  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 It\^o
equations have a piecewise linear drift, and are driven by either (1) a
Brownian motion and a pure-jump L\'evy process, or (2) an anisotropic L\'evy
process with independent one-dimensional symmetric $\alpha$-stable components,
or (3) an anisotropic L\'evy process as in (2) and a pure-jump L\'evy process.
We also study the class of models driven by a subordinate Brownian motion,
which contains an isotropic (or rotationally invariant) $\alpha$-stable L\'evy
process as a special case. We identify conditions on the parameters in the
drift, the L\'evy 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.
