# 5 rearranging we have ebn 1 1 0 1 36 note

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Unformatted text preview: equilibrium, so EZn stays bounded, and hence Zn /n ! 0. The details of completing this proof are too complicated to give here. However, in Example 3.6 we will give a simple proof of this. 107 3.2. APPLICATIONS TO QUEUEING THEORY 3.2.2 Cost equations In this subsection we will prove some general results about the GI/G/1 queue that come from very simple arguments. Let Xs be the number of customers in the system at time s. Let L be the long-run average number of customers in the system: Z 1t L = lim Xs ds t!1 t 0 Let W be the long-run average amount of time a customer spends in the system: n 1X W = lim Wm n!1 n m=1 where Wm is the amount of time the mth arriving customer spends in the system. Finally, let a be the long-run average rate at which arriving customers join the system, that is, a = lim Na (t)/t t!1 where Na (t) is the number of customers who arrive before time t and enter the system. Ignoring the problem of proving the existence of these limits, we can assert that these quantities are rel...
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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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