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Steady state and scaling limit for a traffic congestion model

Published online by Cambridge University Press:  29 October 2010

Ilie Grigorescu and
Min Kang
Ilie Grigorescu*
Affiliation:
Department of Mathematics, University of Miami, Coral Gables, FL, 33124-4250, USA
Min Kang*
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, USA
*
Corresponding authors: igrigore@math.miami.edu, kang@math.ncsu.edu
Corresponding authors: igrigore@math.miami.edu, kang@math.ncsu.edu
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Abstract

In a general model (AIMD) of transmission control protocol (TCP)used in internet traffic congestion management, the time dependentdata flow vector x(t) > 0 undergoes a biased random walk ontwo distinct scales. The amount of data of each component x i (t)goes up to x i (t)+a with probability 1-ζ i (x) ona unit scale or down to γ x i (t), 0 < γ < 1 withprobability ζ i (x) on a logarithmic scale, where ζ i depends on the joint state of the system x. Weinvestigate the long time behavior, mean field limit, and the oneparticle case. According to c = lim inf|x|→∞ |x|ζ i (x), the process drifts to ∞ in thesubcritical c < c + (n, γ) case and has an invariantprobability measure in the supercritical case c > c + (n, γ).Additionally, a scaling limit is proved when ζ i (x)and a are of order N –1 and t → N t, in the form of acontinuum model with jump rate α(x).

Keywords

TCPAIMDfluid limitmean field interaction

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Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 14 , 2010, pp. 271 - 285
Copyright
© EDP Sciences, SMAI, 2010

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