Estimation of system reliability from stress-strength relationship

In this paper, we estimate the reliability of parallel system with two components. We assume that strengths of these components follow a bivariate exponential (BVE) distribution. These two components are subjected to a common stress which is independent of the strength of the components. If the strengths (X(1), X(2)) are subjected to a common random stress (Y), then the reliability of a system or system reliability (R) is given by R = P[Y < Max(X(1),X(2))]. We estimate R when (X(1),X(2)) have different BVE models proposed by Marshall-Olkin (1967), Block-Basu (1974), Freund (1961) and Proschan-Sullo (1974). The distribution of Y is assumed to be either exponential or gamma. The asymptotic normal (AN) distributions of these estimates are obtained. We present a numerical study for obtaining MLE of R in all four BVE models when the common stress (Y) is exponentially distributed.