Queueing Systems

, Volume 85, Issue 3–4, pp 249–267 | Cite as

Lévy-driven GPS queues with heavy-tailed input

  • Krzysztof Dȩbicki
  • Peng Liu
  • Michel Mandjes
  • Iwona Sierpińska-Tułacz


In this paper, we derive exact large buffer asymptotics for a two-class generalized processor sharing (GPS) model, under the assumption that the input traffic streams generated by both classes correspond to heavy-tailed Lévy processes. Four scenarios need to be distinguished, which differ in terms of (i) the level of heavy-tailedness of the driving Lévy processes as well as (ii) the values of the corresponding mean rates relative to the GPS weights. The derived results are illustrated by two important special cases, in which the queues’ inputs are modeled by heavy-tailed compound Poisson processes and by \(\alpha \)-stable Lévy motions.


Lévy process Fluid model Queue General processor sharing Exact asymptotics 

Mathematics Subject Classification

Primary: 60K25 Secondary: 90B22 60G51 



K. Dȩbicki was partially supported by NCN Grant No. 2015/17/B/ST1/01102 (2016-2019), whereas P. Liu was partially supported by the Swiss National Science Foundation Grant 200021-166274. M. Mandjes’ research is partly funded by the NWO Gravitation project Networks, Grant Number 024.002.003. He is also affiliated to (A) CWI, Amsterdam, The Netherlands; (B) Eurandom, Eindhoven University of Technology, Eindhoven, The Netherlands; and (C) Amsterdam Business School, Faculty of Economics and Business, University of Amsterdam, Amsterdam, The Netherlands.


  1. 1.
    Andersen, L.N., Asmussen, S., Glynn, P., Pihlsgard, M.: Lévy processes with two-sided reflection. In: Lévy Matters V. Lecture Notes in Mathematics, vol. 2149, pp. 67–182. Springer, Cham (2015)Google Scholar
  2. 2.
    Asmussen, S.: Subexponential asymptotics for stochastic processes: extremal behavior, stationary distributions and first passage probabilities. Ann. Appl. Probab. 8, 354–374 (1998)CrossRefGoogle Scholar
  3. 3.
    Asmussen, S., Albrecher, H.: Ruin Probabilities. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2010)Google Scholar
  4. 4.
    Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation, volume 27 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1989)Google Scholar
  5. 5.
    Borst, S., Boxma, O., Jelenković, P.: Induced burstiness in general processor sharing queues with long-tailed traffic flows. In: Proceedings of the 37th Annual Allerton Conference on Communication, Control, and Computing, pp. 316–325 (1999)Google Scholar
  6. 6.
    Borst, S., Boxma, O., Jelenković, P.: Reduced-load equivalence and induced burstiness in GPS queues with long-tailed traffic flows. Queueing Syst. Theor. Appl. 43, 273–306 (2003)CrossRefGoogle Scholar
  7. 7.
    Borst, S., Mandjes, M., van Uitert, M.: Generalized processor sharing queues with heterogeneous traffic classes. Adv. Appl. Probab. 35, 806–845 (2003)CrossRefGoogle Scholar
  8. 8.
    Borst, S., Mandjes, M., van Uitert, M.: Generalized processor sharing with light-tailed and heavy-tailed input. IEEE/ACM Trans. Netw. 11, 821–834 (2003)CrossRefGoogle Scholar
  9. 9.
    Dȩbicki, K., Mandjes, M.: A note on large-buffer asymptotics for generalized processor sharing with Gaussian inputs. Queueing Syst. Theor. Appl. 55, 251–254 (2007)CrossRefGoogle Scholar
  10. 10.
    Dȩbicki, K., Mandjes, M.: Queues and Lévy Fluctuation Theory. Springer, Cham (2015)Google Scholar
  11. 11.
    Dȩbicki, K., Mandjes, M., van Uitert, M.: A tandem queue with Lévy input: a new representation of the downstream queue length. Probab. Eng. Inf. Sci. 21, 83–107 (2007)Google Scholar
  12. 12.
    Dȩbicki, K., van Uitert, M.: Large buffer asymptotics for generalized processor sharing queues with Gaussian inputs. Queueing Syst. Theor. Appl. 54, 111–120 (2006)CrossRefGoogle Scholar
  13. 13.
    Demers, A., Keshav, S., Shenker, S.: Analysis and simulation of a fair queueing algorithm. ACM Sigcomm Comput. Commun. Rev. 19, 1–12 (1989)CrossRefGoogle Scholar
  14. 14.
    Lelarge, M.: Asymptotic behavior of generalized processor sharing queues under subexponential assumptions. Queueing Syst. Theor. Appl. 62, 51–73 (2009)CrossRefGoogle Scholar
  15. 15.
    Lieshout, P., Mandjes, M.: Asymptotic analysis of Lévy-driven tandem queues. Queueing Syst. Theor. Appl. 60, 203–226 (2008)CrossRefGoogle Scholar
  16. 16.
    Mandjes, M.: Large deviations for complex buffer architectures: the short-range dependent case. Stoch. Mod. 22, 99–128 (2006)CrossRefGoogle Scholar
  17. 17.
    Mandjes, M., van Uitert, M.: Sample-path large deviations for generalized processor sharing queues with Gaussian inputs. Perform. Eval. 61, 225–256 (2005)CrossRefGoogle Scholar
  18. 18.
    Reich, E.: On the integrodifferential equation of Takács I. Ann. Math. Stat. 29, 563–570 (1958)CrossRefGoogle Scholar
  19. 19.
    Samorodnitsky, G., Taqqu, M.S.: Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman & Hall, New York (1994)Google Scholar
  20. 20.
    Whitt, W.: Stochastic-Process Limits. Springer-Verlag, New York (2002)Google Scholar
  21. 21.
    Willekens, E.: On the supremum of an infinitely divisible process. Stoch. Proc. Appl. 26, 173–175 (1987)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Krzysztof Dȩbicki
    • 1
  • Peng Liu
    • 1
    • 2
  • Michel Mandjes
    • 3
  • Iwona Sierpińska-Tułacz
    • 1
  1. 1.Mathematical InstituteUniversity of WrocławWrocławPoland
  2. 2.Department of Actuarial ScienceUniversity of Lausanne, UNIL-DorignyLausanneSwitzerland
  3. 3.Korteweg-de Vries Institute for MathematicsUniversity of AmsterdamAmsterdamThe Netherlands

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