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 S₂ connected in series, where the reliability of each subsystem is of general stress-strength type, defined by R₁ = P(A ^TX > B^TY). A & B are column-constant vectors, and strength X & stress Y are multigamma random vectors, i.e. (X, Y) ~ MG (alpha, beta), where alpha and beta are k-dimensional constant vectors. A Bayesian approach is adopted for R₂ = P(B ^TW > 0), where W is multinormal, i.e. W ~ MN(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