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1、PerformanceEvaluation18(1992)149-171149North-HollandTheMarkov-modulatedPoissonprocess(MMPP)cookbookWolfgangFischer*SiemensAG,Munich,GermanyKathleenMeier-HellsternAT&TBellLaboratories,Holmdel,NJ,USAReceived8March1991AbstractFischer,W.andK.Meier-Hellstern,TheMarkov-modulatedPoissonprocess(MMPP)cookbo
2、ok,PerformanceEvaluation18(1993)149-171.Pointprocesseswhosearrivalratesvaryrandomlyovertimeariseinmanyapplicationsofinterest,notablyincommunica-tionsmodeling.TheMarkov-modulatedPoissonprocesshasbeenextensivelyusedformodelingtheseprocesses,becauseitqualitativelymodelsthetime-varyingarrivalrateandcap
3、turessomeoftheimportantcorrelationsbetweentheinterarrivaltimeswhilestillremaininganalyticallytractable.ThepurposeofthispaperistocollectanumberofusefulresultsaboutMarkov-modulatedPoissonprocessesandqueueswithMarkov-modulatedinput.Itisintendedfornon-expertsorforpeoplewhoarefamiliarwiththebasicconcept
4、sandalgorithms,butwouldlikeasummaryofrecentdevelopments.Derivationsandproofshavebeenintentionallyomitted.Theycanbefoundinthepublicationslistedattheendofeachsection.Manyoftheresultsandsomeofthetexthavebeentakendirectlyfromthereferences,andthereaderisencouragedtoconsulttherelevantreferencesformoredet
5、ailedinformation.Keywords:Markov-modulatedPoissonprocess;Phase-typedistributionI.IntroductionPointprocesseswhosearrivalratesvaryrandomlyovertimeariseinmanyapplicationsofinterest,notablyincommunicationsmodeling.Examplesincludetheoverflowfromafinitetrunkgroup,thesuperpositionofpacketizedvoiceprocesse
6、s,andpacketdata[12,13,14,17,25,29,33,34,46,57].TheMarkov-modulatedPoissonprocess(MMPP)hasbeenextensivelyusedformodelingtheseprocesses,becauseitqualitativelymodelsthetime-varyingarrivalrateandcapturessomeoftheimportantcorrelationsbetweentheinterarrivaltimeswhilestillremaininganalyticallytractable.Th
7、efirstuseoftheMMPPinqueueingtheorywasinNaorandYechiali[36],followedbyNeuts[37].Sincethen,anumberofexcellentpapersandbookshavebeenwrittenonthesubject.Afewgeneralreferencesare[30,31,38,39,43,47,51].Adiscussio
