Queueing Systems

, Volume 66, Issue 4, pp 313–350 | Cite as

Modeling network traffic by a cluster Poisson input process with heavy and light-tailed file sizes

Article

Abstract

We consider a cluster Poisson model with heavy-tailed interarrival times and cluster sizes as a generalization of an infinite source Poisson model where the file sizes have a regularly varying tail distribution function or a finite second moment. One result is that this model reflects long-range dependence of teletraffic data. We show that depending on the heaviness of the file sizes, the interarrival times and the cluster sizes we have to distinguish different growths rates for the time scale of the cumulative traffic. The mean corrected cumulative input process converges to a fractional Brownian motion in the fast growth case. However, in the intermediate and the slow growth case we can have convergence to a stable Lévy motion or a fractional Brownian motion as well depending on the heaviness of the underlying distributions. These results are contrary to the idea that cumulative broadband network traffic converges in the slow growth case to a stable process. Furthermore, we derive the asymptotic behavior of the cluster Poisson point process which models the arrival times of data packets and the individual input process itself.

Keywords

Cluster Poisson model Covariance function Cumulative input process Fluid queue Fractional Brownian motion Heavy tails Input model Long-range dependence Self-similarity Stable Lévy motion Regular variation Scaling Teletraffic 

Mathematics Subject Classification (2000)

90B22 60F05 60F10 60F17 60G52 

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References

  1. 1.
    Anderson, K.K., Athreya, K.B.: A strong renewal theorem for generalized renewal functions in the infinite mean case. Probab. Theory Relat. Fields 77, 471–479 (1988) CrossRefGoogle Scholar
  2. 2.
    Baccelli, F., Brémaud, P.: Elements of Queueing Theory: Palm Martingale Calculus and Stochastic Recurrences. Springer, Heidelberg (2003) Google Scholar
  3. 3.
    Billingsley, P.: Convergence of Probability and Measures, 1st edn. Wiley, New York (1968) Google Scholar
  4. 4.
    Billingsley, P.: Probability and Measure, 2nd edn. Wiley, New York (1986) Google Scholar
  5. 5.
    Billingsley, P.: Convergence of Probability and Measures, 2nd edn. Wiley, New York (1999) CrossRefGoogle Scholar
  6. 6.
    Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge University Press, Cambridge (1987) Google Scholar
  7. 7.
    Crovella, M., Bestavros, A.: Explaining world wide web traffic self-similarity. Preprint (1995) Google Scholar
  8. 8.
    Crovella, M., Bestavros, A.: Self-similarity in world wide web traffic: evidence and possible causes. IEEE/ACM Trans. Netw. 5, 835–846 (1997) CrossRefGoogle Scholar
  9. 9.
    Crovella, M., Taqqu, M., Bestavros, A.: Heavy-tailed probability distributions in the world wide web. In: Adler, R., Epstein, R. (eds.) A Practical Guide to Heavy Tails: Statistical Techniques for Analysing Heavy Tailed Distributions, pp. 3–26. Birkhäuser, Boston (1999) Google Scholar
  10. 10.
    Daley, D.J., Vere-Jones, D.: An Introduction to the Theory of Point Processes. Vol. I: Elementary Theory and Methods, 2nd edn. Springer, New York (2003) Google Scholar
  11. 11.
    Daley, D.J., Vere-Jones, D.: An Introduction to the Theory of Point Processes. Vol. II: General Theory and Structure, 2nd edn. Springer, New York (2008) CrossRefGoogle Scholar
  12. 12.
    Doney, R.: One-sided local large deviation and renewal theorems in the case of infinite mean. Probab. Theory Relat. Fields 107, 451–465 (1997) CrossRefGoogle Scholar
  13. 13.
    Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling Extremal Events for Insurance and Finance. Springer, Berlin (1997) Google Scholar
  14. 14.
    Faÿ, G., González-Arévalo, B., Mikosch, T., Samorodnitsky, G.: Modeling teletraffic arrivals by a Poisson cluster process. Queueing Syst. 54, 121–140 (2006) CrossRefGoogle Scholar
  15. 15.
    Fasen, V., Samorodnitsky, G.: A fluid cluster Poisson input process can look like a fractional Brownian motion even in the slow growth aggregation regime. Adv. Appl. Probab. 41, 393–427 (2009) CrossRefGoogle Scholar
  16. 16.
    Gaigalas, R., Kaj, I.: Convergence of scaled renewal processes and a packet arrival model. Bernoulli 9, 671–703 (2003) CrossRefGoogle Scholar
  17. 17.
    Guerin, C., Nyberg, H., Perrin, O., Resnick, S., Rootzén, H., Stărică, C.: Empirical testing of the infinite source Poisson data traffic model. Stoch. Models 19, 151–200 (2003) CrossRefGoogle Scholar
  18. 18.
    Hernandes-Campos, H., Marron, S., Samorodnitsky, G., Smith, F.: Variable heavy tailed durations in Internet traffic. Perform. Eval. 58, 261–284 (2004) CrossRefGoogle Scholar
  19. 19.
    Kaj, I., Taqqu, M.S.: Convergence to fractional Brownian motion and to the Telecom process: the integral representation approach. In: Vares, M.E., Sidoravicius, V. (eds.) Out of Equilibrium 2. Progress in Probability, vol. 60, pp. 383–427. Birkhäuser, Boston (2008) CrossRefGoogle Scholar
  20. 20.
    Kallenberg, O.: Foundations of Modern Probability, 2nd edn. Springer, New York (2002) Google Scholar
  21. 21.
    Karr, A.F.: Point Processes and Their Statistical Inference. Marcel Decker, New York (1986) Google Scholar
  22. 22.
    Leland, W.E., Taqqu, M.S., Willinger, W., Wilson, D.V.: On the self-similar nature of Ethernet traffic (extended version). IEEE/ACM Trans. Netw. 2, 1–15 (1994) CrossRefGoogle Scholar
  23. 23.
    Levy, J., Taqqu, M.S.: On renewal processes having stable inter-renewal intervals and stable rewards. Ann. Sci. Math. Québec 11, 95–110 (1987) Google Scholar
  24. 24.
    Levy, J., Taqqu, M.S.: Renewal reward processes with heavy-tailed inter-renewal intervals and stable rewards. Bernoulli 6, 23–44 (2000) CrossRefGoogle Scholar
  25. 25.
    Mikosch, T., Resnick, S., Rootzén, H., Stegeman, A.: Is network traffic approximated by stable Lévy motion or fractional Brownian motion? Ann. Appl. Probab. 12, 23–68 (2002) CrossRefGoogle Scholar
  26. 26.
    Mikosch, T., Samorodnitsky, G.: Scaling limits for cumulative input processes. Math. Oper. Res. 32, 890–919 (2007) CrossRefGoogle Scholar
  27. 27.
    Omey, E.: Multivariate regular variation and its applications in probability theory. Ph.D. thesis, University of Leuven (1982) Google Scholar
  28. 28.
    Pipiras, V., Taqqu, M.: Slow, fast and arbitrary growth conditions for renewal-reward processes when both the renewals and the rewards are heavy-tailed. Bernoulli 10, 121–163 (2004) CrossRefGoogle Scholar
  29. 29.
    Resnick, S., Van den Berg, E.: Convergence of high-speed network traffic models. J. Appl. Probab. 37, 575–597 (2000) CrossRefGoogle Scholar
  30. 30.
    Resnick, S.I.: Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer, New York (2007) Google Scholar
  31. 31.
    Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999) Google Scholar
  32. 32.
    Taqqu, M.S., Levy, J.: Using renewal processes to generate long-range dependence and high variability. In: Eberlein, E., Taqqu, M.S. (eds.) Dependence in Probability and Statistics, pp. 73–89. Birkhäuser, Boston (1986) Google Scholar
  33. 33.
    Whitt, W.: Limits for cumulative input processes to queues. Tech. Rep., AT&T Labs (1999) Google Scholar
  34. 34.
    Whitt, W.: Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Applications to Queues. Springer, New York (2002) Google Scholar
  35. 35.
    Willinger, W., Taqqu, M., Sherman, R., Wilson, D.: Self-similarity through high-variability: statistical analysis of Ethernet LAN traffic at the source level. IEEE/ACM Trans. Netw. 5, 71–86 (1997) CrossRefGoogle Scholar
  36. 36.
    Zolotarev, V.: Mellin–Stieltjes transforms in probability theory. Theory Probab. Appl. 2, 433–459 (1957) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Center for Mathematical SciencesTechnische Universität MünchenGarchingGermany

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