A shop contains three identical machines that are subject to a failure of a certain kind. Therefore, a maintenance system is provided to perform the maintenance operation (recharging) required by a failed machine. The time required by each operation has an exponential distribution with a mean of 30 minutes. However, with probability 1

3

, the operation must be performed a second time (with the same distribution of time) in order to bring the failed machine back to a satisfactory operational state. The maintenance system works on only one failed machine at a time, performing all the operations (one or two) required by that machine, on a firstcome-first-served basis. After a machine is repaired, the time until its next failure has an exponential distribution with a mean of 3 hours.

(a) How should the states of the system be defined in order to formulate this queueing system as a continuous time Markov chain? (Hint: Given that a first operation is being performed on a failed machine, completing this operation successfully and completing it unsuccessfully are two separate events of interest. Then use Property 6 regarding disaggregation for the exponential distribution.)

(b) Construct the corresponding rate diagram.

(c) Develop the balance equations.