Stochastic

# If this happens then we have at most 3 green arrivals

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Unformatted text preview: ication of this consider Example 2.4. M/G/1 queue. In modeling telephone tra c, we can, as a ﬁrst approximation, suppose that the number of phone lines is inﬁnite, i.e., everyone who tries to make a call ﬁnds a free line. This certainly is not always true but analyzing a model in which we pretend this is true can help us to discover how many phone lines we need to be able to provide service 99.99% of the time. The argument for arrivals at the Great Hall implies that the beginnings of calls follow a Poisson process. As for the calls themselves, while many people on the telephone show a lack of memory, there is no reason to suppose that the duration of a call has an exponential distribution. So we use a general distribution function G with G(0) = 0 and mean µ. Suppose that the system starts empty at time 0. The probability a call started at s has ended by time t is G(t s), so using Theorem 2.12 the number of calls still in progress at time t is Poisson with mean Zt Zt (1 G(t s)) ds = (1 G(r)) dr s=0 r =0 Letting t ! 1 and usi...
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## This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.

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