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By Derek F. Lawden

ISBN-10: 0340045868

ISBN-13: 9780340045862

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Y=l=x Thus, we will want to assume that (B58) p(x)q(x, x) + LP(y)q(y, x) = 0, x*- x*. y=l=x It is not too hard to show that under this condition, v is invariant. See Andjel (1982) for a proof of this type. Distributions of this sort that are invariant for queuing systems are known as product form - see Kelly (1979). Note that by (B56), (B57) and (B58), A= L [p(x)q(x, x*) - q(x*, x)] > L x=l=x' n(x)q(x, x*) x=l=x' + q(x*, x*) = 0. *- Under our assumptions, the queue length is finite for x x*, and infinite for In fact, x* can be thought of as a source and sink for customers in the real queuing system on S\{x*}.

The first question one asks is whether the survival probability is strictly positive or not. Note in this connection that state 0 is absorbing for the process. The answer to this question, and other aspects of the behavior of X n , are basically determined by the mean of I, which we assume is finite. The process is said to be subcritical, critical, or supercritical according to whether m < 1, m = 1 or m > 1. The martingale plays a key role in the theory. e. Theorem B55. (a) The extinction probability 1 - p is the smallest nonnegative solution x of the equation L I(k)x 00 k = x.

T E g. 9;. -t for all bounded continuous functions F, G. -t. Then 24 Background and Tools Ill! 11 f E~g(TJs)dvds 11 f 111 EI(TJo)g(TJs)ds = t o t I(TJ)E~g(TJs)d/1ds 0 ={ = { = f S(s)gdvds gd[{ 11 VS(S)dS]. Therefore, by Theorem B51 and the Markov property, TJI is ergodic if and only if for every such I, 111 (B53) t 0 vS(s )ds ::::} /1, where::::} denotes weak convergence. Now suppose (B54) for two probability measures VI, V2. Then Vi is absolutely continuous with respect to /1, so it may be written as Vi = fi /1, with 0 :s fi :s 2 and II + h = 1.

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A Course in Applied Mathematics, Vol.1, 2 by Derek F. Lawden

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