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Preface

Vladimir ANISIMOV1 and Nikolaos LIMNIOS2

1Amgen Inc., London, United Kingdom

2University of Technology of Compiègne, France

Queueing theory is a huge and very rapidly developing branch of science belonging to probability theory and stochastic modelling that originated a long time ago from the pioneering works by Erlang (1909) on the analysis of the models for telephone communication using Poisson processes. Later on, these results were extended further in different directions in the works of such famous mathematicians as Pollaczek, Khinchin, Kendall, Kleinrock and many others.

Nowadays, queueing theory is rapidly growing in various areas including a theoretical analysis of queueing models and networks of rather complicated structure using rather sophisticated mathematical models and various types of stochastic processes. It also includes very wide areas of modern applications: computing and telecommunication networks, traffic engineering, mobile telecommunications, etc.

The aim of this second volume, together with Volume 1, is to reflect the current cutting-edge thinking and established practices in the analysis and applications of queueing models.

This volume includes 8 chapters written by experts well-known in their areas.

Two chapters, Chapters 1 and 7, are devoted to investigating a stability analysis of some types of multiserver regenerative queueing systems with heterogeneous servers and a regenerative input flow using synchronization of the input and majorizing output flows; and a stability analysis of regenerative queueing systems based on a renewal analysis technique, which is illustrated on classical GI/G/1 and GI/G/m queueing systems.

Chapter 2 considers a few selected queueing models that are useful in service sectors using both analytical and simulation approaches; and highlights the significant role played by the correlated arrivals, which may occur due to the fact that customers/jobs arrive from different sources. Some interesting observations based on the analytical models of some well-known queueing systems are also reported.

Chapter 3 is devoted to the discussion of similarity between probability distributions and random processes related to queueing and reliability models and their use in economics, industry, demography, environmental studies. Some open and challenging new problems for further research directions are marked, as well as various possible areas of practical applications.

Chapter 4 is devoted to an important problem of how a social planner or a monopolist should act to incite customers to adopt a desirable behaviour, that is, to increase the social welfare or the monopolist’s revenue/profit, respectively, using the concepts of free market. Several techniques for the control of information in queueing systems and their impact on strategic customer behaviour using different mechanisms, e.g. pricing structures, priority systems and non-standard queueing disciplines, are considered and illustrated using various examples.

Chapter 5 is devoted to applications of Renyi’s and Tsallis’s non-extensive maximum entropy methods of inductive inference for the analysis of state probabilities of the stable M/G/1 queue with long-range interactions (heavy tails) of order q (0.5 < q > 1). Consequently, novel q-dependent state probabilities of the M/G/1 queue with heavy tails are derived by maximising the respective non-extensive maximum entropy functionals. Some numerical examples are considered.

In Chapter 6 a survey of the inventory models with positive service time including perishable items, stock supplied using various control policies such as (s; S); (r; Q), random reorder quantity, etc. is presented. Stochastic decomposition results are also discussed. A brief description of the work done in queues with the requirement of additional items for service, is also provided.

Chapter 8 examines a transient analysis of Markovian queueing systems focussing on the derivation of closed-form solutions for the main transient state distributions and the development of numerical techniques with the aim to position and underline the role of the uniformization technique.

The two volumes of Queueing Theory will be useful for graduate and PhD students, lecturers, and also the researchers and developers working in mathematical and stochastic modelling and various applications in computer and communication networks, science and engineering in the departments of Mathematics & Applied Mathematics, Statistics, or Operations Research at universities and in various research and applied centres.