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Communications in Statistics

As the reader will quickly discover, this article is a short survey – from my personal perspective – of 32 years of research, teaching on the modeling, analysis, and applications of queueing systems. My choice of topics is far from exhaustive; I have focused on those research achievements that I believe have been some of the most significant in their contributions to queueing theory and to its applications. They are the contributions that I have admired and appreciated the most over the course of my teaching and research activities. Another author would undoubtedly have made different choices, as they did in several survey papers on queueing theory. The selection has not been easy at all since there are so many nice results. My *Research is partially supported by Hungarian Scientific Research Fund-OTKA K 60698/2006. The work is supported by the TAMOP 4.2.1./B-09/1/KONV-2010-0007 project. The project is implemented through the New Hungary Development Plan, co-financed by the European Social Fund and the European Regional Development Fund. 9 10 J. Sztrik aim is very simple, I would like to draw the attention of readers to a very unpleasant activity, namely waiting. I have collected some sayings or Murphy’s Laws on Queueing. Here you are: • “If you change queues, the one you have left will start to move faster than the one you are in now. • Your queue always goes the slowest. • Whatever queue you join, no matter how short it looks, will always take the longest for you to get served.” 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