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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 longrun average number of customers in
the system:
Z
1t
L = lim
Xs ds
t!1 t 0
Let W be the longrun 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 longrun 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).
 Spring '10
 DURRETT
 The Land

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