Renewal-reward theorem

For independent identically distributed regenerative cycles with $0<\mathbb{E}{\xi }_1<\infty$ and $\mathbb{E}|R_1|<\infty$, the long-run reward rate isalmost surely, provided the unfinished-cycle reward is negligible after division by $t$.

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