(复杂系统的性能评价与优化课件资料）LNProb

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1、LectureNotes#2ProbabilityandQueueingTheory(Version3.5,2007-9-17)(Seealso[CL99]§6.7,8.2,and8.5,Chapter10.1-10.7)1・ProbabilityFundamentals1.1ThememorylesstriangleThesinglemostimportantfacttocommittomemory:Thememorylesstriangle:(i)Theinte「airivaltimedistributionofthe

2、customersisexponential.LetTbeanexponentiallydistributedrandomvariablewithmean〃入i.e.,p(T)=Ae^.(WeusecapitalPtoindicatetheprobability,andsmallptoindicatetheprobabilitydensityfunction.)ThentheconditionalprobabilityofT,giventhatTisgreaterthananyfinitevalueZqisthesamea

3、stheoriginaldistribution,i.e.,P(T<

4、T>to)=l-e^=Pg).(D=>p⑴=Theintuitiveexplanationofequation(1)is:P(atleastonearrivalinthenexttseconds

5、noarrivalintoseconds)=1-P(noarrivalintseconds)=P(atleastonearrivalintseconds).Thisproperty,illustratedinFig.1，iscalledthememoryless

6、property.Theshadedportionoftheunconditionaldensityfunctionisre-scaledtogivetheconditionaldistributionFig.1TheMemorylessPropertyoftheExponentialDistributionToprove(1)consider/、P(T

7、AzisequaltoAAf+whereQistheso-calledconstantarrivalrate(私/istheprobabilityofLectureNotes#2ProbabilityandQueueingTheory(Version3.5,2007-9-17)(Seealso[CL99]§6.7,8.2,and8.5,Chapter10.1-10.7)1・ProbabilityFundamentals1.1ThememorylesstriangleThesinglemostimportantfacttoc

8、ommittomemory:Thememorylesstriangle:(i)Theinte「airivaltimedistributionofthecustomersisexponential.LetTbeanexponentiallydistributedrandomvariablewithmean〃入i.e.,p(T)=Ae^.(WeusecapitalPtoindicatetheprobability,andsmallptoindicatetheprobabilitydensityfunction.)Thenthe

9、conditionalprobabilityofT,giventhatTisgreaterthananyfinitevalueZqisthesameastheoriginaldistribution,i.e.,P(T<

10、T>to)=l-e^=Pg).(D=>p⑴=Theintuitiveexplanationofequation(1)is:P(atleastonearrivalinthenexttseconds

11、noarrivalintoseconds)=1-P(noarrivalintseconds)=P(atleast

12、onearrivalintseconds).Thisproperty,illustratedinFig.1，iscalledthememorylessproperty.Theshadedportionoftheunconditionaldensityfunctionisre-scaledtogiveth