# Self-Similar Markovian Sources
## 3 Experimental Results  
The traffic models presented in Section 2 were programmed using Python and used in its discrete event simulation module SimPy [46]. We have used the aggregated variance method, briefly described in Section 1, to determine the value of the Hurst parameter for the generated samples. We have plotted the values of the normalized variance versus the size of the time slots used in the aggregation process in the log-log scale (see Equation (4)). The relationship between the slope \(-\beta\) of the plot and the Hurst parameter \(H\) is presented in Equation (3). Note that for a steeper slope of the plot, the corresponding Hurst parameter takes smaller values. With the increase in the value of the slope, the value of the Hurst parameter decreases. The slope of the plot can slightly vary within a few first time-scales. This means that the value of the Hurst parameter within this range is non-identical for different time-scales. On the other hand, the visible decrease of the slope after the initial time-scales means that a process is no longer able to maintain self-similar properties. Such processes are self-similar only in a finite number of time-scales. We have fitted a straight-line for each of the plots using the least squares technique. At this point, it is necessary to mark that the values of the vertical axis differ between the figures. As a result, it is possible to observe that the slope of the straight-line fits differs between the figures. For every plot, we have computed the average and the maximum distance between a straight-line fit and the actual plot. The results of these computations have been presented in Table 1. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Method** & **Max Distance** & **Avg Distance** \\ \hline SSMP4 & 1.0722 & 0.5264 \\ SSMP5 & 0.8912 & 0.3974 \\ SSMP6 & 0.7088 & 0.2664 \\ MMPP5 & 0.3999 & 0.1087 \\ MMPP22 & 0.3257 & 0.0991 \\ PPBP & 0.1790 & 0.0542 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The comparison of distances between calculated values, and the fitted values provided by the method of least squares. SSMP: special semi-Markov process; MMPP: modulated Markov Poisson process; PPBP: Poisson Pareto burst process. Figures 4 and 5 show the results achieved using the MMPP model with two different numbers of active ON-OFF sources (\(d\)). For the MMPP with \(d=5\), we were able to achieve a sample with \(H=0.7\). The corresponding plot is shown in Figure 4. It was necessary to increase the number of active MMPP sources to achieve samples with higher values of the Hurst parameter. In Figure 5, we show the plot for the sample with \(H=0.9\), which is a very high value of the Hurst parameter. We can observe in Table 1 that, for a more complex MMPP model, the values of the average and the maximum distance between the straight-line fit and the actual plot decrease, which means that the approximation reflects the actual plot more closely. In Figure 6, we present a plot for a non-Markovian model--PPBP. This is a model which is frequently used in network traffic analysis. The PPBP process consists of a simultaneous generation of packets by a larger number of Poisson sources. This number was changed during the experiment. Using this model, we obtained a sample with \(H=0.8\). We can observe that the generated sample displays self-similarity over most of the measured time scales. The maximum and average distances between the actual plot and the straight-line fit for this model (presented in Table 1) are the smallest. Figure 4: The aggregate variance of the sample in several time scales with the MMPP model, \(d=5\). Figure 3: The aggregate variance of the sample in several time scales with the SSMP model, \(n=6\). The sample with the highest value of the Hurst parameter was achieved using a model with 22 active MMPPs. It is important to note that by generating this sample, the Markovian model has outperformed the PPBP model in terms of achieving the sample with the highest value of the Hurst parameter.