We study the ergodic properties of a class of multidimensional piecewise Ornstein-Uhlenbeck
processes with jumps, which contains the limit of the queueing processes arising in
multiclass many-server queues with heavy-tailed arrivals and/or asymptotically negligible
service interruptions in the Halfin-Whitt regime as special cases. In these queueing
models, the Ito equations have a piecewise linear drift, and are driven by either
(1) a Brownian motion and a pure-jump Levy process, or (2) an anisotropic Levy process
with independent one-dimensional symmetric alpha-stable components or (3) an anisotropic
Levy process as in (2) and a pure jumpLevy process. We also study the class of models
driven by a subordinate Brownian motion, which contains an isotropic (or rotationally
invariant) alpha-stable Levy process as a special case. We identify conditions on
the parameters in the drift, the Levy measure and/or covariance function which result
in subexponential and/or exponential ergodicity. We show that these assumptions are
sharp, and we identify some key necessary conditions for the process to be ergodic.
In addition, we show that for the queueing models described above with no abandonment,
the rate of convergence is polynomial, and we provide a sharp quantitative characterization
of the rate via matching upper and lower bounds.