FAST TCP: Some fluid models, stability and Hopf bifurcation

We study FAST TCP, a new transmission control protocol that uses queueing delay as its feedback measure. We highlight two continuous-time models proposed for FAST that represent two operating regimes: (i) queueing delay forms a large component of the end-to-end delay, (ii) propagation delay is the dominant component of the end-to-end delay. These models when coupled with the integrator model for the queue, are shown to yield qualitatively similar results. We then study one of these models in different queueing regimes. In the scenario where the queue can be modelled as an integrator, we conduct a detailed local stability analysis. This yields strict bounds, on the system parameters and round-trip time, to ensure local stability. We show that the system undergoes a Hopf bifurcation, when these bounds are violated, leading to the emergence of limit cycles in the system dynamics. As limit cycles could be detrimental to network performance, we conduct a detailed Hopf bifurcation analysis using Poincaré normal forms and center manifold theory. This enables us to characterise the type of the Hopf bifurcation and determine the orbital stability of the limit cycles. We then consider a regime with smaller queues, where end systems react primarily to packet loss. In this regime, larger thresholds could lead to instability. Packet-level simulations corroborate our analytical insights; non-linear oscillations are indeed observed in the queue size.