Measuring the Initial Transient: Reflected Brownian Motion

We analyze the convergence to equilibrium of one-dimensional reflected Brownian motion (RBM) and compute a number of related initial transient formulae. These formulae are of interest as approximations to the initial transient for queueing systems in heavy traffic, and help us to identify settings in which initialization bias is significant. We conclude with a discussion of mean square error for RBM. Our analysis supports the view that initial transient effects for RBM and related models are typically of modest size relative to the intrinsic stochastic variability, unless one chooses an especially poor initialization.