Question: 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 exponential with mean 1/λ, independent of repair times for other machines;

(iii) at times when all repairpersons are busy repairing machines, the newly broken down machines will wait for repair; and

(iv) when a machine breaks down, one of the spares will be used unless there is no spare machine available. Let X(t ) be the number of machines operating or ready to operate at time t . Show that { X(t ) : t ≥ 0 % is a birth and death process and ?nd the birth and death rates.