VerifyingQuantitativeProp ertiesofContinuousProbabilisticReal-TimeGraphsMartaKwiatkowska1,GethinNormanRob ertoSegala2andJeremySproston1UniversityofBirmingham,BirminghamB152TT,UK2UniversitadeBologna,MuraAuteoZamb oni7,40127ItalyDecemb er16,1998AbstractWeconsideranextensionofprobabilisticreal-timegraphswithcontinuous-timeproba-bilitydistributions,basedonthetimedautomataof[3]augmentedwithdiscreteprobabilitdistributionsin[13].Similarlyto[2],wherenon-determinismisnotconsidered,wemo delrandomdelaysbyprobabilitdistributionswith niteintervalsupp ort.Wemo difythestandardregiongraphconstructionbysub dividingcertainunitintervalstobuilda niterepresentationforcontinuousprobabilisticreal-timegraphs.Byapplyingthealgorithmof[7,13]weobtainamo delcheckingmetho dforsuchsystemsagainstformulaeoftheProba-bilisticTimedComputationTreeLogic(anextensionofTCTL[3]),basedonapproximatingtheprobabilitytowithinaninterval.Ourmetho dimprovesonpreviouslyknowntech-niquesinthatitallowstheveri cationofquantitativeprobabilityb ounds,asopp osedtoonlyqualitative(i.e.withprobability0or1).Keywords:probabilisticandreal-timesystems,mo delchecking,veri cation,mo daltemp orallogics1Intro ductionBackground:Formalmetho dsandtechniquesforthereasoninganalysisofreal-timesystems,suchascommunicationproto cols,digitalcircuitswithuncertaindelaylengths,andmediasynchronizationproto cols,havereceivedmuchattentionrecently.Thiscanb epartlyexplainedbytheadvancesoftheoryautomaticallyverifyingtimedautomataagainstprop ertiesofreal-timetemp orallogic,togetherwiththedevelopmentasso ciatedsoftwareto ols[8,9]andthesuccessfulapplicationofthoseinindustrialcasestudies[12 ].Traditionalapproachestoreal-timesystemsdescrib etheirb ehaviourpurelyintermsofnon-determinism.However,itmayb edesirabletoexpresstherelativelikelihoodofcertainb ehaviouro ccurring.Forexample,wemaywishtomo delasysteminwhichaneventistriggeredafterarandom,continuouslydistributeddelay(uniform,normal,exp onential,etc).Thisnotionisparticularlyimp ortantwhenconsideringenvironmentswithunpredictableb ehaviour,suchascomp onentfailureandcustomerarrivalsinanetwork.Furthermore,emayalsowishtorefertothelikeliho o dofcertaintemp orallogicprop ertiesb eingsatis edbyreal-timesystem,andtohaveamo delcheckingalgorithmforerifyingthetruthoftheseassertions.Supp ortedinpartbyEPSRCgrantsGR/M04617andGR/M13046.1
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