Critical Scale for a Continuous AIMD Model

Abstract

A scaled version of the general AIMD model of transmission control protocol (TCP) used in Internet traffic congestion management leads to a Markov process x(t) representing the time dependent data flow that moves forward with constant speed on the positive axis and jumps backward to γx(t), 0 < γ < 1 according to a Poisson clock whose rate α(x) depends on the interval swept in between jumps. We give sharp conditions for Harris recurrence and analyze the convergence to equilibrium on multiple scales (polynomial, fractional exponential, exponential) identifying the critical case xα(x) ∼ β. Criticality has different behavior according to whether it occurs at the origin or infinity. In each case, we determine the transient (possibly explosive), null—and positive—recurrent regimes by comparing β to ( − ln γ)^{− 1}.

Keywords:

• AIMD
• Criticality
• Geometric ergodicity
• Harris recurrence
• Local Doeblin condition
• TCP

Mathematics Subject Classification:

• Primary 60K30
• Secondary 60J25, 90B20