We have a factory that can process jobs. Each job takes an hour to complete. Jobs arrive according to a Poisson arrival process, with a mean of $\lambda$ jobs per hour. If the factory is free when a job arrives, it accepts the job with probability $p$, independently of other jobs. Over the long run, what is the average proportion of time that the factory is busy?
I'm not sure how to set up the calculation for this. I think we have to calculate the amount of time that, starting from any point where the factory is free, we need to wait until the next job is accepted. If a job is always accepted when the machine is free ($p=1$), then the expected waiting time should be $1/\lambda$. But here matters are complicated because we have a probability $p\leq 1$.