On Stochastic Processing Networks∗

Stochastic processing networks arise as models in manufacturing, telecommunications, computer systems and the service industry. Common characteristics of these networks are that they have entities, such as jobs, customers or packets, that move along routes, wait in buffers, receive processing from various resources, and are subject to the effects of stochastic variability through such quantities as arrival times, processing times, and routing protocols. Networks arising in modern applications are often highly complex and heterogeneous. Typically, their analysis and control present challenging mathematical problems. One approach to these challenges is to consider approximate models. In the last 15 years, significant progress has been made on using approximate models to understand the stability and performance of a class of stochastic processing networks called open multiclass HL queueing networks. Here HL stands for a non-idling service discipline that is head-of-the-line, i.e., jobs are drawn from a buffer in the order in which they arrived. Examples of such disciplines are FIFO (firstin-first-out) and static priorities. First order (functional laws of large numbers) approximations called fluid models have been used to study the stability of these networks, and second order (functional central limit theorem) approximations which are diffusion models, have been used to analyze the performance of heavily congested networks. The interplay between these two levels of approximation has been an important theme in this work. In contrast to this progress, optimal control of multiclass HL queueing networks is an active area of research, and performance analysis and optimal control of more general stochastic processing networks are still in their early stages of development. These lectures will progress from motivating the study of stochastic processing networks, through describing some significant developments of the last 15 years, and will end with some current research topics.