EfficientSvstemsReliabilityAnalysisANALYSISOFCOMPLEXSYSTEMSSurvival signaturesystems availability interms of numberof working components》statevectorX = (X1..,Xm)Xi=1:componentiis workingX,=O:componentiisNoTworking》structurefunction((≤) = (X1*,X)p(x)=1 : system is workingp(×)=O : system is NOTworking》exampleβ(×1 =1,X2 = 0,× = 0)= ((1,0,0)= 0((× = 1, ×, = 1,×, = 0) = (P(1,1,0) = 1Michael Beer6/24
Michael Beer 6 / 24 ANALYSIS OF COMPLEX SYSTEMS Survival signature Efficient Systems Reliability Analysis • systems availability in terms of number of working components » state vector xx x = ( 1 m ,., ) xi = 1: component i is working xi = 0: component i is NOT working » structure function ϕ =ϕ (x xx ) ( 1 m ,., ) : system is working : system is NOT working ϕ = (x 1 ) ϕ = (x 0 ) » example ϕ = = = =ϕ = (x 1x 0x 0 100 0 12 3 , , ) ( , , ) ϕ = = = =ϕ = (x 1x 1x 0 110 1 123 , , ) ( , , )
EfficientSystemsReliabilityAnalysisANALYSISOFCOMPLEXSYSTEMSSurvival signaturecompressed representationof systemavailability》systemwithmcomponentsof the sametype》Ioutof mcomponentsareworkingD(Z(α)Φ(0) = 0, @(m) = 1XESprobability that system works whenI out of m components are working》examplem=3, 1=2:S, = ((1,1,0),(1,0,1), (0,1,1)(β(1,1,0) = 1, (1,0,1) = 1, β(0,1,1) = 0(区)=号(1+1+0)=号D(2and further: @(0) = 0, Φ(1) = 0, Φ(3) = 1MichaelBeer7/24
Michael Beer 7 / 24 ANALYSIS OF COMPLEX SYSTEMS Survival signature Efficient Systems Reliability Analysis • compressed representation of system availability ( ) ( ) − ∈ Φ= ⋅ϕ ∑ l 1 x S m l x l » system with m components of the same type » l out of m components are working Φ= Φ = (0 0 m 1 ) , ( ) » example ϕ =ϕ =ϕ = (110 1 101 1 011 0 , , , , , , ) ( ) ( ) m = 3, l = 2: S 110 101 011 l = {( , , , , , , ) ( ) ( )} ( ) ( ) ( ) − ∈ Φ = ⋅ ϕ = ⋅ ++ = ∑ l 1 x S 3 1 2 2 x 110 2 3 3 and further: Φ=Φ=Φ= (0 0 1 0 3 1 ) , , ( ) ( ) probability that system works when l out of m components are working