Exact Failure Frequency Calculations for Extended Systems

TL;DR

This paper introduces a method to efficiently compute steady-state availability and failure frequency for large systems using matrix products, with applications to specific system types and asymptotic analysis.

Contribution

It presents a unified, single-pass calculation method for availability and failure frequency in large systems expressed as matrix products, including new formulas and asymptotic results.

Findings

Failure rate of large systems is asymptotically proportional to system size.

The matrix product approach simplifies calculations for complex systems.

Derived generating functions for systems with identical component availabilities.

Abstract

This paper shows how the steady-state availability and failure frequency can be calculated in a single pass for very large systems, when the availability is expressed as a product of matrices. We apply the general procedure to $k$-out-of-$n$:G and linear consecutive $k$-out-of-$n$:F systems, and to a simple ladder network in which each edge and node may fail. We also give the associated generating functions when the components have identical availabilities and failure rates. For large systems, the failure rate of the whole system is asymptotically proportional to its size. This paves the way to ready-to-use formulae for various architectures, as well as proof that the differential operator approach to failure frequency calculations is very useful and straightforward.