On Reliability Analysis of $k$ -Out-of-$n$ Systems Consisting of Heterogeneous Components With Discrete Lifetimes

Abstract

In this paper, we study reliability of $k$ -out-of- $n$ systems with possibly heterogeneous components. We focus on the case when the lifetimes of components have a joint discrete distribution. The basic tool we use is a new closed-form formula describing the joint probability mass function of any subset of order statistics arising from not necessarily independent nor identically distributed discrete variates. This formula and some its modifications enable us to establish explicit expressions for unconditional and some conditional probabilities of a failure of the $k$ -out-of- $n$ system as well as expressions describing survival functions of various types of residual lifetimes of a used system.