11Birnbaummeasures:ExampleConsiderthesimplesystem:We want to determine Birnbaum's measurefor component 3.NTNU-TrondheimNorwegian University of梦Science and Technologywww.ntnu.edu
11 Consider the simple system: We want to determine Birnbaum’s measure for component 3. Birnbaum measures: Example
12Birnbaummeasures:ExampleConsiderthesimplesystem:WewanttodetermineBirnbaum'smeasureforcomponent3.1.When we know that component 3 isfunctioning (i.e.,X(t)=1),component 2 isirrelevant and the system reliability ish(p(t))=pi(t).2.When weknowthat component3is not functioning (i.e.,X3(t)=O), the system reliabilityis h(p(t)) = pi(t)p2(t).Birnbaum'smeasureistherefore:IB(3/t) = h(13,p(t)) -h(03,p(t)) = p1(t) -P1(t)p2(t)NTNU-TrondheimDNorwegianLUniversityofScience and Technologywww.ntnu.edu
12 Consider the simple system: We want to determine Birnbaum’s measure for component 3. 1. When we know that component 3 is functioning (i.e., ܺଷሺݐሻ ൌ 1), component 2 is irrelevant and the system reliability is ݄ ݐ ൌଵ ݐ. 2. When we know that component 3 is not functioning (i.e., ܺଷሺݐሻ ൌ 0), the system reliability . ݐ ଶ ݐ ଵൌ ݐ ݄ is Birnbaum’s measure is therefore: ݐ ଶ ݐ ଵെ ݐ ଵൌ ݐ ,0ଷ݄ െ ݐ ,1ଷ݄ ൌ ݐ3 ܫ Birnbaum measures: Example
13BirnbaummeasuresNotethatBirnbaum'smeasurejB(ilt)ofcomponent ionlydependsonthestructureofthesystemand thereliabilitiesof theothercomponents.B(ilt)isindependentoftheactual reliabilityp;(t)ofcomponenti.IB (ilt) = E[(1i,X(t)] - E[(Oi,X(t)] = E[(1i,X(t) - (Oi, X(t))= Pr(β(1i,X(t)) - β(Oi,X(t)) = 1)This istosaythatB(ilt)is equaltotheprobabilitythat (1i,X(t))isacritical pathvectorforcomponentiattimet.Birnbaum'smeasure isthereforetheprobabilitythatthe system is in sucha stateattimetthatcomponentiiscriticalforthesystem.NTNU-TrondheimNorwegian University of莎Science and Technologywww.ntnu.edu
13 Birnbaum measures Note that Birnbaum’s measure ܫ ݅ ݐ of component ݅ only depends on the structure of the system and the reliabilities of the other components. ܫ ݅ ݐ is independent of the actual reliability ݐ of component ݅. ݐ ࢄ ,0߮ െ ݐ ࢄ ,1 ߮ ܧൌ ݐ ࢄ ,0 ߮ ܧെ ݐ ࢄ ,1 ߮ ܧൌ ݐ݅ ܫ ൌ Pr ሺ߮ 1, ࢄ ݐ െ߮ 0, ࢄ ݐ ൌ 1ሻ This is to say that ܫ ݅ ݐ is equal to the probability that (1, ࢄ ݐ ( is a critical path vector for component ݅ at time ݐ. Birnbaum’s measure is therefore the probability that the system is in such a state at time ݐ that component ݅ is critical for the system
14Birnbaum measuresAssumethatcomponentihasfailurerateAi.Insomesituationswemaybeinterestedinmeasuringhowmuchthesystemreliabilitywill changebymakingasmall changetoAi.Thesensitivityofthesystemreliabilitywithrespecttochangesincanobviouslybemeasuredbydpi(t)ah(p(t))ah(p(t))pi(t)B(ilt)anianandp;(t)Asimilarmeasurecanbeusedforallparametersrelatedtothecomponentreliabilityp;(t),fori=1,2,..,n.Insomecases,several components inasystemwill havethesamefailurerate 入.TofindthesensitivityofthesystemreliabilitywithrespecttochangesinA,wecanstilluseah(p(t))aaiNTNU-TrondheimNorwegian University ofScienceand Technologywww.ntnu.edu
14 Birnbaum measures Assume that component ݅ has failure rate ߣ .In some situations we may be interested in measuring how much the system reliability will change by making a small change to ߣ .The sensitivity of the system reliability with respect to changes in ߣ can obviously be measured by ߲݄ሺሺݐሻሻ ߣ߲ ൌ ߲݄ሺሺݐሻሻ ሻݐሺ߲ · ݐ ߲ ߣ߲ ݐ ߲ · ݐ݅ ܫ ൌ ߣ߲ A similar measure can be used for all parameters related to the component reliability ሺݐሻ, for ݅ ൌ 1, 2, . . . , ݊. In some cases, several components in a system will have the same failure rate λ. To find the sensitivity of the system reliability with respect to changes in λ, we can still use డሺሺ௧ሻሻ డఒ
15Birnbaum measuresInapractical reliabilitystudyofacomplexsystem,oneofthemosttime-consumingtasksisto find adequate estimatesfor the inputparameters(failurerates,repair rates,etc.).Insomecases,wemaystartwithratherroughestimates,calculateBirnbaum'smeasureofimportanceforthevarious components,ortheparameter sensitivities,and then spendingthemostofthetimefindinghigh-qualitydataforthemost importantcomponents.ComponentswithaverylowvalueofBirnbaum'smeasurewill haveanegligibleeffectonthe system reliability,and extraefforts findinghigh-quality data for such components maybeconsideredawasteoftime.NTNU-TrondheimNorwegian University ofScienceand Technologywww.ntnu.edu
15 Birnbaum measures In a practical reliability study of a complex system, one of the most time‐consuming tasks is to find adequate estimates for the input parameters (failure rates, repair rates, etc.). In some cases, we may start with rather rough estimates, calculate Birnbaum’s measure of importance for the various components, or the parameter sensitivities, and then spending the most of the time finding high‐quality data for the most important components. Components with a very low value of Birnbaum’s measure will have a negligible effect on the system reliability, and extra efforts finding high‐quality data for such components may be considered a waste of time