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Statistical modeling and assessment of software reliability

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Title:
Statistical modeling and assessment of software reliability
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Camara, Louis Richard
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University of South Florida
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Subjects / Keywords:
Logistic regression
Software failure data
Cumulative number of software faults
Time between failure
Bayesian linear regression
Dissertations, Academic -- Mathematics -- Doctoral -- USF
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bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
ABSTRACT: The present study is concerned with developing some statistical models to evaluate and analyze software reliability. We have developed the analytical structure of the logistic model to be used for testing and evaluating the reliability of a software package. The proposed model has been shown to be useful in the testing and debugging stages of the developmental process of a software package. It is important that prior to releasing a software package to marketing that we have achieved a target reliability with an acceptable degree of confidence. The proposed model has been evaluated and compared with several existing statistical models that are commonly used. Real software failure data was used for the comparison of the proposed logistic model with the others. The proposed model gives better results or it is equally effective. The logistic model was also used to model the mean time between failure of software packages. Real failure data was used to illustrate the usefulness of the proposed statistical procedures. Using the logistic model to characterize software failures we proceed to develop Bayesian analysis of the subject model. This modeling was based on two different difference equations whose parameters were estimated with Bayesian regressions subject to specific prior and mean square loss function.
Thesis:
Dissertation (Ph.D.)--University of South Florida, 2006.
Bibliography:
Includes bibliographical references.
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Statement of Responsibility:
by Louis Richard Camara.
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Title from PDF of title page.
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Document formatted into pages; contains 112 pages.
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Includes vita.

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aleph - 001910362
oclc - 173363632
usfldc doi - E14-SFE0001699
usfldc handle - e14.1699
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SFS0026017:00001


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Keywords:Logisticregression,SoftwareFailureData,CumulativeNumberofSoftwareFaults,TimeBetweenFailure,BayesianlinearregressioncCopyright2006,LouisRichardCamara

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ListofFigures..........................................iv Abstract.............................................v Chapter1SoftwareReliability................................1 1.1Introduction......................................1 1.2ParametersEstimationforSoftwareReliabilityusingLogisticRegression.....2 1.3AnEarlyEstimationmethodforSoftwareReliabilityAssessment..........3 1.4ReliabilityGrowthModelforSoftwareReliabilityAnalysis.............3 1.5DiscreteLogisticModelsusingBayesianProceduresforSoftwareReliability...3 1.6FutureResearch....................................4 Chapter2EstimationofparametersforSoftwareReliabilityusingLogisticRegression..6 2.1Introduction......................................6 2.1.1StatisticalAbbreviationsandNotations...................7 2.2Preliminaries.....................................8 2.2.1TheConventionalModel...........................9 2.2.2SatohandYamadaModels..........................10 2.2.3MitsuruOhba'smodels............................13 2.2.4Huang,Kuo,Chen,Lo,andLyu'smodels..................14 2.3DevelopmentofTheproposedModel........................17 2.3.1TheLogisticRegressionModel.......................17 2.3.2ModelDescription..............................18 2.3.3ParametersEstimation............................19 2.4ComparisonsofModels:NumericalApplicationtoSoftwareFailureData.....22 2.4.1PL/IsoftwareFailureData..........................22 2.4.2Tohma'sSoftwareFailureData........................26 2.4.3TheF11-Dprogramtestdata........................30 2.4.4Misra'sSpaceShuttleSoftwareFailureData................34 2.4.5Musa'sSystemT1softwareFailureData..................38 2.4.6Ohba'sOn-linedataentrysoftwaretestData................42 2.4.7Tohma'ssoftwarepackagetestData.....................46 2.5Conclusions......................................51i

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3.1Introduction......................................53 3.1.1StatisticalAbbreviationsandNotations...................54 3.2preliminaries.....................................54 3.2.1SatohandYamada'smodelsandconclusions................54 3.2.2Remarks...................................55 3.3DevelopmentofTheProposedModel:EarlyEstimation...............55 3.3.1ModelDescription..............................56 3.3.2ParametersEstimation............................56 3.4ComparisonsofModels:NumericalApplicationtoSoftwareFailureData.....59 3.5Conclusions......................................64 Chapter4ReliabilityGrowthModelForSoftwareReliability...............66 4.1Introduction......................................66 4.1.1StatisticalAbbreviationsandNotations...................67 4.2Preliminaries.....................................68 4.2.1Bayes,Empirical-BayesModel.......................68 4.2.2Suresh-RaoSRGM..............................69 4.2.3Quiao-TsokosModels............................71 4.3DevelopmentofTheproposedModel........................71 4.3.1TheLogisticModel..............................72 4.3.2ParameterEstimates.............................73 4.3.3DerivationofourproposedLRMTBFModel..............73 4.4ComparisonsofModels:NumericalApplicationstoSoftwareFailureData.....75 4.4.1Apollo8softwarefailuredata:Comparisonofthemodels.........75 4.4.2Musa'sProject14CsoftwarefailureData:Comparisonofmodels.....81 4.5Conclusion......................................87 Chapter5LogisticModelsusingBayesianproceduresforSoftwareReliability......88 5.1Introduction......................................88 5.2LogisticCurveModelandBayesianRegressionModels...............89 5.3PropertiesofModelII................................90 5.4ImplementingBayesianprocedures..........................92 5.4.1BayesianproceduresonModelI.......................92 5.4.2BayesianproceduresonModelII......................92 5.4.3DerivationoftheBayesianestimates.....................93 5.5OurBayesianprocedureonMorishitadifferenceequation..............97 5.6OurBayesianprocedureonHirotadifferenceequation...............99 5.7NumericalApplication:Summaryofthedatadependentquantities.........101 5.8Conclusion......................................103 Chapter6FutureWorkandExtensions...........................105 References............................................107 AbouttheAuthor.....................................EndPageii

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2MSEasACNOFincreases-PL/IsoftwareFailureData.................24 3Summaryofmodelsestimations:PL/IsoftwareFailureData..............25 4Tohma'sSoftwareFailureData..............................27 5MSEasACNOFincreases-Tohma'sSoftwareFailureData..............28 6Summaryofmodelsestimations:Tohma'sSoftwareFailureData............29 7TheF11-Dprogramtestdata...............................31 8MSEasACNOFincreases-TheF11-Dprogramtestdata...............31 9Summaryofmodelsestimations:TheF11-Dprogramtestdata.............32 10Misra'sSpaceShuttleSoftwareFailureData.......................35 11MSEasACNOFincreases-Misra'sSpaceShuttleSoftwareFailureData.......36 12Summaryofmodelsestimations:Misra'sSpaceShuttleSoftwareFailureData.....37 13SystemT1.........................................38 14MSEasACNOFincreases-SystemT1.........................39 15Summaryofmodelsestimations:SystemT1.......................41 16Ohba'sOn-linedataentrysoftwaretestData.......................43 17MSEasACNOFincreases-Ohba'sOn-linedataentrysoftwaretestData.......44 18Summaryofmodelsestimations:Ohba'sOn-linedataentrysoftwaretestdata.....45 19Tohma'ssoftwarepackagetestData...........................47 20MSEasACNOFincreases-Tohma'ssoftwarepackagetestData............48 21Summaryofmodelsestimations:Tohma'ssoftwarepackagetestdata..........49 22OverlappingChain1ofSizeN=2-PL/IDatabaseApplicationSoftware.......60 23OverlappingChain1ofSizeN=2-PL/IDatabaseApplicationSoftware.......61 24OverlappingChain1ofSizeN=3-PL/IDatabaseApplicationSoftware.......61 25OverlappingChain1ofSizeN=5-PL/IDatabaseApplicationSoftware.......62 26Summaryofmodelsestimations:PL/IDatabaseApplicationSoftware.........63 27Apollo8FailureData...................................76 28MSEasACNOFincreases-Apollo8data........................77 29ActualandPredictedMTBFontheApollo8data....................79 30Musa'sProject14CData.................................82 31MSEasACNOFincreases-Musa'sProject14CData..................83 32ActualandPredictedMTBFonMusa'sProject14CData................85iii

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2ComparisonOfthePL/ISoftwareFailureDataandourPredicted............26 3PlotoftheMSEasACNOFincreases-Tohma'sSoftwareFailureData.........28 4ComparisonofTohma'sSoftwareFailureDataandourPredicted............30 5PlotoftheMSEasACNOFincreases-TheF11-Dprogramtestdata.........32 6ComparisonofTheF11-DprogramtestdataandourPredicted.............33 7MSEasACNOFincreases-Misra'sSpaceShuttleSoftwareFailureData.......36 8ComparisonofMisra'sSpaceShuttleSoftwareFailureDataandourPredicted.....37 9PlotoftheMSEasACNOFincreases-SystemT1...................40 10ComparisonoftheSystemT1datasetandourPredicted................42 11PlotoftheMSEasACNOFincreases-Ohba'sOn-linedataentrysoftwaretestData.44 12ComparisonOfohba'sOn-linedataentrysoftwaretestDataandourPredicted.....45 13PlotoftheMSEasACNOFincreases-Tohma'ssoftwarepackagetestData......48 14ComparisonofTohma'ssoftwarepackagetestDatasetandourPredicted.......50 15PlotoftheMSEasACNOFincreases-Apollo8Failuredata..............78 16ComparisonOftheApollo8datasetandourPredicted.................78 17ComparisonOftheApollo8actualandPredictedMTBF................80 18PlotoftheMSEasACNOFincreases-Musa'sProject14CData............84 19ComparisonOfMusa'sProject14CDataandourPredicted...............84 20ComparisonOftheMusa'sProject14CactualandPredictedMTBF..........86iv

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Thepresentstudyisconcernedwithdevelopingsomestatisticalmodelstoevaluateandanalyzesoftwarereliability. Wehavedevelopedtheanalyticalstructureofthelogisticmodeltobeusedfortestingandevaluatingthereliabilityofasoftwarepackage.Theproposedmodelhasbeenshowntobeusefulinthetestinganddebuggingstagesofthedevelopmentalprocessofasoftwarepackage.Itisimportantthatpriortoreleasingasoftwarepackagetomarketingthatwehaveachievedatargetreliabilitywithanacceptabledegreeofcondence.Theproposedmodelhasbeenevaluatedandcomparedwithseveralexistingstatisticalmodelsthatarecommonlyused.Realsoftwarefailuredatawasusedforthecomparisonoftheproposedlogisticmodelwiththeothers.Theproposedmodelgivesbetterresultsoritisequallyeffective. Thelogisticmodelwasalsousedtomodelthemeantimebetweenfailureofsoftwarepackages.Realfailuredatawasusedtoillustratetheusefulnessoftheproposedstatisticalprocedures. UsingthelogisticmodeltocharacterizesoftwarefailuresweproceedtodevelopBayesiananal-ysisofthesubjectmodel.ThismodelingwasbasedontwodifferentdifferenceequationswhoseparameterswereestimatedwithBayesianregressionssubjecttospecicpriorandmeansquarelossfunction.v

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Becauseofthecomplexityoftheproblem,thedenitionofSoftwareReliabilityvariesfromau-thortoauthor.Someofthemostcommonlyaccepteddenitionsofsoftwarereliability,reportedin[44],arethefollowing:TheInstituteofElectricalandElectronicEngineers(IEEE)denessoftwarereliabilityas:theprobabilitythatsoftwarewillnotcauseasystemfailureforaspeciedtimeunderspeciedcondi-tions.Theprobabilityisafunctionoftheinputsto,anduseof,thesystemaswellasfunctionoftheexistenceoffaultsinthesoftware.Theinputstothesystemdeterminewhetherexistingfaults,ifany,areencountered.JohnMusaofATandTBellLaboratoriesdenessoftwarereliabilityas:theprobabilitythatagivensoftwaresystemoperatesfromsometimeperiodwithoutsoftwareerror,onthemachineforwhichitwasdesigned,giventhatitisusedwithindesignlimits.Dr.MartinShoomanoftheNewYorkPolytechnicalUniversitydenessoftwarereliabilityas:the1

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Thedenitionofsoftwarereliabilitythatwewillbeusingisthefollowing:Softwarereliabilityistheprobabilitythatagivensoftwaresysteminagivenenvironmentwilloperatecorrectlyforaspeciedperiodoftime.,[33]. Softwarereliabilityisoneofthemostimportanttopicsinthesubjectarea.Neufelder,[44],givesthefollowingfourmajorreasonswhysoftwarereliabilityhasbecomeaveryimportantissueinthelastdecade:Systemsarebecomingmoresoftwareintensivethanhardwareintensive,manysoftware-intensivesystemsaresafetycriticalormissioncriticalorfailureisextremelycostlynan-cially,customersarerequiringmorereliablesoftwares,softwarefailuresarenotbeingtoleratedbyendusersorbyclientsofendusers,andthecostofdevelopingsoftwareisincreasing. Someoftherelevantpapersinthesubjectareaare:[1;518;24;26;33;44;45],amomgothers. InthepresentstudywewilladdressvariousaspectsofthesubjectareaincludingBayesianap-proachtosoftwarereliability.GivenbelowisabriefintroductionontheproblemswestudiedineachoftheChaptersthatencom-passthisdissertation.1.2ParametersEstimationforSoftwareReliabilityusingLogisticRegression

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InChapter2,wehavedevelopedofasimple,realistic,andeasytoimplementsoftwarereliabilitygrowthmodelthatprovidesadecisionruleasofwhentostopthetestinganddebuggingphaseandreleasethesoftwareforuse,forS-shapedcumulativesoftwarefaults.WecanuseaBayesianorQuasi-BayesianprocedureinthepresentdevelopmentoftheproposedModel. UsingthemainfeatureofourproposedModelinChapter2-itsinectionpoint,inChapter3,wehaveproposedaneffectivemethodforestimatingthenumberoffaultsinthesoftware,atanearlystageofthetestinganddebuggingphase.Ourearlyestimationofthenumberoffaultinthesoftwareenablethesoftwaredeveloperstoplanthesoftwaredevelopmentprocess,managetheirresourcesbyavoidingcostduetoovertesting,makeasoftwarewithhigherreliabilityanddecidewhentoshipitforuse.WeneedtodevelopacostreductionanalysisassociatedwithourEarlyEstimationproposedModel. InChapter4,assumingalogisticmodel,wedevelopaprocedureforpredictingthemeantimebetweenfailure,afterthelastcorrection,ofasoftwarepackagecreatingaconnectionbetweenpre-dictingtheMTBFandcountingthecumulativefailuresexperienceduptoagiventime.ForthepredictionoftheMTBF,onepossibleextensionistointroduceaBayesianorQuasi-BayesianprocedureinthedevelopmentoftheproposedModel. InChapter5,wehavedeveloptheoreticalstructureandgivenparametersestimationoftwomodelusingBayesianprocedures.IllustratingtheBayesianapproachtoreliabilityisveryusefulmodelinginunderstandingthenalevaluationofasoftwarepackage.Oneofthekeydifcultiesisidentifyingandjustifyingthechoiceoftheprior.Thus,weproposetodevelopanempiricalBayesapproachtosoftwarereliabilityandthusby-passinghavingtoassumethechoiceoftheprior.Furthermore,weextendtoaddressthesameproblemfromanonparametricpointofviewbyutilizingKernelDensityestimationprocedurestocharacterizethebehavioroftheprior.Wealsobelievethatformulatingnonparametricsoftwarereliabilitymodelsusingthekerneldensityapproachtocharacterizesoftware4

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Asoftwarefaultisdenedasanunacceptabledepartureofprogramoperationcausedbyasoft-warefaultremaininginthesystem,[1].Softwarereliabilityistheprobabilitythatasoftwarefaultwhichcausesdeviationfromtherequiredoutputbymorethanthespeciedtolerances,inaspeciedenvironment,doesnotoccurduringaspeciedexposureperiod. Themainassumptionthatwemakehereisthatonceasoftwarefaultisfounditiscorrectedforgoodanditscorrectiondoesnotintroduceanynewsoftwarefaults.Asaresult,thereliabilityofthesoftwareincreases,andwerefertosuchamodelasthereliabilitygrowthmodel.Finally,whenassessingthereliabilityofasoftware,duringthedebuggingandtestingphase,anyfailuresotherthensoftwarefaultscouldpreventitseffectiveness. Theobjectiveofthischapteristodeveloparealisticsoftwarereliabilitygrowthmodelthatusesthemeansquareerrorasacriteriatoprovidingadecision-makingruleastowhentostopthetestinganddebuggingphaseandreleasethesoftwareorwhentocontinuewiththedebuggingprocess.Theproposedmodelwillallowustoestimatetheproportionofthesoftwaremistakesareleftinthesystem.Thatis,weclaimthatalmostallthesoftwarefaultshavebeenfoundandcorrectedbefore6

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kL(t)(kL(t));L(0)=K

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kL(t)(kL(t))canbewrittenas1 kL(t)(2.1) Let8>>>><>>>>:tn=nLn=L(n)Yn=Ln+1Ln1 kLn+ k

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kLn+1(kLn)(2.2)andisreferredtoasMorishita'sdifferenceequation[1].Thesolutionoftheabovedifferenceequa-tionisgivenbyLn=k tn=n kLn+1(2.3) kLn+1(2.4) kLn+1(2.5)10

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k(1)Ln+1+1 1(2.6) andyn=BLn+1+A k(1)A=1 1:(2.7) UsingtheLeastsquareregression,theestimates^Aand^Bareobtainedasfollows: ^A^m=PNn=1(^kLn) kLn(kLn+1) 1+)tn 11

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kLn+1+(+1)andyn=BLn+1+A KA=+1: 1+^dh)nbc=1

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(1+Aexp[t])2(2.9)14

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(2.10) ThisSRGMmodelisbasedonthefollowingassumtions:(i)ThefaultremovalprocessfollowstheNon-HomogeneousPoissonProcess(NHPP)(ii)Thesoftwaresystemissubjecttofailuresatrandomtimescausedbyfaultsremaininginthesystem.(iii)Themeannumberoffaultsdetectedinthetimeinterval(t,t+dt]bythecurrenttest-effortisproportionaltothemeannumberofremainingfaultsinthesystem.(iV)Theproportionalityisconstantovertime.(V)Theconsumptioncurveoftestingeffortismodeledbyalogistictesting-effortfunction.(Vi)Eachtimeafailureoccurs,thefaultthatcauseditisimmediatelyremoved,andnonewfaultsareintroduced.Thus,thedifferentialequationisgivenbydm(t) describesanalyticallythetesting-based-effort.Thesolutionoftheabovedifferentialequation,undertheboundaryconditionm(0)=0ism(t)=a(1er(W(t)W(0)))(2.12)wherem(t)istheexpectednumberoffaultsdetectedontheinterval(0,t],w(t)thecurrenttesting-effortconsumptionattimet,atheexpectednumberofinitialfaults,andrtheerrordetectionrateperunittesting-effortattestingtimetthatsatisesr>0.15

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kL(t)(kL(t));t0(2.18)where(>0)andk(k>0)areconstantparameterstobeestimatedbyregressionanalysis[1].LetP(t)=L(t) 1+m;mk

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1+met(2.19)wheremistheconstantofintegration.NotethatthegraphofP(t)isS-shapedand0P(t)1.Asoftwarefaultfoundandcorrectedateachinstantisabinaryoutcomevariable.Weassumethatthebinomialdistributiondescribesthedistributionoftheerrorsandwillbethestatisticaldistributionuponwhichtheanalysisisbased,[4].Fewersoftwarefaultsarefoundearlyinthetestingandlongintothetestinganddebuggingphasewhenmostofthefaultsarefoundandcorrectedandonlyafewfaultsareleftinthesoftware.ForsuchcasesweproposeaLogisticLikeModeltoestimatetheparametersk,mandofP(t).2.3.3ParametersEstimation 1+met

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1+met 1+met

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Havinganestimateofk,bk=im,wesetupourbesttlogisticlikeregressionmodelintheminimalmeansquareerrorsense.proposingdP(t)=e^0im+^1imt Byidenticationmethodweobtain1 1+mete^0im+^1imt 1+1 1+e^0ime^1imt(2.20) NowcomparingtherstandthelasttermofEquation2.20,weobtainthefollowingestimatesofmand:8<:bm=e^0imb=^1im 1+bmebt=1 1+e^0ime^1imt21

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FinallythecumulativefailurebehavioroftheproposedmodelforagivensoftwareisgivenbydL(t)=^k

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Timeofobservation(week) Cumulativeexecutiontime Cumulativenumberoffailures 1 2.45 15 2 4.9 44 3 6.86 66 4 7.84 103 5 9.52 105 6 12.89 110 7 17.1 146 8 20.47 175 9 21.43 179 10 23.35 206 11 26.23 233 12 27.67 255 13 30.93 276 14 34.77 298 15 38.61 304 16 40.91 311 17 42.67 320 18 44.66 325 19 47.65 328 FromTable2,onpage24,wenotethatthemeansquareerrorisminimalwhentheassumedcumu-lativenumberoffailuresis348.Thus,ourestimatebk=348andweestimate20remainingfaultsinthesoftwareafterthelastcorrection.23

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ACNOF MSE ACNOF MSE ACNOF MSE 328 140.8832368 343 93.92986795 358 99.52222464 329 135.0976346 344 93.13746872 359 100.9549485 330 129.7745802 345 92.54189896 360 102.3613104 331 124.8980566 346 92.15235569 361 104.0786848 332 120.4513938 347 91.94171328 362 105.5920188 333 116.4264294 91.92380892 107.4049262 334 112.8021575 349 92.04011389 364 109.167073 335 109.3976453 350 92.31760154 365 111.255242 336 106.5299609 351 92.77149211 366 113.0563834 337 103.8659894 352 93.40226984 367 115.1903413 338 101.5493506 353 94.12959495 368 117.2057907 339 99.44696198 354 94.92442684 369 119.3523414 340 97.75058646 355 95.97719787 370 121.6077276 341 96.24790601 356 96.98046447 342 94.95536474 357 98.23939824

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Models aork MSE AE(percent) Ourmodel 348 91.92380892 2.79 ExistingSRGMs HLMModelGroupA,withLogisticfunction 394.076 118.29 10.06 HLMModelGroupA,withWeibullfunction 565.35 122.09 57.91 HLMModelGroupA,withRayleighfunction 459.08 268.42 28.23 HLMModelGroupA,withExponential 828.252 140.66 131.35 HLMModelGroupB,withLogisticfunction 337.41 163.095 5.75 HLMModelGroupB,withWeibullfunction 345.686 91.0226 3.43 HLMModelGroupB,withRayleighfunction 371.438 158.918 3.75 HLMModelGroupB,withExponential 352.521 83.998 1.53 HLMModelGroupC,withLogisticfunction 430.662 103.03 20.11 HLMModelGroupC,withWeibullfunction 385.39 87.5831 7.65 HLMModelGroupC,withRayleighfunction 379.947 406.71 6.13 HLMModelGroupC,withExponential 385.179 83.3452 7.69 HLMModelGroupD,withLogisticfunction 582.538 96.9321 62.72 HLMModelGroupD,withWeibullfunction 958.718 124.399 167.79 HLMModelGroupD,withRayleighfunction 702.693 247.84 96.09 HLMModelGroupD,withExponential 1225.66 169.72 242.36 G-OModel 562.8 157.75 56.98 InectionS-ShapedModel 389.1 133.53 8.69 DelayedS-ShapedModel 374.05 168.67 4.48 ExponentialModel 455.371 206.93 27.09 HGDM 387.71 138.12 8.3 LogarithmicPoissonModel NA 171.23

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Figure2showsthatourLR-Model'sestimatedcumulativenumberofsoftwarefaultsfoundandcorrectedtsverywelltheactualcumulativesoftwarefailuredata.2.4.2Tohma'sSoftwareFailureData

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Days Times(hour) CumulativeTime Faults CumulativeFaults 1 1 1 1 1 2 2 3 5 6 3 2 5 8 14 4 3 8 5 19 5 4 12 3 22 6 4 16 8 30 7 2 18 4 34 8 4 22 3 37 9 5 27 9 46 10 7 34 8 54 11 7 41 11 65 12 4 45 3 68 13 2 47 0 68 14 3 50 0 68 15 17 67 8 76 16 3 70 3 79 17 5 75 1 80 18 2 77 0 80 19 4 81 3 83 20 4 85 1 84 21 4 89 0 84 22 4 93 2 86 27

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ACNOF MSE 7.402113064 8.094494788 88 9.158227746 89 10.48334332 90 11.97355841 91 13.64098606 92 15.38582403 93 17.16645175 94 19.04502719 95 20.92112638 InTable5weobservethatthemeansquareerrorisminimalwhenassumedcumulativenumberoffailuresis86,whichcoincideherewiththetotalthenumberoffaultsdetectedduringdebugging.Thus,ourestimateisbkis86andweestimate0remainingfaultsinthesoftwareafterthelastcorrection.28

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Models aork MSE Ourmodel 86 7.40211306 ExistingSRGMs HLMModelGroupA,withLogisticfunction 88.8931 25.2279 HLMModelGroupA,withWeibullfunction 87.0318 7.772 HLMModelGroupA,withRayleighfunction 86.1616 3.91643 HLMModelGroupB,withLogisticfunction 89.4528 14.06603 HLMModelGroupB,withWeibullfunction 87.3126 18.956772 HLMModelGroupB,withRayleighfunction 87.3472 20.4568 HLMModelGroupC,withLogisticfunction 97.5332 7.354363 HLMModelGroupC,withWeibullfunction 97.6841 6.5909 HLMModelGroupC,withRayleighfunction 112.182 6.60318 HLMModelGroupD,withLogisticfunction 106.1 7.33727 HLMModelGroupD,withWeibullfunction 114.52 6.36531 HLMModelGroupD,withRayleighfunction 112.183 6.60318 G-OModel 137.072 25.33 DelayedS-ShapedModel 88.6533 6.31268 HGDM 88.6533 6.31268 S-YModels 73.11837775 196.7452124

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Figure4showsthatourLR-Model'spredictivecumulativenumberofsoftwarefaultsfoundandcorrectedtsverywelltheactualcumulativesoftwarefailuredata.Thus,ourproposedmodelwhichiseasiertouseandfreeofanymajorassumptiongivesverygoodifnotbetterresultsthenothermodelsusedinindustry.2.4.3TheF11-Dprogramtestdata

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I.N. Date N.E.D.I.N C.N.E,N M.L.EofN 1 1/12 8 8 9 2 1/15 7 15 16 3 1/16 1 16 17 4 1/17 8 24 24 5 1/18 16 40 43 6 1/19 18 58 60 7 1/22 13 71 73 8 1/23 8 79 81 9 1/24 9 88 90 10 1/25 2 90 92 11 1/26 6 96 99 12 1/27 3 99 100 13 1/29 3 102 102 14 1/30 2 104 104 15 1/31 3 107 107 ACNOF MSE 9.009282277 10.74342705 109 12.77994731 110 14.98197705 111 17.37213712 31

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Table8,onpage31,showsthatthemeansquareerrorisminimalwhenassumedcumulativenumberoffailuresisto107whichagreeswiththetotalthenumberoffaultsdetectedduringactualdebugging.Thus,ourestimatebk=107andweestimate0remainingfaultsinthesoftwareafterthelastmodication.Table9:Summaryofmodelsestimations:TheF11-Dprogramtestdata Models korN MSE Ourmodel 107 9.009282277 ExistingSRGMs F-SModel 107 2.8 S-YModels 107.4860452 8.955939003 Table9providesacomparativesummaryofourestimatebkalongwithafewotherestimatesfromsomepre-existingmodelsandthemeansquareerrorforeachmodel.Table9alsoshowsthatbypredicting107faultsassociatedwithameansquareerrorof9:009282277ourproposedmodel,LR-Model,ttingwelltroughtheK-Sgoodness-of-t(D=0:0667;P=1:00),demonstratesaverygoodt.OurLR-Model,althoughdoesnotdemonstratesbetterperformancethanmostofthepre-existingmodelsinthemeansquareerrorsense,conrmsFormanandSingpurwalla'sempiricalstoppingrule32

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Notethat,forthisactualdataset,Table11,onPage36,showsthatthemeansquareerrorisminimalwhenassumedcumulativenumberoffailuresis237.Thus,ourestimatebk=237andweestimate6remainingfaultsinthesoftwareafterthelastcorrection. Table12,onPage37,providesacomparativesummaryofourestimatebkalongwithafewotherestimatesfromsomepre-existingmodelsandthemeansquareerrorforeachmodel.Table12alsoshowsthatbypredicting237faultsassociatedwithameansquareerrorof38:38456266ourproposedmodel,LR-Model,ttingwelltroughtheK-Sgoodness-of-t(D=0:0526;P=1:00),demonstratesbetterperformancethanmostofthepre-existingmodels. Figure8,onPage37,showsthatourLR-Model'spredictivecumulativenumberofsoftwarefaultsfoundandcorrectedtsverywelltheactualcumulativesoftwarefailuredata.Thus,ourproposedmodelwhichiseasiertouseandfreeofanymajorassumptiongivesverygoodifnotbetterresultsthenothermodelsusedinindustry.34

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Week Cr.E Ma.E Mi.E C.E Week Cr.E Ma.E Mi.E C.E 1 0 6 9 15 20 0 2 3 136 2 0 2 4 21 21 0 1 1 138 3 0 1 7 29 22 0 3 2 143 4 1 1 6 37 23 0 2 4 149 5 0 3 5 45 24 0 4 5 158 6 0 1 3 49 25 0 1 0 159 7 0 2 2 53 26 0 2 2 163 8 0 3 5 61 27 0 2 0 165 9 0 2 4 67 28 0 2 2 169 10 0 0 2 69 29 0 1 3 173 11 0 3 4 76 30 1 2 6 182 12 0 1 7 84 31 1 2 3 188 13 0 3 0 87 32 0 0 1 189 14 0 0 5 92 33 0 2 1 192 15 0 2 3 97 34 0 2 4 198 16 0 5 3 105 35 0 3 3 204 17 0 5 3 113 36 0 1 2 207 18 0 2 4 119 37 1 2 11 221 19 0 2 10 131 38 0 1 9 231

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ACNOF MSE 231 39.24889498 232 38.96808896 233 38.73239432 234 38.55970054 235 38.4544495 236 38.39724536 38.38456266 38.4183368 239 38.49035903 240 38.6077261 241 38.75830898 242 38.94298088 243 39.16746369

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Models aorK MSE Ourmodel 237 38.38456266 G-OModel 597.887 78.87 S-YModels 200.8486903 265.3335203

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Intervalofobservation(intervallength=5Days) Cumulativenumberoffailures 1 3 2 4 3 6 4 16 5 16 6 22 7 29 8 29 9 31 10 42 11 48 12 63 13 78 14 92 15 105 16 122 17 132 18 135 19 136 38

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ACNOF MSE ACNOF MSE 136 59.73849038 164 20.74581198 137 56.45913482 165 20.43461528 138 53.51272497 166 20.16955946 139 50.74510692 167 19.93233847 140 48.05754858 168 19.73619002 141 45.68399539 169 19.56559153 142 43.41103347 170 19.42895933 143 41.32847537 171 19.3171211 144 39.34891981 172 19.2292499 145 37.56874349 173 19.16859798 146 35.97196455 174 19.1305098 147 34.38348791 19.11480503 32.98043088 176 19.11803102 149 31.58854155 177 19.14235957 150 30.37887505 178 19.18716939 151 29.25654819 179 19.24351235 152 28.22982752 180 19.31848929 153 27.28001268 181 19.41178997 154 26.36265956 182 19.52408042 155 25.52487927 183 19.63968954 156 24.77120468 184 19.77192253 157 24.09448225 185 19.91998986 158 23.44652547 186 20.08486024 159 22.90736243 187 20.24602725 160 22.35927548 188 20.4212037 161 21.90723966 189 20.61078576 162 21.45572338 190 20.81366771 163 21.0866153 39

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Table14,onpage39,showsthattheMeanSquareErrorisminimalwhenassumedcumulativenumberoffailuresisto175.Thus,ourestimatebk=175andweestimate39remainingfaultsinthesoftwareafterthelastcorrect. Table15,onpage41,providesacomparativesummaryofourestimatebkalongwithafewotherestimatesfromsomepre-existingmodelsandthemeansquareerrorforeachmodel.Table15alsoshowsthatbypredicting175faultsassociatedwithameansquareerrorof19:11480503ourproposedmodel,LR-Model,ttingwelltroughtheK-Sgoodness-of-t(D=0:1053;P=1:00),demonstratesbetterperformancethanmostofthepre-existingmodels.40

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Models aorK MSE Ourmodel 175 19.11480503 ExistingSRGMs HLMModelGroupA,withLogisticfunction 138.026 62.41 HLMModelGroupA,withRayleighfunction 866.94 89.24095 HLMModelGroupB,withLogisticfunction 137.759 14.6442 HLMModelGroupB,withRayleighfunction 150.047 12.137 HLMModelGroupB,withExponentialfunction 187.537 19.73719 HLMModelGroupC,withLogisticfunction 142.567 13.4266 HLMModelGroupC,withRayleighfunction 156.715 10.9726 HLMModelGroupC,withExponentialfunction 173.064 48.5971 HLMModelGroupD,withLogisticfunction 164.106 38.121 HLMModelGroupD,withRayleighfunction 1543.47 89.7666 ExponentialModel 137.2 3019.66 G-OModel 142.32 2438.3 DelayedS-ShapedModel 237.196 245.246 S-YModels 145.2562193 120.5350962 41

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Timeofobservation(day) Cumulativenumberoffaults 1 2 2 3 3 4 4 5 6 9 7 11 8 12 9 19 10 21 11 22 12 24 13 26 14 30 15 31 16 37 17 38 18 41 19 42 20 45 21 46 43

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ACNOF MSE 46 2.877909807 47 2.439920792 48 2.139457102 49 1.945403281 50 1.832603369 1.784785998 1.786831674 53 1.828631846 54 1.900772326 55 1.998562288 56 2.115459903 57 2.24590523 58 2.387993439 59 2.541360386 60 2.699959644

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Table18providesacomparativesummaryofourestimatebkalongwithafewotherestimatesfromsomepre-existingmodelsandthemeansquareerrorforeachmodel.Table18alsoshowsthatbypredicting51faultsassociatedwithameansquareerrorof1:784785998ourproposedmodel,LR-Model,ttingtheactualdataverywelltroughtheK-Sgoodness-of-t(D=0:1000;P=1:00).Table18:Summaryofmodelsestimations:Ohba'sOn-linedataentrysoftwaretestdata Models aorK MSE Ourmodel 51 1.784785998 DelayedS-ShapedModel[10] 71.7 1.694955709 TheFigure12,onpage45,showsthatourLR-Model'spredictivecumulativenumberofsoftwarefaultsfoundandcorrectedtsverywelltheactualcumulativesoftwarefailuredata.Thus,our45

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FromTable20,onpage48,wenotethatforthisrealdataset,themeansquareerrorisminimalwhenassumedcumulativenumberoffailuresisto481.Thus,ourestimatebk=481andweestimate0remainingfaultsinthissoftwarepackageafterthelastcorrection. Table21,onpage49,providesacomparativesummaryofourestimatebkalongwithseveralotherestimatesfromtheHypergeometricmodel(HGDM)andthemeansquareerrorforeachmodel.413:9032446.Althoughourproposedmodelisassociatedwitharelativelylargemeansquareerrorof413:9032446,respectivetotheHypergeometricmodels,ittstheactualdataverywelltroughtheK-Sgoodness-of-t(D=0:1712;P=0:069). TheFigure14,onpage50,showsthatourproposedmodel'spredictivecumulativenumberofsoftwarefaultsfoundandcorrectedtsverywelltheactualcumulativesoftwarefailuredata.46

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Days Cum.F Days Cum.F Days Cum.F Days Cum.F Days Cum.F 1 5 1 234 1 417 67 465 90 475 2 10 2 236 2 425 68 466 91 475 3 15 3 240 3 430 69 467 92 475 4 20 4 243 4 431 70 467 93 475 5 26 5 252 5 433 71 467 94 475 6 34 6 254 6 435 72 468 95 475 7 36 7 259 7 437 73 469 96 476 8 43 31 263 8 444 74 469 97 476 9 47 32 264 9 446 75 469 98 476 10 49 33 268 10 446 76 469 99 476 11 80 34 271 11 448 77 470 100 477 12 84 35 277 12 451 78 472 101 477 13 108 36 290 13 453 79 472 102 477 14 157 37 309 14 460 80 473 103 478 15 171 38 324 61 463 81 473 104 478 16 183 39 331 62 463 82 473 105 478 17 191 40 346 63 464 83 473 106 479 18 200 41 367 64 464 84 473 107 479 19 204 42 375 65 465 85 473 108 479 20 211 43 381 66 465 86 473 109 480 21 217 44 401 67 465 87 475 110 480 22 226 45 411 68 466 88 475 111 481 23 230 46 414 69 467 89 475

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ACNOF MSE 413.9032446 420.9482805 483 430.0166841 484 441.7300299 485 457.7926775 486 473.4347958 487 492.8117519 488 513.0814357 489 533.5721618 490 557.2539898

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Models aorK MSE Ourmodel 481 413.9032446 HGDM(a)[17] 479 274.3 HGDM(b)[17] 497 242.4 HGDM(c)[17] 479 273.5 HGDM(d)[17] 497 242.5 HGDM(e)[17] 531 298.3 HGDM(f)[17] 476 566.3 HGDM(g)[17] 546 977.1 HGDM(h)[17] 497 364.3 HGDM(i)[17] 496 353.6 HGDM(j)[17] 498 371.8 HGDM(k)[17] 482 253.2 HGDM(l)[17] 486 328.7 S-YModels 444.1965400 3843.008443 49

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Havinganestimateofk,wenameitbk=iearly,wesetupourbesttlogisticlikeregressionmodelintheminimalmeansquareerrorsense.proposingdP(t)=e^0iearly+^1iearlyt Byidenticationmethodweobtain1 1+mete^0iearly+^1iearlyt 1+1 1+e^0iearlye^1iearlyt(3.1) NowcomparingtherstandthelasttermofEquation3.1,weobtainthefollowingestimatesofmand:8<:bm=e^0iearlyb=^1iearly 1+bmebt=1 1+e^0iearlye^1iearlyt58

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Itwillbeshownthatourproposedmodelyieldsaccurateparametersestimatesearlyintothetestinganddebuggingphase,givesaverygoodmodelingoftheS-shapedcumulativenumberofsoftwarefaultsfoundandcorrectedcurvesduringtestinganddebuggingphasegivingthesoftwaredevelopersanearlyprotocolasfaraswhentostopthetestinganddebuggingphaseandreleasethesoftwareforuse,cuttingcostonmaintenance,andduetoovertesting.OurEarlyEstimationonthePL/ISoftwareFailureData Table26,onpage63,providesacomparativesummaryofourestimatebKalongwithseveralotherestimatesfromsomepre-existingmodelsandthemeansquareerrorforeachmodel.Table26alsoshowsthatbypredicting350faultsassociatedwithanoverallmeansquareerrorof92:31760154ourproposedmodel,LR-Model,withoutanymajorassumptions,ttingwelltroughtheK-Sgoodness-of-t,demonstratesbetterperformancethanmostofthepre-existingmodelswhendatapointsabitaftertheinectionpointsareavailable.59

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RUNS [Ti,T(i+1)][Ni,N(i+1)] slopeofRL r RUN1 [1,3];[15,66] 25.5 1 RUN2 [2,4];[44,103] 29.5 1 RUN3 [3,5];[66,105] 19.5 1 RUN4 [4,6];[103,110] 3.5 1 RUN5 [5,7];[105,146] 20.5 1 RUN6 [6,8];[110,175] 32.5 1 RUN7 [7,9];[146,179] 16.5 1 Notethatonlythedatapointsuptothe9thweekareusetopredictthetotalnumberoffaultsinthesoftware.60

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N MSE Pairnumber N=88 6.61E-09 2 N=132 15.30759212 3 N=206 16.17930627 4 N=220 179.7683445 6 N=292 179.2839406 7 165.1157652 8 165.259913 N=358 183.1532424 9 N=412 189.5550172 10 N=466 189.8825614 11 RUNS [Ti,T(i+1)][Ni,N(i+1)] slopeofRL r RUN1 [1,2,3];[15,44,66] 25.5 r:=.9968748929 RUN2 [2,4,5];[44,103,105] 21.64285714 r:=.9539628928 RUN3 [4,6,7];[103,110,146] 12.78571429 r:=.8464894644 RUN4 [6,8,9];[110,175,179] 24.35714286 r:=.9605519477 RUN5 [8,10,11];[175,206,233] 18.78571429 r:=.9887218045 Notethatonlythedatapointsuptothe11thweekareusetopredictthetotalnumberoffaultsinthesoftware.61

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RUNS [Ti,T(i+1)][Ni,N(i+1)] slopeofRL r RUN1 [1,2,3,4,5];[15,44,66,103,105] 23.9 r:=.9778998350 RUN2 [4,6,7,8,9];[103,110,146,175,179] 17.33783784 r:=.9416620919 RUN3 [8,10,11,12,13];[175,206,233,255,276] 20.60810811 r:=.9952196625 Notethatonlythedatapointsuptothe13thweekareusetopredictthetotalnumberoffaultsinthesoftware.62

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Models aorK MSE AE Ourearlyestimationmodel 350 92.31760154 2.23 OurLR-Model 348 91.92380892 2.79 HLMModelGroupA,withLogisticfunction 394.076 118.29 10.06 HLMModelGroupA,withWeibullfunction 565.35 122.09 57.91 HLMModelGroupA,withRayleighfunction 459.08 268.42 28.23 HLMModelGroupA,withExponential 828.252 140.66 131.35 HLMModelGroupB,withLogisticfunction 337.41 163.095 5.75 HLMModelGroupB,withWeibullfunction 345.686 91.0226 3.43 HLMModelGroupB,withRayleighfunction 371.438 158.918 3.75 HLMModelGroupB,withExponential 352.521 83.998 1.53 HLMModelGroupC,withLogisticfunction 430.662 103.03 20.11 HLMModelGroupC,withWeibullfunction 385.39 87.5831 7.65 HLMModelGroupC,withRayleighfunction 379.947 406.71 6.13 HLMModelGroupC,withExponential 385.179 83.3452 7.69 HLMModelGroupD,withLogisticfunction 582.538 96.9321 62.72 HLMModelGroupD,withWeibullfunction 958.718 124.399 167.79 HLMModelGroupD,withRayleighfunction 702.693 247.84 96.09 HLMModelGroupD,withExponential 1225.66 169.72 242.36 G-OModel 562.8 157.75 56.98 InectionS-ShapedModel 389.1 133.53 8.69 DelayedS-ShapedModel 374.05 168.67 4.48 ExponentialModel 455.371 206.93 27.09 HGDM 387.71 138.12 8.3 LogarithmicPoissonModel NA 171.23 63

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InthisChapterweshallfocusonpredictingthetimeintervalbetweenfailuresforagivensoftwarepackage.TheMeanTimeBetweenFailure(MTBF)isthetimedifferencebetweentheexpectednextfailuretimeandcurrentfailuretime.Whenworkingwithsoftwarespackagesduringtheirdevelopmentprocess,themeantimebetweenfailure(MTBF)isaveryimportantconceptwhenassessingthereliabilityofagivenpackageaftereachmodication.Asaresult,anaccurateesti-mationoftheMTBFtopredictthefailuretimesofagivensystemiscrucialwhenitcomestoplanningcorrectivestrategies.Assoftwarefaultsarefoundandcorrectedduringthetestinganddebuggingphase,withtheas-sumptionsdiscussedinChapter2,thereliabilityofthegivensoftwarepackageincreasesmeaningthatthetimebetweenfailureofthesoftwareisexpectedtoberelativelyincreasinginthelongrun.Thus,thelargerthetimebetweenfailure(TBF)themorereliablethesoftwarebecomes.Mostoftheexistingmodelsassumethatthetimebetweenfailuresareexponentiallydistributedwhichmakethemindependentoftime,[47].Whenadependencyontimeisexhibited,authorsproposedthe66

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InSection4:2wepresentanoverviewofafewmodelsthatarecommonlyused:Horigome-Singpurwallamodel,Mazucchi-Soyermodel,Suresh-Raomodel[47],andQuiao-Tsokosmodel,[45].InSection4:3,wedevelopourlogisticmodeltocharacterizebehaviorofthefailuresgrowth.ComparisonofourproposedmodelthatwewillrefertoasLR-MTBF-ModelwiththecommonlyusedmodelsismadeusingdatafromtworealworldproblemsinSection4:4.Finally,ourconclu-sionsandrecommendationsaregiveninSection4:5.4.1.1StatisticalAbbreviationsandNotations

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withfailureratesdependingonthestageoftesting.2.Giventhefailureratesateachstageoftesting,thelife-lengthsofthesoftwareateachstagearestatisticallyindependent.3.Thefailureratesateachstageoftestingarerandomvariables.Thepdfofiisgivenby:g(ij;)= 4.Giventheparametersand,thefailureratesateachstageoftestingarestatisticallyinde-pendent.5.GiventhebackgroundinformationH,uncertaintyofandisexpressedviatheprior68

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and(jH)=ba whereu>0,a>0,andb>0areknownquantities.6.Givenbackground,theinformationH,andarestatisticallyindependent.7.Given,,and(1;2;:::;n),theYi'sarestatisticallyindependentwitheachYistatisticallyindependentof,,andall'sotherthani. Computationoftheposteriorfailuresratesleadstointegralswhichcannotbeexpressedinclosedform,Lindley'sapproximation,[47]isusedtoapproximatetheintegrals.MazucchiandSoyer'smodelwasreportedtobeanimprovementoverthelittlewood/Verrallmodel,afterapplyingthismodeltosomeactualsoftwarefailuredatarstreportedin[47].4.2.2Suresh-RaoSRGM Startinginthetestingphaseofthedevelopmentofasoftware,thefailureintensityisgivenby:(t)=t1;t>0;>0;>0(4.5) LettingYi=TiTi1;i=1;2;:::;bethetimesbetweensuccessivefailuresofthesoftware.ThetimetorstfailureistheWeibulldistributionwithfailurerateasabove[47].Thenfor<1,thesoftwareisimprovingwithtime.TheconditionaldistributionofTkgivenTk1,ifntk1;tkisthe69

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T^n(4.7) Theunbiasedestimateofisderivedtobe: Tn

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Theveryimportantofourproposedmodelallowssoftwaredeveloperstoconnecttimeintervalbetweenfailuresandcumulativefailuresexperienceduptoagiventestinganddebuggingtime,givingthemmoredetailsaboutthesoftwarefailuredataunderstudy,topredictwhenasoftwareisreadytobereleasedforuse.71

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kL(t)(kL(t));t0(4.8)where(>0)andk(k>0)areconstantparameterstobeestimatedbyregressionanalysis,[1].LetP(t)=L(t) 1+m;mk 1+met(4.9)wheremistheconstantofintegration.NotethatthegraphofP(t)isS-shapedand0P(t)1.AsdevelopedinChapter2,weproposeaLogisticModeltoestimatetheparametersk,mandofP(t).72

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Havingtheestimateofk,bk=im,weproposedP(t)=e^0im+^1imt 1+bmebt=1 1+e^0ime^1imtFinallythecumulativefailurebehavioroftheproposedmodelforagivensoftwarewasderivetobe:dL(t)=^k anddividingbothsidesbyk,weobtainthefollowingapproximation:L(t+h) whichisequivalenttoP(t+h)P(t)+P0(t)h(4.12)73

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Ontheotherhand,theprobabilitythatonesoftwarefaultisfoundbetweentandt+hisgivenby:P(t+h)P(t)=1 Combiningtheaboveequationsgives1 fromwhichwederivethefollowingestimates:1 ^k=\P(t+h)dP(t)[P0(t)h(4.16) andobtainh1 ^k[P0(t)(4.17) NotethatsincedP(t)=e^0im+^1imt ^k\P0(Tn)=1 ^k\P(Tn)0=1+e^0im+^1imTn2 NotethatourMTBTpredictiondependsonourestimate^kofthetotalnumberoffaultsintheSoftware.Whenattheendofatestingdebuggingphaseweestimatethatallthekfaultsinthesoftwarewerefoundandcorrected,theMTBFgetsignicantlylarge;inthiscasestoptheMTBFpredictionatLRMTBF(Tk2).Itwillbeshownthatourproposedestimateofthemeantimesbetweenfailuregivesgoodresultsincomparisontotheothermodelsthatarecommonlyused.74

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FailureNumber ActualTBF ActualFailureTime 1 9 9 2 12 21 3 11 32 4 4 36 5 7 43 6 2 45 7 5 50 8 8 58 9 5 63 10 7 70 11 1 71 12 6 77 13 1 78 14 9 87 15 4 91 16 1 92 17 3 95 18 3 98 19 6 104 20 1 105 21 11 116 22 7 123 23 2 125 24 1 126 76

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ACNOF MeanSquareError 24 0.638528862 25 0.444854629 26 0.336156119 27 0.283498583 0.269812501 0.28087377 30 0.308914428 31 0.345979846 32 0.391210652 33 0.441151908 34 0.492927989 35 0.548889331 36 0.601757717 37 0.656688524 38 0.710669697 39 0.761844255 40 0.812992029 41 0.862548918 42 0.91197982 77

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FailureNumber ActualTBF BEBE SRGM LRCRL 1 9 ..... ..... ..... 2 12 10.53 ..... 17.94 3 11 11.84 8.89 12.15 4 4 11.79 9.01 8.75 5 7 9.64 6.12 7.83 6 2 9.15 5.92 6.53 7 5 7.85 4.47 6.23 8 8 7.44 4.3 5.57 9 5 7.55 4.81 4.77 10 7 7.27 4.64 4.41 11 1 7.27 4.87 4.06 12 6 6.66 4.13 4.02 13 1 6.62 4.26 3.85 14 9 6.16 3.72 3.84 15 4 6.39 4.24 3.8 16 1 6.23 4.11 3.86 17 3 5.89 3.7 3.88 18 3 5.71 3.55 3.96 19 6 5.56 3.43 4.06 20 1 5.59 3.58 4.36 21 11 5.35 3.3 4.42 22 7 6.84 3.85 5.35 23 2 9.73 4.04 6.24 24 1 9.39 3.84 6.55 MeanSquareError 19.486 11.41 MeanSquareError 20.27 10.33 10.32 79

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F.N. F.I.L. Cum.F.T. F.N. F.I.L. Cum.F.T. 1 191520 191520 19 228315 5631060 2 2078820 2270340 20 51480 5682540 3 514560 2784900 21 44820 5727360 4 1140 2786040 22 850080 6577440 5 3120 2789160 23 361860 6939300 6 327480 3116640 24 39300 6978600 7 15420 3132060 25 545280 7523880 8 60000 3192060 26 256980 7780860 9 140160 3332220 27 396780 8177640 10 937620 4269840 28 91260 8268900 11 72240 4342080 29 1225620 9494520 12 737700 5079780 30 120 9494640 13 250680 5330460 31 1563300 11057940 14 2965 5333425 32 513000 11570940 15 196 5333621 33 177660 11748600 16 65173 5398794 34 2469000 14217600 17 2370 5401164 35 1678260 15895860 18 1581 5402745 36 170760 16066620

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ACNOF MeanSquareError 3.094882808 3.959599606 38 4.928868157 39 5.931329159 40 6.942521785 41 7.92560927 42 8.878038857 43 9.791517534 44 10.66027927 45 11.47428189 46 12.25633147 83

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F.N. TBF S-RSRGM LR-MTBF F.N. TBF S-RSRGM LR-MTBF 1 191520 19 228315 237626.3298 235431.213 2 2078820 20 51480 233864.8431 232246.5118 3 514560 1332150014 478061.3697 21 44820 214988.3137 231726.4976 4 1140 1.61403E+11 402889.9732 22 850080 198190.1049 231332.4906 5 3120 1.597E+11 402744.7631 23 361860 248290.2288 234115.9895 6 327480 5605186548 402347.7938 24 39300 255329.8335 241329.7101 7 15420 1.21607E+11 364141.6983 25 545280 237020.0068 242340.8523 8 60000 91508318446 362503.4189 26 256980 257738.5758 261292.4014 9 140160 46698576008 356258.5331 27 396780 256089.2429 273653.5196 10 937620 3.54226E+11 342450.7742 28 91260 264275.4747 297622.1508 11 72240 40820287309 274280.3111 29 1225620 251414.6349 304049.2101 12 737700 2.18242E+11 270535.9194 30 120 314021.1506 432154.5279 13 250680 67724173.19 242450.5302 31 1563300 293094.1114 432171.6521 14 2965 54651909080 236742.4777 32 513000 373699.4164 771808.0491 15 196 55927419308 236685.7869 33 177660 382993.8163 953889.7591 16 65173 29415353580 236682.0481 34 2469000 370730.9139 1028352.278 17 2370 54349168455 235499.0811 35 1678260 502013.8426 3113131.758 18 1581 54698599781 235458.3178 36 170760 579874.5015 ..... S-RSRGM LR-LRT MSE 6.68654E+11 2.79596E+11

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^k\P0(Tn)=1 ^k\P(Tn)0=1+e^0im+^1imTn2 where^0imand^1imaretheMaximumLikelihoodEstimatorsoftheparameters0and1,usingbk=imandimthepositiveintegerfromN;N+1;N+2;:::atwhich\MSEiisminimal,aspre-sentedinChapter2Thismodelnotonlygivesverygoodpredictionsofthemeantimetonextfailure,butitiseasytoapplyanditisfreeofassumptions.Thenewmodelwascomparedwiththefollowingcommonlyusedmodelstointhesubjectareai)MazzuchiandSoyerwillbereferredtoastheBayesEmpirical-BayesExponential(BEBE)model.ii)Suresh-RaoSRGM Weusedtwodifferentsetsofactualdata,namely,i)theApollo8SoftwareFailureDataii)Musa'sProject14CsoftwarefailureData Themeansquareerrorcriteriawasusedtocomparetheresultsofourproposedmodelwithothermodelsstatedabove.Theresultspresentedintablesandgraphicalformssupportthefactthatthenewmodelismoreeffectiveinestimatingthemeantimebetweenfailureduringthetestinganddebuggingphaseofagivensoftware.87

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kL(t)(kL(t)) x)+B(xi with:8>>><>>>:0=A+B x1=Bzi=xi

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(Pn1z2i)2Pn1z2iVar(yi)=1 (Pn1z2i)2Pn1z2i2=2

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issuchthat:yiN(0+1zi;2) (2)4=21 2 (0^0)2 20B@(1^1)2 1 (2)4=21 2 (0^0)2 20B@(1^1)2 2 (0^0)2 20B@(1^1)2 2 (0^0)2 20B@(1^1)2 2(0^0)2 2(1 Dening1=1

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weobtain:1 2"(0^0)2 21"202 2(^20 21"01 21(^0 2(^20 21"01 20@(1^1)2 2(1 Dening2=1 and2=2 weobtain:1 2"(1^1)2 22"212 2(^21 22"11 22(^1 2(^21 22"11

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2101 2(20+2)d0R1e1 2211 2(21+2)d1 2101 2(20+2)d0R1e1 2211 2(21+2)d1(5.15) Whichsimplifyto: 2101 2211 2101 2211 equivalently,weobtainb0B=R10e1 2101 2211 2 then,b0B=p 2101 2 then,b0B=p 2 whichequate: Recallingthat:

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Finally,aftersimplication,weobtainthefollowingBayesianestimateb0Bof0: kLn+1(kLn)RewritingasModelI:Ln+1 k(1)Ln+1+1 1(5.23)(5.24)yn=BLn+1+A(5.25)where:8>>><>>>:yn=Ln+1 k(1)A=1 1:(5.26)97

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1 k(1) k(1)Ln+1 where:8>>><>>>:0=1 1 k(1) k(1)zn=Ln+1 Giventheestimatesoftheregressioncoefcients,underourproposedBayesianprocedure,b0Bandb1Boftheparameters0and1respectively,wecanwrite: 1^^ where,^and^k,respectively,ourproposedestimatesfortheparametersandkcanbederiveasfollows:Startingwith^0B=1 1^+^1B wederiveourestimate^oftheparameterasfollows:1 1^=^0B^1B ^0B^1B ^0B^1B ^0B^1B

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weobtainourestimate^koftheparameterk^k=^ Weuse,underMorishita,thefollowingestimatesofmandc(alphacontinous)asin[1]:^m=PNn=1(^kLn) kLn(kLn+1)RewritingasModelI:Ln+1 kLn+1+(+1)(5.41)yn=BLn+1+A(5.42)where:8>>><>>>:yn=Ln+1 kA=+1(5.43) RewritingasModelIILn+1 k Ln+1 kLn+1

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k kzn=Ln+1 Giventheestimatesoftheregressioncoefcients,underourproposedBayesianprocedure,b0Bandb1Boftheparameters0and1respectively,wecanwrite: where,^and^k,respectively,ourproposedestimatesfortheparametersandkcanbederiveasfollows:Startingwith^0B=^+1+^1B wederiveourestimate^oftheparameterasfollows:^=^0B^1B Knowing^andusing^1B=^ weobtainourestimate^koftheparameterk^k=^ Weuse,underHirota,thefollowingestimatesofmandc(alphacontinous)asin[1]: 1+^)n(5.53)bc=1

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i)PL/IsoftwareFailureDatabyOhba,[10]ii)Tohma'sSoftwareFailureData,[11]iii)theF11-DprogramtestdatabyFormanandSingpurwalla,[13]iV)Misra'sSpaceShuttleSoftwareFailureData[15]V)Musa'sSystemT1softwareFailureDataVii)Tohma'seldreporttestData[11;16;18]wehavelistedthevaluesneededforthecalculationofourBayesianestimatesvaluestobeusedintheBayesianestimatesforDataset1

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104

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InChapter2,wehavedevelopedofasimple,realistic,andeasytoimplementsoftwarereliabilitygrowthmodelthatprovidesadecisionruleasofwhentostopthetestinganddebuggingphaseandreleasethesoftwareforuse,forS-shapedcumulativesoftwarefaults.WecanuseaBayesianorQuasi-BayesianprocedureinthepresentdevelopmentoftheproposedModel. InChapter3,usingthemainfeatureofourproposedModelinChapter2-itsinectionpoint,wehaveproposedaneffectivemethodforestimatingthenumberoffaultsinthesoftware,atanearlystageofthetestinganddebuggingphase.Ourearlyestimationofthenumberoffaultinthesoftwareenablethesoftwaredeveloperstoplanthesoftwaredevelopmentprocess,managetheirresourcesbyavoidingcostduetoovertesting,makeasoftwarewithhigherreliabilityanddecidewhentoshipitforuse.WeneedtodevelopacostreductionanalysisassociatedwithourEarlyEstimationproposedModel. InChapter4,assumingalogisticmodel,wedevelopaprocedureforpredictingthemeantimebetweenfailure,afterthelastcorrection,ofasoftwarepackagecreatingaconnectionbetweenpre-dictingtheMTBFandcountingthecumulativefailuresexperienceduptoagiventime.ForthepredictionoftheMTBF,onepossibleextensionistointroduceaBayesianorQuasi-BayesianprocedureinthedevelopmentoftheproposedModel. InChapter5,UsingBayesianprocedures,wehavedevelopedtwodiscretesoftwarereliabilitygrowthmodelsbasedontwodifferenceequationsdiscreteanalogofthelogisticcurvemodelre-spectivelyproposedbyMorishitaandHirota.weshallillustratedthattheBayesianApproachtoreliabilityisveryusefulmodelinginunderstandingthenalevaluationofasoftwarepackage.Oneofthekeydifcultiesistoidentifyingandjustifyingthechoiceoftheprior.Thus,weproposeto105

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D.SatohandS.Yamada,ParameterEstimationofDiscreteLogisticCurvesModelsforSofwareReliabilityAssessement,JapanJ.Indust.Appl.Math.,19(2002);3953[2] F.Morishita,Thettingofthelogisticequationtotherateofincreaseofpopulationdensity,Res.Popul.Ecol.VII(1965);5255[3] Computerscienceandstatistics:proceedingsoftheSixteenthSymposiumontheInterface,Atlanta,1984.[4] D.W.HosmerandS.Lemeshow,Appliedlogisticregression,JohnWileyandSons,NY,1989.[5] C.Y.,J.H.Lo,andS.Y.Kuo,PragmaticstudyofParametricDecompositionModelsforEsti-matingSoftwareReliabilityGrowth,Proc.9thInt'l.Symp.SoftwareReliabilityEngineering(ISSRE'98),1998,pp.111123.[6] C.Y.Huang,S.Y.Kuo,andI.Y.Chen,AnalysisofaSoftwareReliabilityGrowthModelwithLogisticTesting-EffortFunction,Proc.8thInt'l.Symp.SoftwareReliabilityEngineering(ISSRE'97),1997,pp.378388.[7] C.Y.Huang,S.Y.Kuo,andM.R.Lyu,Effort-Index-BasedSoftwareReliabilityGrowthModelsandPerformanceAssessment,Proc.24thAnn.Int'l.ComputerSoftwareandAppli-cationsConf.(COMPSAC2000),2000.[8] R.H.Hou,S.Y.Kuo,andY.P.Chang,ApplyingVariousLearningCurvestoHyper-GeometricDistributionSoftwareReliabilityGrowthModel,Proc.5thInt'l.Symp.SoftwareReliabilityEngineering,1994,pp.716.107

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ABSTRACT: The present study is concerned with developing some statistical models to evaluate and analyze software reliability. We have developed the analytical structure of the logistic model to be used for testing and evaluating the reliability of a software package. The proposed model has been shown to be useful in the testing and debugging stages of the developmental process of a software package. It is important that prior to releasing a software package to marketing that we have achieved a target reliability with an acceptable degree of confidence. The proposed model has been evaluated and compared with several existing statistical models that are commonly used. Real software failure data was used for the comparison of the proposed logistic model with the others. The proposed model gives better results or it is equally effective. The logistic model was also used to model the mean time between failure of software packages. Real failure data was used to illustrate the usefulness of the proposed statistical procedures. Using the logistic model to characterize software failures we proceed to develop Bayesian analysis of the subject model. This modeling was based on two different difference equations whose parameters were estimated with Bayesian regressions subject to specific prior and mean square loss function.
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Logistic regression.
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