Computing system signatures through reliability functions

TL;DR

This paper presents an efficient method to compute system signatures using derivatives of the reliability function, improving over traditional formulas, and applies it to modular systems with known signatures.

Contribution

It introduces a new approach to compute system signatures more efficiently from the reliability function's diagonal derivatives, especially for modular systems.

Findings

Efficient computation of the Samaniego signature from reliability function derivatives.

Application of the method to systems partitioned into modules with known signatures.

Reduction in computational complexity compared to existing formulas.

Abstract

It is known that the Barlow-Proschan index of a system with i.i.d. component lifetimes coincides with the Shapley value, a concept introduced earlier in cooperative game theory. Due to a result by Owen, this index can be computed efficiently by integrating the first derivatives of the reliability function of the system along the main diagonal of the unit hypercube. The Samaniego signature of such a system is another important index that can be computed for instance by Boland's formula, which requires the knowledge of every value of the associated structure function. We show how the signature can be computed more efficiently from the diagonal section of the reliability function via derivatives. We then apply our method to the computation of signatures for systems partitioned into disjoint modules with known signatures.