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.
"Positive recurrence of piecewise Ornstein–Uhlenbeck processes and common quadratic Lyapunov functions." Ann. Appl. Probab. 23 (4) 1291 - 1317, August 2013. https://doi.org/10.1214/12-AAP870