In this technical note, fractional-order congestion control systems are introduced for the first time. In comparison with the conventional integer-order dual congestion control algorithms, the fractional control algorithms are more accurate and versatile. Bifurcation theory in fractional-order differential equations is still an outstanding problem. Sufficient conditions for the occurrence of Hopf bifurcations are extended from integer-order dynamical systems to fractional-order cases. Then, these conditions are used to establish the existence of Hopf bifurcations for the delayed fractional-order model of dual congestion control algorithms proposed in this note. Finally, the onsets of bifurcations are identified, where Hopf bifurcations occur and a family of oscillations bifurcate from the equilibrium. Illustrative examples are also provided to demonstrate the theoretical results.