Transient Instant-of-time Mean, Variance trs

Chapter4SolversThischapterdescribesthevariousanalyticandsimulationsolversthatareavailable.ThesolversareselectedthroughtheSolvebutton,andmustberunafterthemodelhasbeenfullyspeciedbyapplyingthevariouseditorsundertheitemEdit(seeChapter3).Generalsuggestionsaboutwhichsolverstoapplyforspecicmodelandmeasuresarepresentedprecedingthedetaileddescriptionofthespecicsolvers.ThischapterusestheclassicationofmeasuresandmodelsasintroducedinSection1.4ofChapter1.4.1ChoosinganAppropriateSolverApplicabilityofboththeanalyticsolversandsimulatorsdependsonthetypeofper-formabilitymeasureandmodelclassconsidered.Allmeasuresandmodelclassesarede-scribedindetailinChapter1.Givenamodelandmeasure,achoicebetweenanalyticsolutionandsimulationhastobemade.Therearesomebasicruleswhenanalyticsolversandsimulatorscanbeapplied:AnalyticsolversonlycanbeappliedtothetwomodeltypesinSection1.4ofChapter1,i.e.,tomodelswithallexponentialactivitiesorwithatmostoneenableddeterministicactivityatthetime.Simulationcanbeappliedtoallmodelclasses.Modelsmusthaveanitestatespaceforanalyticsolution.Ifactivityvariablesareused,onlysimulationcanbeapplied.Table4.1givesthemeasureandmodelclassesforwhichanalyticsolverscanbeapplied,andTable4.2givesthisinformationforthesimulators.Usethetablesasfollows.First,identifywhichmodelclassamodelbelongstoandwhichmeasuretypeistobederived.Thesolverthatcanbeappliedcanthenbefoundintheright-sidecolumn.So,forinstance,thearssolvergivesthemeanoftransientinterval-of-timemeasuresformodelswhereall4-14-2CHAPTER4.SOLVERSAnalyticSolvers(forrewardvariablesonly)Steady-stateInstant-of-timeMean,ApplicableororVarianceorAnalyticModelClassTransientInterval-of-timeDistributionSolverAllactivitiesSteady-stateInstant-of-timeaMean,VariancedssandissexponentialandDistributionTransientInstant-of-timeMean,VariancetrsandDistributionInterval-of-timeMeanarsDistributionpdfExponentialandSteady-stateInstant-of-timebMean,dissdeterministicVarianceactivitiesandDistributionaifonlyraterewardsareused,thetime-averagedinterval-of-timesteady-statemeasureisidenticaltotheinstant-of-timesteady-statemeasure(ifbothexist).bprovidedtheinstant-of-timesteady-statedistributioniswell-dened.Otherwise,thetime-averagedinterval-of-timesteady-statevariableiscomputedandonlyresultsforraterewardsshouldbederived.Table4.1:Modelsandmeasuresversustheapplicableanalyticsolver.Simulators(forallmodels)Steady-stateInstant-of-timeMean,ororVarianceorApplicableTransientInterval-of-timeDistributionVariableSimulatorTransientInstant-of-timeMeanRewardvariableTSimandandandITSimInterval-of-timeVarianceActivityVariableTSimSteady-stateInstant-of-timeMeanandRewardVariableSSimVarianceandActivityVariableTable4.2:Measuresobtainablewithdierentsimulationsolvers.4.1.CHOOSINGANAPPROPRIATESOLVER4-3timedactivitiesareexponentiallydistributed.TheabbreviationsusedinTable4.1fortheanalyticsolvershavethefollowingmeaning(wegivethenamesastheyappearinthemenuitemsofthecontrolpanel):dss:Solve-DirectSteady-StateSolveriss:Solve-IterativeSteady-StateSolverdiss:Solve-DeterministicIterativeSteady-StateSolvertrs:Solve-TransientSolverars:Solve-AccumulatedRewardSolverpdf:Solve-ProbabilityDistributionSolverForthesimulatorsinTable4.2weusethefollowingabbreviations:SSim:Solve-Steady-StateSimulatorTSim:Solve-TerminatingSimulatorITSim:Solve-ISTerminatingSimulator(ImportanceSampling).Animportantdierencebetweenhowtheanalyticsolversareappliedandhowthesimulatorsareappliedistheprecisespecicationoftheperformabilitymeasure.Foranalyticsolutions,thechoiceofthesolverdeterminesthespecicperformabilitymeasurewhichwillbecomputed.Inotherwords,givenaspeciedrewardstructure,theperformabilitymeasure(includingaspectsasmeanorvariance,transientorsteady-state,etc.)isspeciedbythechoiceofthesolver.Forsimulation,ontheotherhand,theprecisespecicationoftheperformabilityvariableiscarriedoutseparatelyunderEdit-PerformabilityVariables,byselectingEditStatsinthePerformabilityVariableSpecificationwindow.Onlywhethertransientorsteady-stateresultswillbeobtainedisdeterminedbythechoiceofthesimulator.ItmustbeunderstoodthattheTables4.1and4.2givethepotentialuseofthesolvers;itisnottosaythattheresultswillalwaysbederivedinareasonabletimespanorcanbetintostandardsizememory.Theseconsiderationsareofprimaryimportanceforthepracticalapplicationofthesolversandarethereforediscussedinsomemoredetail.AdvantagesanddisadvantagesoftheuseofthedierentsolversprovidedbyUltraSANarepresentedinfourlists:advantagesanddisadvantagesofanalyticsolutionsandadvantagesanddisadvantagesofsimulation.Thelistsarenotclaimedtobeexhaustive,buttrytocapturetheproblemsmostlikelytobeencounteredinthepracticalapplicationoftheUltraSANsolvers.4-4CHAPTER4.SOLVERSAdvantagesofanalyticsolutionExactcomputationofasolutioniscarriedout,i.e.,opposedtosimulationtheoutcomedoesnotdependonageneratedstreamofpseudo-randomnumbers.Fortheinstant-of-timeperformabilityvariables,distributionscanbeobtainedwithoutextracostinadditiontotheirmeanandvariance.Accuracyofthesolutioncan,formostsolvers,beincreasedwithoutexcessiveincreaseinth