Problem 1.

A certain shop has three identical machines that are operated continuously except when they are broken down. Because they break down fairly frequently, the top- priority assign- ment for a full-time maintenance person is to repair them whenever needed.

The mean failure rate of each machine is 3 (three times per day), and the repair person is able to repair the broken down machines at a rate of 2 per day, on average.

(a) Develop the rate diagram for this Markov chain.

(b) Write down time-dependent ordinary differential equations for this Markov chain.

(c) Construct the steady-state equations.

(d) Determine the the steady-state probabilities.