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Stochastic order for redundancy allocations in series and parallel systems. (English) Zbl 0748.62053

Consider a system with \(n\) components where the lifetimes \(T_ 1,\dots,T_ n\) of components \(1,\dots,n\), respectively, are independent. Assume that another component with lifetime \(T\), which is independent of \(T_ 1,T_ 2,\dots,T_ n\), is also available as a parallel or as a standby spare. The authors are concerned with the problem as to what component should the spare be allocated in order to stochastically maximize the system lifetime.
First they consider the case of a \(k\)-out-of-\(n\) system and parallel redundancy. Let \(\tau^{(i)}_ k\) denote the lifetime of such a system when the spare is allotted to the ith component. They show that if \(T_ 1\leq_{st} T_ 2\leq_{st}\dots\leq_{st} T_ n\) then \(\tau_ k^{(1)}\geq_{st}\tau_ k^{(2)}\geq_{st}\dots\geq_{st}\tau_ k^{(n)}\), where \(\leq_{st}\) denotes the usual stochastic order. Thus, the optimal policy is to allocate the spare to the weakest component. Next they consider the case of series [respectively, parallel] system and standby redundancy. Let \(\tau_ n^{[i]}\) [respectively, \(\tau_ 1^{[i]}\)] denote now the lifetime of the series [parallel] system when the spare is allotted to the \(i\)th component. The authors show that if \(T_ 1\leq_{lr} T_ 2\leq_{lr}\dots \leq_{lr} T_ n\) then \(\tau_ n^{[1]}\geq_{st}\tau_ n^{[2]}\geq_{st}\dots\geq_{st} \tau_ n^{[n]}\) [\(\tau_ 1^{[1]}\leq_{st}\tau_ 1^{[2]}\leq_{st}\dots\leq_{st}\tau_ 1^{[n]}\)], where \(\leq_{lr}\) denotes the likelihood ratio order. Thus, in the series [parallel] system case, the optimal policy is to allocate the spare to the weakest [strongest] component. The authors derive also some results regarding optimal allotment of standby redundancies.
Reviewer: M.Shaked (Tucson)

MSC:

62N05 Reliability and life testing
60K10 Applications of renewal theory (reliability, demand theory, etc.)
90B25 Reliability, availability, maintenance, inspection in operations research
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