# Replace-after-Fixed-or-Random-Time Extensions of the Poisson Process

## TL;DR

This paper studies extensions of the Poisson process where interarrival times are capped at a fixed or random time, modeling replacement scenarios in engineering with potential applications in reliability analysis.

## Contribution

It introduces and analyzes replace-after-fixed-time and replace-after-random-time processes, extending classical Poisson processes to incorporate fixed or stochastic replacement times.

## Key findings

- Derived properties of replace-after-fixed-time processes.
- Extended the model to include random replacement times.
- Applicable to reliability and maintenance modeling.

## Abstract

We analyze extensions of the Poisson process in which any interarrival time that exceeds a fixed value $r$ is counted as an interarrival of duration $r$. In the engineering application that initiated this work, one part is tested at a time, and $N(t)$ is the number of parts that, by time $t$, have either failed, or if they have reached age $r$ while still functioning, have been replaced. We refer to $\{N(t), t \geq 0\}$ as a replace-after-fixed-time process. We extend this idea to the case where the replacement time for the process is itself random, and refer to the resulting doubly stochastic process as a replace-after-random-time process.