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HW4Soln

# HW4Soln - IEOR 161 Introduction to Stochastic Processes...

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IEOR 161 - Introduction to Stochastic Processes Spring 2010 HW4 Solutions ** Note that the numbering is from Ross 9th Edition 5.20 Denote X = the service time of the new customer, A = the service time of customer A, and B = the service time of customer B. (a) P A = P { service time at server 1 < service time at server 2 for A } = P { X < A } = μ 1 μ 1 + μ 2 . (b) P B = P { service time at server 1 < service time at server 2 for A and B } = P { X < A + B } = P { X < A + B | X < A } P { X < A } + P { X < A + B | X > A } P { X > A } = P { X < A } + P { X < B } P { X > A } by memoryless of exponential distribution = μ 1 μ 1 + μ 2 + μ 1 μ 1 + μ 2 μ 2 μ 1 + μ 2 = μ 2 1 + 2 μ 1 μ 2 ( μ 1 + μ 2 ) 2 . (c) E[ T ] = E[ S 1 + S 2 + W A + W B ] = E[ S 1 ] + E[ S 2 ] + E[ S A ] P A + E[ S B ] P B = 1 μ 1 + 1 μ 2 + 1 μ 2 μ 1 μ 1 + μ 2 + 1 μ 2 μ 2 1 + 2 μ 1 μ 2 ( μ 1 + μ 2 ) 2 . 5.21 Let T denote the total time spent in the system, W i be time spent waiting for the previous customer to leave server i , and S i be time spent in service by server i . E [ T ] = E [ W 1 ] + E [ S 1 ] + E [ W 2 ] + E [ S 2 ] = 1 μ 1 + 1 μ 1 + 1 μ 2 μ 1 μ 1 + μ 2 + 1 μ 2 = 2 μ 1 + 1 μ 2 1 + μ 1 μ 1 + μ 2 1

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HW4Soln - IEOR 161 Introduction to Stochastic Processes...

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