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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 setup 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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 Spring '10
 DURRETT
 The Land

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