Dynamics of the Internet TCP–RED congestion control system

In this paper, the ideal case for the important congestion control algorithms, i.e., the TCP (transmission control protocol) algorithm and the RED (random early detection) algorithm, is analyzed, and the following results are found. First, mathematical analysis proves the existence of two equilibria of this dynamical system (of DDEs—delay differential equations), which has not been established in previous works. Second, reduction of the round-trip delay leads to the optimal design of the TCP–RED congestion control. Unfortunately, a drawback of TCP–RED is that package dropping and congestion are induced. The dynamics of the DDEs are considered for when congestion does not take place and the averaged queue length is between the minimum threshold and the maximum one. Stability and Hopf bifurcation of the DDEs are considered. We find that if the time delays are sufficiently large, Hopf bifurcation of the two equilibria will appear, and thus stationary motions with approximately constant rates of arrival, averaged queue length and oscillations with periodically varying forms will arise. Simulations illustrate the richness of the dynamics of the DDEs.