# hw12sln - IEOR 161 Operations Research II University of...

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IEOR 161 Operations Research II University of California, Berkeley Spring 2008 Homework 12 Solution Chapter 8. 4. (a) Let N be the number of other customers in the system when the customer arrived (since x > 0, N 1), and W Q denote the amount of time that customer spends waiting in queue. Pr { N = n | W Q = x } = Pr { W Q = x | N = n } Pr { N = n } Pr { W Q = x } = μe - μx ( μx ) n - 1 ( n - 1)! ( λ μ ) n (1 - λ μ ) f W Q ( x ) = K ( λx ) n - 1 ( n - 1)! where K = μe - μx λ μ (1 - λ μ ) f W Q ( x ) 1 = n =1 Pr { N = n | W Q = x } = K n =1 ( λx ) n - 1 ( n - 1)! = Ke λx K = e - λx Pr { N = n | W Q = x } = e - λx ( λx ) n - 1 ( n - 1)! Thus, N-1 is Poisson with mean λx . Also Pr { W Q = x } = μe - μx λ μ (1 - λ μ ) K = λ μ ( μ - λ ) e - ( μ - λ ) x (b) x = 0 P ( W Q = 0) = P 0 = 1 - λ μ x > 0 P ( W Q x ) = P ( W Q = 0) + x 0 P ( W Q = y ) dy = 1 - λ μ + λ μ (1 - e - ( μ - λ ) x ) 6. Let the state be the idle server. The balance equations are: Rate Out = Rate In (1) ( μ 2 + μ 3 ) P 1 = μ 1 ( P 3 + P 2 ) (2) ( μ 1 + μ 3 ) P 2 = μ 2 ( P 1 + P 3 ) (3) ( μ 1 + μ 2 ) P 3 = μ 3 ( P 1 + P 2 ) P 1 + P 2 + P 3 = 1 P 1 = μ 1 , P 2 = μ 2 , P 3 = μ 3 1

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10. (a) The state is the number of customers in the system. The balance equations are: Rate Out = Rate In
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• Spring '08
• Lim
• Operations Research, p1, AirTrain Newark, State State State, server server server

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