  This paper develops the first method for the exact simulation of reflected
Brownian motion (RBM) with non-stationary drift and infinitesimal variance. The
running time of generating exact samples of non-stationary RBM at any time $t$
is uniformly bounded by $\mathcal{O}(1/\bar\gamma^2)$ where $\bar\gamma$ is the
average drift of the process. The method can be used as a guide for planning
simulations of complex queueing systems with non-stationary arrival rates
and/or service time.
