Stability and non-linear dynamics of Dual congestion control schemes with two delays

In this paper, we analyze some local stability and local bifurcation properties of the Proportionally fair, TCP fair, and the Delay-based dual algorithms in the presence of two distinct time delays. In particular, our focus is on the interplay between different notions of fairness, stability, and bifurcation theoretic properties. Different notions of fairness give rise to different non-linear models for the class of Dual algorithms. One can devise conditions for local stability, for each of these models, but such conditions do not offer clear design recommendations on which fairness criteria is desirable. With a bifurcation-theoretic analysis, we have to take non-linear terms into consideration, which helps to learn additional dynamical properties of the various systems. In the case of TCP fair and Delay dual algorithms, with two delays, we present evidence that they can undergo a sub-critical Hopf bifurcation, which has not been previously revealed through analysis of the single delay variants of these algorithms. A sub-critical Hopf bifurcation can result in either large amplitude limit cycles or unstable limit cycles, and hence should be avoided in engineering applications. In the case of the Proportionally fair algorithm, we provide strong evidence to suggest that all one should expect is the occurrence of a super-critical Hopf bifurcation, which leads to stable limit cycles with small amplitude. Thus, from a design perspective, our analysis favors the use of Proportional fairness in the class of dual congestion control algorithms. To best of our knowledge, this is the first study that presents evidence to suggest that fluid models representing Internet congestion control algorithms may undergo a sub-critical Hopf bifurcation.