# At the repair facility machines are worked on by a

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Unformatted text preview: 1 computer left it places an order for 2 more computers. The order takes an exponentially distributed amount of time with mean 1 week to arrive. Of course, while the store is waiting for delivery, sales may reduce the inventory to 1 and then to 0. (a) Write down the matrix of transition rates Qij and solve ⇡ Q = 0 to ﬁnd the stationary distribution. (b) At what rate does the store make sales? 4.3. Consider two machines that are maintained by a single repairman. Machine i functions for an exponentially distributed amount of time with rate i before it fails. The repair times for each unit are exponential with rate µi . They are repaired in the order in which they fail. (a) Formulate a Markov chain model for this situation with state space {0, 1, 2, 12, 21}. (b) Suppose that 1 = 1, µ1 = 2, 2 = 3, µ2 = 4. Find the stationary distribution. 4.4. Consider the set-up of the previous problem but now suppose machine 1 is much more important than 2, so the repairman will always service 1 if it is broken. (a) Form...
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