Steady state and scaling limit for a traffic congestion model

¹ Department of Mathematics, University of Miami, Coral Gables, FL, 33124-4250, USA
² Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, USA

Corresponding authors: Corresponding authors: igrigore@math.miami.edu, kang@math.ncsu.edu

Received: 28 September 2007
Revised: 27 June 2008

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 x_i(t) goes up to x_i(t)+a with probability 1-ζ_i(x) on a unit scale or down to γx_i(t), 0 < γ < 1 with probability ζ_i(x) on a logarithmic scale, where ζ_i 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|ζ_i(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 ζ_i(x) and a are of order N^–1 and t → Nt, in the form of a continuum model with jump rate α(x).