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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 that originated a long time ago from the pioneering works by Erlang (1909) on the analysis of the models for telephone communication.

Now, it is growing in various directions 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 applications: computing and telecommunication networks, traffic engineering, mobile telecommunications, etc.

The aim of this book is to reflect the current state-of-the-art and some contemporary directions of the analysis of queueing models and networks including some applications.

The first volume of the book consists of 10 chapters written by world-known experts in these areas. These chapters cover a large spectrum of theoretical and asymptotic results for various types of queueing models, including different applications.

Chapter 1 is devoted to the investigation of some theoretical problems for non-classical queueing models including the analysis of queues with inter-dependent arrival and service times.

Chapter 2 deals with the analysis of some characteristics of fluid queues including busy period, congestion analysis and loss probability.

Some contemporary tendencies in the asymptotic analysis of queues are reflected in the following three survey Chapters 3, 7 and 10.

Chapter 3 includes the results on the average and diffusion approximation of Markov queueing systems and networks with a small series parameter ε including applications to some Markov state-dependent queueing models and some other type of models, in particular, repairman problem, superposition of Markov processes and semi-Markov type queueing systems.

Diffusion and Gaussian limits for multi-channel queueing networks with rather general time-dependent input flow and under heavy traffic conditions including some applications to networks with semi-Markov or renewal type input and Markov service are considered in Chapter 7.

Chapter 10 is devoted to the asymptotic analysis of time-varying queues using the large deviations principle for two-time-scale non-homogeneous Markov chains including the analysis of the queue length process and some characterizations of the quality and the efficiency of the system.

The analysis of so-called retrial queueing models is reflected in two Chapters – 4 and 8.

In Chapter 4, two models that provide some modifications of “First-Come First- Served” retrial queueing system introduced by Laszlo Lakatos are investigated.

Chapter 8 gives a review of recent results on single server finite-source retrial queueing systems with random breakdowns and repairs and collisions of the customers.

The analysis of transient behavior of the infinite-server queueing models with a mixed arrival process and Coxian service times and of the Markov-modulated infinite¬server queue with general service times is considered in Chapter 5.

Chapter 6 deals with the applications of fast simulation methods used in queueing theory to solve some high-dimension combinatorial problems in case the other approaches fail.

A survey on the analysis of a strong stability method and its applications to queueing systems and networks and some perspectives are considered in Chapter 9.

The second volume of the book will include the additional chapters devoted to some other contemporary directions of the analysis of queueing models.

The volumes 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.

Dedication to Igor Mykolayovych Kovalenko who died shortly after the writing of this book

Ukrainian and world science is mourning the loss of a brilliant scientist, Professor Igor Mykolayovych Kovalenko, who died on October 19, 2019, after a difficult fight with heart disease.

Prof. Igor Kovalenko was a prominent Ukrainian mathematician in the field of probability theory and its practical applications, a disciple and associate of Boris Gnedenko and Vladimir Korolyuk. He became famous worldwide for his book Introduction to Queueing Theory, written together with Gnedenko. He founded a scientific school in the theory of reliability, queueing theory and cryptography, well known in Ukraine and all over the world.

Igor Kovalenko was born on March 16, 1935 in Kyiv, Ukraine. After graduating from the Faculty of Mechanics and Mathematics of Kyiv Taras Shevchenko University, he worked at the Computing Centre of the Academy of Sciences of Ukraine. From 1962 till 1971, Kovalenko worked in Moscow, where he headed a laboratory at the Moscow Institute of Electronic Engineering, and together with other Gnedenko’s disciples, was the head of the seminar on queueing theory at Moscow State University. Many leading scientists of the former Soviet Union and foreign countries attended this seminar.

Based on the model of piecewise linear Markov processes developed by him, Kovalenko built a mathematical model of a complex defence system reliability and developed numerical algorithms for its implementation based on the method of a small parameter.

In 1964, Igor Kovalenko became a Doctor of Technical Sciences. He formulated the principle of monotonous failures, which, while maintaining high accuracy, significantly simplified the calculations of system reliability. In 1970, Kovalenko was awarded the degree of Doctor of Physics and Mathematics for another thesis on the probabilistic theory of systems of random Boolean equations. Being a doctor twice over is a very rare practice in the scientific world.

After returning to Kyiv in 1971, Prof. Kovalenko founded and headed the Department of Mathematical Methods of the Theory of Complex Systems Reliability at the V.M. Glushkov Institute of Cybernetics. Two areas of research formed the mainstream of investigations: approximate combined analytical and statistical methods of reliability analysis, and theoretical and applied cryptography, systems and methods for data protection. Under his guidance, the first national standard in the field of cryptographic information security was developed in Ukraine.

Prof. Kovalenko is the author of 25 monographs and more than 200 articles. He was elected as an Academician of the National Academy of Sciences of Ukraine in 1978 (Corresponding Member since 1972). He was an extremely hard-working, honest and sincere person, a competent manager and, thanks to his human qualities, professional experience and knowledge, highly respected among his colleagues.

Prof. Igor Kovalenko left many disciples, among them there are many professors and associate professors. All of them preserve in their memory the unforgettable days of joining the science and independent creativity under the guidance of a Great Scientist and Teacher, hours of direct communication with a person of great erudition and high culture.