Nonlinear Instabilities in Computer Network Dynamics

TL;DR

This paper investigates complex nonlinear behaviors in computer network models, especially TCP-RED interactions and rate control, revealing bifurcations and stability conditions that explain observed practical phenomena.

Contribution

It develops a nonlinear discrete-time model for TCP-RED interactions and analyzes bifurcations, and studies delay-based rate control stability using delay-differential equations.

Findings

TCP-RED networks can undergo period doubling bifurcations.

Limited buffer space leads to border collision bifurcations.

Delay-independent stability bounds are derived for rate control.

Abstract

This work studies two types of computer networking models. The primary focus is to understand the different dynamical phenomena observed in practice due to the presence of severe nonlinearities, delays and widely varying operating conditions. The first models considered are of senders running TCP (Transmission Control Protocol) and traffic passing through RED (Random Early Detection) gateways. Building on earlier work, a first order nonlinear discrete-time model is developed for the interaction scenario between transport protocols like TCP and UDP (User Datagram Protocol) and Active Queuing Management schemes like RED. It is shown that the dynamics resulting from the interaction with TCP is consistent with various dynamical behaviors and parameter sensitivities observed in practice. Using bifurcation-theoretic ideas it is shown that TCP-RED type networks may lose their stability through a period doubling bifurcation followed by border collision bifurcations. The nonlinear dependence of the throughput function of TCP-type flows on drop probability is found to be responsible for the period doubling bifurcation, whereas limited buffer space and lack of sufficient damping results in border collision bifurcations. A second class of models studied in this work deals with optimal rate control in networks and are based on the rate-control framework proposed by Kelly. Using the results on delay-differential equation stability, the stability and its lack thereof is studied through an underlying map which arises naturally in time delay systems. An invariance property of this map is used to prove delay-independent stability and to compute bounds on periodic oscillations.