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1、Chapter4:Continuous-TimeMarkovChains4.1Continuous-TimeMarkovChainsDefinition:Supposewehaveacontinuous-timestochasticprocess??,?≥0takingonvaluesinthesetofnonnegativeintegers,wesaythattheprocess??,?≥0isacontinuous-timeMarkovchain,ifforall?,?≥0andnonnegativeinte
2、gers?,?,??,0≤?
3、??=?,??=??,0≤?
4、??=?Inaddition,if???+?=?
5、??=?=???=?
6、?0=?,i.e.,independentof?,thenthecontinuous-timeMarkovchainissaidtohavestationaryorhomogeneoustransitionprobabilities.TheImplicationofMarkovianPropertySupposethatacontinuous-t
7、imeMarkovchainentersstate?atsometime,saytime0,andsupposethattheprocessdoesnotleavestate?(thatis,atransitiondoesnotoccur)duringthenexttenminutes.Whatistheprobabilitythattheprocesswillnotleavestate?duringthefollowingfiveminutes?Let??denotetheamountoftimethatthe
8、processstaysinstate?beforemakingatransitionintoadifferentstate,then???>15
9、??>10=???>5oringeneral,???>?+?
10、??>?=???>?,forall?,?≥0.AlternativeDefinitionAcontinuous-timeMarkovchainisastochasticprocesshavingthepropertiesthateachtimeitentersstate?1.Theamountoftimei
11、tspendsinthatstatebeforemakingatransitionintoadifferentstate1isexponentiallydistributedwithmean,say,??2.Whentheprocessleavesstate?,itnextentersstate?withsomeprobability,say,???,???=0,forall?,????=1forall?Example:AshoeshineshopConsiderashoeshineestablishmentco
12、nsistingoftwochairs-chair1andchair2.Acustomeruponarrivalgoesinitiallytochair1wherehisshoesarecleanedandpolishisapplied.Afterthisisdonethecustomermovesontochair2wherethepolishisbuffed.Theservicetimesatthetwochairsareassumedtobeindependentrandomvariablesthatare
13、exponentiallydistributedwithrespectiverates?1and?2.SupposethatpotentialcustomersarriveinaccordancewithaPoissonprocesshavingrate?,andthatapotentialcustomerwillenterthesystemonlyifbothchairsareempty.Howtoanalyzetheprecedingmodelwithacontinuous-timeMarkovchain?T
14、hecontinuous-timeMarkovchainmodelStatesExplanations0Nocostumersintheshop1Onecostumeratchair12Onecostumeratchair2Transitionprobability:?01=?12=?20=1Waitingtimedistribution:?0=?,?1=?1,?2=?2
