There are m machines in a factory operating independently. Each machine works for a time period that is exponentially distributed with mean 1/µ. Then it breaks down. The time that it takes to repair...

There are m machines in a factory operating independently. Each machine works for a time period that is exponentially distributed with mean 1/µ. Then it breaks down. The time that it takes to repair an out-of-order machine is exponential with mean 1/λ, independent of repair times for and operating times of other machines. For 0 ≤ i ≤ m, find the long-run proportion of time that there are exactly i machines operating