FILE SIZE DISTRIBUTION AND TRAFFIC SELF-SIMILARITY in Visual Studio .NET

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14.4 FILE SIZE DISTRIBUTION AND TRAFFIC SELF-SIMILARITY
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Fig. 14.3 Hurst parameter estimates (TCP run): R=S and variance time for a 1:05, 1.35, 1.65 and 1.95. (a) Base run, (b) large bandwidth=large buffer, and (c) large buffer.
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idealized line for a close to 1, and above the line for a close to 2. Figure 14.3(b) shows similar results for the case in which there is no signi cant limitation in bandwidth (155 Mb=s) leading to zero packet loss. There is noticeably more spread among the estimates, which we believe to be the result of more variability in the traf c patterns since traf c is less constrained by bandwidth limitations. Figure 14.3(c) shows the results when bandwidth is limited, as in the baseline case, but buffer sizes at the switch are increased (64 kB). Again, a roughly linear relationship between the heavy-tailedness of le size distribution (a) and self-similarity of link traf c (H) is observed. To verify that this relationship is not due to speci c characteristics of the TCP Reno protocol, we repeated our baseline simulations using TCP Tahoe and TCP Vegas. The results, shown in Figure 14.4, were essentially the same as in the TCP Reno baseline case, which indicates that speci c differences in implementation of
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Fig. 14.4 Hurst parameter estimates for (a) TCP Tahoe and (b) TCP Vegas runs with a 1.05, 1.35, 1.65, 1.95.
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THE PROTOCOL STACK AND ITS MODULATING EFFECT
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TCP's ow control between Reno, Tahoe, and Vegas do not signi cantly affect the resulting traf c self-similarity. Figure 14.5 shows the relative le size distribution of client=server interactions over the 10,000 second simulation time interval, organized into le size buckets (or bins). Each le transfer request is weighted by its size in bytes before normalizing to yield the relative frequency. Figure 14.5(a) shows that the Pareto distribution with a 1:05 generates le size requests that are dominated by le sizes above 64 kB. On the other hand, the le sizes for Pareto with a 1:95 (Fig. 14.5(b)) and the exponential distribution (Fig. 14.5(c)) are concentrated on le sizes below 64 kB, and in spite of ne differences, their aggregated behavior (cf. Figure 14.2) is similar with respect to self-similarity. We note that for the exponential distribution and the Pareto distribution with a 1:95, the shape of the relative frequency graph for the weighted case is analogous to the nonweighted (i.e., one that purely re ects the frequency of le size requests) case. However, in the case of Pareto with a 1:05, the shapes are ``reversed'' in the sense that the total number of requests are concentrated on small le sizes even though the few large le transfers end up dominating the 10,000 second simulation run. This is shown in Figure 14.6.
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(Weighted) file size distribution: Pareto 1.05 (Weighted) file size distribution: Pareto 1.95 (Weighted) file size distribution: exponential
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Fig. 14.5 Relative frequency of weighted le size distributions obtained from three 10,000 second TCP runs Pareto (a) with a 1:05 and (b) with a 1:95; (c) exponential distribution.
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Fig. 14.6 Relative frequency of unweighted le size distributions of TCP runs with Pareto (a) with a 1.05 and (b) with a 1.95; (c) exponential distribution.
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14.4 FILE SIZE DISTRIBUTION AND TRAFFIC SELF-SIMILARITY
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Effect of Idle Time Distribution
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All the runs thus far were obtained with an exponential idle time distribution with mean 600 ms. Figure 14.7(a) and (b) show the H-estimates of the baseline con guration when the idle time distribution is exponential with mean 0.6 s and Pareto with a 1:05 and mean 1.197 s. The le size distribution remained Pareto. As the H-estimates show, the effect of a Pareto-modeled heavy-tailed idle time distribution is to boost long-range dependence when a is close to 2, decreasing in effect as a approaches 1. This phenomenon may be explained as follows. For le size distributions with a close to 2, the correlation structure introduced by heavy-tailed idle time is signi cant relative to the contribution of the le size distribution, thus increasing the degree of self-similarity as re ected by H. As a approaches 1, however, the tail mass of the le size distribution becomes the dominating term, and the contribution of idle time with respect to increasing dependency becomes insigni cant in comparison. Figure 14.7(c) shows the Hurst parameter estimates when the le size distribution was exponential with mean 4.1 kB, but the idle time distribution was Pareto with a ranging between 1.05 and 1.95 and mean 1.197 s at a 1:05. As the idle time distribution is made more heavy-tailed a 3 1 , a positive trend in the H-estimates is discernible. However, the overall level of H-values is signi cantly reduced from the case when the le size distribution was Pareto, indicating that the le size distribution is the dominating factor in determining the self-similar characteristics of network traf c. 14.4.4 Effect of Traf c Mixing
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Figure 14.8 shows the effect of making one of the le size distributions heavy-tailed a 1:05 and the other one exponential in the 2-server system. Downstream throughput is plotted against time where the aggregation level is 100 seconds. Figure 14.8(a) shows the case when both servers are Pareto with a 1:05. Figure 14.8(c)
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Fig. 14.7 TCP run: exponential idle time versus Pareto idle time with Pareto le size distributions (a) variance time, (b) R=S, (c) Pareto idle times with exponential le size distribution (right).
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