We study the joint control of arrival and service rates in queueing systems with the objective of minimizing long-run expected cost minus revenue. Although the objective function is non-convex, first-order methods have been empirically observed to converge to globally optimal solutions. This paper provides a theoretical foundation for this empirical phenomenon by characterizing the optimization landscape and identifying a hidden convexity: the problem admits a convex reformulation after an appropriate change of variables. Leveraging this hidden convexity, we establish the Polyak-Lojasiewicz-Kurdyka (PLK) condition for the original control problem, which excludes spurious local minima and supports global convergence guarantees for first-order methods. Our analysis applies to a broad class of $GI/GI/1$ queueing models, including those with Gamma-distributed interarrival and service times, as well as $GI/M/1$ queues with log-concave interarrival times. As a key ingredient in the proof, we establish a new convexity property of the expected queue length under a square-root transformation of the traffic intensity.