A single repair person looks after two machines. Suppose that the amount of time each machine runs before breaking down is exponentially distributed with rate and suppose the amount of time it takes the repair person to fix a machine is exponentially distributed with rate . When a machine breaks down, a repair process begins immediately if the repair person is free. If the repair person is busy the repair process doesnt begin until the other machine is fixed. Define a continuous time Markov chain representing the number of working machines at a given time.

(a) Starting with two working machines find the expected time until both machines are broken for the first time.

b) Find the average number of working machines in the long-run.