hw11sln - 97-(28 37 = 0 45 customers per hour 17 Let the...

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IEOR 161 Operations Research II University of California, Berkeley Spring 2008 Homework 11 Solution Chapter 6. 13. Let the number of customers in the shop be the state. We then have a birth and death process with λ 0 = λ 1 = 3 μ 1 = μ 2 = 4 Therefore P 1 = 3 4 P 0 P 2 = 3 4 P 1 = ( 3 4 ) 2 P 0 Solve 2 i =0 P i = 1 to get P 0 = 16 37 . (a) P 1 + 2 P 2 = 30 / 37. (b) P 0 + P 1 = 28 / 37 (c) Now μ = 8, and solve the problem again to get P 0 0 = 64 / 97. The proportion of customers who enter the shop is P 0 0 + P 0 1 = 88 / 97. The business he does would improve by λ (88 /
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Unformatted text preview: 97)-λ (28 / 37) = 0 . 45 customers per hour. 17. Let the state be 0 if the machine is up, i when it is down due to a type i failure, i = 1 , 2. The balance equations are λP = μ 1 P 1 + μ 2 P 2 μ 1 P 1 = λpP μ 2 P 2 = λ (1-p ) P P + P 1 + P 2 = 1 Solve them to get P = (1 + λp μ 1 + λ (1-p ) μ 2 )-1 P 1 = λpP μ 1 P 2 = λ (1-p ) P μ 2 1...
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