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

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 $\{N(t),t\ge 0\}$ 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.