Critical scale for a continuous AIMD model. (English) Zbl 1323.60120

Grigorescu, Ilie; Kang, Min

Stoch. Models 30, No. 3, 319-343 (2014).

Summary: 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 \(\gamma x(t)\), \(0 < \gamma < 1\) according to a Poisson clock whose rate \(\alpha(x)\) 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 \(x\alpha(x) \sim \beta\). 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 {\(\beta\)} to \((-\ln \gamma)^{- 1}\).

MSC:

60K30	Applications of queueing theory (congestion, allocation, storage, traffic, etc.)
60J25	Continuous-time Markov processes on general state spaces
90B20	Traffic problems in operations research

Keywords:

AIMD model; transmission control protocol; internet traffic congestion management; criticality; geometric ergodicity; Harris recurrence; local Doeblin condition

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