A print shop has three critical machines that essentially deter-mine the productivity of the shop. They are each subject to failure every day.Assume for simplicity that if a machine fails, it is at the moment it is turned on in the morning. Let the probability that an operable machine fails on any given day be 0.1, independently of the other machines. A mechanic is called in whenever there is a failure and can always repair one machine that same day (so it is available for the next), but cannot do more than one a day. Machine A feeds both machines B and C. Thus, if machine A fails, there is no production that day. Consequently, the policy is always to repair A if it is down. If both B and C are down, and A is up, the policy is to repair B. Of course, units that are down, remain down until repaired. Units that are operable today may be down tomorrow, whether or not the shop is able to produce anything. But a unit that is repaired today is certain to operate tomorrow.

(a) Construct a Markov chain to model the productivity of the shop. Hint: There are 8 possible states corresponding to the combinations of the up and down states of the three machines. Number these in accordance to the binary numbers. That is, state 0 (corresponds to 000) means all machines are working, state 1 (corresponds to 001) means A and B are working but

C is down, state 2 (corresponds to 010) mean A and C are working but B is down, and so on.

(b) Give the communication classes for this Markov chain.

(c) Is the Markov chain irreducible?

(d) Assume that all machines are up yesterday, what is the probability that all machines are up today and tomorrow?

(e) Assume that all machines are up today, what is the expected number of days over the next five days (including today, i.e. over four transitions) for which the system will be operational?