Time-varying delayed feedback control for an internet congestion control model
Shu Zhang Jian Xu
Discrete & Continuous Dynamical Systems - B 2011, 16(2): 653-668 doi: 10.3934/dcdsb.2011.16.653
A proportionally-fair controller with time delay is considered to control Internet congestion. The time delay is chosen to be a controllable parameter. To represent the relation between the delay and congestion analytically, the method of multiple scales is employed to obtain the periodic solution arising from the Hopf bifurcation in the congestion control model. A new control method is proposed by perturbing the delay periodically. The strength of the perturbation is predicted analytically in order that the oscillation may disappear gradually. It implies that the proved control scheme may decrease the possibility of the congestion derived from the oscillation. The proposed control scheme is verified by the numerical simulation.
keywords: delayed differential equation time-varying delay. Internet congestion control method of multiple scales Hopf bifurcation
The Filippov equilibrium and sliding motion in an internet congestion control model
Shu Zhang Yuan Yuan
Discrete & Continuous Dynamical Systems - B 2017, 22(3): 1189-1206 doi: 10.3934/dcdsb.2017058

We consider an Internet congestion control system which is presented as a group of differential equations with time delay, modeling the random early detection (RED) algorithm. Although this model achieves success in many aspects, some basic problems are not clear. We provide the result on the existence of the equilibrium and the positivity and boundedness of the solution. Also, we implement the model by route switch mechanism, based on the minimum delay principle, to model the dynamic routing. For the simple network topology, we show that the Filippov solution exists under some restrictions on parameters. For the case with a single user group and two alternative links, we prove that the discontinuous boundary, or equivalently the sliding region, always exists and is locally attractive. This result implies that for some cases this type of routing may deviate from the purpose of the original design.

keywords: Internet congestion control dynamic routing switch system Filippov solution sliding motion

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