The Annals of Applied Probability

Self-similar communication models and very heavy tails

Sidney Resnick and Holger Rootzén

Full-text: Open access

Abstract

Several studies of file sizes either being downloaded or stored in the World Wide Web have commented that tails can be so heavy that not only are variances infinite, but so are means. Motivated by this fact, we study the infinite node Poisson model under the assumption that transmission times are heavy tailed with infinite mean. The model is unstable but we are able to provide growth rates. Self-similar but nonstationary Gaussian process approximations are provided for the number of active sources, cumulative input, buffer content and time to buffer overflow.

Article information

Source
Ann. Appl. Probab., Volume 10, Number 3 (2000), 753-778.

Dates
First available in Project Euclid: 22 April 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1019487509

Digital Object Identifier
doi:10.1214/aoap/1019487509

Mathematical Reviews number (MathSciNet)
MR1789979

Zentralblatt MATH identifier
1083.60521

Subjects
Primary: 60K25: Queueing theory [See also 68M20, 90B22]
Secondary: 60F05: Central limit and other weak theorems 60F10: Large deviations 60F17: Functional limit theorems; invariance principles 60G18: Self-similar processes 60G55: Point processes

Keywords
Heavy tails regular variation Pareto tails self-similarity scaling data communication traffic modeling infinite source Poisson connections

Citation

Resnick, Sidney; Rootzén, Holger. Self-similar communication models and very heavy tails. Ann. Appl. Probab. 10 (2000), no. 3, 753--778. doi:10.1214/aoap/1019487509. https://projecteuclid.org/euclid.aoap/1019487509


Export citation

References

  • [1] Arlitt, M. and Williamson, C. L. (1996). Web server workload characterization: the search for invariants (extended version). In Proceedings of the ACM Sigmetrics Conference Philadelphia. Available from mfa126,carey @cs.usask.ca.
  • [2] Asmussen, S. (1987). Applied Probability and Queues. Wiley, Chichester.
  • [3] Beirlant, J., Vynckier, P. and Teugels, J. (1996). Tail index estimation, Pareto quantile plots, and regression diagnostics. J. Amer. Statist. Assoc. 91 1659-1667.
  • [4] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, NewYork.
  • [5] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge Univ. Press.
  • [6] Crovella, M. and Bestavros, A. (1995). Explaining world wide web traffic self-similarity. Preprint TR-95-015 from crovella,best @cs.bu.edu.
  • [7] Crovella, M. and Bestavros, A. (1996). Self-similarity in world wide web traffic: evidence and possible causes. In Proceedings of the 1996 ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Systems 24 160-169.
  • [8] Crovella, M., Bestavros, A. and Taqqu, M. (1999). Heavy-tailed probability distributions in the world wide web. In A Practical Guide to Heavy Tails: Statistical Techniques for Analyzing Heavy Tailed Distributions (M. S. Taqqu, R. Adler and R. Feldman, eds.) Birkh¨auser, Boston.
  • [9] Crovella, M., Kim, G. and Park, K. (1996). On the relationship between file sizes, transport protocols, and self-similar network traffic. In Proceedings of the Fourth International Conference on Network Protocols 171-180.
  • [10] Crovella, M., Park, K. and Kim, G. (1997). On the effect of traffic self-similarity on network performance. In Proceedings of the SPIE International Conference on Performance and Control of Network Systems.
  • [11] Cunha, A. and Crovella, M. (1995). Characteristics of www client-based traces. Preprint BU-CS-95-010. Available from crovella,best @cs.bu.edu.
  • [12] Durrett, R. and Resnick, S. (1977). Weak convergence with random indices. J. Stochastic Process. Appl. 5 213-220.
  • [13] Erramilli, A., Narayan, O. and Willinger, W. (1996). Experimental queueing analysis with long-range dependent packet traffic. IEEE/ACM Trans. Network Comput. 4 209-223.
  • [14] Geluk, J. L. and de Haan, L. (1987). Regular Variation, Extensions and Tauberian Theorems. CWI Tract. Stichting Math. Centrum, Amsterdam.
  • [15] Harrison, J. M. (1985). Brownian Motion and Stochastic Flow Systems. Wiley, NewYork.
  • [16] Heath, D., Resnick, S. and Samorodnitsky, G. (1999). Howsystem 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.
  • [17] Hill, B. M. (1975). A simple general approach to inference about the tail of a distribution. Ann. Statist. 3 1163-1174.
  • [18] Jelenkovi´c, P. and Lazar, A. (1998). Subexponential asymptotics of a Markov-modulated random walk with queueing applications. J. Appl. Probab. 35 325-347.
  • [19] Jelenkovi´c, P. and Lazar, A. (1999). Asymptotic results for multiplexing subexponential on-off processes. Adv. in Appl. Probab. 31 394-421.
  • [20] Konstantopoulos, T. and Lin, S. J. (1998). Macroscopic models for long-range dependent network traffic. Queueing Systems Theory Appl. 28 215-243.
  • [21] Kratz, M. and Resnick, S. (1996). The qq-estimator and heavy tails. Stochastic Models. 12 699-724.
  • [22] Lamperti, J. W. (1962). Semi-stable stochastic processes. Trans. Amer. Math. Soc. 104 62-78.
  • [23] Leland, W. E., Taqqu, M. S., Willinger, W. and Wilson, D. V. (1993). On the self-similar nature of Ethernet traffic. ACM/SIGCOMM Comput. Comm. Rev. 183-193.
  • [24] Leland, W. E., Taqqu, M. S., Willinger, W. and Wilson, D. V. (1994). On the self-similar nature of Ethernet traffic (extended version). IEEE/ACM Trans. Networking 2 1-15.
  • [25] Leland, W. E., Taqqu, M. S., Willinger, W. and Wilson, D. V. (1993). Statistical analysis of high time-resolution ethernet Lan traffic measurements. In Proceedings of the 25th Symposium on the Interface between Statistics and Computer Science (M. E. Tarter and M. D. Lock, eds.) 25 146-155.
  • [26] Prabhu, N. U. (1998). Stochastic Storage Processes: Queues, Insurance Risk, Dams and Data Communication. Springer, NewYork.
  • [27] Resnick, S. I. (1986). Point processes, regular variation and weak convergence. Adv. in Appl. Probab. 18 66-138.
  • [28] Resnick, S. I. (1987). Extreme Values, Regular Variation and Point Processes. Springer, New York.
  • [29] Resnick, S. and Samorodnitsky, G. (1997). Performance decay in a single server exponential queueing model with long range dependence. Oper. Res. 45 235-243.
  • [30] Resnick, S. and St aric a, C. (1997). Smoothing the Hill estimator. Adv. in Appl. Probab. 29 271-293.
  • [31] Taqqu, M., Willinger, W. and Sherman, R. (1997). Proof of a fundamental result in selfsimilar traffic modeling. Comput. Comm. Rev. 27 5-23.
  • [32] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes with Applications to Statistics. Springer, NewYork.
  • [33] Vervaat, W. (1972). Success Epochs in Bernoulli Trials (with Applications in Number Theory). Math. Centrum, Amsterdam.
  • [34] Willinger, W., Taqqu, M. S., Leland, M. and Wilson, D. (1995). Self-similarity in highspeed packet traffic: analysis and modelling of ethernet traffic measurements. Statist. Sci. 10 67-85.
  • [35] Willinger, W., Taqqu, M. S., Leland, M. and Wilson, D. (1995). Self-similarity through high variability: statistical analysis of ethernet lan traffic at the source level. Comput. Comm. Rev. 25 100-113.
  • [36] Willinger, W., Taqqu, M. S., Leland, M. and Wilson, D. (1997). Self-similarity through high variability: statistical analysis of ethernet lan traffic at the source level (extended version). IEEE/ACM Trans. Networking 51 71-96.