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Unformatted text preview: ISyE 3232 Stochastic Manufacturing and Service Systems Fall 2007 Professors H. Ayhan and J. G. Dai Solutions to Homework 12 1. For all the subproblem below, the state space will always be { , 1 , 2 , 3 } . (a) The transition diagram is as the following Solve the stationary distribution using “cuts”, we get π = (1 / 15 , 2 / 15 , 4 / 15 , 8 / 15) So the throughput is λ eff = 1(1 π 3 ) = 7 / 15. To find the average waiting time, we use Little’s law. Note that L = 1 π 2 + 2 π 3 = 20 / 15, so W = L λ eff = 20 / 7 . (b) The transition diagram is as the following Solve the stationary distribution using “cuts”, we get π = (1 / 7 , 2 / 7 , 2 / 7 , 2 / 7) So the throughput is λ eff = 1(1 π 3 ) = 5 / 7. To find the average waiting time, we use Little’s law. Note that L = 1 π 3 = 2 / 7, so W = L λ eff = 2 / 5 . (c) The transition diagram is as the following Solve the stationary distribution using “cuts”, we get π = (3 / 19 , 6 / 19 , 6 / 19 , 4 / 19) 1 So the throughput is...
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 Fall '07
 Billings
 Poisson Distribution, Probability theory, Exponential distribution, average waiting time, stationary distribution, ﬁnd Wq

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