Steady state and scaling limit for a traffic congestion model
ESAIM: Probability and Statistics, Tome 14 (2010), pp. 271-285.

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).

DOI : 10.1051/ps:2008029
Classification : 60K30, 60J25, 90B20
Mots clés : TCP, AIMD, fluid limit, mean field interaction 
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Grigorescu, Ilie; Kang, Min. Steady state and scaling limit for a traffic congestion model. ESAIM: Probability and Statistics, Tome 14 (2010), pp. 271-285. doi : 10.1051/ps:2008029. http://www.numdam.org/articles/10.1051/ps:2008029/

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