Call Us: US - +1 845 478 5244 | UK - +44 20 7193 7850 | AUS - +61 2 8005 4826

The Annals of Mathematical Statistics.

papers written in the next 20 years contain some of the most important concepts and techniques; the notion of statistical equilibrium and the method of writing down balance of state equations (later called Chapman-Kolmogorov equations) are two such examples. Erlang’s motivation was to develop tools for the analysis and design of telephone systems) an application that continues to the present day to motivate research in queueing theory. It should be noted that in Erlang’s work, as well as the work done by others in the twenties and thirties, the motivation has been the practical problem of congestion. The trend toward the analytical study of the basic stochastic processes of the system continued, and queueing theory proved to be a fertile field for researchers who wanted to do fundamental research on stochastic processes involving mathematical models. Mathematical modeling is a process of approximation. A probabilistic model brings it a little bit closer to reality; nevertheless it cannot completely represent the real world phenomenon because of involved uncertainties. Therefore, it is a matter of convenience where one can draw the line between the simplicity of the model and the closeness of the representation. Renewed interest in queueing theory and its potential applications outside of telecommunications came with the codification of the field of operations research in the early 1950’s. And in the early 1960’ s queueing theory was rediscovered by researchers interested in the performance analysis of time-shared computer systems. The use of queueing theory as a tool for the performance evaluation of computer systems and components of computer systems is now well established with the result that most undergraduate computer science majors receive at least some exposure to its models and methodology. By the mid-70’s researchers interested in modeling the performance of computer systems had discovered the Jackson network and its variants and come to appreciate the versatility and applicability of such models. In those days, the emphasis was on systems consisting of a mainframe computer with disk storage and satellite terminals, a precursor to what we would now call a local area network (LAN). The idea of a communication network tying together widely separated computers was just a gleam in the eye of Leonard Kleinrock and a few other visionaries. It was Kleinrock who, more than anyone else, was responsible for spreading the word among computer scientists about Jackson networks in particular and queueing theory in general. Recently web-based research dominates the main directions. It is difficult to list all the main trends, but in my opinion the followings certainly belong to them: long-range dependence, numerical problems of stochastic processes, time-dependent solutions, modeling tools, retrial systems, approximations, simulations, statistical inference. For a more detailed discussions about ongoing