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

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Unformatted text preview: 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( ). Pr { X > T } = F ( T ) = e- T (b). Let W be the waiting time, condition on the next arrival time X. E [ W ] = Z E [ W | X = x ] e- x dx = Z T E [ W | X = x ] e- x dx + Z T E [ W | X = x ] e- x dx = Z T E [ W | X = x ] e- x dx = Z T ( x + E [ W ]) e- x dx = (1- e- x dx ) E [ W ] + Z T xe- x dx E [ W ] = e T Z T xe- x dx 47. (a). Let T be the time until the next customer enters the system. E [ T ] = 1 2 + 1 (b). Let T 2 be the time until both servers are busy starting empty, then E [ T 2 ] = 1 |{z} time for arrival + 1 + | {z } time for the next event + + | {z } additional time if its arrival + + E [ T 2 ] | {z } additional time if its departure Solve for E [ T 2 ] to get E [ T 2 ] = + 2 2 1 (c). Let T 1 be the time until both servers are busy starting with one server busy,...
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hw5sln - IEOR 161 Operations Research II University of...

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