Conditional Residual Lifetimes of $(n-k+1)$-out-of-$n$ Systems with Mixed Erlang Components

Abstract

We consider an $(n-k+1)$-out-of-$n$ system with component lifetimes being correlated. The main objective of this paper is to study the conditional residual lifetime of an $(n-k+1)$-out-of-$n$ system, given that at a fixed time a certain number of components have failed, assuming that the component lifetimes follow a multivariate Erlang mixture. Comparison studies of the stochastic ordering of the $(n-k+1)$-out-of-$n$ system are presented. Several examples are presented to illustrate and confirm our results.