Fitting Measured Data Sets

The fitting method described in the previous section was applied to approximate three real measured traffic traces:

• The first trace contains local-area network (LAN) traffic collected in 1989 on an Ethernet at the Bellcore Morristown Research and Engineering facility. It may be downloaded from the WEB site collecting traffic traces [80]. The trace is the first trace of the collection called BC at [80]. The trace is analyzed in [28].

• The second trace is the first data set of the collection called dec-pkt at [80]. It was measured at Digital Equipment Corporation from a wide-area network (WAN) in 1995. The trace is analyzed in [62].

• The third trace is collected in 1996 at FIX West which is an FDDI LAN medium that serves as a network inter-exchange point among several service providers, both national and regional. The trace is available at [79].

Variance-time plots of the traffic generated by the MAPs resulted from fitting for the first trace are depicted in Figure 8.5. The length of the interval ∆ that is used to generate the series X = {Xi, i = 0, 1, ...} equals the expected interarrival time. The curve signed by (x¹, x2) belongs to the fitting when the first (second) time point of fitting the IDC value, t1 (t2), is x1 (x2) times the expected interarrival time.

The Hurst parameter of this trace consisting of one million arrivals (approximated by the variance-time plot) is 0.83. The interval (L1, L2) is (10, 10⁶) for the first two fitting, while it is (500, 5 · 10⁶) for the other two fittings. For the last two fittings the interval had to be changed because the time point at which the IDC is set is so high that the IPP destroys the pseudo self-similar nature of the PH arrival process and the algorithm can not provide the desired Hurst parameter.

Setting the IDC at a time point implies that the variance of the aggregated process is set at that time point as well. It can be observed in Figure 8.5 that the method is not capable of setting the IDC at t1. The variation of this traffic trace for low values of t1 is lower than the limit of this structure. The IDC at t1 was set as close as possible to the IDC of the real source.

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Variance

Figure 8.5: Variance-time plots of MAPs with dif-ferent time points of IDC matching for the first trace

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Variance

Figure 8.6: Variance-time plots of MAPs with dif-ferent time points of IDC matching for the second trace

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Variance

Figure 8.7: Variance-time plots of MAPs with dif-ferent time points of IDC matching for the third trace

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log10(R/S(n))

Figure 8.8: R/S plots of MAPs with different time points of IDC matching for the first trace

The second trace consisting of about 2 million arrivals has Hurst parameter 0.80 given by the variance-time test. The interval (L1, L2) is set as for the first trace. For this data set the algorithm is able to set the IDC at both time points t1 and t2 exactly. Figure 8.6 shows the variance-time plots for the MAPs resulted from the fitting.

The third and longest trace with more than 11 million arrivals has Hurst parameter 0.72. The interval (L1, L2) is set to (10, 10⁷). The algorithm sets the IDC at both timepoints to the desired values. Figure 8.7 shows the variance-time plots for the MAPs resulted from the fitting.

R/S plots for both the real traffic trace and the traffic generated by the approximating MAPs are given in Figure 8.8, 8.9 and 8.10. For the third trace the R/S plots of the approximating traces show higher degree of pseudo self-similarity than the R/S plot of the original trace does. The reason for this is that approximating the degree of self-similarity of the third trace by R/S plot gives a lower value for H

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log10(R/S(n))

Figure 8.9: R/S plots of MAPs with different time points of IDC matching for the second trace

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Figure 8.10: R/S plots of MAPs with different time points of IDC matching for the third trace

Visual inspection suggests that for both the variance-time and the R/S characteristics lower values of t1and t2 result in a closer fitting of the traffic trace behavior.

Regarding the third centralized moment of the number of arrivals in (0, t), M (t), the algorithm is able to set it for the first and the second traffic trace. Its value is set at the expected interarrival time.

For the third traffic trace M (t3) is too low to reach it. As it is mentioned in Section 8.1.2 we may only increase the third centralized moment by transforming the IPP into a two-state MMPP. For the third trace the superposition of the IPP and the PH arrival process results in a higher value than the desired one. As an example the third centralized moment of the third trace at the expected interarrival time is 1.008 while this value for the first approximating model is 1.52.

The goodness of fitting of the traces were tested for the queuing behavior of the •/D/1 queue, as well.

The results are depicted in Figure 8.11, 8.12 and 8.13. The •/D/1 queue was analyzed by applying matrix analytic methods (see e.g., [45] or [52]) with 80 % utilization of the server. As one may observe the lower t1and t2the longer the queue length distribution follows the original one. The experiments suggest that the pair E[Yi], 2E[Yi] is a good choice for t1 and t2. For the third trace which has the lowest but still notable estimated Hurst parameter the results show satisfying correspondence with the real traffic trace.

The cumulative distribution functions (cdf), which are important when calculating loss probabilities, are depicted in Figure 8.14, 8.15 and 8.16.

For what concerns the queueing behavior, the fitting is rather poor for the first two traces while it is good for the third one. The reason can be the quite irregular long-term behavior of the first two traces.

Since a single parameter is used to characterize the long-term behavior, this irregularity is not captured by the model.

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Figure 8.11: Queue length distribution for the first trace

Figure 8.12: Queue length distribution for the sec-ond trace

Figure 8.13: Queue length distribution for the third trace

Figure 8.14: cdf of the queue length for the first trace

Figure 8.15: cdf of the queue length for the second

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Figure 8.16: cdf of the queue length for the third