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Asymptotic independence and a network traffic model

Part of: Special processes Operations research and management science

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.
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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 

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