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Queueing Systems

, 69:29 | Cite as

The on–off network traffic model under intermediate scaling

  • Clément Dombry
  • Ingemar Kaj
Article

Abstract

The result provided in this paper helps complete a unified picture of the scaling behavior in heavy-tailed stochastic models for transmission of packet traffic on high-speed communication links. Popular models include infinite source Poisson models, models based on aggregated renewal sequences, and models built from aggregated on–off sources. The versions of these models with finite variance transmission rate share the following pattern: if the sources connect at a fast rate over time the cumulative statistical fluctuations are fractional Brownian motion, if the connection rate is slow the traffic fluctuations are described by a stable Lévy motion, while the limiting fluctuations for the intermediate scaling regime are given by fractional Poisson motion. In this paper, we prove an invariance principle for the normalized cumulative workload of a network with m on–off sources and time rescaled by a factor a. When both the number of sources m and the time scale a tend to infinity with a relative growth given by the so-called ’intermediate connection rate’ condition, the limit process is the fractional Poisson motion. The proof is based on a coupling between the on–off model and the renewal type model.

Keywords

On–off process Workload process Renewal process Intermediate scaling Fractional Poisson motion Fractional Brownian motion Lévy motion Heavy tails Long-range dependence 

Mathematics Subject Classification (2000)

60G22 60F05 

References

  1. 1.
    Asmussen, S.: Ruin Probabilities. World Scientific, Singapore (2000) CrossRefGoogle Scholar
  2. 2.
    Beghin, L., Orsingher, E.: Fractional Poisson processes and related planar random motions. Electron. J. Probab. 14, 1790–1826 (2009) Google Scholar
  3. 3.
    Beghin, L., Orsingher, E.: Poisson-type processes governed by fractional and higher-order recursive differential equations. Electron. J. Probab. 15, 684–709 (2010) Google Scholar
  4. 4.
    Biermé, H., Estrade, A., Kaj, I.: Self-similar random fields and rescaled random balls models. J. Theor. Probab. 23, 1110–1141 (2010) CrossRefGoogle Scholar
  5. 5.
    Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley, New York (1968) Google Scholar
  6. 6.
    Bingham, N.H., Goldie, C.M., Teugels, J.H.: Regular Variation. Cambridge University Press, Cambridge (1987) Google Scholar
  7. 7.
    Mikosch, T., Resnick, S., Rootzen, H., Stegeman, A.: Is network traffic approximated by stable Lévy motion or fractional Brownian motion. Ann. Appl. Probab. 12(1), 23–68 (2002) CrossRefGoogle Scholar
  8. 8.
    Mikosch, T., Samorodnitsky, G.: Scaling limits for cumulative input processes. Math. Oper. Res. 32(4), 890–918 (2007) CrossRefGoogle Scholar
  9. 9.
    Gaigalas, R.: A Poisson bridge between fractional Brownian motion and stable Lévy motion. Stoch. Process. Appl. 116, 447–462 (2006) CrossRefGoogle Scholar
  10. 10.
    Gaigalas, R., Kaj, I.: Convergence of scaled renewal processes and a packet arrival model. Bernoulli 9(4), 671–703 (2003) CrossRefGoogle Scholar
  11. 11.
    Jumarie, G.J.: Fractional master equation: non-standard analysis and Liouville–Rieman derivative. Chaos Solitons Fractals 12, 2577–2587 (2001) CrossRefGoogle Scholar
  12. 12.
    Kaj, I.: Stochastic Modeling in Broadband Communications Systems. SIAM Monographs on Mathematical Modeling and Computation, vol. 8. SIAM, Philadelphia (2002) CrossRefGoogle Scholar
  13. 13.
    Kaj, I.: Limiting fractal random processes in heavy-tailed systems. In: Levy-Lehel, J., Lutton, E. (eds.) Fractals in Engineering, New Trends in Theory and Applications, pp. 199–218. Springer, London (2005) Google Scholar
  14. 14.
    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.) An Out of Equilibrium 2. Progress in Probability, vol. 60, pp. 383–427. Birkhäuser, Basel (2008) CrossRefGoogle Scholar
  15. 15.
    Lévy, J.B., Taqqu, M.S.: Renewal reward processes with heavy-tailed interrenewal times and heavy-tailed rewards. Bernoulli 6, 23–44 (2000) CrossRefGoogle Scholar
  16. 16.
    Mainardi, F., Gorenflo, R., Scalas, E.: A fractional generalization of the Poisson processes. Vietnam J. Math. 32, 53–64 (2005) Google Scholar
  17. 17.
    Petrov, V.V.: Sums of Independent Random Variables. Springer, New York (1975) Google Scholar
  18. 18.
    Pipiras, V., Taqqu, M.S.: The limit of a renewal-reward process with heavy-tailed rewards is not a linear fractional stable motion. Bernoulli 6, 607–614 (2000) CrossRefGoogle Scholar
  19. 19.
    Pipiras, V., Taqqu, M.S., Lévy, L.B.: Slow, fast and arbitrary growth conditions for renewal reward processes when the renewals and the rewards are heavy-tailed. Bernoulli 10, 121–163 (2004) CrossRefGoogle Scholar
  20. 20.
    Resnick, S.I.: Heavy-Tail Phenomena, Probabilistic and Statistical modeling. Springer Series in Operations Research and Financial Engineering. Springer, New York (2007) Google Scholar
  21. 21.
    Taqqu, M.S., Willinger, W., Sherman, R.: Proof of a fundamental result in self-similar traffic modeling. Comput. Commun. Rev. 27, 5–23 (1997) CrossRefGoogle Scholar
  22. 22.
    Wang, X.-T., Wen, Z.-X.: Poisson fractional process. Chaos Solitons Fractals 18, 169–177 (2003) CrossRefGoogle Scholar
  23. 23.
    Wang, X.-T., Wen, Z.-X., Zhang, S.-Y.: Fractional Poisson processes (II). Chaos Solitons Fractals 28, 143–147 (2006) CrossRefGoogle Scholar
  24. 24.
    Wang, X.-T., Wen, Z.-X., Fan, S.: Nonhomogeneous fractional Poisson processes. Chaos Solitons Fractals 31, 236–141 (2007) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Laboratoire LMAUniversité de PoitiersFuturoscope-Chasseneuil cedexFrance
  2. 2.Department of MathematicsUppsala UniversityUppsalaSweden

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