I have this exercise problem in stochastic process textbook.
A worker has a number of machines to repair. Each time a repair is completed a new one is begun. Each repair independently takes an exponential amount of time with rate $μ$ to complete. However, independent of this, mistakes occur according to a Poisson process with rate $λ$. Whenever a mistake occurs, the item is ruined and work is started on a new item. In the long run how often are jobs completed?
At first I thought this would just be $\frac{t}{N(t)-M(t)}$ where $N(t)$ is the Poisson process for fixing machines and $M(t)$ for the Poisson process of mistakes. But then I realized the repairing would stop one a mistake happens, so it’s not really $N(t)$. What should I do?
Let $T_i$ represent the time spent repairing the $i^\text{th}$ machine. Then $T_i=\min\{X_i,Y_i\}$ where $X_i\sim\text{Exp}(\mu)$ and $Y_i\sim\text{Exp}(\lambda)$ for all $i$. Since $T_i$ are iid the question is just asking for $\mathbb{P}(T_i=X_i)$ which is $\mu/(\mu+\lambda)$.