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Fig. 18.7 Transformation of the heavy-tailedness of le size distribution property at the application layer via the action of the transport layer into its manifestation as self-similar aggregated traf c at the link layer.
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maximize throughput. For any reasonable assignment of bandwidth, buffer size, mean le request size, and other system parameters, we found that by either adjusting the number of clients or the mean of the idle time distribution between successive le transfers appropriately, any target contention level could be achieved. In a typical con guration, the rst 32 connections served as ``background traf c,'' transferring les from clients to servers (or sinks), where the le sizes were drawn from Pareto distributions with shape parameter a 1:05, 1.35, 1.65, 1.95. As in Park et al. [33], there was a linear relation between a and the long-range dependence of aggregate traf c observed at the bottleneck link G1 ; G2 as captured by the Hurst parameter H. H was close to 1 when a was near 1, and H was close to 1 when a 2 was near 2. The 33rd connection acted as an in nite source trying to maximize throughput by running the generic feedback control, with or without SAC. In other settings, the number of congestion-controlled in nite sources was increased to observe their mutual interaction and the impact on fairness and ef ciency. A typical run lasted for 10,000 or 20,000 seconds (simulated time) with traces collected at 10 ms granularity.
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Network con guration with bottleneck link G1 ; G2 .
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Per-Connection On-Line Predictability
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18.5.2.1 Predictability Structure One of the rst items to test was estimation of the conditional probability densities PrfL2 jL1 g using the per-connection, on-line method described in Section 18.4.2.1. We observe the same skewed diagonal shift characteristics as seen in the off-line case for a 1:05 traf c and the relatively invariant shape of the probability densities for a 1:95 traf c (we omit the 3D plots due to space constraints). Also, as in the off-line case, as we increase the time scale (i.e., T1 ) from 500 ms to 1 s and higher, for a 1:05 traf c the probability densities becomes more concentrated, thus increasing the accuracy of prediction. Figure 18.9(a) shows the shifting effect of the conditional probabilities for a 1:05 traf c via a 2D projection that shows the marginal densities. Whereas the shifting effect is evident for a 1:05 traf c, for a 1:95 traf c (Fig. 18.9(b)) the probability densities stay largely invariant. 18.5.2.2 Time Scale In Section 18.3.2, in connection with the off-line estimation, we showed using entropy calculations that the conditional probability densities became more concentrated as time scale was increased, attening out eventually near the 4 5 s mark. We observe a similar behavior with respect to the entropy curve in the on-line case. Locating the knee of the entropy curve is of import for on-line prediction and its use in congestion control since the size of the time scale will in uence whether a certain procedure for exploiting predictability will be effective or not. As an extreme case in point, if the time scale of prediction were, say, 1000 s, it is dif cult to imagine a mechanism that would be able to effectively exploit this information for congestion control purposes too many changes may be occurring during such a time period that may be both favorable and detrimental to a constant control action yielding a zero net gain. On the other hand, if control actions capable of spanning large time frames such as bandwidth reservation or pricing-based admission control were made part of the
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Fig. 18.9 (a) Shifting effect of conditional probability densities P L2 jL1 1 and P L2 jL1 8 under a 1:05 background traf c. (b) Shifting effect of conditional probability densities P L2 jL1 1 and P L2 jL1 8 under a 1:95 traf c.
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