Transient behavior of regulated Brownian Motion, l: Starting at the origin.

This memorandum is the first in a series of memoranda investigating the time-dependent behavior of queues and related stochastic models. The principal goal is to obtain easily comprehensible closed-form approximate descriptions of the moments of the number of customers in the system as functions of time. These approximations show how the moments approach their steady- state limits as time evolves. This first memorandum focuses on regulated or reflected Brownian motion, which is the standard diffusion process used to approximate queueing processes. In Part l attention is restricted to regulated Brownian motion starting at the origin. With this special initial condition, the moments as functions of time are increasing, so that after normalization by steady-state limits, the moment functions become cdf's (cumulative distribution functions), which can be studied probabilistically. The first two moment cdf's are approximated by hyperexponential cdf's (mixtures of two exponential cdf's), which are obtained by matching the first three moments.