# 5 recalling that esi 1 we have a simple proof that the

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Unformatted text preview: hat the queue is stable if the arrival rate is smaller than the long-run service rate. Theorem 3.5. Suppose < µ. If the queue starts with some ﬁnite number k 1 customers who need service, then it will empty out with probability one. Furthermore, the limiting fraction of time the server is busy is /µ. Proof. Let Tn = t1 + · · · + tn be the time of the nth arrival. The strong law of large numbers, Theorem 3.2 implies that 1 Tn ! n Let Z0 be the sum of the service times of the customers in the system at time 0 and let si be the service time of the ith customer to arrive after time 0. The strong law of large numbers implies Z0 + Sn 1 ! n µ The amount of time the server has been busy up to time Tn is Z0 + Sn . Using the two results Z0 + Sn ! Tn µ The actual time spent working in [0, Tn ] is Z0 + Sn Zn where Zn is the amount of work in the system at time Tn , i.e., the amount of time needed to empty the system if there were no more arrivals. To argue that equality holds we need to show that Zn /n ! 0. Intuitively, the condition < µ implies the queue reaches...
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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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