Estimation of Reliability in a Multicomponent Stress-Strength Model Based on a Marshall-Olkin Bivariate Weibull Distribution

Abstract

In this paper, we consider a system which has $k$ s-independent and identically distributed strength components, and each component is constructed by a pair of s-dependent elements. These elements $(X_{1},Y_{1}),(X_{2},Y_{2}),\ldots ,(X_{k},Y_{k})$ follow a Marshall-Olkin bivariate Weibull distribution, and each element is exposed to a common random stress $T$ which follows a Weibull distribution. The system is regarded as operating only if at least $s$ out of $k (1\leq s\leq k)$ strength variables exceed the random stress. The multicomponent reliability of the system is given by $R_{s,k}=P$ (at least $s$ of the $(Z_{1},\ldots ,Z_{k})$ exceed $T$ ) where $Z_{i}=\min (X_{i},Y_{i})$ , $i=1,\ldots ,k$ . We estimate $R_{s,k}$ by using frequentist and Bayesian approaches. The Bayes estimates of $R_{s,k}$ have been developed by using Lindley's approximation, and the Markov Chain Monte Carlo methods, due to the lack of explicit forms. The asymptotic confidence interval, and the highest probability density credible interval are constructed for $R_{s,k}$ . The reliability estimators are compared by using the estimated risks through Monte Carlo simulations.