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16) A print shop has three critical machines that essentially determine 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, as shown in Figure 2.16 (reproduced below). B A 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 stay 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. Develop a Markov chain model to assess the productivity of the shop. Hint: There are eight possible states corresponding to the combinations of up versus down conditions of the three machines. Number these in accordance with the binary numbers. That is, state 0 corresponds to (000), which means all machines are working; state 1 corresponds to (001), which means A and B are working and C is down; state 2 corresponds to (010), which means A and C are working and B is down; and so on.

16) A print shop has three critical machines that essentially determine 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, as shown in Figure 2.16 (reproduced below).
B
A
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 stay 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. Develop a Markov chain model to assess the productivity of the shop. Hint: There are eight possible states corresponding to the combinations of up versus down conditions of the three machines. Number these in accordance with the binary numbers. That is, state 0 corresponds to (000), which means all machines are working; state 1 corresponds to (001), which means A and B are working and C is down; state 2 corresponds to (010), which means A and C are working and B is down; and so on.

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