Annals of Operations Research

, Volume 170, Issue 1, pp 199–216 | Cite as

Bivariate statistical analysis of TCP-flow sizes and durations



We approximate the distribution of the TCP-flow rate by deriving it from the joint bivariate distribution of the flow sizes and flow durations of a given access network. The latter distribution is represented by a bivariate extreme value distribution using the Pickand’s dependence A-function. We estimate the A-function to measure the dependencies of random pairs: TCP-flow size and duration, the rate of TCP-flow and size, as well as the rate and duration. We provide a method to test that the achieved estimate of A-function is good and perform the analysis with one concrete data example.


TCP-flow Extreme value distribution Pickands function 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Beirlant, J., Goegebeur, Y., Teugels, J., & Segers, J. (2004). Statistics of extremes. New York: Wiley. CrossRefGoogle Scholar
  2. Brockwell, P., & Davis, R. (1991). Time series: theory and methods. Berlin: Springer. Google Scholar
  3. Capéraà, P., Fougères, A.-L., & Genest, C. (1997). Estimation of bivariate extreme value copulas. Biometrika, 84, 567–577. CrossRefGoogle Scholar
  4. Castellana, J. V., & Leadbetter, M. R. (1986). On smoothed probability density estimation for stationary processes. Stochastic Processes and their Applications, 21, 179–193. CrossRefGoogle Scholar
  5. Coles, S. (2001). An introduction to statistical modeling of extreme values. Berlin: Springer. Google Scholar
  6. D’Auria, B., & Resnick, S. (2006). Data network models of burstiness. Advances in Applied Probability, 38, 373–404. CrossRefGoogle Scholar
  7. Davis, R., & Resnick, S. (1985). Limit theory for moving averages of random variables with regularly varying tail probabilities. Annals of Probability, 13, 179–195. CrossRefGoogle Scholar
  8. Davydov, Y., Paulauskas, V., & Račkauskas, A. (2000). More on p-stable convex sets in Banach spaces. Journal of Theoretical Probability, 13(1), 39–64. CrossRefGoogle Scholar
  9. Embrechts, P., Klüppelberg, C., & Mikosch, T. (1997). Modelling extremal events for finance and insurance. Berlin: Springer. Google Scholar
  10. Fougères, A.-L. (2004). Multivariate extremes. In Extreme values in finance, telecommunications and the environment (pp. 373–388). London: Chapman & Hall. Google Scholar
  11. Hall, P., & Tajvidi, N. (2000). Distribution and dependence-function estimation for bivariate extreme-value distributions. Bernoulli, 6, 835–844. CrossRefGoogle Scholar
  12. Hall, P., Lahiri, S. N., & Truong, Y. K. (1995). On bandwidth choice for density estimation with dependent data. Annals of Statistics, 23(6), 2241–2263. CrossRefGoogle Scholar
  13. Kendall, M. (1970). Rank correlation methods (4th edn.). London: Griffin. Google Scholar
  14. Kettani, H., & Gubner, J. A. (2006). A novel approach to the estimation of the long-range dependence parameter. IEEE Transactions on Circuits and Systems-II: Express Briefs, 53(6), 463–467 CrossRefGoogle Scholar
  15. Kilpi, J., & Lassila, P. (2006). Micro- and macroscopic analysis of RTT variability in GPRS and UMTS networks. In F. Boavida (Eds.), LNCS: Vol. 3976. Networking 2006 (pp. 1176–1181). Berlin: Springer. Google Scholar
  16. Leadbetter, M. R. (1983). Extremes and local dependence in stationary sequences. Probability Theory and Related Fields, 65(2), 291–306. Google Scholar
  17. Markovich, N. M. (2005). On-line estimation of the tail index for heavy-tailed distributions with applications to WWW-traffic. In Proceedings of the EuroNGI first conference on NGI: traffic engineering, Rome, Italy. Google Scholar
  18. Markovitch, N. M., & Krieger, U. R. (2002). The estimation of heavy-tailed probability density functions, their mixtures and quantiles. Computer Networks, 40(3), 459–474. CrossRefGoogle Scholar
  19. Markovich, N. M., & Krieger, U. R. (2006). Inspection and analysis techniques for traffic data arising from the Internet. In Proceedings of the HETNETs’04 2nd international working conference on performance modelling and evaluation of heterogeneous networks, Ilkley, West Yorkshire (pp. 72/1–72/9). Google Scholar
  20. Mikosch, T. (2002). Modeling dependence and tails of financial time series (Technical Report Working Paper No. 181). University of Copenhagen, Laboratory of Actuarial Mathematics. Google Scholar
  21. Resnick, S. (1997). Heavy tail modeling and teletraffic data. Annals of Statistics, 25, 1805–1869. With discussion and a rejoinder by the author. CrossRefGoogle Scholar
  22. Resnick, S. (2006). Heavy-tail phenomena. Probabilistic and statistical modeling. Berlin: Springer. Google Scholar
  23. Ribatet, M. (2006). A user’s guide to the POT package (Version 1.0). Google Scholar
  24. Taqqu, M., & Teverovsky, V. (1998). On estimating the intensity of long-range dependence in finite and infinite variance time series. In R. J. Adler, F. E. Feldman, & M. S. Taqqu (Eds.), A practical guide to heavy tails (pp. 177–217). Basel: Birkhäuser. Google Scholar
  25. Tstat (2007). Tstat: TCP statistic and analysis tool.
  26. van de Meent, R., & Mandjes, M. (2005). Evaluation of ‘user-oriented’ and ‘black-box’ traffic models for link provisioning. In Proceedings of the 1st EuroNGI conference on next generation Internet networks traffic engineering, Rome, Italy. New York: IEEE Press. Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Institute of Control SciencesRussian Academy of SciencesMoscowRussia
  2. 2.VTTVTTFinland

Personalised recommendations