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PARTIIIMARKOVCHAINSOTHERWISE,IFNOSUCHSTATEJEXISTS,THENSTATEIISRECURRENTIRREDUCIBLEMARKOVCHAINAMARKOVCHAINISIRREDUCIBLEIFTHEREISONLYONECOMMUNICATINGCLASS28EXAMPLE121THEMARKOVCHAINISWITHEACHBRANCHPIJ0HOWMANYCOMMUNICATINGCLASSESSOL021654329EXAMPLE122CONSIDERTHEFIVESTATEDISCRETERANDOMWALKWITHMARKOVCHAINWHATISTHEPERIODOFEACHSTATEISOL24310P1PP1PP111P30THEOREM121COMMUNICATINGSTATESHAVETHESAMEPERIODPF31EXAMPLE123THEMARKOVCHAINISWITHEACHBRANCHPIJ0FINDTHEPERIODICITYOFEACHCOMMUNICATINGCLASSSOL021654332THEOREM122IFIISRECURRENTANDIJ,THENJISRECURRENTSOL33EXAMPLE124THEMARKOVCHAINISWITHEACHBRANCHPIJ0IDENTIFYEACHCOMMUNICATINGCLASSANDINDICATEWHETHERITISTRANSIENTORRECURRENTSOL24310534THEOREM123IFSTATEIISTRANSIENT,THENNI,THENUMBEROFVISITSTOSTATEIOVERALLTIME,HASEXPECTEDVALUEENI0IFTHESYSTEMSTARTSINSTATEJC10,1,2,THESYSTEMNEVERLEAVESC1IFTHESYSTEMSTARTSINCOMMUNICATINGCLASSC34,5,THESYSTEMNEVERLEAVESC3IFTHESYSTEMSTARTSINTHETRANSIENTSTATE3,THENINTHEFIRSTSTEPTHEREISARANDOMTRANSITIONTOEITHERSTATE2ORTOSTATE4ANDTHESYSTEMTHENREMAINSFOREVERINTHECORRESPONDINGCOMMUNICATINGCLASS24310537PARTIIIMARKOVCHAINSOTHERWISESTATEIISTRANSIENT57EXAMPLE137ASYSTEMWITHSTATES0,1,2,HASMARKOVCHAINNOTETHATFORANYSTATEI0,PI,01/I1ANDPI,I1I/I1ISSTATE0TRANSIENTORRECURRENTSOL243101/21/32/31/43/41/511/24/558THEOREM138THEEXPECTEDNUMBEROFVISITSTOSTATEIOVERALLTIMEISPF59THEOREM139STATEIISRECURRENTIFANDONLYIFENIIPF60EXAMPLE138THEDISCRETERANDOMWALKINTRODUCEDINEXAMPLE104HASSTATESPACE,1,0,1,ANDMARKOVCHAINISSTATE0RECURRENTSOL02112P1PP1PP1PP1PP1PP1P61POSITIVERECURRENCEANDNULLRECURRENCEDEFINITIONARECURRENTSTATEIISPOSITIVERECURRENTIFETII0ANDPII1FORSTATESIWITHI0DEFINITIONTHECOMMUNICATINGCLASSESOFACONTINUOUSTIMEMARKOVCHAINAREGIVENBYTHECOMMUNICATINGCLASSESOFITSEMBEDDEDDISCRETETIMEMARKOVCHAIN69IRREDUCIBLECONTINUOUSTIMEMARKOVCHAINDEFINITIONACONTINUOUSTIMEMARKOVCHAINISIRREDUCIBLEIFTHEEMBEDDEDDISCRETETIMEMARKOVCHAINISIRREDUCIBLEDEFINITIONANIRREDUCIBLECONTINUOUSTIMEMARKOVCHAINISPOSITIVERECURRENTIFFORALLSTATESI,THETIMETIITORETURNTOSTATEISATISFYINGETII121012230134378THEOREM143FORABIRTHDEATHQUEUEWITHARRIVALRATEIANDSERVICERATEI,THESTATIONARYPROBABILITIESPISATISFYPF79THEOREM144FORABIRTHDEATHQUEUEWITHARRIVALRATEIANDSERVICERATEI,LETII/I1THELIMITINGSTATEPROBABILITIES,IFTHEYEXIST,AREPF80THEM/M/1QUEUETHEARRIVALSAREAPOISSONPROCESSOFRATE,INDEPENDENTOFTHESERVICEREQUIREMENTSOFTHECUSTOMERSTHESERVICETIMEOFACUSTOMERISANEXPONENTIALRV,INDEPENDENTOFTHESYSTEMSTATEM/M/1MARKOVIANARRIVALPOISSONARRIVALSMARKOVIANSERVICETIMEEXPONENTIALRVONESERVER81THEOREM145THEM/M/1QUEUEWITHARRIVALRATE0ANDSERVICERATE,HASLIMITINGSTATEPROBABILITIES,WHERE/PF82EXAMPLE145CARSARRIVEATANISOLATEDTOLLBOOTHASAPOISSONPROCESSWITHARRIVALRATE06CARSPERMINUTETHESERVICEREQUIREDBYACUSTOMERISANEXPONENTIALRVWITHEXPECTEDVALUE1/03MINUTESWHAT
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