Stochastic

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Unformatted text preview: 3 + 2/9 + 4/27 = 19/27 in agreement with the previous calculation. Example 4.22. Return to o ce hours. With the machinery just developed we can give a simple solution to one of the exercises in Chapter 2. Ron, Sue, and Ted arrive at the beginning of a professor’s o ce hours. The amount of time they will stay is exponentially distributed with means of 1, 1/2, and 1/3 hour, i.e., rates 1, 2, and 3. What is the expected time until all three students are gone? If we describe the state of the Markov chain by the rates of the students that are left, with ; to denote an empty o ce, then the Q-matrix is 123 12 13 23 1 2 3 123 6 0 0 0 0 0 0 12 3 3 0 0 0 0 0 13 2 0 4 0 0 0 0 23 1 0 0 5 0 0 0 1 0 2 3 0 1 0 0 2 0 1 0 3 0 2 0 3; 00 00 10 20 01 02 33 Letting R be the previous matrix with the last column deleted, the first row of R 1 is 1/6 1/6 1/12 1/30 7/12 2/15 1/20 The sum is 63/60, or 1 hour and 3 minutes. The first term is the 1/6 hour until the first student leaves. The next three are 11 · 23 11 · 34 11 · 65 which are the probability we visit...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.

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