A proportionally fair controller with small Drop-Tail buffers: Local bifurcation analysis with two delays

We study a proportionally fair congestion controller with small Drop-Tail buffers. The model considers two sets of users, each with a different round-trip time, feeding into a single bottleneck link. Using a suitably motivated exogenous bifurcation parameter, we computationally exhibit the existence of a Hopf bifurcation. We then employ Poincaré normal forms and the center manifold theory, to provide an analytical basis to determine the type of the Hopf bifurcation. Using the analysis, one can also verify the orbital stability of the bifurcating limit cycles. Some numerical computations complement the theoretical analysis.