On signature-based expressions of system reliability

TL;DR

This paper explores how the signature concept can be extended to non-i.i.d. component lifetimes in system reliability analysis, identifying conditions for different signature-based representations.

Contribution

It extends the signature framework to non-i.i.d. components, establishing exchangeability conditions and alternative representations for system reliability.

Findings

Signature representation holds for exchangeable component states.

Conditions identified for alternative reliability representations.

Both representations coexist under specific system conditions.

Abstract

The concept of signature was introduced by Samaniego for systems whose components have i.i.d. lifetimes. This concept proved to be useful in the analysis of theoretical behaviors of systems. In particular, it provides an interesting signature-based representation of the system reliability in terms of reliabilities of k-out-of-n systems. In the non-i.i.d. case, we show that, at any time, this representation still holds true for every coherent system if and only if the component states are exchangeable. We also discuss conditions for obtaining an alternative representation of the system reliability in which the signature is replaced by its non-i.i.d. extension. Finally, we discuss conditions for the system reliability to have both representations.