Bernoulli

  • Bernoulli
  • Volume 7, Number 2 (2001), 191-210.

Steady-state distribution of the buffer content for M/G/∞ input fluid queues

Sidney Resnick and Gennady Samorodnitsky

Full-text: Open access

Abstract

We consider a fluid queue with on periods initiated by a Poisson process and having a long-tailed distribution. This queue has long-range dependence, and we compute the asymptotic behaviour of the steady-state distribution of the buffer content. The tail of this distribution is much heavier than the tail of the buffer content distribution of a queue which does not possess long-range dependence and which has light-tailed on periods and the same traffic intensity.

Article information

Source
Bernoulli, Volume 7, Number 2 (2001), 191-210.

Dates
First available in Project Euclid: 25 March 2004

Permanent link to this document
https://projecteuclid.org/euclid.bj/1080222087

Mathematical Reviews number (MathSciNet)
MR1828502

Zentralblatt MATH identifier
0994.60085

Keywords
fluid queue heavy tails large deviations long-range dependence M/G/∞queue performance of a queue random walk steady-state distribution

Citation

Resnick, Sidney; Samorodnitsky, Gennady. Steady-state distribution of the buffer content for M/G/∞ input fluid queues. Bernoulli 7 (2001), no. 2, 191--210. https://projecteuclid.org/euclid.bj/1080222087


Export citation

References

  • [1] Agrawal, R., Makowski, A. and Nain, P. (1999) On a reduced load equivalence for fluid queues under subexponentiality. Queueing Systems Theory Appl., 33, 5-41. Abstract can also be found in the ISI/STMA publication
  • [2] Arlitt, M. and Williamson, C. (1996) Web servers workload characterization: The search for invariants (extended version). In Proceedings of the ACM Sigmetrics International Conference on Measurement and Modeling of Computer Systems. New York: Association for Computing Machinery.
  • [3] Asmussen, S. (1987) Applied Probability and Queues. Chichester: Wiley.
  • [4] Beran, J., Sherman, R., Willinger, W. and Taqqu, M. (1995) Long-range dependence in variable-bitrate video traffic. IEEE Trans. Commun., 43, 1566-1579.
  • [5] Boxma, O. (1997) Regular variation in a multi-source fluid queue. In V. Ramaswami and P. Wirth (eds), Teletraffic Contributions for the Information Age, pp. 391-402. Amterdam: North-Holland.
  • [6] Boxma, O. and Dumas, V. (1998) Fluid queues with long-tailed activity period distributions. Comput. Commun., 21, 1509-1529.
  • [7] Choudhury, G.L. and Whitt, W. (1997) Long-tail buffer-content distributions in broadband networks. Perform. Eval., 30, 177-190.
  • [8] Cline, D. and Hsing, T. (1991) Large deviation probabilities for sums and maxima of random variables with heavy or subexponential tails. Preprint, Texas A&M University.
  • [9] Cohen J. (1997) The M/G/1 fluid model with heavy-tailed message length distributions. Technical Report PNA-R9714, Centrum voor Wiskunde en Informatica.
  • [10] Crovella, M. and Bestavros, A. (1996) Self-similarity in World Wide Web traffic: evidence and possible causes. Perform. Eval. Rev., 24, 160-169.
  • [11] Cunha, C., Bestavros, A. and Crovella, M. (1995) Characteristics of www client-based traces. Preprint BU-CS-95-010, Boston University.
  • [12] Embrechts, P. Veraverbeke, N. (1982) Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance Math. Econom., 1, 55-72.
  • [13] Embrechts, P., Goldie, C. and Veraverbeke, N. (1979): Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitstheorie Verw. Geb., 49, 335-347.
  • [14] 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. Abstract can also be found in the ISI/STMA publication
  • [15] 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.
  • [16] Heath, D., Resnick, S. and Samorodnitsky, G. (1999) 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, 352-375. Abstract can also be found in the ISI/STMA publication
  • [17] Jelenkovic, P. and Lazar, A. (1999) Asymptotic results for multiplexing subexponential on-off sources. Adv. Appl. Probab., 31, 394-421. Abstract can also be found in the ISI/STMA publication
  • [18] Leland, W., Taqqu, M., Willinger, W. and Wilson, D. (1994) On the self-similar nature of Ethernet traffic (extended version). IEEE/ACM Trans. Networking, 2, 1-15.
  • [19] Likhanov, N. and Mazumdar, R. (1999) Cell loss asymptotics for buffers fed with a large number of independent stationary sources. J. Appl. Probab., 36, 86-96. Abstract can also be found in the ISI/STMA publication
  • [20] Liu, Z., Nain, P., Towsley, D. and Zhang, Z.-L. (1999) Asymptotic behavior of a multiplexer fed by a long-range dependent process. J. Appl. Probab., 36, 105-118. Abstract can also be found in the ISI/STMA publication
  • [21] Mikosch, T. and Samorodnitsky, G. (1999) Ruin probability with claims modeled by a stationary ergodic stable process. Preprint, Cornell University. Abstract can also be found in the ISI/STMA publication
  • [22] Mikosch, T. and Samorodnitsky, G. (2000) The supremum of a negative drift random walk with dependent heavy-tailed steps. Ann. Appl. Probab., 10, 1025-1064. Abstract can also be found in the ISI/STMA publication
  • [23] Nagaev, A. (1969) Limit theorems for large deviations where Cramér's conditions are violated. Izv. Akad. Nauk UzSSR Ser. Fiz.-Mat. Nauk, 6, 17-22 (in Russian).
  • [24] Paxson, V. and Floyd, S. (1994) Wide area traffic: the failure of Poisson modelling. IEEE/ACM Trans. Networking, 3, 226-244.
  • [25] Prabhu, N. (1998) Stochastic Storage Processes: Queues, Insurance Risk, Dams, and Data Communication. New York: Springer-Verlag.
  • [26] Resnick, S. (1987) Extreme Values, Regular Variation and Point Processes. New York: Springer- Verlag.
  • [27] Resnick, S. and Rootzén, H. (2000) Self-similar communication models and very heavy tails. Ann. Appl. Probab., 10, 753-778. Abstract can also be found in the ISI/STMA publication
  • [28] Resnick, S. and Samorodnitsky, G. (1999) Activity periods of an infinitte server queue and performance of certain heavy tailed fluid queues. Queueing Systems Theory Appl., 33, 43-71.
  • [29] Resnick, S. and Samorodnitsky, G. (2000) Fluid queues, leaky buckets, on-off processes and teletraffic modeling with high variable and correlated Inputs. In K. Park and W. Willinger (eds), Self-similar Network Traffic and Performance Evaluation. New York: Wiley.
  • [30] Vamvakos, S. and Anantharam, V. (1998) On the departure process of a leaky bucket system with long-range dependent input traffic. Queueing Systems Theory Appl., 28, 191-214. Abstract can also be found in the ISI/STMA publication
  • [31] Whitt, W. (1999) The reflection map is Lipschitz with appropriate Skorohod M-metrics. Preprint, AT&T Labs Research, Florham Park, NJ.
  • [32] Zwart, A. (2000) A fluid queue with a finite buffer and subexponential input. Adv. Appl. Probab., 32, 221-243. Abstract can also be found in the ISI/STMA publication