The Influence of the Traffic Self-similarity on the Choice of the Non-integer Order PI\(^\alpha \) Controller Parameters

  • Adam Domański
  • Joanna Domańska
  • Tadeusz CzachórskiEmail author
  • Jerzy Klamka
  • Dariusz Marek
  • Jakub Szyguła
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 935)


The article discusses the problem of choosing the best parameters of the non-integer order \(PI^{\alpha }\) controller used in IP routers Active Queue Management for TCP/IP traffic flow control. The impact of the self-similarity of the traffic on the controller parameters is investigated with the use of discrete event simulation. We analyze the influence of these parameters on the length of the queue, the number of rejected packets and waiting times. The results indicate that the controller parameters strongly depend on the value of the Hurst parameter.


Active queue management Network congestion control Parameters of the non-integer order \(PI^{\alpha }\) controller 



This work was partially financed by National Science Center project no. 2017/27/ B/ST6/00145.


  1. 1.
    Leszczyński, J., Ciesielski, M.: A numerical method for solution of ordinary differential equations of fractional order. In: Wyrzykowski, R., Dongarra, J., Paprzycki, M., Waśniewski, J. (eds.) PPAM 2001. LNCS, vol. 2328, pp. 695–702. Springer, Heidelberg (2002). Scholar
  2. 2.
    Cox, D.: Long-range dependance: a review. In: Statistics: An Appraisal, pp. 55–74. Iowa State University Press (1984)Google Scholar
  3. 3.
    Domańska, J., Augustyn, D., Domański, A.: The choice of optimal 3-rd order polynomial packet dropping function for NLRED in the presence of self-similar traffic. Bull. Pol. Acad. Sci Tech. Sci. 60, 779–786 (2012)Google Scholar
  4. 4.
    Domańska, J., Domański, A., Augustyn, D., Klamka, J.: A RED modified weighted moving average for soft real-time application. Int. J. Appl. Math. Comput. Sci. 24(3), 697–707 (2014)CrossRefGoogle Scholar
  5. 5.
    Domańska, J., Domański, A., Czachórski, T.: Fluid flow analysis of RED Algorithm with modified weighted moving average. In: Dudin, A., Klimenok, V., Tsarenkov, G., Dudin, S. (eds.) BWWQT 2013. CCIS, vol. 356, pp. 50–58. Springer, Heidelberg (2013). Scholar
  6. 6.
    Domańska, J., Domański, A., Czachórski, T., Klamka, J.: Fluid flow approximation of time-limited TCP/UDP/XCP streams. Bull. Pol. Acad. Sci. Tech. Sci. 62(2), 217–225 (2014)Google Scholar
  7. 7.
    Domańska, J., Domański, A., Czachórski, T., Klamka, J.: Use of a non integer order PI controller to active queue management mechanism. Int. J. Appl. Math. Comput. Sci. 26(4), 777–789 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Domańska, J., Domański, A., Czachórski, T., Klamka, J.: Self-similarity traffic and AQM mechanism based on non-integer order \(PI^{\alpha }D^\beta \) controller. In: Proceedings of 24th International Conference on Computer Networks, CN 2017, Ladek Zdroj (2017)Google Scholar
  9. 9.
    Domańska, J., Domański, A., Czachórski, T., Klamka, J., Szyguła, J.: The AQM dropping packet probability function based on non-integer order \(PI^{\alpha }D^\beta \) controller. In: International Conference on Non-Integer Order Calculus and its Applications (2017)Google Scholar
  10. 10.
    Domański, A., Domańska, J., Czachórski, T.: Comparison of AQM control systems with the use of fluid flow approximation. In: Kwiecień, A., Gaj, P., Stera, P. (eds.) CN 2012. CCIS, vol. 291, pp. 82–90. Springer, Heidelberg (2012). Scholar
  11. 11.
    Floyd, S., Jacobson, V.: Random early detection gateways for congestion avoidance. IEEE/ACM Trans. Netw. 1(4), 397–413 (1993)CrossRefGoogle Scholar
  12. 12.
    Hooke, R., Jeeves, T.: Direct search solution of numerical and statistical problems. J. ACM 8(2), 212–229 (1961). (USA)CrossRefGoogle Scholar
  13. 13.
    Karagiannis, T., Molle, M., Faloutsos, M.: Long-range dependence: ten years of internet traffic modeling. IEEE Internet Comput. 8(5), 57–64 (2004)CrossRefGoogle Scholar
  14. 14.
    Kelley, C.: Iterative methods for optimization. Front. Appl. Math. (18) (1999) (SIAM, Philadelphia)Google Scholar
  15. 15.
    Latawiec, K., Stanisławski, R., Łukaniszyn, M., Czuczwara, W., Rydel, M.: Fractional-order modeling of electric circuits: modern empiricism vs. classical science. In: Progress in Applied Electrical Engineering (PAEE) (2017)Google Scholar
  16. 16.
    Lopez-Ardao, J., Lopez-Garcia, C., Suarez-Gonzalez, A., Fernandez-Veiga, M., Rodriguez-Rubio, R.: On the use of self-similar processes in network simulation. ACM Trans. Model. Comput. Simul. 10(2), 125–151 (2000)CrossRefGoogle Scholar
  17. 17.
    Mandelbrot, B., Ness, J.: Fractional brownian motions, fractional noises and applications. SIAM Rev. 10, 422–437 (1968)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Miller, K., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)zbMATHGoogle Scholar
  19. 19.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego, USA (1999)zbMATHGoogle Scholar
  20. 20.
    Samorodnitsky, G., Taqqu, M.: Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman and Hall, New York (1994)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Adam Domański
    • 1
  • Joanna Domańska
    • 2
  • Tadeusz Czachórski
    • 2
    Email author
  • Jerzy Klamka
    • 2
  • Dariusz Marek
    • 1
  • Jakub Szyguła
    • 1
  1. 1.Institute of InformaticsSilesian University of TechnologyGliwicePoland
  2. 2.Institute of Theoretical and Applied InformaticsPolish Academy of SciencesGliwicePoland

Personalised recommendations