Analysis of the Distribution of the Statistic of a Test for Discriminating Correlated Processes

  • M. E. Sousa-Vieira
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6869)


In this paper, we analyze the distribution of the statistic of a test for identifying the type of correlated time series. The rule for selecting a model suitable to the data is based on the comparison between the normalized prediction errors of the Whittle estimator applied to the candidate models. We consider one application of the test: assessing the significance of increasing the number of parameters within a given class of models. The results obtained demonstrate that the Weibull distribution is a good approximation for the distribution of the test statistic.


Correlated processes Whittle estimator Model selection Traffic modeling 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Beran, J.: Statistics for Long-Memory Processes. Chapman and Hall, Boca Raton (1994)zbMATHGoogle Scholar
  2. 2.
    Beran, J., Shreman, R., Taqqu, M.S., Willinger, W.: Long-Range Dependence in Variable-Bit-Rate video traffic. IEEE Transactions on Communications 43(2/4), 1566–1579 (1995)CrossRefGoogle Scholar
  3. 3.
    Cox, D.R., Isham, V.: Point Processes. Chapman and Hall, Boca Raton (1980)zbMATHGoogle Scholar
  4. 4.
    Cox, D.R.: Long-Range Dependence: A review. In: Statistics: An Appraisal, pp. 55–74. Iowa State University Press, Iowa (1984)Google Scholar
  5. 5.
    Crovella, M.E., Bestavros, A.: Self-similarity in World Wide Web traffic: Evidence and possible causes. IEEE/ACM Transactions on Networking 5(6), 835–846 (1997)CrossRefGoogle Scholar
  6. 6.
    Duffield, N.: Queueing at large resources driven by long-tailed M/G/∞ processes. Queueing Systems 28(1/3), 245–266 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Eliazar, I.: The M/G/∞ system revisited: Finiteness, summability, long-range dependence and reverse engineering. Queueing Systems 55(1), 71–82 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Erramilli, A., Narayan, O., Willinger, W.: Experimental queueing analysis with Long-Range Dependent packet traffic. IEEE/ACM Transactions on Networking 4(2), 209–223 (1996)CrossRefGoogle Scholar
  9. 9.
    Garrett, M.W., Willinger, W.: Analysis, modeling and generation of self-similar VBR video traffic. In: Proc. ACM SIGCOMM 1994, London, UK, pp. 269–280 (1994)Google Scholar
  10. 10.
    Hurst, H.E.: Long-term storage capacity of reservoirs. Transactions of the American Society of Civil Engineers 116, 770–799 (1951)Google Scholar
  11. 11.
    Jiang, M., Nikolic, M., Hardy, S., Trajkovic, L.: Impact of self-similarity on wireless data network performance. In: Proc. IEEE ICC 2001, Helsinki, Finland, pp. 477–481 (2001)Google Scholar
  12. 12.
    Krunz, M., Makowski, A.: Modeling video traffic using M/G/∞ input processes: A compromise between Markovian and LRD models. IEEE Journal on Selected Areas in Communications 16(5), 733–748 (1998)CrossRefGoogle Scholar
  13. 13.
    Leland, W.E., Taqqu, M.S., Willinger, W., Wilson, D.V.: On the self-similar nature of Ethernet traffic (extended version). IEEE/ACM Transactions on Networking 2(1), 1–15 (1994)CrossRefGoogle Scholar
  14. 14.
    Li, S.Q., Hwang, C.L.: Queue response to input correlation functions: Discrete spectral analysis. IEEE/ACM Transactions on Networking 1(5), 317–329 (1993)Google Scholar
  15. 15.
    Likhanov, N., Tsybakov, B., Georganas, N.D.: Analysis of an ATM buffer with self-similar (“fractal”) input traffic. In: Proc. IEEE INFOCOM 1995, Boston, MA, USA, pp. 985–992 (1995)Google Scholar
  16. 16.
    López, J.C., López, C., Suárez, A., Fernández, M., Rodríguez, R.F.: On the use of self-similar processes in network simulation. ACM Transactions on Modeling and Computer Simulation 10(2), 125–151 (2000)CrossRefGoogle Scholar
  17. 17.
    Norros, I.: A storage model with self-similar input. Queueing Systems 16, 387–396 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Novak, M.: Thinking in patterns: Fractals and related phenomena in nature. World Scientific, SingaporeGoogle Scholar
  19. 19.
    Paxson, V., Floyd, S.: Wide-area traffic: The failure of Poisson modeling. IEEE/ACM Transactions on Networking 3(3), 226–244 (1995)CrossRefGoogle Scholar
  20. 20.
    Resnick, S., Rootzen, H.: Self-similar communication models and very heavy tails. Annals of Applied Probability 10(3), 753–778 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Sousa, M.E., Suárez, A., López, C., Fernández, M., López, J.C., Rodríguez, R.F.: Fast simulation of self-similar and correlated processes. Mathematics and Computers in Simulation 80(10), 2040–2061 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Sousa, M.E., Suárez, A., Rodríguez, R.F., López, C.: Flexible adjustment of the short-term correlation of LRD M/G/∞-based processes. Lecture Notes in Theoretical Computer Science 261, 131–145 (2010)CrossRefGoogle Scholar
  23. 23.
    Sousa, M.E., Suárez, A., Fernández, M., López, J.C., López, C., Rodríguez, R.F.: Application of a hypothesis test for discriminating long-memory processes to the M/G/∞ process. In: Proc. Statistical Methods of Signal and Data Processing, Kiev, Ukraine (2010)Google Scholar
  24. 24.
    Suárez, A., López, J.C., López, C., Fernández, M., Rodríguez, R.F., Sousa, M.E.: A new heavy-tailed discrete distribution for LRD M/G/∞ sample generation. Performance Evaluation 47(2/3), 197–219 (2002)CrossRefzbMATHGoogle Scholar
  25. 25.
    Tsoukatos, K.P., Makowski, A.M.: Heavy traffic analysis for a multiplexer driven by M/G/∞ input processes. In: Proc. 15th International Teletraffic Congress, Washington, DC, USA, pp. 497–506 (1997)Google Scholar
  26. 26.
    Whittle, P.: Estimation and information in stationary time series. Arkiv Matematick 2(23), 423–434 (1953)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • M. E. Sousa-Vieira
    • 1
  1. 1.Department of Telematics EngineeringUniversity of VigoSpain

Personalised recommendations