  We study the ergodic properties of a class of controlled stochastic
differential equations (SDEs) driven by $\alpha$-stable processes which arise
as the limiting equations of multiclass queueing models in the Halfin-Whitt
regime that have heavy-tailed arrival processes. When the safety staffing
parameter is positive, we show that the SDEs are uniformly ergodic and enjoy a
polynomial rate of convergence to the invariant probability measure in total
variation, which is uniform over all stationary Markov controls resulting in a
locally Lipschitz continuous drift. We also derive a matching lower bound on
the rate of convergence (under no abandonment). On the other hand, when all
abandonment rates are positive, we show that the SDEs are exponentially ergodic
uniformly over the above-mentioned class of controls. Analogous results are
obtained for L\'evy-driven SDEs arising from multiclass many-server queues
under asymptotically negligible service interruptions. For these equations, we
show that the aforementioned ergodic properties are uniform over all stationary
Markov controls. We also extend a key functional central limit theorem
concerning diffusion approximations so as to make it applicable to the models
studied here.
