  We study ergodic properties of Markovian multiclass many-server queues which
are uniform over scheduling policies, as well as the size n of the system. The
system is heavily loaded in the Halfin-Whitt regime, and the scheduling
policies are work-conserving and preemptive. We provide a unified approach via
a Lyapunov function method that establishes Foster-Lyapunov equations for both
the limiting diffusion and the prelimit diffusion-scaled queueing processes
simultaneously.
  We first study the limiting controlled diffusion, and we show that if the
spare capacity (safety staffing) parameter is positive, then the diffusion is
exponentially ergodic uniformly over all stationary Markov controls, and the
invariant probability measures have uniform exponential tails. This result is
sharp, since when there is no abandonment and the spare capacity parameter is
negative, then the controlled diffusion is transient under any Markov control.
In addition, we show that if all the abandonment rates are positive, the
invariant probability measures have sub-Gaussian tails, regardless whether the
spare capacity parameter is positive or negative.
  Using the above results, we proceed to establish the corresponding ergodic
properties for the diffusion-scaled queueing processes. In addition to
providing a simpler proof of the results in Gamarnik and Stolyar [Queueing Syst
(2012) 71:25-51], we extend these results to the multiclass models with renewal
arrival processes, albeit under the assumption that the mean residual life
functions are bounded. For the Markovian model with Poisson arrivals, we obtain
stronger results and show that the convergence to the stationary distribution
is at an exponential rate uniformly over all work-conserving stationary Markov
scheduling policies.
