TCP: Local stability and Hopf bifurcation

In this paper we analyze a fluid model of TCP with an approximation of drop tail using tools from control and bifurcation theory. The focus of our analysis and experiments lies in a regime where the buffer sizes are small, as recently advocated by Appenzeller, Keslassy and McKeown [G. Appenzeller, I. Keslassy, N. McKeown, Sizing router buffers, in: Proceedings of ACM SIGCOMM, 2004].We find that to ensure local stability of TCP with drop tail it is necessary and sufficient that the arrival rate be greater than capacity by a certain factor, which does not depend on the round-trip time. This factor is found to be 1.1415.The next natural question to ask is: what if these conditions of local stability are just violated? This entails conducting a local bifurcation theoretic analysis (at the point of linear instability), from which we conclude that the corresponding nonlinear system undergoes a supercritical Hopf bifurcation. So as stability of the equilibrium is just lost, it is regained by a stable limit cycle.The analysis is complemented by simulations at the packet level performed using the Network Simulator, ns2.