A linear m

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