# Dependent Competing Failure Processes in Reliability Systems

## TL;DR

This paper studies how different types of shocks affect system reliability, modeling failure due to extreme shocks, cumulative critical shocks, or closely spaced harmful shocks.

## Contribution

A new analytical model for dependent competing failure processes using generalized random walks and discrete operational calculus.

## Key findings

- A closed-form joint distribution for time-to-failure and related metrics is derived.
- The reliability function is obtained from the marginal distribution of failure time.
- Formulas are validated through Monte Carlo simulations.

## Abstract

This paper deals with a reliability system hit by three types of shocks ranked as harmless, critical, or extreme, depending on their magnitudes, being below H1, between H1 and H2, and above H2, respectively. The system’s failure is caused by a single extreme shock or by a total of N critical shocks. In addition, the system fails under occurrences of M pairs of shocks with lags less than some δ (δ-shocks) in any order. Thus, the system fails when one of the three named cumulative damages occurs first. Thus, it fails due to the competition of the three associated shock processes. We obtain a closed-form joint distribution of the time-to-failure, shock count upon failure, δ-shock count, and cumulative damage to the system on failure, to name a few. In particular, the reliability function directly follows from the marginal distribution of the failure time. In a modified system, we restrict δ-shocks to those with small lags between consecutive harmful shocks. We treat the system as a generalized random walk process and use an embellished variant of discrete operational calculus developed in our earlier work. We demonstrate analytical tractability of our formulas which are also validated, through Monte Carlo simulation.