Long-run availability under periodic inspection
= Long-run availability under periodic inspection
If a repaired machine has independent exponential lifetimes of rate $\lambda$ and is inspected and repaired every $m$ time units after each repair, then its cycle length is $m\lceil X/m\rceil$ and its working reward is $X$. Its long-run working proportion is
$$
\frac{1-e^{-\lambda m}}{\lambda m}.
$$