hmwk10sol+_2_ - ISyE 3232 Stochastic Manufacturing and...

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ISyE 3232 Stochastic Manufacturing and Service Systems Fall 2011 H. Ayhan Solutions to Homework 10 1. Let S 1 and S 2 be independent and exponentially distributed random variables with means 1 μ 1 = 3 and 1 μ 2 = 6. They corresponds to the service time of the first and the second teller. (a) Since A starts service immediately after arrival, P ( T A 6) = P ( S 1 6) = e - 1 2 × 6 = 0 . 0498. (b) By the same reason, E ( T A ) = E ( S 1 ) = 3. (c) This amount, in mathematical expression, will be E ( T A | T A > 6) = E ( S 1 | S 1 > 6) = 6 + E ( S 1 ) // by memoryless property of exponential random variables = 9 . (d) P ( T A < T B ) = P ( S 1 < S 2 ) = μ 1 μ 1 + μ 2 = 2 / 3 . (e) The time from noon till a customer leaves is min { T A ,T B } . Its expectation will be the same as E (min { S 1 ,S 2 } ) = 1 μ 1 + μ 2 = 2 . (f) If a customer leaves, one of the tellers will be able to serve C . So it is the same as time till one departure, which is computed in (e). (g) The total time C spends in the system will be the summation of the waiting time min { T A ,T B } and its service time. The service time will depend on at which teller the
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This note was uploaded on 12/07/2011 for the course ISYE 3232 taught by Professor Billings during the Fall '07 term at Georgia Institute of Technology.

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hmwk10sol+_2_ - ISyE 3232 Stochastic Manufacturing and...

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