This chapter starts by communicating how various aspects of our lives involve interacting with queues. It then provides a brief history of the main inception of queueing theory and its main governing princples, and discusses how it has impacted various aspects of our lives. It educates the reader about the main ideas and principles in queueing theory and also elaborates on the psychological aspects of waiting in queues. Showcasing various examples of how the main ideas in queueing theory have enabled important improvements, ranging from what happened during Queen Elizabeth II’s memorial, to the creation of the internet and modern telephones, to our experiences in airports or on roads, the chapter presents queueing theory as a potent branch of analytics science that has enabled scholars to make the world a better place. The chapter also discusses the vital interplays between queueing theory, public policy, and technology.

This chapter is devoted to the spectral analysis of birth–death processes on nonnegative integers, which are the most basic and important continuous-time Markov chains. These processes will be characterized by an infinitesimal operator which is a tridiagonal matrix whose spectrum is always contained in the negative real line (including 0). The Karlin–McGregor integral representation formula of the transition probability functions of the process is obtained in terms of orthogonal polynomials with respect to a probability measure with support inside a positive real interval. Although many of the results are similar or equivalent to those of discrete-time birth–death chains, the methods and techniques are quite different. The chapter gives an extensive collection of examples related to orthogonal polynomials, including the M/M/k queue for any k servers, the continuous-time Ehrenfest and Bernoulli–Laplace urn models, a genetics model of Moran and linear birth–death processes. As in the case of discrete-time birth–death chains, the Karlin–McGregor formula is applied to the probabilistic aspects of birth–death processes, such as processes with killing, recurrence, absorption, the strong ratio limit property, the limiting conditional distribution, the decay parameter, quasi-stationary distributions and bilateral birth–death processes on the integers.