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

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