Positive recurrence of piecewise Ornstein-Uhlenbeck processes and common quadratic Lyapunov functions

摘要

We study the positive recurrence of piecewise Ornstein-Uhlenbeck (OU)diffusion processes, which arise from many-server queueing systems withphase-type service requirements. These diffusion processes exhibit differentbehavior in two regions of the state space, corresponding to "overload"(service demand exceeds capacity) and "underload" (service capacity exceedsdemand). The two regimes cause standard techniques for proving positiverecurrence to fail. Using and extending the framework of common quadraticLyapunov functions from the theory of control, we construct Lyapunov functionsfor the diffusion approximations corresponding to systems with and withoutabandonment. With these Lyapunov functions, we prove that piecewise OUprocesses have a unique stationary distribution.