A linear m-consecutive-k-out-of-n system with sparse d of non-homogeneous Markov-dependent components

Abstract

Consider non-homogeneous Markov-dependent components in an m-consecutive- k-out-of- n:F (G) system with sparse [Formula: see text], which consists of [Formula: see text] linearly ordered components. Two failed components are consecutive with sparse [Formula: see text] if and if there are at most [Formula: see text] working components between the two failed components, and the m-consecutive- k-out-of- n:F system with sparse [Formula: see text] fails if and if there exist at least [Formula: see text] non-overlapping runs of [Formula: see text] consecutive failed components with sparse [Formula: see text] for [Formula: see text]. We use conditional probability generating function method to derive uniform closed-form formulas for system reliability, marginal reliability importance measure, and joint reliability importance measure for such the F system and the corresponding G system. We present numerical examples to demonstrate the use of the formulas. Along with the work in this article, we summarize the work on consecutive- k systems of Markov-dependent components in terms of system reliability, marginal reliability importance, and joint reliability importance.