Analysis for joint importance of components in a coherent system

JRI (Joint Reliability Importance) of two components is a measure of interaction of two components in a system for their contribution to the system reliability. It is defined as the rate at which the system reliability improves as the reliabilities of the two components improve. But, sometimes we may improve system reliability through improving reliabilities of three or more components. This article extends the concepts of JRI & JFI (Joint Failure Importance) of two components to multi-components, and establishes some relationships between JRI & JRI, JFI & JFI, and JFI & JRI. The paper also investigates the concept of Conditional Reliability Importance while the working states of certain components are known. Finally, the JRI of multi-components and Conditional Reliability Importance are analyzed in detail for a k-out-of-n:G system.