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1、ClosedSystems•Therearesingle-servernodeswithservicetimebeingexponentiallydistributedwithrate•Afixedgroupofcustomersmovearoundthesenodes.Nocustomersenterintoorexitfromthenetwork.•Acustomer,uponcompletionofservicebyserver,goestothesingle-serverwiththeprobabilit
2、y•isaMarkovtransitionprobabilitymatrix,whichweassumeisirreducible.•LetdenotethestationaryprobabilitiesforthisMarkovchain,thatisTheAverageArrivalRateLetdenotetheaveragearrivalrateatserver,thenmustsatisfyHence,weconcludethatwhereJackson’sTheoremLetdenotethelimi
3、tingprobabilitiesthen,ೕೕೕWhereೕభೖೕ//SummationovervectorsTheArrivalTheoremIntheclosednetworksystemwithcustomers,thestationarypopulationdistributionofthesystemasseenbyarrivalstoserveristhetime-averagedistributioninthesamenetworksystemwhenthereareonly-customers.
4、Intuition:•FromJackson’stheorem,arrivalstothestationaryclosednetworklooklikePoisson•Ontheotherhand,Poissonarrivalseestimeaverage•Acustomerarrivingatanodealwaysseesatotalof-othercustomersinthenetworkProof:Forwith,wecalculatethefollowingଵଶୀଵposterioriprobab
5、ilityasinBayesrule:ଵଵᇱᇱᇱᇱᇲା⋯ାᇲୀିଵଵభೖೕୀଵᇲೕᇲା⋯ାᇲୀିଵୀଵభೖೕୀଵᇲೕᇲା⋯ାᇲୀିଵୀଵభೖିଵଵଶMeanvalueanalysisoftheclosedsystemNotations:•Populationinaclosednetworkofqueues•Averagearrivalrateatserver•=Systemthroughput•Station
6、arystateoftheimbeddeddiscrete-timeMarkovchain•Averagedelayacustomerspendsatserverwhenhevisitsserver•AveragenumberofcustomersatserverjTheRecursiveEquationUponconditioningonthenumberofcustomersfoundatserverbyanarrivaltothatserver,itfollowsthatByLittle’sformula,
7、Hence,Usingthefactthat,WehaveorHence,weobtaintherecursionExample:kcyclicqueuesConsidera-servernetworkinwhichcustomersmoveinacyclicpermutation.ThatisTheservicerateforeachserverisFindouttheaveragenumberofcustomersatserverwhentherearetwocustomersinthesystemDueto
8、symmetry,thelimitingdistributionisandwiththeobservationthatweobtainHence,and5.3TheM/G/1QueueingSystemPreliminaries:WorkandAnotherCostIdentityForanarbitraryqueueingsystem,letusdefinethewor
