# Influence of Model and Traffic Pattern on Determining the Self-Similarity in IP Networks
## 5 Analysis of Network Processes for High-Intensity Traffic  
The RStudio program performs statistical analysis for the overall traffic volume and determines the basic statistical parameters of the collected data, as shown in Table 1.  
A detailed analysis of Hurst's parameters was performed using the hurstexp command. There were used several methods, such as  
* Estimation of R/S;
* Corrected R/S estimation by Hurst's exponent;
* Improved Hurst empirical exponent;
* Hurst theoretical exponent. This function is part of the pracma library, which contains a large number of functions for statistical, mathematical, and numerical analysis. The hurstexp command uses R/S (long-range) analysis to determine the Hurst factor. The results obtained are summarized in Table 2.  
The resulting simulations show that regardless of the time window's size, for the R/S estimation method, the value of the Hurst coefficient was constant. The empirical Hurst exponent changed. It increased to 1.2 for window 256 and then decreases to 0.84 for window 2048. Improved empirical Hurst exponent kept similar relations. It took lower values than the empirical Hurst exponent. The theoretical Hurst exponent reached the highest value of 0.6 for the smallest window and then decreased with the increase in the time window size. On the basis of the values of Hurst exponent, we carried out a study on the convergence of the results obtained by spectral regression [35]. Using the hurstSpec command, we determined the Hurst coefficient according to three methods. The hurstSpec command determines the time series's Hurst coefficient by linear regression of the spectrum logarithm concerning the frequency logarithm. Frequency points are accumulated in equal width for the logarithmic scale. Spectrum values are averaged over each frame. The hurstSpec command uses the following methods:  
* Standard method estimates the Hurst exponent of the time series by performing a linear logarithmic regression of the spectral power density (SDF) in relation to the frequency's logarithmic function. * Smoothed method is based on the standard method, but the only change is to divide the frequency into blocks of equal width, while the spectral power density (SDF) is averaged for each block. * Robinson method estimates the Hurst factor of a time series using [PERSON]'s integration method. This method uses low frequency to the end of a block of spectral power density (SDF) [1,29]. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Minimum** & **1 st Quarter** & **Median** & **Average** & **3 rd Quarter** & **Maximum** \\ \hline
0 & 23,326 & 29,799 & 27,678 & 35,970 & 65,159 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Statistical analysis for general network traffic. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**Windows Size** & **2** & **4** & **8** & **16** & **32** & **64** & **128** & **256** & **512** & **1024** & **2048** \\ \hline R/S estimation & 0.85 & 0.85 & 0.85 & 0.85 & 0.85 & 0.85 & 0.85 & 0.85 & 0.85 & 0.85 & 0.85 & 0.85 \\ \hline R/S improved estration & 0.99 & 0.99 & 0.99 & 0.9 Evaluation of the Hurst coefficient by means of spectral regression showed that for the standard method, the H-value was 1.042; for the smoothed method, 1.207; and the Robinson method, 0.999. The discrepancy was therefore about 21%. The Robinson method returned a factor identical to that of the corrected R/S estimation. The hurstSpec command also offers the possibility to draw logarithmic graphs of spectral regression for standard and smooth methods. It is not possible to draw diagrams for the Robinson method. It is possible to draw diagrams according to four methods of spectral power density (SDF): direct, lag window, wax, multitaper [29; 35]. Detailed results of analyses are shown in Figures 5-8. Each figure has two charts--(a) shows the standard method, (b) shows the smoothed method. From the graphs (Figures 5-8), it can be concluded that the determined Hurst factor depends on the chosen method of spectral power density (SDF).