homework10 - ORIE 361/523 Homework 9 Instructor Mark E...

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ORIE 361/523 – Homework 9 Instructor: Mark E. Lewis due April 16, 2008 (drop box) 1. Potential customers arrive at a a single-server station (which works at rate μ ) in accordance with a Poisson process with rate λ . However, if a customer finds n people already in the station, then he will enter the system with probability 1 / ( n +1). Find the limiting distribution of the number of customers in the system. 2. Customers arrive at a two-server station according to a Poisson process with rate λ . Upon arriving, they join a single queue. Whenever a server completes a service, the person first in line enters service. The service times of server i are exponential with rate μ i , i = 1 , 2 , where μ 1 + μ 2 > λ. An arrival finding both servers free is equally likely to go to either one. Define an appropriate CTMC for this model and find the limiting probabilities. 3. A total of N customers move about among r servers in the following manner. When a customer is served by i, he then goes to server j, j 6 = i , with probability 1 r - 1 . If the server he goes to
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