Queueing Systems

, Volume 54, Issue 2, pp 121–140

Modeling teletraffic arrivals by a Poisson cluster process

  • Gilles Faÿ
  • Bárbara González-Arévalo
  • Thomas Mikosch
  • Gennady Samorodnitsky


In this paper we consider a Poisson cluster process N as a generating process for the arrivals of packets to a server. This process generalizes in a more realistic way the infinite source Poisson model which has been used for modeling teletraffic for a long time. At each Poisson point Γj, a flow of packets is initiated which is modeled as a partial iid sum process \(\Gamma_j+\sum_i=1^kX_ji, k\le K_j\), with a random limit Kj which is independent of (Xji) and the underlying Poisson points (Γj). We study the covariance structure of the increment process of N. In particular, the covariance function of the increment process is not summable if the right tail P(Kj > x) is regularly varying with index α∊ (1, 2), the distribution of the Xji’s being irrelevant. This means that the increment process exhibits long-range dependence. If var(Kj) < ∞ long-range dependence is excluded. We study the asymptotic behavior of the process (N(t))t≥ 0 and give conditions on the distribution of Kj and Xji under which the random sums \(\sum_{i=1}^{K_j}X_{ji}\) have a regularly varying tail. Using the form of the distribution of the interarrival times of the process N under the Palm distribution, we also conduct an exploratory statistical analysis of simulated data and of Internet packet arrivals to a server. We illustrate how the theoretical results can be used to detect distribution al characteristics of Kj, Xji, and of the Poisson process.


Teletraffic Poisson cluster model Long-range dependence Palm distribution Regular variation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    G. ALSMEYER, On generalized renewal measures and certain first passage times, Ann. Probab. 20 (1992) 1229–1247.Google Scholar
  2. 2.
    F. BACCELLI AND P. BRÉMAUD, Elements of Queueing Theory. Palm Martingale Calculus and Stochastic Recurrences, 2nd edition (Springer, New York, 2003).Google Scholar
  3. 3.
    J.M. Bardet, G. Lang, E. Moulines, and P. Soulier, Wavelet estimator of long-range dependent processes. Stat. Inference Stoch. Process. 3 (2000) 85–99. 19th “Rencontres Franco-Belges de Statisticiens” (Marseille, 1998).Google Scholar
  4. 4.
    M.S. BARTLETT, The spectral analysis of point processes es. J. Roy. Statist. Soc., Series B 25 (1963) 264–296.Google Scholar
  5. 5.
    P. BILLINGSLEY, Convergence of Probability Measures. (Wiley, New York, 1968).Google Scholar
  6. 6.
    N.H. BINGHAM, C.M. GOLDIE, AND J.L. TEUGELS, Regular Variation (Cambridge University Press, Cambridge (UK), 1987).Google Scholar
  7. 7.
    P.J. BROCKWELL AND R.A. DAVIS, Time Series: Theory and Methods, (2nd edition) (Springer, New York, 1991).Google Scholar
  8. 8.
    D.B.H. CLINE AND T. HSING, Large deviation probabilities for sums of random variables with heavy or subexponential tails. Technical Report, Texas A& M University (1998).Google Scholar
  9. 9.
    M. CROVELLA AND A. BESTAVROS, Self-similarity in world wide web traffic: evidence and possible causes. In Proceedings of the 1996 ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Systems (1996) vol. 24, pp. 160–169.Google Scholar
  10. 10.
    M. CROVELLA, A. BESTAVROS, AND M.S. TAQQU, Heavy-tailed probability distributions in the world wide web. A Practical Guide to Heavy Tails: Statistical Techniques for Analysing Heavy Tailed Distributions, R., Adler, R. Feldman, and M.S., Taqqu (eds.) (Birkhäuser, Boston, 1998) pp. 3–26.Google Scholar
  11. 11.
    D. DALEY, Asymptotic properties of stationary point processes es with generalized clusters. Z. Wahrscheinlichkeitstheorie verw. Geb. 21 (1972) 65–76.CrossRefGoogle Scholar
  12. 12.
    D. DALEY AND R. VESILO, Long range dependence of point processes, with queueing examples. Stochastic Processes and Their Applications 70 (1997) 265–282.CrossRefGoogle Scholar
  13. 13.
    D. DALEY AND D. VERE-JONES, An Introduction to the Theory of Point Processes (Springer, New York, 1988).Google Scholar
  14. 14.
    P. EMBRECHTS, C. KLÜppelberg, and T. Mikosch, Modelling Extremal Events for Insurance and Finance (Springer, Berlin, 1997).Google Scholar
  15. 15.
    P. EMBRECHTS AND E. OMEY, On subordinated distribution s and random record processes. Proc. Camb. Phil. Soc. 93 (1983) 339–353.CrossRefGoogle Scholar
  16. 16.
    G. Faÿ, F. Roueff, and P. Soulier, Estimation of the memory parameter of the infinite source Poisson process. Preprint (2005). Available at http://arxiv.org/abs/math/0509371.
  17. 17.
    A. GUT, Stopped Random Walks (Springer, New York, 1988).Google Scholar
  18. 18.
    N.R. Hansen, Markov Controlled Excursions, Local Alignment and Structure. PhD thesis, Institute of Mathematics, University of Copenhagen (2003) .Google Scholar
  19. 19.
    D. HEATH, S. RESNICK, AND G. SAMORODNITSKY, Heavy tails and long range dependence in ON/OFF processes and associated fluid models. Math. Oper. Res. 23 (1998) 145–165.Google Scholar
  20. 20.
    N. Hohn and D. Veitch, Inverting sampled traffic. ACM/SIGCOMM Internet Measurement Conference, (Miami, USA, 2003) pp. 222–233.Google Scholar
  21. 21.
    N. Hohn, D. Veitch, and P. Abry, Cluster processes: a natural language for network traffic. IEEE Trans. Signal Process. 51 (2003), 2229–2244.Google Scholar
  22. 22.
    O. Kallenberg, Foundations of Modern Probability, 2nd edition (Springer, New York, 2001).Google Scholar
  23. 23.
    W.E. LELAND, M.S. TAQQU, W. WILLINGER, AND D.V. WILSON, On the self-similar nature of Ethernet traffic. ACM/SIGCOMM Computer Communications Review (1993) 183–193.Google Scholar
  24. 24.
    P.A.W. LEWIS, A branching Poisson process model for the analysis of computer failure patterns. J. Roy. Statist. Soc., Series B 26 (1964) 398–456.Google Scholar
  25. 25.
    T. MIKOSCH AND G. SAMORODNITSKY, The supremum of a negative drift random walk with dependent heavy-tailed steps. Ann. Appl. Probab. 10 (2000) 1025–1064.CrossRefGoogle Scholar
  26. 26.
    T. MIKOSCH AND G. SAMORODNITSKY, Scaling limits for cumulative input processes. Math. Oper. Research (2006) To appear.Google Scholar
  27. 27.
    T. MIKOSCH, S. RESNICK, H. ROOTZÉN, AND A. STEGEMAN, Is network traffic approximated by stable Lévy motion or fractional Brownian motion? Ann. Appl. Probab. 12 (2002) 23–68.Google Scholar
  28. 28.
    E. Moulines, F. Roueff, and M. Taqqu, A wavelet Whittle estimator of the memory parameter of a non-stationary Gaussian time series. Preprint (2005). Available at http://arxiv.org/abs/math.ST/0601070.
  29. 29.
    A.V. NAGAEV, Limit theorems for large deviations where Cramér’s conditions are violated (in Russian). Izv. Akad. Nauk UzSSR, Ser. Fiz.–Mat. Nauk 6 (1969a) 17–22.Google Scholar
  30. 30.
    A.V. NAGAEV, Integral limit theorems for large deviations when Cramér’s condition is not fulfilled I,II. Theory Probab. Appl. 14 (1969b) 51–64, 193–208.Google Scholar
  31. 31.
    S.V. NAGAEV, Large deviations of sums of independent random variables. Ann. Probab. 7 (1979) 745–789.Google Scholar
  32. 32.
    V.V. PETROV, Limit Theorems of Probability Theory (Oxford University Press, Oxford, 1995).Google Scholar
  33. 33.
    S.I. RESNICK, Point processes, regular variation and weak convergence . Adv. Appl. Probab. 18 (1986) 66–138.CrossRefGoogle Scholar
  34. 34.
    S.I. RESNICK, Extreme Values, Regular Variation, and Point Processes. (Sprin-ger, New York, 1987).Google Scholar
  35. 35.
    G. SAMORODNITSKY AND M.S. TAQQU, Stable Non-Gaussian Random Processes. Stochastic Models with Infinite Variance (Chapman & Hall, New York, 1994).Google Scholar
  36. 36.
    A.J. STAM, Regular variation of the tail of a subordinated probability distribution . Adv. Appl. Probab. 5 (1973) 287–307.CrossRefGoogle Scholar
  37. 37.
    M. WESTCOTT, On existence and mixing results for cluster point processes. J. Royal Stat. Soc. Series B 33 (1971) 290–300.Google Scholar
  38. 38.
    W. WILLINGER, M.S. TAQQU, R. SHERMAN, AND D. WILSON, Self-similarity through high variability: statistical analysis of ethernet LAN traffic at the source level. Proceedings of the ACM/SIGCOMM’95, Cambridge, MA. Computer Communications Review 25 (1995) 100–113.Google Scholar
  39. 39.
    G.W. Wornell and A.V. Oppenheim, Estimation of fractal signals from noisy measurements using wavelets. IEEE Transactions on Signal Processing 40 (1992) 611–623.CrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  • Gilles Faÿ
    • 1
  • Bárbara González-Arévalo
    • 2
  • Thomas Mikosch
    • 3
  • Gennady Samorodnitsky
    • 4
  1. 1.Laboratoire Paul-PainlevéUniversité Lille 1Villeneuve d’Ascq cedexFrance
  2. 2.Department of Mathematics and Actuarial SciencesRoosevelt UniversityChicagoUSA
  3. 3.Laboratory of Actuarial MathematicsUniversity of CopenhagenCopenhagenDenmark
  4. 4.School of Operations Research and Industrial EngineeringCornell UniversityIthacaUSA

Personalised recommendations