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upon the size of the buffer, the CLP and the precise mix of traf c that is multiplexed at the operating point. If any of these change, then the most probable time to over ow also changes. For example, t tends to zero as the size of buffer, B, tends to zero. The value of the space parameter s (perhaps measured in kb 1 ) measures the degree to which advantage can be gained from statistical multiplexing. In particular, for links with capacity much greater than the sum of the mean rates of the multiplexed sources, s tends to zero and j .s; t/ approaches the mean rate of a source of type j (e.g. as lims!0 [s 1 log. 1 eas C 1 ebs / D 1 .a C b/, for a source producing either a or b, with equal 2 2 2 probabilities, in a window of length t). For links with capacity not much greater than the sum of the mean rates of the sources, there can be little statistical multiplexing gain. Intuitively, buffering is crucial and increasing the size of the buffer will make a large reduction in the CLP, and thus s D d =d B will be large. As s ! 1, we nd that j .s; t/ tends to N N a value, say j .1; t/ D X j [0; t]=t, where X j [0; t] D supfa : P.X j [0; t] a/ > 0g, i.e., the least upper bound on the value that X j [0; t] takes with positive probability (e.g. as lims!1 [s 1 log. 1 eas C 1 ebs / D maxfa; bg). For sources that do not have 2 2 N maximum peak rates, such as a Gaussian one, X j [0; t] D 1. Note that this gives the appropriate effective bandwidth for deterministic multiplexing (i.e. for CLP D 0), since if P j x j j .1; t/ C, where t is given the value that maximizes the left-hand side of this P inequality, then j x j X j [0; t] Ct with probability 1 for all t. We pursue this further in Section 4.12. There is also a mathematical interpretation for s. Conditional on an over ow event happening, the empirical distributions of the inputs just prior to that event differ from their unconditional distributions. For example, they have greater means than usual and realize a total rate of C C B=t over the time t. The so-called exponentially tilted distribution with parameter s , speci es the distribution of the sources most probable behaviour leading up to an over ow event. The single constraint (4.6) is a good approximation to the boundary of the acceptance region if the values of s and t remain fairly constant on that boundary of A and so j .s; t/ does not vary much. In practice, the values of x might be expected to lie within some small part of the acceptance region boundary (perhaps because the network tries to keep x near some point where social welfare or revenue is maximized). In this case it is only important for (4.6) to give a good approximation to A on this part of its boundary. The motivation for the above approach comes from a large deviations analysis of a model of a single link. Here we simply state the main result. Let C be the capacity of the link and B be the size of its buffer. Suppose the operating point is x (dropping the bar for simplicity). Consider an asymptotic regime in which there are many sources , in which link capacity is C D N C .0/ , buffer size is B D N B .0/ , the operating point is x D N x .0/ , and N tends to in nity. It can be shown that lim log.CLP.N // " k X .0/ D sup inf st x j j .s; t/
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