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INFINITE-SERIESREPRESENTATIONSOFLAPLACETRANSFORMSOFPROBABILITYDENSITYFUNCTIONSFORNUMERICALINVERSIONJournaloftheOperationsResearchSocietyofJapan42(1999)268–285(May6,1999)AbstractInordertonumericallyinvertLaplacetransformstocalculateprobabilitydensityfunctions(pdf’s)andcumulativedistributionfunctions(cdf’s)inqueueingandrelatedmodels,weneedtobeabletocalculatetheLaplacetransformvalues.InmanycasesthedesiredLaplacetransformvalues(e.g.,ofawaiting-timecdf)canbecomputedwhentheLaplacetransformvaluesofcomponentpdf’s(e.g.,ofaservice-timepdf)canbecomputed.However,therearefewexplicitexpressionsforLaplacetransformsofcomponentpdf’savailablewhenthepdfdoesnothaveapureexponentialtail.Inordertoremedythisproblem,weproposetheconstructionofinfinite-seriesrepresentationsforLaplacetransformsofpdf’sandshowhowtheycanbeusedtocalculatetransformvalues.WeusetheLaplacetransformsofexponentialpdf’s,LaguerrefunctionsandErlangpdf’sasbasiselementsintheseriesrepresentations.Wedevelopseveralspecificparametricfamiliesofpdf’sinthisinfinite-seriesframework.Weshowhowtodeterminetheasymptoticformofthepdffromtheseriesrepresentationandhowtotruncatesoastopreservetheasymptoticformforatimeofinterest.1.IntroductionThispaperisconcernedwithprobabilitydensityfunctions(pdf’s)fonR+(i.e.,nonnegativefunctionsfonthepositivehalflineforwhichR∞0f(t)dt=1)andtheirLaplacetransformsˆf(s)=Z∞0e−stf(t)dt.(1.1)Weintroduceandinvestigatepdf’swhoseLaplacetransformscanberepresentedasinfiniteseries.Ourmotivationcomesfromnumericaltransforminversion.Wehavefoundthatmanydescriptivequantitiesofinterestinqueueingmodelsandotherprobabilitymodelsinopera-tionsresearchcanbeeffectivelycomputedbynumericallyinvertingLaplacetransforms;e.g.,byusingtheFourier-seriesmethodortheLaguerre-seriesmethod;seeHosono[13]and[3],[4].Themostdifficultstepinperformingthenumericalinversion,ifthereisanydifficultyatall,isusuallycomputingtheLaplacetransformvalues;e.g.,seetheapplicationtopollingmodelsin[10].Inmanycases,numericalinversionisstraightforwardprovidedthatLaplacetransformsareavailableforcomponentpdf’s.Afamiliarexampleisthesteady-statewaiting-timeintheM/G/1queue.NumericalinversioncanbeapplieddirectlytotheclassicalPollaczek-KhintchinetransformprovidedthattheLaplacetransformoftheservice-timepdfisavail-able.Moregenerally,thesteady-statewaiting-timedistributionintheGI/G/1queuecanbecomputedbynumericaltransforminversionprovidedthattheLaplacetransformsofboththeinterarrival-timepdfandtheservice-timepdfareavailable[2].IntheGI/G/1case,anextranumericalintegrationisrequiredtocalculatetherequiredwaiting-timetransformvalues.Fortheseinversionalgorithmstobeeffective,thepdf’salsoneedtobesuitablysmooth;otherwisepreliminarysmoothingmayneedtobeperformed.Inthispaper,allpdf’swillbecontinuous.Ofcourse,therearenumerouspdf’sonR+withconvenientLaplacetransforms,butalmostallofthesepdf’shavea(pure)exponentialtail,i.e.,theyhavetheasymptoticformf(t)∼Ae−αtast→∞,(1.2)whereAandαareconstantsandf(t)∼g(t)meansthatf(t)/g(t)→1ast→∞.Forexample,allphase-typepdf’ssatisfy(1.2).Thusweareinterestedinobtaininginfinite-seriesrepresentationsofLaplacetransformsofpdf’sthatdonotsatisfy(1.2).Forexample,thesealternativepdf’smayhaveasemi-exponentialtail,i.e.,f(t)∼At−βe−αtast→∞(1.3)orapowertail,i.e.,f(t)∼At−αast→∞.(1.4)Weareinterestedinpdf’sofclassIIandIIIintheterminologyof[2],[6].ApdfisofclassIifitsLaplacetransformsˆfhasrightmostsingularity−s∗0andˆf(−s∗)=∞.ApdfisofclassIIifagain−s∗0butˆf(−s∗)∞.ApdfisofclassIIIif0istherightmostsingularityofˆf.ExamplesofclassIIandIII,respectively,are(1.3)and(1.4).ClassIIIpdf’sareinterestingbecausetheyhavelong(orheavy)tails.ClassIIpdf’sarelesscommon,buttheydooccur;e.g.,theyplayaprominentroleinpriorityqueues[6].AtypicalapplicationoftheseriesrepresentationswouldbetoseehowtheperformanceofaqueueingsystemdependsonaclassIIorIIIservice-timedistribution,asin[2].(Theeffectcanbedramatic,butwedonotdiscussqueueingmodelshere.)WesuggestageneralapproachforconstructingclassIIandIIIpdf’sforwhichtheLaplacetransformvaluescanbecomputed.WesuggestrepresentingtheLaplacetransformasaninfiniteseriesandthennumericallycalculatingthesum,usingaccelerationmethodsifnecessary.Fortheinversion,wedonotactuallyneedaconvenientclosed-formexpressionforthetransform.Itsufficestohaveanalgorithmtocomputethetransformvalueˆf(s)fortherequiredargumentss.Thusaninfiniteseriescanbeasatisfactoryrepresentation.Wealsoshowhowtoworkwiththeseinfinite-seriesrepresentations.Wediscussthreeinfinite-seriesrepresentations:exponentialseries(Section2),Laguerreseries(Sections4–5)andErlangseries(Sections6–7).2.Exponential-SeriesRepresentationsAnaturalwaytoobtainalong-tail(classIII)pdf,where0istherightmostsingularityofitsLaplacetransformˆf,istoconsideraninfinite-seriesofexponentialpdf’s,wherethemeansoftheexponentialpdf’scanbearbitrarilylarge.Anexponential-seriesrepresentationisjustacountablyinfinitemixtureofexponentialpdf’s,i.e.,f(t)=∞Xk=1pke−t/akak,t≥0andˆf(s)=∞Xk=1pk(1+sak)−1,(2.1)where{pk:k≥1}isaprobabilitymassfunction(pmf)and{ak:k≥1}isthesequenceofmeansofthecomponentexponentialpdf’s.Thestandardcaseisakak+1and