# 13.Sol.Fall - L = ρ 1 / (1-ρ 1 ) + ρ 2 / (1-ρ 2 ) + ρ...

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ISyE 3232 Stochastic Manufacturing and Service Systems Fall 2007 Professors Ayhan and J. Dai Solutions to Homework 13 1. (a) Let λ i be the total arrival rate to station i , i = 1 , 2 , 3. Then we have the following traﬃc equations: λ 1 = 1 + (0 . 1) λ 2 + (0 . 05) λ 3 , λ 2 = λ 1 + (0 . 1) λ 3 , λ 3 = (0 . 9) λ 2 . Solving the equations, one has λ 1 = 1 . 1895, λ 2 = 1 . 3072 and λ 3 = 1 . 1765. Thus, ρ 1 = λ 1 m 1 = 0 . 9516 , ρ 2 = 0 . 9150 , ρ 3 = 0 . 9412 . Let X i ( t ) be the number of jobs at station i at time t . Then the long-run fraction of time that there are 2 jobs at station 1, 1 job at station 2 and 4 jobs at station 3 is equal to P { X 1 ( ) = 2 ,X 2 ( ) = 1 ,X 3 ( ) = 4 } = P { X 1 ( ) = 2 } P { X 2 ( ) = 1 } P { X 3 ( ) = 4 } = (1 - ρ 1 ) ρ 2 1 (1 - ρ 2 ) ρ 2 (1 - ρ 3 ) ρ 4 3 = 1 . 5718 × 10 - 4 . (b) ρ 3 / (1 - ρ 3 ) = 16 . 0. (c) The total long-run average system size is
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Unformatted text preview: L = ρ 1 / (1-ρ 1 ) + ρ 2 / (1-ρ 2 ) + ρ 3 / (1-ρ 3 ) = 46 . 4449 jobs . Using Little’s law, L = λW , the average time in system per job is 46 . 4449 hours, since the arrival rate into the system is 1 job per hour. (d) The average size of the system is L = 21 . 985 jobs. Thus, the average time in system per job is 21 . 985 hours. 5% of reduction of the rework cuts the average cycle time to less than half. This is consistent with Goldratt theory in managing bottlenecks: a small help to a bottleneck machine, the great reduction to the cycle time. 1...
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## This note was uploaded on 11/26/2009 for the course ISYE 3232 taught by Professor Billings during the Fall '07 term at Georgia Institute of Technology.

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