# A Production Model with History Based Random Machine Failures

## TL;DR

This paper develops a continuous-time production model incorporating machine failures influenced by historical production data, using PDMPs to analyze the system's reliability and failure dynamics.

## Contribution

It introduces a novel PDMP-based framework linking past production to failure probabilities, enabling Markovian analysis of failure-repair processes.

## Key findings

- Model captures history-dependent failure behavior
- Numerical analysis demonstrates system reliability insights
- Sample means and repair frequencies validate the model

## Abstract

In this paper, we introduce a time-continuous production model that enables random machine failures, where the failure probability depends historically on the production itself. This bidirectional relationship between historical failure probabilities and production is mathematically modeled by the theory of piecewise deterministic Markov processes (PDMPs). On this way, the system is rewritten into a Markovian system such that classical results can be applied. In addition, we present a suitable solution, taken from machine reliability theory, to connect past production and the failure rate. Finally, we investigate the behavior of the presented model numerically in examples by considering sample means of relevant quantities and relative frequencies of number of repairs.