Steady state and scaling limit for a traffic congestion model
Department of Mathematics, University of Miami, Coral Gables, FL, 33124-4250, USA
2 Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, USA
Revised: 27 June 2008
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).
Mathematics Subject Classification: 60K30 / 60J25 / 90B20
Key words: TCP / AIMD / fluid limit / mean field interaction
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