Consider non-homogeneous Markov-dependent components in an m-consecutive-k-out-of-n:F (G) system with sparse , which consists of linearly ordered components. Two failed components are consecutive with sparse if and if there are at most working components between the two failed components, and the m-consecutive-k-out-of-n:F system with sparse fails if and if there exist at least non-overlapping runs of consecutive failed components with sparse for . 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.
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