Measuring the Initial Transient: Reflected Brownian Motion

TL;DR

This paper analyzes the convergence behavior of reflected Brownian motion, providing formulas for initial transient effects relevant to queueing systems, and discusses the impact of initialization bias and mean square error.

Contribution

It introduces new formulas for initial transient analysis of reflected Brownian motion and discusses their implications for queueing system initialization bias.

Findings

Initial transient effects are generally modest compared to stochastic variability.

Poor initialization can lead to significant transient bias.

Formulas help identify when initialization bias is important.

Abstract

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.