System stress-strength reliability: The multivariate case

Present day complex systems with dependence between their components require more advanced models to evaluate their reliability. We compute the reliability of a system consisting of two subsystems S-1, and S-2 connected in series, where the reliability of each subsystem is of general stress-strength type, defined by R-1 = P(A(T)X > (BY)-Y-T). A & B are column-constant vectors, and strength X & stress Y are multigamma random vectors, i.e. (X,Y) similar to MG(alpha,beta), where alpha and beta are k-dimensional constant vectors. A Bayesian approach is adopted for R-2 = P((BW)-W-T >= 0), where W is multinormal, i.e.W similar to M N(mu, T), with the mean vector mu, and the precision matrix T having a joint s-normal-Wishart prior distribution. Final computations are carried out by simulation, an approach which plays a major role in this article. The results obtained show that the approach adopted can deal effectively with the dependence between components of X & Y.