On the extensions of Barlow-Proschan importance index and system signature to dependent lifetimes

TL;DR

This paper extends the Barlow-Proschan importance index and system signature concepts to systems with dependent component lifetimes, providing explicit formulas and probabilistic interpretations.

Contribution

It offers an explicit expression for the importance index under dependent lifetimes and introduces a probabilistic interpretation and symmetry index for such systems.

Findings

Explicit formula for importance index with dependent lifetimes

Probabilistic interpretation of the importance index

Introduction of a symmetry index for systems

Abstract

For a coherent system the Barlow-Proschan importance index, defined when the component lifetimes are independent, measures the probability that the failure of a given component causes the system to fail. Iyer (1992) extended this concept to the more general case when the component lifetimes are jointly absolutely continuous but not necessarily independent. Assuming only that the joint distribution of component lifetimes has no ties, we give an explicit expression for this extended index in terms of the discrete derivatives of the structure function and provide an interpretation of it as a probabilistic value, a concept introduced in game theory. This enables us to interpret Iyer's formula in this more general setting. We also discuss the analogy between this concept and that of system signature and show how it can be used to define a symmetry index for systems.