Reliability and Joint Reliability Importance in a Consecutive-<formula formulatype="inline"> <tex Notation="TeX">$k$</tex> </formula>-Within-<formula formulatype="inline"> <tex Notation="TeX">$m$</tex> </formula>-out-of-<formula formulatype="inline"> <tex Notation="TeX">$n$</tex> </formula>:F System With Markov-Dependent Components

Abstract

This paper studies a consecutive- $k$ -within- $m$ -out- of- $n$ :F system with Markov-dependent components; that is, the reliability of a component depends on its neighbors. Using probability generating functions, the closed-form formula for reliability of the consecutive- $k$ -within- $m$ -out-of- $n$ :F system with Markov-dependent components and a closed-form formula for joint reliability importance (JRI) of two components in such a system are derived. The JRI of two components evaluates the interaction effect between the components on contributing to system reliability. A formula of the JRI of more than two components are also derived and presented. Many real systems and procedures, such as radar detection systems, pipeline systems, quality inspection procedures, and so on, can be modeled as a consecutive- $k$ -within- $m$ -out-of- $n$ :F system, in which components are Markov-dependent. The present results can evaluate the reliability of these systems or the accuracy of the procedures as well as the contributions of components to the system reliability or the accuracy of the procedures. The applications of the present formulas are demonstrated through the numerical examples. The examples show the changes of system radiabilities and the changes among the JRI values of different pairs of components in consecutive- $k$ -within- $m$ -out-of- $n$ :F systems. The JRI values of Markov-dependent components are also compared to the JRI values of $s$ -independent components.