Joint Reliability Importance in a Consecutive- $k$ -out-of-$n$ :F System and an $m$-Consecutive- $k$-out-of-$n$:F System for Markov-Dependent Components

The Joint Reliability Importance (JRI) of two components evaluates the interaction effect between the components on system reliability. This paper focuses on the JRI of components in a consecutive- k-out-of- n:F system, and an m-consecutive- k- out-of- n:F system, both with Markov-dependent components. We derive the closed-form formulas of the JRI of two components using probability generating functions, and extend the results to the JRI of three and more components. We further provide exact conditional distributions of random variables which are used in probability generating functions for determining the JRI of two components. Our numerical examples and tests demonstrate the use of derived formulas, and provide further insights about the JRI for Markov-dependent components.