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Unformatted text preview: 1. Customers arrive at a 2-server 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 continuous-time Markov Chain for this model and find the limiting probabilities. (b) (5 marks) Is this a time reversible continuous-time 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 continuous-time 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) k-2 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
- Probability theory, Exponential distribution, Tn, Markov chain