# B suppose that 1 1 1 2 2 3 2 4 find the stationary

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Unformatted text preview: tation examples, so we omit it. Example 4.29. At a government agency entering customers always go to server 1. After completing service there, 30% leave the system while 70% go to server 2. At server 2, 50% go to server 3, 20% of the customers have to return to server 1, and 30% leave the system. From server 3, 20% go back to server 2 but the other 80% can go. That is, the routing matrix is 1 0 .2 0 1 p= 2 3 2 .7 0 .2 3 0 .5 0 Suppose that arrivals from outside only occur at server 1 and at rate 1 = 3.8 per hour. Find the stationary distribution if the service rates are µ1 = 9, µ2 = 7, and µ3 = 7. The ﬁrst step is to solve the equations rj = j + 3 X ri p(i, j ) i=1 By (4.30) the solution is r = (I (I p) 1 p) 1 , where 0 1 45/38 35/38 35/76 = @ 5/19 25/19 25/38A 1/19 5/19 43/38 so multiplying the ﬁrst row by 1 , we have r1 = 9/2, r2 = 7/2, and r3 = 7/4. Since each ri &lt; µi the stationary distribution is: ⇡ (n1 , n2 , n3 ) = 3 (1/2)n1 (1/2)n2 (1/4)n3 16 It is easy to see that Little...
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