Types of signature analysis in reliability based on Hilbert series

TL;DR

This paper introduces a novel algebraic approach using Hilbert series and lcm-filtration of monomial ideals to analyze multiple failure modes and signatures in coherent systems, providing deeper insights into system reliability.

Contribution

It develops the concept of lcm-filtration of monomial ideals and applies it to enhance the analysis of system failures and signatures, including explicit resolution computations.

Findings

Explicit resolutions for all ideals in the lcm-filtration are computed.

The approach reveals deeper algebraic structures influencing system behavior.

Computational examples demonstrate the method's effectiveness and challenges.

Abstract

The present paper studies multiple failure and signature analysis of coherent systems using the theory of monomial ideals. While system reliability has been studied using Hilbert series of monomial ideals, this is not enough to understand in a deeper sense the ideal structure features that reflect the behavior of the system under multiple simultaneous failures and signature. Therefore, we introduce the lcm-filtration of a monomial ideal, and we study the Hilbert series and resolutions of the corresponding ideals. Given a monomial ideal, we explicitly compute the resolutions for all ideals in the associated lcm-filtration, and we apply this to study coherent systems. Some computational results are shown in examples to demonstrate the usefulness of this approach and the computational issues that arise. We also study the failure distribution from a statistical point of view by means of the algebraic tools described.