Review of Random Processes

Abstract

In this chapter, we review some of the key results from the theory of random processes that are needed in the study of queueing systems. In the first section, we provide a brief review of probability. We begin with a definition of the elements of a statistical experiment and conclude with a discussion of computing event probabilities via conditioning. In the second section, we discuss random variables, their distributions, and manipulation of distributions. In the third and fourth sections, we discuss the exponential distribution and the Poisson process, respectively, which play a key role in queueing analysis, and we develop some of their key properties. In the fifth section, we provide a brief review of discrete and continuous parameter Markov chains defined on the nonnegative integers. While the materials presented here are, for the most part, self-contained and a mastery of the materials presented here would provide an adequate basis for understanding queueing systems, our experience is that these materials cannot be used as a substitute for good courses on probability theory and random processes. Rather, our presentation is intended primarily as review and reinforcement of a subset of the ideas and principles from probability theory that are useful in understanding queueing systems. As as example, in courses on stochastic processes the distinction between discrete time and discrete parameter stochastic processes is often mentioned briefly and then ignored. But, in the study of queueing systems, this difference is significant, and we reinforce that fact herein. Similarly, the relationship between frequency-averaged probabilities and time-averaged probabilities is addressed in detail at the end of this chapter.