On the rate of convergence to equilibrium for two-sided reflected Brownian motion and for the Ornstein–Uhlenbeck process

This paper studies the rate of convergence to equilibrium for two diffusion models that arise naturally in the queueing context: two-sided reflected Brownian motion and the Ornstein–Uhlenbeck process. Specifically, we develop exact asymptotics and upper bounds on total variation distance to equilibrium, which can be used to assess the quality of the steady state as an approximation to finite-horizon performance quantities. Our analysis relies upon the simple spectral structure that these two processes possess, thereby explaining why the convergence rate is “pure exponential,” in contrast to the more complex convergence exhibited by one-sided reflected Brownian motion.