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Unformatted text preview: ng (A.22) we see that in the long run the number of calls
in the system will be Poisson with mean
Z1
(1 G(r)) dr = µ
r =0 That is, the mean number in the system is the rate at which calls enter times
their average duration. In the argument above we supposed that the system
starts empty. Since the number of initial calls still in the system at time t
decreases to 0 as t ! 1, the limiting result is true for any initial number of
calls X0 . 2.4.2 Superposition Taking one Poisson process and splitting it into two or more by using an i.i.d. sequence Yi is called thinning. Going in the other direction and adding up a lot
of independent processes is called superposition. Since a Poisson process can
be split into independent Poisson processes, it should not be too surprising that
when the independent Poisson processes are put together, the sum is Poisson
with a rate equal to the sum of the rates.
Theorem 2.13. Suppose N1 (t), . . . Nk (t) are independent Poisson processes
with rates 1 , . . . , k , then N1 (t) + · · · + Nk (t) is a Poisson process with rate
1 + · · · + k....
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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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