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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 dependent data flow vector x(t) > 0 undergoes a biased random walk on two distinct scales. The amount of data of each component xi(t) goes up to xi(t)+a with probability 1-ζi(x) on a unit scale or down to γxi(t), 0 < γ < 1 with probability ζi(x) on a logarithmic scale, where ζi depends on the joint state of the system x. We investigate the long time behavior, mean field limit, and the one particle case. According to c = lim inf|x|→∞ |x|ζi(x) , the process drifts to ∞ in the subcritical c < c+(n, γ) case and has an invariant probability measure in the supercritical case c > c+(n, γ). Additionally, a scaling limit is proved when ζi(x) and a are of order N–1 and t → Nt, in the form of a continuum model with jump rate α(x).

Keywords

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

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