Local Stability and Hopf Bifurcation Analysis for Compound TCP

We conduct a local stability and Hopf bifurcation analysis for Compound transmission control protocol (TCP), with small Drop-Tail buffers, in three topologies. The first topology consists of two sets of TCP flows having different round-trip times and feeding into a core router. The second topology consists of two distinct sets of TCP flows, regulated by a single-edge router and feeding into a core router. The third topology comprises two distinct sets of TCP flows, regulated by two separate edge routers and feeding into a core router. In each case, we conduct a local stability analysis and obtain conditions on the network and protocol parameters to ensure stability. If these conditions get marginally violated, we show that the underlying systems lose local stability via a Hopf bifurcation. After exhibiting a Hopf, a key concern is to determine the asymptotic orbital stability of the bifurcating limit cycles. We then present a detailed analytical framework to address the stability of the limit cycles and the type of the Hopf bifurcation by invoking Poincaré normal forms and the center manifold theory. We finally conduct packet-level simulations to corroborate our analytical insights.