In a factory, there are m operating machines and s machines used as spares and ready to operate. The factory has k repairpersons, and each repairperson repairs one machine at a time. Suppose that, (i)...

In a factory, there are m operating machines and s machines used as spares and ready to operate. The factory has k repairpersons, and each repairperson repairs one machine at a time. Suppose that, (i) each machine works, independent of other machines, for a time period which is exponentially distributed with mean 1/µ, then it breaks down; (ii) the time that it takes to repair an out-of-order machine is  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.