Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 27K / Solution

A renewal-reward process consists of i.i.d. regenerative cycles $({\xi }_n,R_n)$, where ${\xi }_n>0$ is the $n$th cycle length and $R_n$ is the reward accumulated during that cycle. If $0<\mathbb{E}{\xi }_1<\infty$ and $\mathbb{E}|R_1|<\infty$, the renewal-reward theorem states that the cumulative reward $Z(t)$ satisfiesalmost surely, with the usual negligible-remainder condition for an unfinished cycle.

For the machine, let $X\sim \mathrm{Exp}(\lambda )$ be its working lifetime after a repair. The next repair occurs at the first inspection following failure, so one repair-to-repair cycle has lengthwhile its working-time reward is $R=X$. Since an exponential distribution of rate $\lambda$ has mean $1/\lambda$,By the tail-sum formula for the positive integer-valued variable $\lceil X/m\rceil$,Thereforeand the long-run proportion of time for which the machine is working is