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3044hw3s09sols

3044hw3s09sols - 2 ISyE 3044/Spring 2009/Solutions to...

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2 ISyE 3044/Spring 2009/Solutions to Homework #3 1. The tool crib is modeled by an M/M/c queue ( λ =1 / 4 =1 / 3 ,c = 1 or 2). Given that attendants are paid \$6 per hour and mechanics are paid \$10 per hour, Mean cost per hour = \$10 c + \$15 L. assuming that mechanics impose cost on the system while in the queue and in service. CASE 1: one attendant - M/M/1 ( c =1 = λ/μ = . 75). L = ρ/ (1 - ρ ) = 3 mechanics. Mean cost per hour = \$10(1) + \$15(3) = \$55. CASE 2: two attendants - M/M/2 ( c =2 = λ/cμ = . 375). L = + ± ( ) c +1 P 0 ² / ± c ( c !)(1 - ρ ) 2 ² = . 8727 , where P 0 = ³´ c - 1 µ n =0 ( ) n /n ! + [( ) c (1 /c !)(1 / (1 - ρ ))] · - 1 = . 4545 . Mean cost per hour = \$10(2) + \$15(.8727) = \$33.09. It would be advisable to have a second attendant because long run costs are reduced by \$21.91 per hour. 2. The single landing strip airport is modeled by an M/M/1 queue ( μ =2 / 3). The maximum arrival rate, λ , such that the average wait,

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3044hw3s09sols - 2 ISyE 3044/Spring 2009/Solutions to...

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