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Unformatted text preview: 1. Customers arrive at a 2server station in accordance with a Poisson process having rate 1. Upon arriving, a customer joins the queue when both servers are busy. The service times of servers A and B are exponential with respective rates 2 and 1. Whenever a server completes a service, the person first in line enters service. An arrival finding both servers free goes to server A. (a) (20 marks) Define an appropriate continuoustime Markov Chain for this model and find the limiting probabilities. (b) (5 marks) Is this a time reversible continuoustime Markov Chain? Why? (c) (Bonus 5 marks) Modify the system as follows: When server A is idle and server B is busy, the customer will immediately shift to A. Does this modified system make a time reversible continuoustime Markov Chain? Why? Suggested solution: (a) Let the state be the number of customers in the system. We have the following transition rate diagram. Let μ 1 = 2, μ 2 = 1, λ = 1. Forming the flow balance equations, we have P = 2P 1A + P 1B // at state 0 3P 1A = P + P 2 // at state 1 A 2P 1B = 2P 2 // at state 1 B P k = (1/3) k2 P 2 . // at state k for k ≥ 2 We can solve from the first three equations and obtain P 1A...
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 Spring '08
 Choi
 Probability theory, Exponential distribution, Tn, Markov chain

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