Hostname: page-component-848d4c4894-nr4z6 Total loading time: 0 Render date: 2024-05-05T09:13:25.224Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic independence and a network traffic model

Part of: Operations research and management science Special processes

Published online by Cambridge University Press:  14 July 2016

Krishanu Maulik ,
Sidney Resnick  and
Holger Rootzén
Krishanu Maulik*
Affiliation:
Cornell University
Sidney Resnick*
Affiliation:
Cornell University
Holger Rootzén*
Affiliation:
Chalmers University of Technology
*
∗ Current address: EURANDOM, PO Box 513, 5600 MB Eindhoven, The Netherlands.
∗∗ Postal address: School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY 14853, USA. Email address: sid@orie.cornell.edu
∗∗∗∗ Postal address: Department of Mathematics, Chalmers University of Technology, S-412 96 Gothenburg, Sweden.
Get access Rights & Permissions [Opens in a new window]

Abstract

The usual concept of asymptotic independence, as discussed in the context of extreme value theory, requires the distribution of the coordinatewise sample maxima under suitable centering and scaling to converge to a product measure. However, this definition is too broad to conclude anything interesting about the tail behavior of the product of two random variables that are asymptotically independent. Here we introduce a new concept of asymptotic independence which allows us to study the tail behavior of products. We carefully discuss equivalent formulations of asymptotic independence. We then use the concept in the study of a network traffic model. The usual infinite source Poisson network model assumes that sources begin data transmissions at Poisson time points and continue for random lengths of time. It is assumed that the data transmissions proceed at a constant, nonrandom rate over the entire length of the transmission. However, analysis of network data suggests that the transmission rate is also random with a regularly varying tail. So, we modify the usual model to allow transmission sources to transmit at a random rate over the length of the transmission. We assume that the rate and the time have finite mean, infinite variance and possess asymptotic independence, as defined in the paper. We finally prove a limit theorem for the input process showing that the centered cumulative process under a suitable scaling converges to a totally skewed stable Lévy motion in the sense of finite-dimensional distributions.

Keywords

Asymptotic independenceM/G/∞ queuenetwork modelinginfinite source Poisson modelheavy tailslong-range dependencerandom ratesPoisson input

MSC classification

Primary: 90B15: Network models, stochastic
Secondary: 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 39 , Issue 4 , December 2002 , pp. 671 - 699
Copyright
Copyright © Applied Probability Trust 2002 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Basrak, B. (2000). The sample autocorrelation function of non-linear time series. Doctoral Thesis, Rijksuniversiteit Groningen.Google Scholar
Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York.CrossRefGoogle Scholar
Breiman, L. (1965). On some limit theorems similar to the arc-sin law. Theory Prob. Appl. 10, 323–331.Google Scholar
Cunha, C. A., Bestavros, A., and Crovella, M. E. (1995). Characteristics of WWW client-based traces. Tech. Rep. TR-95-010, Computer Science Department, Boston University. Available at http://www.cs.bu.edu/techreports/.Google Scholar
De Haan, L., and de Ronde, J. (1998). Sea and wind: multivariate extremes at work. Extremes 1, 7–45.Google Scholar
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.CrossRefGoogle Scholar
Guerin, C. A. et al. (1999). Empirical testing of the infinite source Poisson data traffic model. Tech. Rep. 1257, School of ORIE, Cornell University. Available at http://www.orie.cornell.edu/trlist/.Google Scholar
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. Prob. 9, 352–375.Google Scholar
Jelenković, P. R., and Lazar, A. A. (1996). A network multiplexer with multiple time scale and subexponential arrivals. In Stochastic Networks: Stability and Rare Events (Lecture Notes Statist. 117), eds Glasserman, P., Sigman, K. and Yao, D. D., Springer, New York, pp. 215–235.Google Scholar
Jelenković, P. R., and Lazar, A. A. (1999). Asymptotic results for multiplexing subexponential on–off processes. Adv. Appl. Prob. 31, 394–421.Google Scholar
Konstantopoulos, T., and Lin, S.-J. (1998). Macroscopic models for long-range dependent network traffic. Queueing Systems 28, 215–243.CrossRefGoogle Scholar
Levy, J. B., and Taqqu, M. S. (2000). Renewal reward processes with heavy-tailed inter-renewal times and heavy-tailed rewards. Bernoulli 6, 23–44.CrossRefGoogle Scholar
Maulik, K., Resnick, S. and Rootzén, H. (2000). A network traffic model with random transmission rate. Tech. Rep. 1278, School of ORIE, Cornell University. Available at http://www.orie.cornell.edu/trlist/.Google Scholar
Mikosch, T., Resnick, S., 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.CrossRefGoogle Scholar
Pipiras, V., and Taqqu, M. S. (2000). The limit of a renewal reward process with heavy-tailed rewards is not a linear fractional stable motion. Bernoulli 6, 607–614.CrossRefGoogle Scholar
Pipiras, V., Taqqu, M. S., and Levy, J. B. (2000). Slow and fast growth conditions for renewal reward processes with heavy tailed renewals and either finite-variance or heavy tailed rewards. Preprint.Google Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.CrossRefGoogle Scholar
Resnick, S. (1997). Heavy tail modeling and teletraffic data. Ann. Statist. 25, 1805–1869.CrossRefGoogle Scholar
Resnick, S. and Rootzén, H. (2000). Self-similar communication models and very heavy tails. Ann. Appl. Prob. 10, 753–778.CrossRefGoogle Scholar
Resnick, S., and Samorodnitsky, G. (2000). A heavy traffic approximation for workload processes with heavy tailed service requirements. Manag. Sci. 46, 1236–1248.Google Scholar
Resnick, S. and Stărică, C. (1997). Smoothing the Hill estimator. Adv. Appl. Prob. 29, 271–293.CrossRefGoogle Scholar
Resnick, S. and van den Berg, E. (2000). Weak convergence of high-speed network traffic models. J. Appl. Prob. 37, 575–597.Google Scholar
Skorohod, A. V. (1956). Limit theorems for stochastic processes. Theory Prob. Appl. 1, 261–290.CrossRefGoogle Scholar
Stărică, C. (1999). Multivariate extremes for models with constant conditional correlations. J. Empirical Finance 6, 515–553.CrossRefGoogle Scholar
Whitt, W. (1980). Some useful functions for functional limit theorems. Math. Operat. Res. 5, 67–85.Google Scholar
Whitt, W. (1999a). On the Skorohod M-topologies. Preprint, AT&T Labs, Florham Park, NJ.Google Scholar
Whitt, W. (1999b). The reflection map is Lipschitz with appropriate Skorohod M-metrics. Preprint, AT&T Labs, Florham Park, NJ.Google Scholar
Whitt, W. (2000). Limits for cumulative input processes to queues. Preprint, AT&T Labs, Florham Park, NJ. Prob. Eng, Inf. Sci. 14, 123–150.Google Scholar
Whitt, W. (2002). Stochastic-Process Limits. Springer, New York.CrossRefGoogle Scholar
Willinger, W., and Paxson, V. (1998). Where mathematics meets the internet. Notices Amer. Math. Soc. 45, 961–970.Google Scholar
Willinger, W., Taqqu, M. S., Leland, M., and Wilson, D. (1995). Self-similarity in high-speed packet traffic: analysis and modeling of Ethernet traffic measurements. Statist. Sci. 10, 67–85.Google Scholar
Willinger, W., Taqqu, M. S., 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. Networking 5, 71–86.Google Scholar