Queueing Systems

, Volume 77, Issue 1, pp 75–96 | Cite as

Generalized sub-Gaussian fractional Brownian motion queueing model

  • R. Yamnenko
  • Yu. Kozachenko
  • D. Bushmitch


It is well known that often the one-dimensional distribution of a queue content is not Gaussian but its tails behave like a Gaussian. We propose to consider a general class of processes, namely the class of \(\varphi \)-sub-Gaussian random processes, which is more general than the Gaussian one and includes non-Gaussian processes. The class of sub-Gaussian random processes contains Gaussian processes also and therefore is of special interest. In this paper we provide an estimate for the queue content distribution of a fluid queue fed by \(N\) independent strictly \(\varphi \)-sub-Gaussian generalized fractional Brownian motion input processes. We obtain an upper estimate of buffer overflow probability in a finite buffer system defined on any finite time interval \([a,b]\) or infinite interval \([0,\infty )\). The derived estimate captures more accurately the performance of the queueing system for a wider-range of input processes.


Fractional Brownian motion Buffer overflow probability  Sub-Gaussian processes 

Mathematics Subject Classification

60K25 60G20 


  1. 1.
    Addie, R., Mannersalo, P., Norros, I.: Most probable paths and performance formulae for buffers with Gaussian input traffic. Eur. Trans. Telecommun. 13(3), 183–196 (2002)CrossRefGoogle Scholar
  2. 2.
    Beran, J., Sherman, R., Taqqu, M.S., Willinger, W.: Long-range dependence in variable-bit-rate video traffic. IEEE Trans. Commun. 43(2/3/4), 1566–1579 (1995)CrossRefGoogle Scholar
  3. 3.
    Boulongne, P., Pierre-Loti-Viaud, D., Piterbarg, V.: On average losses in the ruin problem with fractional Brownian motion as input. Extremes 12, 77–91 (2009)CrossRefGoogle Scholar
  4. 4.
    Buldygin, V.V., Kozachenko, Yu.V.: Metric Characterization of Random Variables and Random Processes. American Mathematical Society, Providence (2000)Google Scholar
  5. 5.
    Choe, J., Shroff, N.B.: Use of supremum distribution of Gaussian processes in queueing analysis with long-range dependence and self-similarity. Commun. Stat. Stoch. Models 16(2), 209–231 (2000)CrossRefGoogle Scholar
  6. 6.
    Duffield, N.G., O’Connell, N.: Large deviations and overflow probabilities for the general single-server queue, with Applications. Math. Proc. Camb. Philos. Soc. 118, 363–374 (1995)CrossRefGoogle Scholar
  7. 7.
    Feder, J.: Fractals. Plenum Press, New York (1988)CrossRefGoogle Scholar
  8. 8.
    Fitzek, F., Reisslein, M.: MPEG-4 and H.263 video traces for network performance evaluation. IEEE Netw. 15(6), 40–54 (2001). Research is available at Accessed 22 Sept 2013Google Scholar
  9. 9.
    Giuliano-Antonini, R., Kozachenko, Yu., Nikitina, T.: Spaces of \(\phi \)-sub-Gaussian random variables, Rendiconti, Academia Nazionale delle Scienze detta dei XL. Memorie di Matematica e Applicazioni, no 121. XXVII (fasc.1), 95–124 (2003)Google Scholar
  10. 10.
    Hernández-Campos, F., Marron, J.S., Samorodnitsky, G., Smith, F.D.: Variable heavy tails in internet traffic. Perform. Eval. 58(2+3), 261–261 (2004)CrossRefGoogle Scholar
  11. 11.
    Krasnoselskii, M.A., Rutitskii, Ya.B.: Convex Functions in the Orlicz spaces. Fizmatgiz, Moskow, 1958(Russian); Noordhoff, Gröningen (1961) (English)Google Scholar
  12. 12.
    Kozachenko, Yu.V., Kovalchuk, Yu.A.: Boundary value problems with random initial conditions, and functional series from \(Sub_\varphi (\Omega )\). I. Ukrainian Math. J. 50(4), 504–515 (1998)Google Scholar
  13. 13.
    Kozachenko, Yu., Sottinen, T., Vasilik, O.: Weakly self-similar stationary increment processes from the space \(SSub_\varphi (\Omega )\). Theory Probab. Math. Stat. 65, 77–88 (2002)Google Scholar
  14. 14.
    Kozachenko, Yu.V., Vasilik, O.I.: On the distribution of suprema of \(Sub_\varphi (\Omega )\) random processes. Theory Stoch. Process. 4(20), no. 1–2, 147–160 (1998)Google Scholar
  15. 15.
    Kozachenko, Yu., Vasylyk, O., Yamnenko, R.: Upper estimate of overrunning by \(\mathit{Sub}_\varphi (\Omega )\) random process the level specified by continuous function. Random Oper. Stoch. Equ. 13(2), 101–118 (2005)Google Scholar
  16. 16.
    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), 1–15 (1994)CrossRefGoogle Scholar
  17. 17.
    Massoulie, L., Simonian, A.: Large buffer asymptotics for the queue with fractional Brownian input. J. Appl. Probab. 36, 894–906 (1999)CrossRefGoogle Scholar
  18. 18.
    Narayan, O.: Exact asymptotic queue length distribution for fractional Brownian traffic. Adv. Perform. Anal. 1(1), 39–63 (1998)Google Scholar
  19. 19.
    Norros, I.: A storage model with self-similar input. Queueing Syst. 16, 387–396 (1994)CrossRefGoogle Scholar
  20. 20.
    Norros, I.: On the use of Fractional Brownian motions in the theory of connectionless networks. IEEE J. Sel. Areas Commun. 13(6), 953–962 (1995)CrossRefGoogle Scholar
  21. 21.
    Vasylyk, O.I., Kozachenko, Yu.V., Yamnenko, R.E.: \(\varphi \)-Subgaussovi Vypadkovi Protsesy: Monographia. VPC Kyiv University, Kyiv (Ukrainian) (2008)Google Scholar
  22. 22.
    Yamnenko, R.: Ruin probability for generalized \(\varphi \)-sub-Gaussian fractional Brownian motion. Theory Stoch. Process. 12(28), no. 3–4, 261–275 (2006)Google Scholar
  23. 23.
    Yamnenko, R., Vasylyk, O.: Random process from the class \(V(\varphi ,\psi )\): exceeding a curve. Theory of Stoch. Process. 13(29), no. 4, 219–232 (2007)Google Scholar
  24. 24.
    Yang, Z., Walden, A.T., McCoy, E.J.: Correntropy: implications of nonGaussianity for the moment expansion and deconvolution. Signal Process. 91(4), 864–876 (2011)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Probability Theory, Statistics and Actuarial MathematicsTaras Shevchenko National University of KyivKievUkraine
  2. 2.Mitre CorporationEatontownUSA

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