# Application of the interacting particle system method to piecewise deterministic Markov processes used in reliability

## TL;DR

This paper introduces IPS+M, an adaptation of the interacting particle system method, which improves variance reduction in reliability assessments of concentrated PDMPs by preventing repetitive trajectories, outperforming the standard IPS.

## Contribution

The paper proposes IPS+M, a modified IPS method that enhances variance reduction for concentrated PDMPs by conditioning propagation to avoid identical trajectories, with proven lower variance.

## Key findings

- IPS+M always yields lower variance estimators than IPS.
- Simulation confirms IPS+M's improved efficiency on a two-component system.
- Concentrated PDMPs pose challenges for standard IPS, which IPS+M addresses.

## Abstract

Variance reduction methods are often needed for the reliability assessment of complex industrial systems, we focus on one variance reduction method in a given context, that is the interacting particle system method (IPS) used on piecewise deterministic Markov processes (PDMP) for reliability assessment . The PDMPs are a very large class of processes which benefit from high modeling capacities, they can model almost any Markovian phenomenon that does not include diffusion. In reliability assessment, the PDMPs modeling industrial systems generally involve low jump rates and jump kernels favoring one safe arrival, we call such model a "concentrated PDMP". Used on such concentrated PDMPs, the IPS is inefficient and does not always provide a variance reduction. Indeed, the efficiency of the IPS method relies on simulating many different trajectories during its propagation steps, but unfortunately concentrated PDMPs are likely to generate the same deterministic trajectories over and over. We propose an adaptation of the IPS method called IPS+M that reduces this phenomenon. The IPS+M consists in modifying the propagation steps of the IPS, by conditioning the propagation to avoid generating the same trajectories multiple times. We prove that, compared to the IPS, the IPS+M method always provides an estimator with a lower variance. We also carry out a quick simulation study on a two-components system that confirms this result.