Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

The Filippov equilibrium and sliding motion in an internet congestion control model

Pages: 1189 - 1206, Volume 22, Issue 3, May 2017      doi:10.3934/dcdsb.2017058

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Shu Zhang - School of Aerospace Engineering and Applied Mechanics, Tongji University, 1239 Siping Road, Shanghai 200092, China (email)
Yuan Yuan - Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's NL, Canada A1C 5S7, Canada (email)

Abstract: 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.
Mathematics Subject Classification:  Primary: 34K11, 34K18; Secondary: 94C99.

Received: September 2015;      Revised: January 2016;      Available Online: January 2017.