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(b) 18 Nov. 8 Dec. 1995 external AT&T
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Fig. 15.7 Histogram (empirical density function) for interarrival time of one CMU and one external AT&T dataset.
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The obvious spikes around 0.146 s for the CMU dataset and around 0.000234 s for the external AT&T dataset are due to protocol dependencies. The spike in the CMU dataset is caused almost entirely by the KSHELL protocol. Each KSHELL invocation establishes two TCP connections within a time period of roughly 0.146 s. Since, on this Ethernet segment, it is quite likely that no other TCP connection request is initiated within such a small time window, this creates a spike in the empirical cumulative distribution. The spike in the external AT&T dataset is caused almost entirely by World Wide Web accesses that correspond to an HTTP client establishing parallel TCP connections to load inlined HTTP objects such as gif les. Since this external AT&T dataset is dominated by HTTP traf c, this spike is much more prevalent here than in the CMU dataset. For the purpose of the analysis discussed in this chapter we did not exclude any of these observations. Next, consider the shape and characteristics of the empirical cumulative distribution of all interarrival times. Figure 15.8(a) shows the empirical cumulative distribution resulting from the empirical density function of Fig. 15.7(a) for a CMU trace together with the tted cumulative distributions of the Weibull, Pareto, lognormal (lnorm), and exponential distributions. Figure 15.8(b) shows the cumulative distributions for the external AT&T trace. It is immediately obvious that the exponential, Pareto, and lognormal distributions result in very poor ts, while the agreement between the tted Weibull distribution and the measured interarrival times is quite good. It is interesting that the exponential model underestimates short as well as long interarrival times, while the lognormal model overestimates both. The model based on the Pareto distribution overestimates the medium length interarrival times and yields a very poor t for the shorter interarrival times. The observation that the tted Weibull distribution yields a much better t than any of the other ones be con rmed the discrepancy measure l2 . Table 15.5 q can q using 2 2 2 ^ ^ ^ ^ ^ ^ summarizes l v l and l v l2 for all ts and three data sets. This corresponds to a 68.26% con dence interval. Figure 15.9 visualizes the data from Table 15.5 for all datasets in a bar plot where the height of the bars corresponds to ^ ^ the value of l2 and the error bars to the standard deviation of l2 or the 68.26% con dence intervals. To visualize the discrepancy better, all values above a threshold of ve are cut off. The Weibull distribution yields statistically signi cant better models for the interarrival times of all ten datasets. For all datasets except the CMU dataset from November 1994 the lognormal model is an order of magnitude better than both the exponential model and the Pareto model, while the three cannot be distinguished for the CMU November dataset. This difference between the datasets is most likely due to the fact that they span signi cantly different time periods and capture different traf c mixes. 15.5.4 Application-Speci c Interarrival Times
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While the overall times are of interest, one expects that different protocols have different characteristics. We compute the discrepancies on a per application basis. Figure 15.10 visualizes their values in a bar plot. Again the height of the bars
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15.5 CHARACTERIZATION OF CONNECTION INTERARRIVAL TIMES
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Fig. 15.8 Empirical and tted cumulative interarrival time distributions of one CMU and one external AT&T dataset. ^ TABLE 15.5 The 68.26% Con dence Intervals for l2 for the Fitted Models of Some Datasets and the Parameters of the Fitted Weibull model Dataset Model Weibull Lognormal Exponential Pareto Weibull c Weibull a CMU 29 30 June 0.06 0.07 0.79 0.82 2.08 2.17 2.25 2.42 0.569 3.175 External AT&T 18 Nov. 8 Dec. 0.08 0.09 0.93 0.95 1.12 1.14 3.25 3.41 0.600 3.894 Internal AT&T 8 23 Dec. 0.18 0.20 1.48 1.58 2.34 2.53 13.6 14.9 0.419 32.582
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