Stochastic

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Unformatted text preview: lity that both servers will stay busy for all time and the queue lengths will tend to inﬁnity. Not covered by (i) or (ii) is the situation in which server 1 can handle her worst case scenario but server 2 cannot cope with his: 1 + p2 µ2 < µ1 and 2 + p1 µ1 > µ2 In some situations in this case, queue 1 will be empty often enough to reduce the arrivals at station 2 so that server 2 can cope with his workload. As we will see, a concrete example of this phenomenon occurs when 1 = 1, µ1 = 4, p1 = 1/2 2 = 2, µ2 = 3.5, p2 = 1/4 To check that for these rates server 1 can handle the maximum arrival rate but server 2 cannot, we note that 1 · 3.5 = 1.875 < 4 = µ1 4 1 · 4 = 4 > 3.5 = µ2 2 + p1 µ1 = 2 + 2 1 + p2 µ2 = 1 + To derive general conditions that will allow us to determine when a twostation network is stable, let ri be the long-run average rate that customers arrive at station i. If there is a stationary distribution, then ri must also be the long run average rate at which customers leave station i or the queue would grow linearly in time. If we want the ﬂow in and out of each of the stations to balance, then we need r1 = 1 + p2 r2 a...
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## This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.

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