2

3

4

5

Time (weeks)

Rela

tive e

rror

minus1


0 5 10 15 20065

07

075

08

085

09

Time (weeks)

FDR


0 5 10 15 20072074076078

08082084086088

09092

Time (weeks)

FDR


0 5 10 15 20074076078

08082084086088

09092094

Time (weeks)

FDR


0 5 10 15 2007

075

08

085

09

095

Time (weeks)

FDR


0 5 10 15 20074076078

08082084086088

09092

Time (weeks)

FDR






Acronyms


Notations










119903(119905)



119894(119905)






119882lowast(119905) 119882(119905) minus 119882(0)






119864(119905) 119882

119864(119905) minus 119882

119864(0)





119877(119905) 119882

119877(119905) minus 119882

119877(0)





119871(119905) 119882

119871(119905) minus 119882

119871(0)









119894


119894



Acknowledgments


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











STGMs 119886 1198871

1198872


0 5 10 15 20Time (weeks)

Cum

ulat

ive T

GE

(faul

t tim

es)

ExponentialRayleigh

LogisticActural

0

50

150

200

250

300

350

100


0 5 10 15 20Time (weeks)

Cum

ulat

ive T

GE

(faul

t tim

es)


0

50

100

150

200

250

300

350


0 5 10 15 20 25

002040608

11214

Time (weeks)

Rela

tive e

rror

minus04

minus02


0 5 10 15 20 25

002040608

Time (weeks)

Rela

tive e

rror

minus1

minus08

minus06

minus04

minus02


0 5 10 15 20 25

010203040506070

Time (weeks)

Rela

tive e

rror

minus10







0 5 10 15 20 25

002040608

Time (weeks)

Rela

tive e

rror

minus1

minus08

minus06

minus04

minus02


0 5 10 15 20 25

0

1

2

3

4

5

Time (weeks)

Rela

tive e

rror

minus1


0 5 10 15 20065

07

075

08

085

09

Time (weeks)

FDR


0 5 10 15 20072074076078

08082084086088

09092

Time (weeks)

FDR


0 5 10 15 20074076078

08082084086088

09092094

Time (weeks)

FDR


0 5 10 15 2007

075

08

085

09

095

Time (weeks)

FDR


0 5 10 15 20074076078

08082084086088

09092

Time (weeks)

FDR






Acronyms


Notations










119903(119905)



119894(119905)






119882lowast(119905) 119882(119905) minus 119882(0)






119864(119905) 119882

119864(119905) minus 119882

119864(0)





119877(119905) 119882

119877(119905) minus 119882

119877(0)





119871(119905) 119882

119871(119905) minus 119882

119871(0)









119894


119894



Acknowledgments


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15 20 25

002040608

Time (weeks)

Rela

tive e

rror

minus1

minus08

minus06

minus04

minus02


0 5 10 15 20 25

0

1

2

3

4

5

Time (weeks)

Rela

tive e

rror

minus1


0 5 10 15 20065

07

075

08

085

09

Time (weeks)

FDR


0 5 10 15 20072074076078

08082084086088

09092

Time (weeks)

FDR


0 5 10 15 20074076078

08082084086088

09092094

Time (weeks)

FDR


0 5 10 15 2007

075

08

085

09

095

Time (weeks)

FDR


0 5 10 15 20074076078

08082084086088

09092

Time (weeks)

FDR






Acronyms


Notations










119903(119905)



119894(119905)






119882lowast(119905) 119882(119905) minus 119882(0)






119864(119905) 119882

119864(119905) minus 119882

119864(0)





119877(119905) 119882

119877(119905) minus 119882

119877(0)





119871(119905) 119882

119871(119905) minus 119882

119871(0)









119894


119894



Acknowledgments


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Acronyms


Notations










119903(119905)



119894(119905)






119882lowast(119905) 119882(119905) minus 119882(0)






119864(119905) 119882

119864(119905) minus 119882

119864(0)





119877(119905) 119882

119877(119905) minus 119882

119877(0)





119871(119905) 119882

119871(119905) minus 119882

119871(0)









119894


119894



Acknowledgments


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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