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 $\Re_{1}=P({\bf A}^{T}{\bf X}>{\bf B}^{T}{\bf Y})$. ${\bf A}$ & ${\bf B}$ are column-constant vectors, and strength ${\bf X}$ & stress ${\bf Y}$ are multigamma random vectors, i.e. $({\bf X},{\bf Y})\sim MG({\mmb \alpha},{\mmb \beta})$, where ${\mmb \alpha}$ and ${\mmb \beta}$ are k-dimensional constant vectors. A Bayesian approach is adopted for $\Re_{2}=P({\bf B}^{T}{\bf W}\geq 0)$, where ${\bf W}$ is multinormal, i.e.${\bf W}\sim MN({\mmb \mu},{\bf T})$, with the mean vector ${\mmb \mu}$, and the precision matrix ${\bf T}$ 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 ${\bf X}$ & ${\bf Y}$.