This is an expanded and updated version of a book that I published in 1985 with John Wiley and Sons, titled Brownian Motion and Stochastic Flow Systems. Like the original, it fits comfortably under the heading of “applied probability,” its primary subjects being (i) stochastic system models based on Brownian motion, and (ii) associated methods of stochastic analysis.

Here the word “system” is used in the engineer's or economist's sense, referring to equipment, people, and operating procedures that work together with some economic purpose, broadly construed. Examples include telephone call centers, manufacturing networks, cash management operations, and data storage centers. This book emphasizes dynamic stochastic models of such man-made systems, that is, models in which system status evolves over time, subject to unpredictable factors like weather, demand shocks, or mechanical failures. Some of the models considered in the book are purely descriptive in nature, aimed at estimating performance characteristics like long-run average inventory or expected discounted cost, given a fixed set of system characteristics. Other models are explicitly aimed at optimizing some measure of system performance, especially through the exercise of a dynamic control capability.

Brownian models of system evolution and dynamic control are increasingly popular in economics and engineering, and in allied business fields like finance and operations. The reason for that popularity is mathematical tractability: In one instance after another, researchers working with Brownian models have been able to derive explicit solutions and clear-cut insights that were unobtainable using conventional models. In the ways that really matter, then, Brownian models provide the simplest possible representations of dynamic, stochastic phenomena.