  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.
