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 $d$ , which consists of $n$ linearly ordered components. Two failed components are consecutive with sparse $d$ if and if there are at most $d$ working components between the two failed components, and the m-consecutive-k-out-of-n:F system with sparse $d$ fails if and if there exist at least $m$ non-overlapping runs of $k$ consecutive failed components with sparse $d$ for $1⩽d⩽n-k$ . 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.