6. There are m machines in a factory operating independently. The factory has k (k < m)

repair persons, and each repair person repairs one machine at a time. Suppose that

(i) each machine works for a time period that 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; and (iii) at times when all repair persons are busy repairing machines, the newly broken down machines will wait for repair. Let X(t) be the number of machines operating at time t.

Show that



X(t) : t ≥ 0

is a birth and death process and find the birth and death rates.