Necessary and sufficient conditions for Hopf bifurcation in exponential RED algorithm with communication delay

In this paper, we investigate a novel congestion control algorithm, i.e., exponential RED algorithm, with communication delay. We derive some necessary and sufficient conditions ensuring Hopf bifurcation to occur for this model. By choosing the delay as a bifurcation parameter, we demonstrated that Hopf bifurcation would occur when the delay exceeds a critical value. A formula for determining the bifurcation direction and stability of bifurcation periodic solutions is given by applying the normal form theory and the center manifold theorem. Some numerical simulations for justifying the theoretical results are also provided.