18 mm1 queue has jump rates q i i 1 for i 0 and q i

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Unformatted text preview: se customers arrive at times of a rate 2 Poisson process, but will leave if both chairs in the waiting room are full. (a) Find the equilibrium distribution. (b) What fraction of customers enter service? (c) What is the average amount of time in the system for a customer who enters service? 132 CHAPTER 4. CONTINUOUS TIME MARKOV CHAINS Solution. We define our state to be the number of customers in the system, so S = {0, 1, 2, 3}. From the problem description it is clear that q (i, i 1) = 3 for i = 1, 2, 3 q (i, i + 1) = 2 for i = 0, 1, 2 The detailed balance conditions say 2⇡ (0) = 3⇡ (1), 2⇡ (1) = 3⇡ (2), 2⇡ (2) = 3⇡ (3) Setting ⇡ (0) = c and solving, we have ⇡ (1) = 2c , 3 ⇡ (2) = 2 4c · ⇡ (1) = , 3 9 ⇡ (3) = 2 8c · ⇡ (2) = 3 27 The sum of the ⇡ ’s is (27 + 18 + 12 + 8)c/27 = 65c/27, so c = 27/65 and ⇡ (0) = 27/65, ⇡ (1) = 18/65, ⇡ (2) = 12/65, ⇡ (3) = 8/65 From this we see that 8/65’s of the time someone is waiting, so that fraction of the arrivals...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

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