TWO MUTUALLY RAREFIED RENEWAL PROCESSES剩余时间过长

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1、APPLICATIONESMATHEMATICAE22,2(1994),pp.267–273I.KOPOCINSKA(Wroclaw)´TWOMUTUALLYRAREFIEDRENEWALPROCESSESAbstract.Letusconsidertwoindependentrenewalprocessesgeneratedbyappropriatesequencesoflifetimes.Wesaythatarenewaltimeisac-ceptedifinthetimebetweenasignalandtheprecedingone,somesignalofthese

2、condprocessoccurs.Ourpurposeistoanalyzethesequencesofacceptedrenewals.Forsimplicityweconsidercontinuousanddiscretetimeseparately.InthefirstcasewemainlyconsidertherenewalprocessrarefiedbythePoissonprocess,inthesecondweanalyzetheprocessgeneratedbythemotionofdraughtsmenmovedbydietossing.1.Thepro

3、blem.Letusconsidertwoindependentrenewalprocesses(i)(i)generatedbysequences{X1,X2,...},i=1,2,ofnonnegative,uniformlydistributed,independentrandomvariables.Introducetheusualnotationfortherenewaltimes,therenewalprocessesandtheresidualtimeprocesses:S(i)=0,S(i)=S(i)+X(i),n=1,2,...,0nn−1nN(t)=max

4、{n≥0:S(i)

5、acceptedtimesofprocess1.NoticethatZ′′=0<120Z′≤Z′′≤Z′≤...,i.e.thesetimesalternateorsometimescoincide(our112notationsandthecoincidenceconceptarederivedfrom[1]).Ourpurposeistoanalyzetheprobabilitydistributionfunctionofthesequenceofrandom1991MathematicsSubjectClassification:Primary60F05.Keywords

6、andphrases:Markovchain,rarefiedrenewalprocess.268I.Kopoci´nskavariables(1)V=Z′−Z′′,U=Z′′−Z′,n=1,2,...nnn−1nnnTheresidualtimeprocesses,therenewaltimesandtheacceptedtimes(dottedlines)areshowninFig.1.Fig.1Letusintroducethestandardnotationfortheprobabilitydistributionfunctionsofthelifetimesofthe

7、renewalprocesses,therenewalfunctionsandtheprobabilitydistributionfunctionoftheresidualtimeprocess:(i)Fi(x)=P(X1