Steady State and Scaling Limit for a Traffic Congestion Model

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).