FBirnbaummeasuresBirnbaum(1969)proposedthefollowingmeasureofthereliabilityimportanceofcomponentiattimet:ah(p(t))[B(ilt) =ap;(t)Birnbaum'smeasure is thereforeobtained as the partial derivativeof the system reliabilityh(p(t))withrespecttop;(t).ThisapproachiswellknownfromclassicalsensitivityanalysisIfB(ilt)islarge,a small changeinthereliabilityof component iresultsinacomparativelylargechangeinthesystemreliabilityattimetWhentakingthisderivative,thereliabilities of theothercomponents remain constant-onlytheeffectofvaryingp;(t)isstudied.NTNU-TrondheimNorwegian UniversityofScienceandTechnologywww.ntnu.edu
6 Birnbaum (1969) proposed the following measure of the reliability importance of component ݅ at time ݐ: ሻݐሺ߲ ሻሻݐሺሺ ߲݄ൌ ݐ݅ ܫ Birnbaum’s measure is therefore obtained as the partial derivative of the system reliability ݄ሺሺݐሻሻ with respect to ሺݐሻ. This approach is well known from classical sensitivity analysis. If ܫ ݅ ݐ is large, a small change in the reliability of component ݅ results in a comparatively large change in the system reliability at time ݐ. When taking this derivative, the reliabilities of the other components remain constant – only the effect of varying ሺݐሻ is studied. Birnbaum measures
BirnbaummeasuresIn the definition of Birnbaum's measure,the system reliability is denoted h(p(t))and thesystem reliability isthereforea function of the component reliabilities only,i.e.,ofp(t):(pi(t),p2(t),...Pn(t)).Thismeansthattheallthencomponentsmustbeindependent.This definition of Birnbaum's measure is therefore not useable when the components aredependent, e.g., when we have common-causefailures.NTNU-TrondheimNorwegianUniversityofScienceandTechnologywww.ntnu.edu
7 In the definition of Birnbaum’s measure, the system reliability is denoted h(p(t)) and the system reliability is therefore a function of the component reliabilities only, i.e., of ݐ ൌ ሺଵ ݐ, ଶ ݐ., ሺݐሻሻ. This means that the all the ݊ components must be independent. This definition of Birnbaum’s measure is therefore not useable when the components are dependent, e.g., when we have common‐cause failures. Birnbaum measures
8BirnbaummeasuresConsider a series structure of two independent components,1 and 2,with componentreliabilitiespi andp2,respectively.Assume that pi>p2,i.e.,component1isthemostreliable of the two.The reliabilityof the seriessystem is h(p(t))=PiP2ah(p)1.Birnbaum'smeasureofcomponent1isB(1)==P2apiah(p)=p12.Birnbaum's measure of component2 is IB(2)=ap2Thismeansthat[B(2)>[B(1)andwecanconcludethatwhenusingBirnbaum'smeasure,themost importantcomponent inaseriesstructureistheonewiththelowestreliability.To improve a series structure,we should therefore improve the"weakest"component, i.e.,the component withthelowest reliability.NTNU-TrondheimDNorwegian University ofScienceandTechnologywww.ntnu.edu
8 Consider a series structure of two independent components, 1 and 2, with component reliabilities ଵ and ଶ, respectively. Assume that ଵ ଶ, i.e., component 1 is the most reliable of the two. The reliability of the series system is ݄ ݐ ൌଵଶ. 1. Birnbaum’s measure of component 1 is ܫ 1 ൌ డሺሻ డభ ൌ ଶ 2. Birnbaum’s measure of component 2 is ܫ 2 ൌ డሺሻ డమ ൌ ଵ This means that ܫ 2 ܫ 1 and we can conclude that when using Birnbaum’s measure, the most important component in a series structure is the one with the lowest reliability. To improve a series structure, we should therefore improve the “weakest” component, i.e., the component with the lowest reliability. Birnbaum measures
0BirnbaummeasuresConsider a parallel structure of two independent components,1 and 2,with componentreliabilitiespi andp2,respectively.Assume thatpi>p2,i.e.,component1isthemostreliableof thetwo.The reliabilityof the seriessystem is h(p(t))=pi+p2-pip2.ah(p)=1-P21.Birnbaum'smeasureofcomponent1isB(1)apiah(p)= 1-p12.Birnbaum'smeasure of component2isB(2)1ap2This means that [B(1)>[B(2) and we can conclude that when using Birnbaum's measure,themost importantcomponent inaparallelstructure istheonewiththehighest reliability.Toimproveaparallel structure,we shouldthereforeimprovethe“strongest"component,i.e.,thecomponentwiththehighestreliabilityNTNU-TrondheimNorwegian University ofScienceandTechnologywww.ntnu.edu
9 Consider a parallel structure of two independent components, 1 and 2, with component reliabilities ଵ and ଶ, respectively. Assume that ଵ ଶ, i.e., component 1 is the most reliable of the two. The reliability of the series system is ݄ ݐ ൌଵ ଶ െ ଵଶ. 1. Birnbaum’s measure of component 1 is ܫ 1 ൌ డሺሻ డభ ൌ1െଶ 2. Birnbaum’s measure of component 2 is ܫ 2 ൌ డሺሻ డమ ൌ1െଵ This means that ܫ 1 ܫ 2 and we can conclude that when using Birnbaum’s measure, the most important component in a parallel structure is the one with the highest reliability. To improve a parallel structure, we should therefore improve the “strongest” component, i.e., the component with the highest reliability. Birnbaum measures
10BirnbaummeasuresBypivotaldecomposition,wehaveh(p(t)) =p;(t)-h(1li,p(t))+ (1 -p;(t)) -h(Oi,p(t))Birnbaum'smeasurecanthereforewewrittenasah(p(t))2 = h(1;,p(t) - h(0i,p(t)[B(ilt) =ap;(t)whereh(1.p(t))isthesystemreliabilitywhenweknowthatcomponentiisfunctioningandh(Op(t))isthesystemreliabilitywhenweknowthat component i isnotfunctioning.Thisleads toa very simple way of calculating IB(ilt)-as illustrated by the example on thenextslide.Most computer programs for fault tree analysis computes Birnbaum's measure by thisapproach.NTNU-TrondheimNorwegian University ofScience and Technologywww.ntnu.edu
10 By pivotal decomposition, we have ݐ ,0·݄ ሻ ݐ െ ሺ1 ݐ ,1 ·݄ ݐ ൌ ݐ ݄ Birnbaum’s measure can therefore we written as ݐ ,0݄ െ ݐ ,1 ݄ ൌ ሻݐሺ߲ ሻሻݐሺሺ ߲݄ൌ ݐ݅ ܫ where ݄ 1, ݐ is the system reliability when we know that component ݅ is functioning and ݄ 0, ݐ is the system reliability when we know that component ݅ is not functioning. This leads to a very simple way of calculating ܫ ݅ ݐ – as illustrated by the example on the next slide. Most computer programs for fault tree analysis computes Birnbaum’s measure by this approach. Birnbaum measures