This book is about stochastic-process limits, i.e., limits in which a sequence of stochastic processes converges to another stochastic process. Since the converging stochastic processes are constructed from initial stochastic processes by appropri- ately scaling time and space, the stochastic-process limits provide a macroscopic view of uncertainty. The stochastic-process limits are interesting and important be- cause they generate simple approximations for complicated stochastic processes and because they help explain the statistical regularity associated with a macroscopic view of uncertainty.
This book emphasizes the continuous-mapping approach to obtain new stochastic- process limits from previously established stochastic-process limits. The continuous- mapping approach is applied to obtain stochastic-process limits for queues, i.e., probability models of waiting lines or service systems. These limits for queues are called heavy-traffic limits, because they involve a sequence of models in which the offered loads are allowed to increase towards the critical value for stability. These heavy-traffic limits generate simple approximations for complicated queueing pro- cesses under normal loading and reveal the impact of variability upon queueing performance. By focusing on the application of stochastic-process limits to queues, this book also provides an introduction to heavy-traffic stochastic-process limits for queues.