Reliability and importance measures for m-consecutive-k, l-out-of-n system with non-homogeneous Markov-dependent components

• m-Consecutive-k, l-out-of-n:F and G systems.
• Non-homogenous Markov-dependent components.
• Formulas for marginal reliability importance and joint reliability importance.
• Probability generating function method and simulation.
• A practical application and the numerical examples.

We study an m-consecutive-k, l-out-of-n system with non-homogenous Markov-dependent components. The m-consecutive-k, l-out-of-n:F system fails if and only if there are at least m runs of k consecutive failed components and each of the runs may have at most l components overlapping with the previous run of k consecutive failed components. Using probability generating function method, we derive closed-form formulas for the reliability of the m-consecutive-k, l-out-of-n:F system, the marginal reliability importance measure of a single component, and the joint reliability importance measure of two or more components when components are non-homogenous Markov-dependent. We also extend these results into an analogous m-consecutive-k, l-out-of-n:G system, which is developed by considering consecutive working components. The results can be simplified to the situations of the homogenous Markov-dependent components and the independent components. We present a practical application in quality control and related numerical examples that demonstrate the use of derived formulas and provide the insights on the m-consecutive-k, l-out-of-n system and the importance measures.