The Annals of Applied Probability

How system performance is affected by the interplay of averages in a fluid queue with long range dependence induced by heavy tails

David Heath, Sidney Resnick, and Gennady Samorodnitsky

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We consider a fluid queue with sessions arriving according to a Poisson process. A long-tailed distribution of session lengths induces long range dependence in the system and causes its performance to deteriorate. The deterioration is due to occurrence of load regimes far from average ones. Nonetheless, the extent of this performance deterioration is shown to depend crucially on the average values of the system parameters.

Article information

Ann. Appl. Probab., Volume 9, Number 2 (1999), 352-375.

First available in Project Euclid: 21 August 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 90B15: Network models, stochastic
Secondary: 60K25: Queueing theory [See also 68M20, 90B22]

Fluid queue heavy tails long range dependence performance of a queue $M/G/\infty$ queue time until overflow large deviations association


Heath, David; Resnick, Sidney; Samorodnitsky, Gennady. How system performance is affected by the interplay of averages in a fluid queue with long range dependence induced by heavy tails. Ann. Appl. Probab. 9 (1999), no. 2, 352--375. doi:10.1214/aoap/1029962746.

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  • Bingham, N., Goldie, C. and Teugels, J. (1987). Regular Variation. Cambridge Univ. Press.
  • Boxma, O. and Dumas, V. (1996). Fluid queues with long-tailed activity period distributions. Special Issue of Stochastic Analy sis and Optimization of Communication Sy stems. To appear.
  • Burton, R. and Way mire, E. (1985). Scaling limits for associated random measures. Ann. Probab. 13 1267-1278.
  • Duffield, M. and O'Connell, N. (1995). Large deviations and overflow probabilities for the general single-server queue, with applications. Math. Proc. Cambridge Philos. Soc. 118 363-374.
  • Embrechts, P. and Goldie, C. (1982). On convolution tails. Stochastic Process. Appl. 13 263-278.
  • Embrechts, P., Kl ¨uppelberg, C. and Mikosch, T. (1997). Modelling Extreme Events for Insurance and Finance. Springer, Berlin.
  • Erramilli, A., Naray an, O. and Willinger, W. (1996). Experimental queueing analysis with long-range dependent packet traffic. IEEE/ACM Trans. on Network Computing 4 209-223.
  • Esary, J., Proschan, F. and Walkup, D. (1967). Association of random variables, with applications. Ann. Math. Statist. 38 1466-1474.
  • Heath, D., Resnick, S. and Samorodnitsky, G. (1997). Patterns of buffer overflow in a class of queues with long memory in the input stream. Ann. Appl. Probab. 7 1021-1057.
  • Heath, D., Resnick, S. and Samorodnitsky, G. (1998). Heavy tails and long range dependence in on/off processes and associated fluid models. Math. Oper. Res. 23 145-165.
  • Jelenkovi´c, P. and Lazar, A. (1998). Subexponential asy mptotics of a Markov-modulated random walk with queueing applications. J. Appl. Probab. 35 1-23.
  • Liu, Z., Nain, P., Towsley, D. and Zhang, Z.-L. (1997). Asy mptotic behavior of a multiplexer fed by a long-range dependent process. Technical Report CMPSRI 97-16, Univ. Massachusetts, Amherst.
  • Resnick, S. (1987). Extreme Values, Regular Variation and Point Processes. Springer, New York.
  • Resnick, S. (1992). Adventures in Stochastic Processes. Birkh¨auser, Boston. Resnick, S. and Samorodnitsky, G. (1997a). Activity periods of an infinite server queue and performance of certain heavy tailed fluid queues. Queuing Sy stems Theory Appl. To appear. Resnick, S. and Samorodnitsky, G. (1997b). Performance decay in a single server exponential queueing model with long range dependence. Oper. Res. 45 235-243.
  • Ry u, B. and Lowen, S. (1995). Modeling, analysis and simulation of self-similar traffic using the fractal-shot-noise-driven Poisson process. In Proceedings of IASTED International Conference on Modeling and Simulation, Pittsburgh, PA.
  • Vamvakos, S. and Anantharam, V. (1998). On the departure process of a leaky bucket sy stem with long-range dependent input traffic. Queuing Sy stems Theory Appl. 28 191-214.
  • Willinger, W., Taqqu, M., Sherman, R. and Wilson, D. (1997). Self-similarity through high variability: statistical analysis of ethernet LAN traffic at the source level (extended version). IEEE/ACM Trans. on Networking 5 71-96.