zbMATH — the first resource for mathematics

On the superposition of heterogeneous traffic at large time scales. (English) Zbl 1291.60197
Summary: Various empirical and theoretical studies indicate that cumulative network traffic is a Gaussian process. However, depending on whether the intensity at which sessions are initiated is large or small relative to the session duration tail [T. Mikosch et al., Ann. Appl. Probab. 12, No. 1, 23–68 (2002; Zbl 1021.60076); I. Kaj and M. S. Taqqu, Prog. Probab. 60, 383–427 (2008; Zbl 1154.60020)] have shown that traffic at large time scales can be approximated by either fractional Brownian motion (fBm) or stable Lévy motion. We study distributional properties of cumulative traffic that consists of a finite number of independent streams and give an explanation of why Gaussian examples abound in practice but not stable Lévy motion. We offer an explanation of how much vertical aggregation is needed for the Gaussian approximation to hold. Our results are expressed as limit theorems for a sequence of cumulative traffic processes whose session initiation intensities satisfy growth rates similar to those used in [Mikosch et al., loc. cit.].
60K30 Applications of queueing theory (congestion, allocation, storage, traffic, etc.)
90B20 Traffic problems in operations research
Full Text: DOI arXiv
[1] Arlitt, M. F. and Williamson, C. L. (1996). Web server workload characterization: The search for invariants (extended version). In Proceedings of the 1996 ACM SIGMETRICS Conference 126-137. ACM, New York, Philadelphia, PA.
[2] Billingsley, P. (1999). Convergence of probability measures , second ed. Wiley Series in Probability and Statistics . John Wiley & Sons, Inc. · Zbl 0944.60003
[3] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular variation . Encyclopedia of Mathematics and its Applications 27 . Cambridge University Press. · Zbl 0617.26001
[4] Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods , Second ed. Springer-Verlag, New York. · Zbl 0709.62080
[5] Cisco Systems, Inc. (2007). Introduction to Cisco IOS NetFlow - A Technical Overview San Jose, CA. USA.
[6] Coles, S. (2001). An Introduction to Statistical Modeling of Extreme Values . Springer Series in Statistics . Springer. · Zbl 0980.62043
[7] Crovella, M. E. and Bestavros, A. (1997). Self-Similarity in World Wide Web Traffic: Evidence and Possible Causes. IEEM/ACM Transactions on Networking 5 845-846.
[8] Cunha, C. R., Bestavros, A. and Crovella, M. E. (1995). Characteristics of WWW Client-based Traces Technical report report No. BU-CS-95-010, Computer Science Department, Boston University.
[9] de Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction . Springer-Verlag, New York. · Zbl 1101.62002
[10] de Haan, L. and Resnick, S. I. (1998). On asymptotic normality of the Hill estimator. Stochastic Models 14 849-866. · Zbl 1002.60519 · doi:10.1080/15326349808807504
[11] Guerin, C. A., Nyberg, H., Perrin, O., Resnick, S. I., Rootzén, H. and Stărică, C. (2003). Empirical testing of the infinite source Poisson data traffic model. Stochastic Models 19 151-200. · Zbl 1048.62080 · doi:10.1081/STM-120020386
[12] Hill, B. M. (1975). A simple general approach to inference about the tail of a distribution. The Annals of Statistics 3 1163-1174. · Zbl 0323.62033 · doi:10.1214/aos/1176343247
[13] Jain, M. and Dovrolis, C. (2005). End-to-end Estimation of the Available Bandwidth Variation Range. In Proceedings of the 2005 ACM SIGMETRICS International Conference on Measurement and Modeling of computer systems 265-276. ACM.
[14] Jin, Y., Bali, S., Duncan, T. E. and Frost, V. S. (2007). Predicting Properties of Congestion Events for a Queueing System With fBm Traffic. IEEM/ACM Transactions on Networking 15 1098-1108.
[15] Kaj, I. and Taqqu, M. S. (2008). In and Out of Equilibrium 2 . Progress in Probability 60 Convergence to Fractional Brownian Motion and to the Telecom Process: the Integral Representation Approach 383-427. Birkhäuser Basel. · Zbl 1154.60020 · doi:10.1007/978-3-7643-8786-0_19
[16] Kallenberg, O. (1984). Random measures . Akademie-Verlag. · Zbl 0345.60032
[17] Kilpi, J. and Norros, I. (2002). Testing the Gaussian approximation of aggregate traffic. In Proceedings of the 2nd ACM SIGCOMM Workshop on Internet measurment . Session 2: modeling 49-61. ACM, Marseilles, France.
[18] Kingman, J. F. C. (1993). Poisson Processes . Oxford Studies in Probability . Oxford University Press. · Zbl 0771.60001
[19] Kortebi, A., Muscariello, L., Oueslati, S. and Roberts, J. (2005). Evaluating the Number of Active Flows in a Scheduler Realizing Fair Statistical Bandwidth Sharing. In SIGMETRICS ’05: Proceedings of the 2000 ACM SIGMETRICS international conference on Measurement and modeling of computer systems 217-228. ACM.
[20] Kurtz, T. G. (1996). Limit theorems for workload input models. In Stochastic Networks: Theory and Applications , ( F. P. Kelly, S. Zachary and I. Ziedins, eds.). Royal Statistical Society Lecture Note Series 4 119-139. Clarendon Press, Oxford. · Zbl 0855.60089
[21] Leland, W. E., Taqqu, M. S., Willinger, W. and Wilson, D. V. (1994). On the self-similar nature of Ethernet traffic (extended version). IEEM/ACM Transactions on Networking 2 1-15.
[22] López-Oliveros, L. and Resnick, S. I. (2011). Extremal dependence analysis of network sessions. Extremes 14 1-28. · Zbl 1329.62234 · doi:10.1007/s10687-009-0096-4
[23] Maulik, K., Resnick, S. I. and Rootzén, H. (2002). Asymptotic independence and a network traffic model. Journal of Applied Probability 39 671-699. · Zbl 1090.90017 · doi:10.1239/jap/1037816012
[24] Mc Neil, A. J., Frey, R. and Embrechts, P. (2005). Quantitative Risk Management . Princeton Series in Finance . Princeton University Press, Princeton, NJ. Concepts, Techniques and Tools.
[25] Mikosch, T., Resnick, S. I., Rootzén, H. and Stegeman, A. (2002). Is Network Traffic Approximated by Stable Lévy Motion or Fractional Brownian Motion? Annals of Applied Probability 12 23-68. · Zbl 1021.60076 · doi:10.1214/aoap/1015961155
[26] Mitrinović, D. S. and Vasić, P. M. (1970). Analytic Inequalities . Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete 165 . Springer-Verlag, Berlin, New York. · Zbl 0199.38101
[27] Resnick, S. I. (1986). Point processes, regular variation and weak convergence. Advances in Applied Probability 18 66-138. · Zbl 0597.60048 · doi:10.2307/1427239
[28] Resnick, S. I. (1987). Extremes Values, Regular Variation and Point Processes . Springer-Verlag. · Zbl 0633.60001
[29] Resnick, S. I. (2003). Modeling Data Networks, in SemStat: Seminaire Europeen de Statistique, Extreme Values in Finance, Telecommunications, and the Environment 287-372. Chapman-Hall, London. · Zbl 1042.90004
[30] Resnick, S. I. (2007). Heavy-Tail Phenomena: Probabilistic and Statistical Modeling . Springer Series in Operations Research and Financial Engineering . Springer-Verlag, New York. · Zbl 1152.62029
[31] Rosiński, J. and Rajput, B. S. (1989). Spectral representations of infinitely divisible processes. Probability Theory and Related Fields 82 451-487. · Zbl 0659.60078 · doi:10.1007/BF00339998
[32] Samorodnitsky, G. and Taqqu, M. S. (1994). Stable non-Gaussian random processes: stochastic models with infinite variance . Stochastic Modeling . Chapman & Hall. · Zbl 0925.60027
[33] Sarvotham, S., Riedi, R. and Baraniuk, R. (2005). Network and user driven alpha-beta on-off source model for network traffic. Computer Networks 48 335-350.
[34] Sarvotham, S., Wang, X., Riedi, R. H. and Baraniuk, R. G. (2002). Additive and multiplicative mixture trees for network traffic modeling. In ICASSP 2002: Proceedings of the International Conference on Acoustics, Speech, and Signal Processing IV 4040-4043. IEEE, Signal processing society, Orlando, Florida.
[35] Shakkottai, S., Brownlee, N. and Claffy, K. (2005). A Study of Burstiness in TCP Flows. In Proceedings of the 6th International Workshop in Passive and Active Network Measurement, PAM 2005 ( C. Dovrolis, ed.). Lecture Notes in Computer Science 3431 13-26. Springer, Boston, MA.
[36] Taqqu, M. S., Willinger, W. and Sherman, R. (1997). Proof of a fundamental result in self-similar traffic modeling. ACM SIGCOMM Computer Communication Review 27 5-23.
[37] van de Meent, R. and Mandjes, M. (2005). Evaluation of ‘user-oriented’ and ‘black-box’ traffic models for link provisioning. In Proceedings of the 1st EuroNGI Conference on Next Generation Internet Networks Traffic Engineering 380-387. IEEE, Rome, Italy.
[38] van de Meent, R., Mandjes, M. and Pras, A. (2006). Gaussian traffic everywhere? In ICC ’06. IEEE International Conference on Communications, 2006. 2 573-578.
[39] Willinger, W., Paxson, V. and Taqqu, M. S. (1998). Self similarity and heavy tails: Structural modeling of network traffic, in A Practical Guide to Heavy Tails. Statistical Techniques and Applications 27-53. Birkhäuser Boston Inc., Boston, MA. · Zbl 0926.90014
[40] Willinger, W., Taqqu, M. S., Leland, W. E. and Wilson, D. V. (1995). Self-Similarity in High-Speed Packet Traffic: Analysis and Modeling of Ethernet Traffic Measurements. Statistical Science 10 67-85. · Zbl 1148.90310 · doi:10.1214/ss/1177010131
[41] Willinger, W., Taqqu, M. S., Sherman, R. and Wilson, D. V. (1997). Self-similarity through high variability: Statistical Analysis of Ethernet LAN traffic at the source level. IEEM/ACM Transactions on Networking 5 71-86.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.