Equivalent or or Note Poisson Process Renewal pr ocess with expected number of arrivals in interval of length Memoryless Pr operty 10 243 Theorem informal statement a Interarrival interval fr om until the 57346rst arrival after is RV with b This ID: 31731
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RenewalProcesses/CountingProcesses/PoissonProcessestN(t) :numberofarrivalsin :Epochof\ntharrival \r :interarrivalintervals RenewalProcess:InterarrivalintervalsarepositiveIIDrandomvariablesprocessesaremoregeneralthanitmightseem)1CountingProcesses :familyofrandomvariables :numberofarrivalsininterval. withprobability1CountingProcess :familyofnon-negativeintegervaluedrandomvariables(oneforeach)withthepropertiesthat and withprobability1.Equivalent: or \r or .Note: \n2PoissonProcess Renewalprocesswith !"#$% &:expectednumberofarrivalsinintervaloflength '( & MemorylessProperty) *+!, *) *!! ) \n-$./1024356 7Theorem(informalstatement):(a)InterarrivalintervalfromuntiltherstarrivalafterisaR.V.with !"#$(b)ThisR.V.isindependentofallarrivalepochsbeforetimeand for 3StationaryIncrementPropertyt :countingprocess8 :9 9 :numberofarrivalsin9,:98 ;9hassamedistributionas 94 IndependentIncrementPropertyt 8 8 independentrandomvariables5DenitionPoissonProcessDenition1:Renewalprocesswith !"#$Denition2:Countingprocess withindependentandstationaryincrementproperties,and ) \n-$./02356 7Denition3:Countingprocess withindependentandstationaryincrementproperties,and 8 +"&+ 8 +"&+ 8 +6CombiningIndependentPoissonProcesses7BernoulliSplittingofPoissonProcessesThetwoprocessesareindependent8