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Communications in Statistics

A queue is a waiting line (like customers waiting at a supermarket checkout counter); queueing theory is the mathematical theory of waiting lines. More generally, queueing theory is concerned with the mathematical modeling and analysis of systems that provide service to random demands. A queueing model of a system is an abstract representation whose purpose is to isolate those factors that relate to the system’s ability to meet service demands whose occurrences and durations are random. Typically, simple queueing models are specified in terms of the arrival process the service mechanism and the queue discipline. The arrival process specifies the probabilistic structure of the way the demands for service occur in time; the service mechanism specifies the number of servers and the probabilistic structure of the duration of time required to serve a customer, and the queue discipline specifies the order in which waiting customers are selected from the queue for service. Selecting or constructing a queueing model that is rich enough to reflect the complexity of the real system, yet simple enough to permit mathematical analysis) is an art. The ultimate objective of the analysis of queueing systems is to understand the behavior of their underlying processes so that informed and intelligent decisions can be made in their management. Then, the mathematical analysis of the models would yield formulas that presumably relate the physical and stochastic parameters to certain performance measures, such as average response/ waiting time, server utilization, throughput, probability of buffer overflow, distribution function of response/waiting time, busy period of server, etc. The art of applied queueing theory is to construct a model that is simple enough so that it yields to mathematical analysis, yet contains sufficient detail so that its performance measures reflect the behavior of the real system. In the course of modeling one could use analytical, numerical, asymptotic, and simulation methods integrated into performance evaluation tools.