An Allocation Scheme for Estimating the Reliability of a Parallel-Series System

We give a hybrid two-stage design which can be useful to estimate the reliability of a parallel-series and/or by duality a series-parallel system. When the components' reliabilities are unknown, one can estimate them by sample means of Bernoulli observations. Let be the total number of observations allowed for the system. When is fixed, we show that the variance of the system reliability estimate can be lowered by allocation of the sample size at components' level. This leads to a discrete optimization problem which can be solved sequentially, assuming is large enough. First-order asymptotic optimality is proved systematically and validated Monte Carlo simulation. 1. Introduction In reliability engineering two crucial objectives are considered: to maximize an estimate of system reliability and to minimize the variance of the reliability estimate. Because system designers and users are risk averse, they generally prefer the second objective which leads to a system design with a slightly lower reliability estimate but a lower variance of that estimate, (e.g., [1]). It provides decision makers efficient rules compared to other designs which have a higher system reliability estimate, but with a high variability of that estimate. In the case of parallel-series and/or by duality series-parallel systems, the variance of the reliability estimate can be lowered by allocation of a fixed sample size (the number of observations or units tested in the system), while reliability estimate is obtained by testing components, see Berry [2]. Allocation schemes for estimation with cost, see, for example, [2–7], lead generally to a discrete optimization problem which can be solved sequentially using adaptive designs in a fixed or a Bayesian framework. Based on a decision theoretic approach, the authors seek to minimize either the variance or the Bayes risk associated to a squared error loss function. The problem of optimal reliability estimation reduces to a problem of optimal allocation of the sample sizes between Bernoulli populations. Such problems can be solved via dynamic programming but this technique becomes costly and intractable for complex systems. In the case of a two component series or parallel system, optimal procedures can be obtained and solved analytically when the coefficients of variation of the associated Bernoulli populations are known, (cf., e.g., [5, 8]). Unfortunately, these coefficients are not known in practice since they depend themselves on the unknown components’ reliabilities of the system. In [9], the author has defined a sequential allocation