exact system reliability

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1、EXACTSYSTEMRELIABILITYWangwei2012.04.16Fourmethodsforcalculationofexactsystemreliabilityatagiventimet0whenthencomponentsareindependent.Tosimplifythenotationwereplacethecomponentreliabilitypj(t0)bypj.•ComputationbasedontheStructureFunction•ComputationbasedonPivotalDecomposition•ComputationbasedonMini

2、malCut(path)sets•TheInclusion-ExclusionPrincipleComputationbasedontheStructureFunctionThemoststraightforwardmethod•SeriesstructuresnStructurefunctionφ(())Xt=∏Xti()i=1Systemvariablearebinary,thenXt()iEXt(())=0Pr(⋅=X0)1Pr(+⋅=X1)=pt()iiiiSystemreliabilitypst()=E((()))φXt=hptpt((),(),...,pt())=ht(())p12

3、nTheseriessystemreliabilityisnnnht(())p=EXt((()))φ=E(∏∏∏Xti())=EXt(())ii=pt()ii=11=i=1ComputationbasedontheStructureFunction•Parallelstructuresnnφ(())1Xt=−−=∏(1Xtii())Xt()i=1i=1nht(())p=EXt((()))1φ=−−∏(1EXt(i()))i=1nn=1−−=∏(1ptii())pt()i=1i=1ComputationbasedonPivotalDecomposition•PivotalDecomposit

4、ionφφ()xx≡xx(1,)+−(1)(0,)φxiiiix=1⇒=φφ()xx1⋅(1,)iix=0⇒=φφ()1(0,)xx⋅iin-dimensionalbinaryvectorsyyjj1−φφ()xy=∑y∏jxxjj(1−)()φφ()xx=xx(1,)+−(1)(0,)φx3333=(x31xxxx2)(45)(1+−xx3)(14xx25x)=ComputationbasedonPivotalDecomposition•Systemreliabilitynyyjj1−Structurefunctionφφ(Xy)=∑y∏XXjj(1−)()j=10wherey

5、isn-dimensionalbinaryvectorsand.01≡yyjj1−ifareindependent,arealsoXX12,,,XnXXjj(1−)independentforallj.Since,theny=0or1iyj11−−yyjjyjEX[(1−=X)]p(1−p)jjjjnyyjj1−ps=E[(φφXy)]=∑∏pjj(1−p)()yj=1First,determinebyexperimentthevalueofφ()yNextputtheobtainedin.(,systemfunctioning)φ()yφ()1y=ComputationbasedonMin

6、imalCut(Path)setswhenalltheminimalcutsetsoralltheKK,,K12kminimalpathsetsaredetermined,thePP,,,P12pstructurefunctioncanbewrittenkpφφ()XXXX=∏∏iior()=j=1iK∈=jj1iP∈jThesystemreliabilityisobtainedbyreplacingalltheXinthestructurefunctionbycorresponding.piiComputationbasedonMinimalCut(Path)setsThestruc

7、tureinfigurehasthereminimalpathsets:PPP={1,2,3},={1,2,4},={1,3,4}1233Henceφ()X=∏Xi=XXX123XXX124XXX134j=1iP∈j=−−1(1XXX)(1−XXX)(1−XXX)123124134==++−XXXXXXXXX2XXXX1231241341234h()p=++−ppppppppp2pppp1