Hopf bifurcation in REM congestion control under gamma-distributed feedback delays

This paper is devoted to the stability analysis and Hopf bifurcation of a REM-based congestion control algorithm with gamma-distributed feedback delays, a more realistic alternative to conventional constant-delay models. By employing the linear chain trick, which allows to transform To capture delay variability due to queuing, routing, and protocol overhead, we model the system using gamma-distributed kernels and convert the resulting integro-differential equations into finite-dimensional ODEs, we show explicit delay thresholds for the onset of bifurcations under two memory regimes, called weak and strong regimes. In the weak memory case, a single critical delay governs stability, whereas strong memory induces stability only within a bounded delay interval. Numerical investigations confirm that distributed-delay models exhibit smoother transitions to instability and generate oscillations consistent with supercritical Hopf bifurcations. Compared to traditional discrete-delay REM models, our formulation offers superior robustness and smoother bifurcation dynamics. These findings provide actionable insights for delay-aware congestion control design in heterogeneous network environments.