hw5sln - IEOR 161 Operations Research II University of...

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IEOR 161 Operations Research II University of California, Berkeley Spring 2008 Homework 5 Suggested Solution Chapter 5. 44. (a). Let X be the time of the next arrival, X EXP( λ ). Pr { X > T } = F ( T ) = e - λT (b). Let W be the waiting time, condition on the next arrival time X. E [ W ] = 0 E [ W | X = x ] λe - λx dx = T 0 E [ W | X = x ] λe - λx dx + T E [ W | X = x ] λe - λx dx = T 0 E [ W | X = x ] λe - λx dx = T 0 ( x + E [ W ]) λe - λx dx = (1 - e - λx dx ) E [ W ] + T 0 λxe - λx dx E [ W ] = e λT T 0 λxe - λx dx 47. (a). Let T be the time until the next customer enters the system. E [ T ] = 1 2 μ + 1 λ (b). Let T 2 be the time until both servers are busy starting empty, then E [ T 2 ] = 1 λ time for arrival + 1 μ + λ time for the next event + λ μ + λ 0 additional time if it’s arrival + μ μ + λ E [ T 2 ] additional time if it’s departure Solve for E [ T 2 ] to get E [ T 2 ] = μ + 2 λ λ 2 1
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(c). Let T 1 be the time until both servers are busy starting with one server busy, L i be the time until a customer is lost when start with i busy servers, i=1,2.
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