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Data network models of burstiness

Part of: Special processes Operations research and management science

Published online by Cambridge University Press:  01 July 2016

Bernardo D'Auria  and
Sidney I. Resnick
Bernardo D'Auria*
Affiliation:
EURANDOM
Sidney I. Resnick*
Affiliation:
Cornell University
*
∗ Postal address: EURANDOM, Den Dolech 2, 5612 A2 Eindhoven, The Netherlands. Email address: bdauria@eurandom.tue.nl
∗∗ Postal address: School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY 14853, USA. Email address: sir1@cornell.edu
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Abstract

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We review characteristics of data traffic which we term stylized facts: burstiness, long-range dependence, heavy tails, bursty behavior determined by high-bandwidth users, and dependence determined by users without high transmission rates. We propose an infinite-source Poisson input model which supplies traffic in adjacent time slots. We study properties of the model as slot width decreases and traffic intensity increases. This model has the ability to account for many of the stylized facts.

Keywords

Bursty trafficM/G/∞ input modelinfinite-source Poisson modelnetwork modelinglimit distributionLévy processGaussian limit

MSC classification

Primary: 90B15: Network models, stochastic 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Secondary: 90B22: Queues and service 90B18: Communication networks
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 38 , Issue 2 , June 2006 , pp. 373 - 404
Copyright
Copyright © Applied Probability Trust 2006 

References

Arlitt, M. F. and Williamson, C. L. (1996). Web server workload characterization: the search for invariants. In Joint International Conference on Measurement and Modeling of Computer Systems (Proc. 1996 ACM SIGMETRICS Internat. Conf., Philadelphia, PA), ed. Gaither, B. D., Association for Computing Machinery, New York, pp. 126–137.Google Scholar
Ben Azzouna, N., Clérot, F., Fricker, C. and Guillemin, F. (2004). A flow-based approach to modeling ADSL traffic on an IP backbone link. Ann. Telecommun. 59, 1260–1299.CrossRefGoogle Scholar
Breiman, L. (1965). On some limit theorems similar to the arc-sin law. Theory Prob. Appl. 10, 323–331.Google Scholar
Campos, F. H. et al. (2005). Extremal dependence: internet traffic applications. Stoch. Models 21, 1–35.CrossRefGoogle Scholar
Duffy, D. E., McIntosh, A. A., Rosenstein, M. and Willinger, W. (1993). Analyzing telecommunications traffic data from working common channel signaling subnetworks. In Computing Science and Statistics (Proc. 25th Symp. Interface, San Diego, CA, 1993), eds Tarter, M. E. and Lock, M. D., Interface Foundation of North America, Fairfax Station, VA, pp. 156–165.Google Scholar
Guerin, C. A. et al. (2003). Empirical testing of the infinite source Poisson data traffic model. Stoch. Models 19, 151–200.Google Scholar
Heath, D., Resnick, S. I. and Samorodnitsky, G. (1998). Heavy tails and long range dependence in on/off processes and associated fluid models. Math. Operat. Res. 23, 145–165.Google Scholar
Heffernan, J. E. and Resnick, S. I. (2005). Hidden regular variation and the rank transform. Adv. Appl. Prob. 37, 393–414.Google Scholar
Kaj, I. and Taqqu, M. S. (2004). Convergence to fractional Brownian motion and to the telecom process: the integral representation approach. Submitted.Google Scholar
Kallenberg, O. (1983). Random Measures, 3rd edn. Akademie-Verlag, Berlin.Google Scholar
Konstantopoulos, T. and Lin, S. J. (1998). Macroscopic models for long-range dependent network traffic. Queueing Systems Theory Appl. 28, 215–243.Google Scholar
Leland, W. E., Taqqu, M. S., Willinger, W. and Wilson, D. V. (1993). Statistical analysis of high time-resolution ethernet Lan traffic measurements. In Computing Science and Statistics (Proc. 25th Symp. Interface, San Diego, CA, 1993), eds Tarter, M. E. and Lock, M. D., Interface Foundation of North America, Fairfax Station, VA, pp. 146–155.Google Scholar
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.CrossRefGoogle Scholar
Levy, J. and Taqqu, M. (2000). Renewal reward processes with heavy-tailed interrenewal times and heavy-tailed rewards. Bernoulli 6, 23–44.CrossRefGoogle Scholar
Maulik, K. and Resnick, S. (2003). The self-similar and multifractal nature of a network traffic model. Stoch. Models 19, 549–577.CrossRefGoogle Scholar
Maulik, K., Resnick, S. I. and Rootzén, H. (2002). Asymptotic independence and a network traffic model. J. Appl. Prob. 39, 671–699.Google Scholar
Mikosch, T., Resnick, S. I., Rootzén, H. and Stegeman, A. W. (2002). Is network traffic approximated by stable Lévy motion or fractional Brownian motion? Ann. Appl. Prob. 12, 23–68.Google Scholar
Neveu, J. (1977). Processus ponctuels. In École d'Été de Probabilités de Saint-Flour, VI, 1976 (Lecture Notes Math. 598), Springer, Berlin, pp. 249–445.Google Scholar
Park, K. and Willinger, W. (2000). Self-similar network traffic: an overview. In Self-Similar Network Traffic and Performance Evaluation, eds Park, K. and Willinger, W., Wiley-Interscience, New York, pp. 1–38.Google Scholar
Pratt, J. W. (1960). On interchanging limits and integrals. Ann. Math. Statist. 31, 74–77.Google Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation and Point Processes. Springer, New York.Google Scholar
Resnick, S. I. (1992). Adventures in Stochastic Processes. Birkhäuser, Boston, MA.Google Scholar
Resnick, S. I. (1998). A Probability Path. Birkhäuser, Boston, MA.Google Scholar
Resnick, S. I. (2003). Modeling data networks. In Extreme Values in Finance, Telecommunications, and the Environment, eds Finkenstädt, B. and Rootzén, H., Chapman and Hall/CRC Press, Boca Raton, FL, pp. 287–372.Google Scholar
Resnick, S. I. (2004). On the foundations of multivariate heavy tail analysis. In Stochastic Methods and Their Applications (J. Appl. Prob. Spec. Vol. 41A), eds Gani, J. and Seneta, E., Applied Probability Trust, Sheffield, pp. 191–212.Google Scholar
Resnick, S. I. and Rootzén, H. (2000). Self-similar communication models and very heavy tails. Ann. Appl. Prob. 10, 753–778.CrossRefGoogle Scholar
Riedi, R. H. and Willinger, W. (2000). Toward an improved understanding of network traffic dynamics. In Self-Similar Network Traffic and Performance Evaluation, eds Park, K. and Willinger, W., Wiley-Interscience, New York, pp. 507–530.Google Scholar
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.Google Scholar
Taqqu, M. S., Willinger, W. and Sherman, R. (1997). Proof of a fundamental result in self-similar traffic modeling. Computer Commun. Rev. 27, 5–23.Google Scholar
Willinger, W. (1998). Data network traffic: heavy tails are here to stay. Presentation at Extremes – Risk and Safety, Nordic School of Public Health, Gothenberg, Sweden, August 1998.Google Scholar
Willinger, W. and Paxson, V. (1998). Where mathematics meets the Internet. Not. Amer. Math. Soc. 45, 961–970.Google Scholar
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, eds Adler, Robert J. et al., Birkhäuser, Boston, MA, pp. 27–53.Google Scholar
Willinger, W., Taqqu, M. S., Leland, M. and Wilson, D. (1995). Self-similarity in high-speed packet traffic: analysis and modelling of ethernet traffic measurements. Statist. Sci. 10, 67–85.CrossRefGoogle Scholar
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 5, 71–96.Google Scholar