2001年度日本オペレーションズ。 1−C−8 リサーチ学会春季研究発表会 Ama且ys五s⑳紆斑ype『ge⑳me屯『五cD五s七『弛Ⅷ舶omS①氏wa『e凪e且五ab組紬yM①de且 T．Dohi†（01307065），N．Wakana†，S．Osaki†（01002265）andKishorS・Trivedi‡ †HiroshimaUniversity，Higashi−Hiroshima，Japan，‡DukeUniversity，NC，USA beenalreadydetectedandremovedbyt（1）．Ingeneral，the numberofnewlydetectedfaults bytheithtestinstance 1．INTRODUCTION ThisarticleglVeSthedetailcdmathematicalresultson t（i），Xi，Canberegardedasapositiverandomvariable・ thehypergeometricsoftwarereliabilitymodel（HGDSRM） Then，thccumulativenumberofnewlydetectedfhultsby proposed by Thoma et al・【1，2トIntheearlierpaper， thetestinstancest（1），‥・，t（i）isCi＝∑；＝1Xj・ Thoma etal．【2】derived arecursiveformulaontheex− Withthisnotation，theprobabilitythatxfaultscanbe pectedcumulativenumberofsoftwarefaultsdetected up newlydetectedbythetestinstancet（i）isgivenby toithtestinstanceintestingphase．Sincetheirmodelwas basedononlythemeanvalueofthecumulativenumberof p〈ざ‘＝∬】立方j＝Cト1 Lhults，itwasimpossibletoestimatethesoftwarereliability aswenastheotherprobabilisticdependabilitymeasures・ Byintroducingtheconceptofcumulativetrialprocesses， Wederivetheprobabilitymassfunctionofthenumberof SOftwarefhultsdetectedatithtestinstanceexplicitly． J＝1 〉中m，帰㈹〉（m￣：ト1）（品） ▼n （ u（i）（1） 2．正丑GDSRM whereO≦x≦min（w（i），m−Ci−1）andciistherealization Suppose that the test ofasoftwareisasetofteslin− StanCeS（such as test runs），Which consists ofinput test ofthe random variable Ci．Since the above expression dat，a a．nd observed test rcsult．Define the soft．ware test． is the hyper−geOmCtric pmf，the sequentialmodelbased onEq．（1）iscal1edtheHGDSRM・FtomEq・（1），themean byD＝（土（i）li＝1，2，…，），Wheret（i）istheithtestin− number ofnewlydetected faults at theith testinstanCe StanCe・Let B＝（b（j）Ij＝1，2，…，m）denoteaset of andit．s variance are faultsremaininginthesoftwareattheinitialtime（i＝0）， m￣Ci＿1 ）坤） Whereb（j）means the faultlabelled byj（＝1，2，…，m）中居弟＝Cトホ（打l j＝1 aIld m（＞0）is theinitialnumber offaults・Ifasoft− WareerrOrCauSedbyb（j）isobservedatthetestinstance and t（i），thefaultb（j）issaidtobesensedbythetestinstance （m−q−1）ci−11〃（哀） L（i）．SupposethatatestinstanCet（i）senscsw（i）software 可ズ‘一宏ズブ＝Cト1］ j＝1 fhults，Wherew（i）iscallcdthesensitivityJhctorandisthe functionofthenumberoftestinstance（ortime）・Make （2）），（3） m−1 thcfo1low皿gaSSumptions： respectively・In（2］，Substitutingci−1＝∑＝シ砧（A−1）Thesoftwarefaultsthatmanifbstthemselvesupon ∑こiE【XkJ＝ErCi−1】intoEq・（2），thefb1lowingrecur− theapplicationofatestinstancet（i）mayberemoved siveformulais obtained：（丘xed）beforethenexttestinstancet（i＋1）isapplied．（4）︸（A−2）Nonewfaultsareintroducedduringthesoftware Efq＝E一弘】（ト慧）…（け testing．This meanS that the software reliabilityis ThiscanbesoIvedbytheinductionasfbllows【2】・ nondecreaslngaSthetestingprogresses． ErG】＝m〈1一口叫一望）﹁■．﹂︸禁（A−3）Arandomsetofw（i）softwarefaultsaresensedby testinstanCet（i）outofthetotalminitialfaults・ t ＝m〈トexp［∑log（ト LFfom these assumptions，itis evident that thenumber ゴ＝1 Offaultsdetected by the6rst testinstance t（1）is w（1）． However，thenumberofnewLydetectedfaultsbyt（2）is Theaboveequationisderivedfromtheheuristicargument， butiscorrectfromtheindependenceoftheBernoullitrials・ notnecessarilyw（2），Sincesomeofw（2）faultsmayhave （5） −64 − Theoreml：FbrtheinitialnumberofremalnlngSOftware faults m，SuppOSe that the sensitivity factorinith test Suppose that theinitialnumber ofdetected払ults at instancet（i）isdefinedbyw（i）・Then，theprobabilitythat i＝OisO・Ofourinterestisthederivationoftheproba− xifaultsaredetectednewlyatithtestinstance，Et（xiI bilityofthenumberofnewlydetectedfaultsbythetest mパ〟（豆）），i岳 instanCe t（i）．Let Xibe thenumberofnewlydetected 3．FURTHER RESUIノーS faultsatithtestinstance．Wbmakethefo11owingaddi− U（2）u（3） w（ト1） tionalassumptions： ∑∑…∑ （B−1）Theinitialnumberoffhultsmremaininginthe エ2＝0＝3＝0 ェi＿1＝0 SOftwareissu侃cientlylargerthanw（i），i・e・m≫w（i）払rau豆＝1，2，3，・‥ （B−2）In the software test，itisimpossible to de− tect allfaults with probability one，i．e． m ＞（10） 1血→∞∑；＝1エト Theorem2：Themeannumberofnewlydetectedfaults LFtom these it assumptions， Canbeseenthatmin（w（i）．m−Ci−1）＝W（i）hasto be at，ith tcstinst，anCeis alwayssatisfied．Infact，theseassumptionsareplausible intuitivelyandareoftenusedinearliermodelingapproach． SupposethattheprobabilityEl（xllm，W（1））thatxl faultsisdetectedby thetestinstancet（1）isthehyper− E【糎（壱卜誓苫1苫’＊エi＝1ェi＿1＝0 （m一石三三当（謀ご・） geometricpmf・Let為（x2fm，W（2））denotetheprobability ．㌻く thatx2fhults aredctected newly at thesecond testin− た＝1 m Tl＝2 （U（れ） StanCe，i．e．i＝2．Thcn，itisstraightforwardtoobtain （m￣与幸一1）（謀ニk） u（1）ヰ2れひ（2）〉＝∑ヰ1−m，叫）〉’ × （11）〇1＝0 ×P〈材m，Clル（2）〉・（6） w（ト1）− u（2）u（3） w（ト1） wbere∑ ＝∑∑…∑．・ FtomLhesimi1armanlpulation，Wehave エi−1＝0 u（ト1）ヰ巾両）〉＝∑ 〇2＝0ェ3＝0 ェi−1＝O Similar to Theoremland Theorem2，We Can derive 王i＿1＝0 analytical1yVar［Ci】and Var［Xi］aLSWellasthesoftware 軋1〈ヱト1■m，坤−1）〉中Im，q−1，㈹） reliabilityP（∑ニ＝j．1Xn＝Olm・Ci−1・W（i））・AIso，it CanbeshownthatEtGl＝∑；＝1E【XjlinEq・（11）・These U（i−1） ×∑彗一1ト1恒（五−1）〉 resultsarenevertrivialandareallprovedbytheinduction・ェi＿1＝0 ×中1m，C砧ひ（豆）〉，（7） Wheretheright−handsideofEq・（7）isdueto（B−1）and REFERENCES 【1】Y・Tblma，K・Tbkunaga，S・NagaBeandY・Murata，（B−2）．Then，theproblemistosoIvetheaboverecursive equationwiththeinitialconditions： Structuralapproach to the estimation ofthe num− beI・Ofresidualsoftware faults based on the hyper− geometricdistribution，m7h7・nS・SpPwarY，En9・ 15（3），345−355（1989）・可姉叩（1）＝1 （8）【2］Y・Tblma，H・Yamano，M・OhbaandR・Jacoby，The estimationofparameters ofthehypergeometricdis− tributionanditsapplicationtothesoftwarereliability and ヰ2lmル（2））＝（m三；1）（u（宗L。。） growthmodel，IEEE7hns．SojtwareEn9・17（5），（9）妄＿∴妄 Thefo1lowlngisthemainresultofthisarticle． −65 − 483−489（1991）．