5 rearranging we have ebn 1 1 0 1 36 note

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

Ask a homework question - tutors are online