Abstract
We study stability properties of two specific diffusion processes: a high dimensional piecewise OU (Ornstein Uhlenbeck) process and a degenerate OU process. They arise from diffusion approximations for many-server queues that have phase-type service time distributions and allow for customer abandonment. We prove that they are positive recurrent and has unique stationary distribution under certain conditions. The proof techniques are based on constructing suitable quadratic Lyapunov functions. This is joint work with Prof. Jim Dai and Prof. Ton Dieker.