  We study the positive recurrence of piecewise Ornstein-Uhlenbeck (OU)
diffusion processes, which arise from many-server queueing systems with
phase-type service requirements. These diffusion processes exhibit different
behavior in two regions of the state space, corresponding to "overload"
(service demand exceeds capacity) and "underload" (service capacity exceeds
demand). The two regimes cause standard techniques for proving positive
recurrence to fail. Using and extending the framework of common quadratic
Lyapunov functions from the theory of control, we construct Lyapunov functions
for the diffusion approximations corresponding to systems with and without
abandonment. With these Lyapunov functions, we prove that piecewise OU
processes have a unique stationary distribution.
