ch4_add_2 - Birth and Death Processes Example: A Barber...

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Birth and Death Processes Example: A Barber Shop A small town barbershop has two barbers and an additional chair for 1 waiting customer. If a customer arrives when there are 3 customers in the shop, the customer leaves. The customers arrive according to a homogeneous Poisson process. The average time between arrivals is 30 minutes. The two barbers are Andy and Bob. Andy completes serving a customer at a rate of 2 per hour. Bob completes a serving a customer at a rate of 1 per hour. Because Andy is faster than Bob, if only one customer is in the store, Andy handles the customer. Assume that inter-arrival times and service times are independent exponential random vari- ables. (a) Draw a state diagram with possible states and corresponding birth/death rates. Solution: 2 2 0 1 2 2 3 3 2 3 E [ B ] = 30min = 1 / 2 hrs. implying rate λ = 2 per hour. Let T A = time until Andy finishes and T B = time until Bob finishes. For k = 1, D = T A ; since μ A = 2 per hour, D Exp (2) For k = 2 , 3, D = min { T A ,T B } giving D Exp (1 + 2 = 3); since μ A = 2 and μ B = 1 (b) What is the ( large t ) probability that the shop is empty? S = 1 + 1 2 + 2 · 2 2 · 3 + 2 · 2 · 2 3 · 3 · 2 = 1 + 1 + 4 6 + 8 18 = 28 9 implying that p 0 = 1 S
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This note was uploaded on 02/01/2012 for the course STAT 330B taught by Professor Zhou during the Spring '11 term at Iowa State.

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ch4_add_2 - Birth and Death Processes Example: A Barber...

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