Queueing theory

Queueing theory is concerned with the quantitative modeling of dynamic systems that generate waiting lines, and the analysis of the behavior of such systems in the short and long time spans. In this chapter, we present a brief overview of the history of queueing theory and the basic concepts of it. The aim of this chapter is to expose the reader to one of the most instrumental and widely applicable topics of stochastic analysis, to provide the basic concepts of it, and to stimulate further interest in it. Some motivating examples of stochastic processes are discussed followed by the main topics in a standard course of stochastic processes. These include the renewal processes, Markov chains, and continuous-time Markov chains. Queueing models are then discussed in more detail, where the focus is mainly restricted to Markovian queues. A more complicated topic of queueing networks is also briefly discussed. Most of the topics discussed are also supported by illustrative examples. The methods are presented mainly under the assumption of a single queue, where a single type of service is provided by possibly several servers under the FIFO (first in first out) service protocol and under the traditional assumption that once a customer enters a queue, she/he stays there until the service has been completed. In an extensive section, we pointed out relaxation of such assumptions and extensions in several directions regarding the queue discipline, service protocol, customer behavior, several service types, estimation of major system parameters, as well as the current research interests in the field. In an appendix, we included a brief review of the background material in probability theory.