Recently, a negative interplay has been shown to arise when scheduling/Active Queue Management (AQM) techniques and low-priority congestion control protocols are used together; namely, AQM resets the relative level of priority among congestion control protocols. This work explores this issue by carrying out a control-theoretic analysis of the dynamical system to prove some fundamental properties that fully characterize the reprioritization phenomenon. In particular, (i) we provide the closed-form solution of the equilibrium in the open loop (i.e., fixing a target loss probability p); (ii) we provide a stability analysis and a characterization of the reprioritization phenomenon when closing the loop with AQM (i.e., that dynamically adjusts the system loss probability). Our results are important as the characterization of the reprioritization phenomenon is not only quantitatively accurate for the specific protocols and AQM considered but also qualitatively accurate for a broader range of congestion control protocol and AQM combinations. Finally, while we find a sufficient condition to avoid the reprioritization phenomenon, we also show, at the same time, such conditions to be likely impractical: Therefore, we propose a simple and practical system-level solution that is able to reinstate priorities among protocols.