Local stability and Hopf bifurcation analysis of a Rate Control Protocol with two delays

There is growing interest in explicit congestion control for congestion management in future high bandwidth-delay communication networks. In the class of explicit congestion control protocols, the Rate Control Protocol (RCP) is a protocol designed to minimize flow completion time which is an important metric for the user. RCP estimates the common fair share rate for all flows by using two forms of feedback: rate mismatch and queue size. An outstanding design question for RCP is whether the feedback based on queue size is useful or not. In an effort to make progress on this question, we study the local stability and Hopf bifurcation properties of RCP with feedback based only on rate mismatch. In particular, we focus on the proportionally fair variant of RCP over a network carrying flows with two different round-trip times. We show that as parameters vary, the system may lose local stability through a Hopf bifurcation which leads to the emergence of limit cycles. Using Poincaré normal forms and the center manifold theorem, we show that the system would give rise to a super-critical Hopf bifurcation and the emerging limit cycles are asymptotically orbitally stable. The analysis is corroborated with some numerical examples and bifurcation diagrams.