# Self-Similar Markovian Sources
## 4 Summary  
The accuracy in creating Internet traffic models is crucial in the design and evaluation of secure and high-performance networks. It is necessary to understand the nature of Internet traffic to successfully optimize its performance and efficiency in the face of growing demands for fast information transfer. This can result in better job scheduling and resource management, ensuring the quality of transmission and preventing system malfunctions.  
The network traffic exhibits self-similarity, which has a significant impact on network performance. The most well-known self-similar traffic models are not based on the Markov approach. Nevertheless, Markov models are traditional tools in the analytical evaluation of computer networks, and they should be adapted to the case of self-similar traffic.  
A few Markov models can be used to create self-similar traffic sources. We have described two of them: the SSMP model using discrete-time Markov chains and the MMPP model using continuous-time Markov chains. Both of them enable us to obtain traffic that is self-similar in a finite number of time-scales. It has been shown that it is necessary to increase the number of states in a corresponding Markov chain (i.e., increase the size \(n\) of the modulator in the case of SSMP or the number \(d\) of sources in an MMPP) to produce traffic with higher values of the Hurst parameter. In this way, we also increase the number of time-scales for which the examined traffic may sustain a relatively stable value of the Hurst parameter and consequently self-similar behavior. These Markov models may also be used in simulations. We also compare these models with a non-Markovian PPBP model, which is widely used in simulations. A more complex MMPP model (with \(d=22\)) can successfully \"compete\" with a PPBP, not only in terms of the quality of provided self-similar process but also in terms of the efficiency of its generation: the generation time of a defined number of packets is several times shorter in case of an MMPP.  
The great advantage of the Markov models is that they may be used in the modeling of transient states, which is important in the evaluation of computer networks as transmitted flows are continually changing due to the variability of transmissions and due to the performance of the flow control algorithms. The evaluation of transient states by simulation is extremely costly as it requires multiple repetitions. The biggest problem of the Markov approach is the large number of Markov chain states and consequently the high computational complexity. In the past, it was impossible to implement the more complex Markov models which are necessary to mimic real traffic data accurately. Due to the advances made in computing technology, complexity is no longer a problem, and Markov models can be successfully used in optimization tasks.  
\begin{table}
\begin{tabular}{c c} \hline
**Method** & **Avg Time [ms]** \\ \hline SSMP4 & 12.1257 \\ SSMP5 & 12.8312 \\ SSMP6 & 13.5641 \\ MMPP5 & 18.1601 \\ MMPP22 & 32.6147 \\ PPBP & 197.1943 \\ \hline \end{tabular}
\end{table}
Table 2: Comparison of time needed to generate 1 million packets using the different methods.  
**Author Contributions:** Investigation, A.D. and J.D.; Methodology, A.D. and T.C.; Software, A.D.; Validation, J.D.; Writing--original draft preparation, K.F., J.S. and T.C.; Writing--review and editing, A.D., J.D., K.F., J.S. and T.C.; Visualization, J.D.; Project administration, J.S.; Formal analysis, J.D. and K.F. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research received no external funding.  
**Conflicts of Interest:** The authors declare no conflict of interest.