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17:40
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where bij is the required bandwidth and pij is the steady-state probability of level j in source i, as in Eq. (17.32). Thus, P L t ! x e I x ; where I x sup yx
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17:41
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It can be shown [11] that such bounds are asymptotically tight (have a large deviation limit) as the number of sources increases, provided the spectrum of
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NETWORK DESIGN USING HEAVY-TAILED DISTRIBUTIONS
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behavior of individual sources is suf ciently regular, yielding the exponential approximation P L t ! x % e I x : 17:42
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Finding the rate function I will in general require numerical solution of the variational expression (17.41). It can be shown that the right-hand side (RHS) of Eq. (17.41) is a concave function of y, and under mild conditions it is differentiable also. Hence, the supremum is achieved at the unique solution y to the Euler Lagrange equation P
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17:43
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Generally, it is not dif cult to numerically determine the supremum in Eq. (17.41) by location of the solution to Eq. (17.43). Example 17.7.1. In special cases the variational problem can be solved explicitly. This is possible in the case of n homogeneous two-level sources. Here we have bij bj with j P f1; 2g, 0 b1 < b2 , and p1 p2 1. For this case, I x n sup yx log p1 eyb1 p2 eyb2 x n log y x log p1 y x b1 = b2 b1 p2 y x b2 = b2 b1 b2 b1 with y x p1 x b1 = p2 b2 x for b1 x 
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17:44  17:45
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We now show how to estimate the time to recover from the high-congestion event, where the high-congestion event is a large initial bandwidth x. We understand recovery to occur when the aggregate bandwidth is again less than or equal to the capacity c. In applications, we suggest examining the function of aggregate bandwidth giving both the probability of reaching that level and the recovery time from that level to assess whether or not capacity is adequate to meet demand. We assume that recovery occurs when the aggregate bandwidth drops below a level c, where x > c > m, with x being the initial level and m being the steady-state mean. Given that we know the current level of each level process, we know that the remaining holding time (and also the age) is distributed according to the levelholding-time stationary-excess distribution in Eq. (17.33). We use the LDP to approximate the conditional distribution of the level process for each source (in the steady state). The idea is to perform the appropriate change of measure (tilting) corresponding to the rare event.
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17.8 A LINEAR APPROXIMATION
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Given that P tion is
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Bi ! x % e I x for I x in Eq. (17.41), the LDP approxima ! !x %  p ij  PJ
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17:46
where y* yields the supremum in Eq. (17.41). Put another way, comparing Eq. P (17.46) with Eq. (17.43) we see that y* is chosen to make the expectation of i Bi  equal to x under the distribution p. In the homogeneous case, equality in Eq. (17.46) in the limit as the number of sources increases is due to the conditional limit theorem of Van Campenhout and Cover [26]. The limit can be extended to cover suitably regular heterogeneity in the bij , for example, nitely many types of source. We thus approximate the conditional bandwidth process by B t jB 0 x % E B t jB 0 x n Ji PP i I i  % E Bij t jBij 0 y dGje y pj
i 1 j 1 0
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17:47
 for pij in Eq. (17.46), which has Laplace transform
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17:48
^ The Laplace transform Pjk s in Eq. (17.48) was derived in Theorem 17.6.1. We can numerically invert it to calculate the conditional mean as a function of time. We then can determine when E B t jB 0 x rst falls below c. In general, this conditional mean need not be a decreasing function, so that care is needed in the de nition, but we expect it to be decreasing for suitably small t because the initial point B 0 is unusually high.