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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
Proof. Let Tn = t1 + · · · + tn be the time of the nth arrival. The strong law of
large numbers, Theorem 3.2 implies that
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
The amount of time the server has been busy up to time Tn is Z0 + Sn . Using
the two results
Z0 + Sn
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
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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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