2010InternationalConferenceonIntelligentComputationTechnologyandAutomationANewAssessmentMethodforSystemDuanReliabilityBasedonDynamicFaultTreeRongxing,WanGuochun,DongDecunUniversity,Shanghai,201Ton萄i804,Chinaduanrongxing@126.comAbstrac手.一AceordingtothedeficiencychainapproachinoftraditionaIMarkovmodularthismethodfaulttreedynamicfaulttreeanalysis.anewreducethecomputationalcomplexityofanalysis.butisstillfacedwithstatespacecanmethodforsystemreliabilityanalysisisproposed.Thispaperfocusessystemintoindependentlinear-timeusingalgorithm,andtheprocessingmethodfordifierentsubtrees:BinarydecisiondiagramsolutionforstaticsubtreesandBayesianNetworksolutionforondividingthefaulttreeofasubtreesexplosionproblembecauseofusingMC.Inordertoovercomethisdisadvantage,anewassessmentmethodforsystemreliabilitybasedonDFTisproposed.InmisP印er,DFTofmesystemismodularizedintoindependentstaticsubtreesanddynamicsubtrees,andsubsequentlysolvedbyBDDandBayesianNetwork门BN)modelseparately.Inaddition,themappingofdynamiclogicgatesintodiscrete・timeBayesiannetworkisalsoillustrated.Atlast,themodularmethodhasbeendemonstratedusingtheDFTmodelingofasatellitekeydevice.whichcanovercomethestateexplosionproblem,andisusefulforassessingthereliabilityoflargeandcomplexsystems.IIJtl’dynamicprovidedsubtrees.respectively.Inaddition.anapproachisalsoformappingsomeassessdynamicaIogicgatesintodiscrete-timeBayesiannetworkAtIast.themodularmethodhasbeenappliedtothereliabilityofsatellitekeYdevieeandtheresultshaveshownthattheproposedstateexplosionmethodusefulcanovercometheproblem,andisforassessingthereliabifityofIargeandcomplexsystems.F4HllK删|rds卸枷icTt雠;眦吣栅删e嘲iDecisionDia膏ram;BayesianBNetworkEERTnaryMODULARIZAT燃OF1”1’FAULTsystems,manybetreatedInthelargeandcomplextheircombinationdon’tinteractthemostwidelyusedmodelforreliabilityanalysis.Itbecomespopularbecauseitisveryeasytouse.presentsthedesigner、析manintuitivehigh—levelFaultTree(Frliscomponentsorthesesodirectly,singly.Intheisdividedintounconnecteddynamicfaultdistinctpartstreecansystem,andcanbeefficientlysolvedusingmanytechniques.However.traditionalfaulttreeanalysisisamethodbasedonstaticlogicandstaticfaultmechanism,soseveralimportanttypesofdynamicbehaviorinadvancedabstractionofthefault-tolerantsystemscannotbeadequatelycaptured,e.g.,fault&errorrecovery.sequence.dependentfailures.andtheindependentstaticsubtreesusingaveryefficientlinear-fimeanalysis,afaulttreesubtreesandindependentdynamicalgorithmf41.TheflowchartofDFTisshowninfigure1.BDDisusedforstaticsubtreesanddynamicsubtreesaretreatedbyconveningdynamiclogicgatesintodiscrete-timeBayesiannetwork.Theindependentmodulesaresolvedseparatelyandtheresultsareintegratedtoobtaintheresultforthewholeuseofspares.Intherealworl正systemfailureoftenhasdynamiccharacteristies.MarkovChainfMC)hasproventobeadynamicfaulttree.Eachprocessedmodelcanbeverysmall.Therefore,thismodulartechniquecanreduceeffectivelythesizeofversatiletoolforcomponentbehavior.Itmodelingcomplexhasbeenextensivelydynamicusedforisdependabilityanalysisofdynamicfacedwiththeinfamousstatespaceexplosionproblemn1wherethestatetobegeneratedgrowsexponentially谢tllthenumberofcomponentscomprisedinthesystem,whichmakesestablishmentandsolutiononMCmodelverysystems.However,MCdi衔cult.So.ateamofresearchersledbyDugan,aprofessorattheUniversityofVirginiahasstudiedthespacestationaswellasairtrafficcontrolcomplexsystemsindepth.andestablisheddynamicfault.treemodelsusingMarkovtheoryandcombinatorialmathematics『2].ButthetraditionalFigure1.TheflowchartofDFTdynamicfault-treemodelsaresolvedbyMarkovmodelwhichalsohastheinfamousstateSl:Iaceexplosionproblem.Toreducethecomplexity.modulartechniqueisusedtoanalyzetheisusedforⅡI.STATICFAULTTREEANALYSISdynamicdynamicfaulttodiagram(BDD)isusedtree(DFT)[3】.Binarydecisionanalyzestaticfaulttree.whileMCABDDisadirectedacyelicgraph.AllpathsthroughtheBDDstartattherootvertexandterminateinoneoftwostates—afaulttree.Theindependentmodulesareseparatelyandtheresultsareintegratedtogettheresultforthewholedynamicfaultflee.Toacertainextent,solvedl-state(systemfailure),orBDDisaSuccess).ThecomposedofterminalO-state(systemandnon・978・0-7695-4077.1/10¥26.00o2010旺EE130110.1109/ICICTA.2010.237219④comIEE峻E南terminalconnectedbybranches.vertices,whichareTerminalverticescorrespondtothefmalstateofthesystemandnon—terminalverticescorrespondtothebasiceventsofInthisPap%adiscrete-timeBNf81fordynamicfaulttreeanalysisisfDTBN)formalismf717applied.Eachnodethefaulttree.TheBDDmethoddoesnotanalyzethefaulttreedirectly,butconvertsthetreetoaBDDthatrepresentstheBooleanequationforthetopevent.AcommonlyusedmethodofconstructingBDDwasdevelopedbyRauzy『51.Thisapproachappliesanif-then-elsetechniquetoeachofthefaulttree.ThecomplexityofcalculationforquantitativeanalysisofafaulttreeislinearwiththesizeoftheBDD.andthesizeofaBDDstructuredependscriticallythebasiceventorderingofafaulttree.A200dvariableorderingcanresultinaveryefficientanalysisandapooronrepresentsasystemcomponent.Asystemcomponentcanbeeitherabasiccomponentorasubsystemdescribingtheinteractionbetweenacollectionofcomponents.Wedivideathetimelineinto,什lintervals.Eachnodevariablehasfinitenumber,什lofstates.Thenfirststatesdividethetimenintervalf0,71(丁isthemissiontime)intoRandomvariablesequalintervals。gatesintheandthelastState,什1representsthetimeintervalIT,∞1.XisinStateorn+lmeansthatthecorrespondingbasiccomponentduringthemissiontime.B.gateoutputdidnotfailcouldn’tgenerateaBDD.TherulesoforderingofbasiceventSproposedinthisParIel-aredescribedorderingevenMappingdynamictogicgatesintoDTBN舔follows【6】:・DFTextendstraditionalfaulttreesbydefiningspecialgatestocapturethecomponents’sequentialandfunctionalConsiderthebasiceventsinThebasiceventsbasiceventsconsideredfirst;top.downordering.inthefrontofafaulttreeareadependencies.Currentlytherearesixtypesofdynamicgatesfunctionaldefined:thedependencygate(FDEP)。thecol正・Thewiththeclosedistanceinthehot,andwarmsparegates{CSP,HSP,WSP),thepriorityANDgate(PAND).thesequenceenforcinggate(SEQ).Here,weoffaulttree,shouldalsokeeptheclosedistanceintheorder;structure・Therepeatingtimesofthebasiceventsdeterminetheirordering.Thebasiceventswhichrepeatthemosttimesareconsideredfirst.gatesasilThefunctionatdependencygateAfunctionaldependencygate饵DEP)ismodelingsituationswhereonebrieflydiscusstheFDEP,theCSP,andtheSEOtheywillbelaterusedinourexamples.usedforcomponent’scorrectⅣ.A.DFTANALYSISBASEDONBNoperationisdependentuponthecorrectoperationofsomeothercomponent.Ithasasingletriggerinput,whichcouldbeanotherbasiceventortheoutputofanothergate,aROll-dependentoutputreflectingthestatusofthetrigger,andoneormoredependentbasicevents.Figure2showsfunctionaldependencygateanditsequivalentDTBN.Theprobabilitydistributionofthenon-rootnodecanbeexpressedas:BayesianNetworkABayesiannetworkisadirectedacyclicgraph(DAG).AlsoitiscalledBayesianbeliefnetwork,isagraphicalmethod.Therandomvariablesaredenotedbynodesandthedirectedarcstel:Iresenttheconditionaldependenciesamongthenodes.EachnodehasaprobabilitydensityfunctionI矿毛蚰(矿6一1),(o<Y<x≤七)associated、砘thit.Thearcemanates行omaparentnodetoachildnode.Anodewithoutanyarcs1inkingintoitisknownasarootnode.anode、加tharrowslinkingintoitisknownaschildnode.Achildnodewithoutanyarcsleadingoutisa1eafnode.Eachchildnodethuscarriesaconditionala={l一∑只∥(o<Y=工≤I|})y<x【o,(o<工<y≤七)probability.ABayesiannetworkismainlycomposedofanetworkstructureandconditionalprobabilities.ThenetworkstructureisthequalitativepartofaBNmodel,andisusedtodescribequalitativelytheprobabilitydependenceamongvariables.Theconditionalprobabilityisthequantitativepartofthemodel.Itreflectsthedegreeofprobabilitydependenceofcorrelatingnodes(variables).InaBN,the{ointprobabilitydistributionofallthenodescanbewrittenastheproductovera11nodesoftheconditionalprobabilityofeachnodegivenitsparents.LetV={n,...,%}bethesetofNnodescomprisingdistributionBayesiannetwork.ThentlleiointprobabilityoverallvariablesrepresentedbythenodesintheaⅣFigure2.FDEPanditsequivalentDTBNgraphisgivenbygateThesequence-enforcinggateforceseventstooccurinaparticularorder.Theinputeventsareconstrainedtooccurintheleft.to.rightorderinwhichtheyappearunderthegate.2)ThesequenceenforcingWhen以矿)=17尸(KIpa(Vj))fWherepa(功isthesetofallparentnodesofnodeK.DFT,itneverhappensthatthefailuresequencetakesplaceindifierentorders.Figure3showssequenceenforcinggateanditsequivalentDTBN.Theprobabilitydistributionofthenon—rootnodecanbeaaSEQisfoundinexpressedas:£,=P(曰=[O一1)A,ya]IA=【0一OA,xa])・ModuleBconsistsoftwodevices.11ismasterand12iSslave;・ModuleCineludesthreedevices.J1ismaster.whileJ2andJ3areslave;ModuleDiscomprisedofthreedevices.ThefailureofdeviceKor0,(O<Y<X≤七)・1(屯一凡)(P毛6一1)’卜瓦20i(da。丽-aJ覆a-面1),(O<y=xSk)(无一五)p讪-1)7thefailureofbothLandMwillresult”。一。7…7型=≠祟芸娑竺’(0<x<y钏inthefailureofmoduleD.Basedonthisanalysis.thefaulttreemodelforthepowercontrollerisshowninFigure5.Asmentionedabove,thefirstst印iStOdividethefaulttreeofsystemintoindependentalgorithm.faulttreeiSaORObviously,Staticfour-inputgate,andiIynarnjcfaultfleesconsistofmoduleA,B,CandD.ThesubtreesusingthelineartimedⅦ枷cSEQistosolvestaticfaulttreeandd”lamicfaulttreesusingBDDandBNrespectively.Atlast,theresultsofvariousindependentsubtreesareintegratedtoobtainnlenextstepreliabilityofthepowercontroller.Figure3.SEQanditsequivalentDTBNcoldsparegateareusedformodelingcoldandhotpooledspares.AsparegatehasoneprimaryinputandoneormorealternateinputsfromcomponentsthatCallreplacetheSparegatesprimarycomponentuponfailure.Sparegatescallf-ail.butusuallyatslowerratesbeforetheirusage.TheCSPgatehaswhichbecomestrueafteralltheinputeventsoccur.Figure4showscoldsparegateanditsequivalentone引砌eoutputDTBN.T1leprobabilitydistributionofthenon-rootnodebeexpressedas"call只.,=P(B=【(y一1)A,yA】lA=【(工一0a,xa])10,0<Y<x≤kFigure5.DFrmodelofpowercontroller一1/i△矿瓢oq),O<x<y≤kA.Solutionfo,.staticsubtreesfour-inputORgateincluding八B,C,ofA,B,C,andDisrandombecauseoforderingtheirequalpositioninthesubtree.SoanequivalentBDDisshowninFigure.6.aStaticsubtreeisandD.TheI譬P’④^丫@Figure4.CSPanditsequivalentDTBNV.APPLICATIONEXAMPLEOFPOWERCONTROLLERREL队BILⅡYInthissection,allexampleofpowercontrollerinasatelliteisprovidedtodemonstrate廿leeffectivenessoftheproposedmethod.ThepowercontrollerconsistsoffourismasterdeviceFigure6.TheequivalentBDDoftheORgateofthesystemcanmodules:A、B、CandD『91.・ModuleAincludestwobatteries:H1device,whileH2iSconsistsofthreebatteries.Iffails.thedevicefails;Thefailureprobabilitybeexpressedas"slavedevice.EachP(.r)=P(彳)+P(丑彳)+烈CBA)+P(DCBA)anybatteryofthree221B.SolutionfordynamicDynamicwecansubtreesACKNOWLEDGMENTsubtreesconsistoffourthemintounits:凡B,C,andDTBNratesD.theallconvertmethodmentionedequivalentabove.ThefailureusingOurworkissupportedby863projectfundofChineseandMinistryofScienceTechnology(ItemNo.2007AAlfor12247).REFERENCEScomponentsofthepowercontrollerarederivedfrom『9].AssumeTl00000.sothefailureprobabilitiesforfourmodulesandthewholesystemcanbesolved.OfCOUrSe,youcan【l】resorttoTableresultstheBNtools『l01tosolvethemifyoulike.1summarizestheresults.whichcoincidewi也theSouza,eSilva,Edmundode,Mejia,Pedro,“Statespaceexplorationinmarkovmodels,”ACMSIGMElRICS,PerformEvalRev.1992。V01.20,No.1.pp.152-166.DuganJB,BavusoSJ,BoydM丸‘'Dynamicfault-treemodelsforfault・tolerantcomputersystems,”正EETransactionsoilReliabill饥in【9].n砸FAILURERATESOFDIFFERENTMODULESANDTABLELTim1992.V01.4l,No.3.PP.363.377.WHOLESYSTEM.ModuleOF【3】Failurerate8.9967e—lO3.9986e.89.9976e.130|3530.353R.Gulati,J.B.Dugan,‘‘AmodularapproachforanalyzingstaticandfaultAnnualdynamicandtrees,”ProceedingsReliabihtyMaintainabilitySymposium,1997,pp.57.63.systemModuleAModuteBModuleCModuleDSystem【4】Y.Dutuit.A.Rauzy,“Alinear-timealgorithmt0findmodulesoffaulttrees,”IEEETransactionsonReliability,1996,V01.45,No.3。pp.422-425.A.Rauzy,‘'NewAlgorithmsforFaultTreeAnalysis.”RtliabilityEngineering【5】【6】andSystemSafety,No.40,1993,lap.203.21.eventM玳Ping,ToNGJiejuan,ⅪShuren,“Basictreeanalysesusingbinarydecisionorderinginfault儆hnology,2005,pp.1646-1649.(inThisPaperhaspresentedanew,effectiveandflexiblemodularmethodforsystemreliabilityanalysisbasedonDFT.Largedynamicfaulttreesaredividedintoindependentusingaveryefficientlineartimealgorithm.Differentmethodsareappliedtosolvedifferentsubtrees.subtreesBDDisusedtoanalyzestailefaultsubtree.whileBNisusedfordynamicfaultsubtree.andtheresultsofdiffbrentsubtreesintegratedtogettheresultforthewholedynamicfaulttree.ThemethodpresentedinthisPaDel"canovercomestatespaceexplosionproblemandreducethecomplexityofsystemanalysis.IthasbeendemonstratedusingtheDFTmodelingofsatellitekeydevice,anditisusefulforassessingthereliabilityoflargeandcomplexsystems.areanddiagram,”TsingIluaScienceandChinese)H.Boudali.J.B.Dugaa,“Adiscrete-timeBayesiannetworkreliabilitymodelingandanalysisframework,”ReliabilityEngineeringSystemSafety,2005。V01.87.1ap.337.349.【8】【9】H.Langseth,L.Portinale,“Bayesiannetworksinreliability.”ReliabilityEngineeringandSystemSafety,2007,V01.92,PP.92-108.ZHANGHUANGXiao-jie,zHA0Hai-tan,MIAOQiang,zHANGwei,ofSatelliteSystemHong-zhong.“ReliabilityBasedoilDynamicFaultTree,”JournalofAstronautic,2009,V01.33。Analysispp.1249一1254.(inChine辩)【io]MushyKP,‘mIcBayesnettoolboxforMatlab,”ComputingScienceandStatistics,200l,V01.33,pp.33l-351.