12. In a factory, there are m operating machines and s machines used as spares and ready to operate. The factory has k repair persons, and each repair person 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-ordermachine is exponential with mean 1/λ, independent of repair times for other machines; (iii) at times when all repair persons 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 find the birth and death rates.