The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

Praise for the Third Edition: "This is one of the best books available. Its excellent organizational structure allows quick reference to specific models and its clear presentation . . . solidifies the understanding of the concepts being presented."IIE Transactions on Operations EngineeringThoroughly revised and expanded to reflect the latest developments in the field, Fundamentals of Queueing Theory, Fourth Edition continues to present the basic statistical principles that are necessary to analyze the probabilistic nature of queues. Rather than presenting a narrow focus on the subject, this update illustrates the wide-reaching, fundamental concepts in queueing theory and its applications to diverse areas such as computer science, engineering, business, and operations research.This update takes a numerical approach to understanding and making probable estimations relating to queues, with a comprehensive outline of simple and more advanced queueing models. Newly featured topics of the Fourth Edition include:Retrial queuesApproximations for queueing networksNumerical inversion of transformsDetermining the appropriate number of servers to balance quality and cost of serviceEach chapter provides a self-contained presentation of key concepts and formulae, allowing readers to work with each section independently, while a summary table at the end of the book outlines the types of queues that have been discussed and their results. In addition, two new appendices have been added, discussing transforms and generating functions as well as the fundamentals of differential and difference equations. New examples are now included along with problems that incorporate QtsPlus software, which is freely available via the book's related Web site.With its accessible style and wealth of real-world examples, Fundamentals of Queueing Theory, Fourth Edition is an ideal book for courses on queueing theory at the upper-undergraduate and graduate levels. It is also a valuable resource for researchers and practitioners who analyze congestion in the fields of telecommunications, transportation, aviation, and management science.

Fundamentals of Queueing Theory

We study the performance of a statistical multiplexer whose inputs consist of a superposition of packetized voice sources and data. The performance analysis predicts voice packet delay distributions, which usually have a stringent requirement, as well as data packet delay distributions. The superposition is approximated by a correlated Markov modulated Poisson process (MMPP), which is chosen such that several of its statistical characteristics identically match those of the superposition. Matrix analytic methods are then used to evaluate system performance measures. In particular, we obtain moments of voice and data delay distributions and queue length distributions. We also obtain Laplace-Stieitjes transforms of the voice and data packet delay distributions, which are numerically inverted to evaluate tails of delay distributions. It is shown how the matrix analytic methodology can incorporate practical system considerations such as finite buffers and a class of overload control mechanisms discussed in the literature. Comparisons with simulation show the methods to be accurate. The numerical results for the tails of the voice packet delay distribution show the dramatic effect of traffic variability and correlations on performance.

A Markov Modulated Characterization of Packetized Voice and Data Traffic and Related Statistical Multiplexer Performance

The Theory of Stochastic Processes. By D. R. Cox and H. D. Miller. Pp. x, 398. 70s. (Methuen)

Automatic linguistic indexing of pictures is an important but highly challenging problem for researchers in computer vision and content-based image retrieval. In this paper, we introduce a statistical modeling approach to this problem. Categorized images are used to train a dictionary of hundreds of statistical models each representing a concept. Images of any given concept are regarded as instances of a stochastic process that characterizes the concept. To measure the extent of association between an image and the textual description of a concept, the likelihood of the occurrence of the image based on the characterizing stochastic process is computed. A high likelihood indicates a strong association. In our experimental implementation, we focus on a particular group of stochastic processes, that is, the two-dimensional multiresolution hidden Markov models (2D MHMMs). We implemented and tested our ALIP (Automatic Linguistic Indexing of Pictures) system on a photographic image database of 600 different concepts, each with about 40 training images. The system is evaluated quantitatively using more than 4,600 images outside the training database and compared with a random annotation scheme. Experiments have demonstrated the good accuracy of the system and its high potential in linguistic indexing of photographic images.

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Automatic Linguistic Indexing of Pictures by a statistical modeling approach

This is a mathematical text suitable for students of engineering and science who are at the third year undergraduate level or beyond. It is a book of applicable mathematics. It avoids the approach of listing only the techniques, followed by a few examples, without explaining why the techniques work. Thus, it provides not only the know-how but also the know-why. Equally, the text has not been written as a book of pure mathematics with a list of theorems followed by their proofs. The authors' aim is to help students develop an understanding of mathematics and its applications. They have refrained from using clichés like “it is obvious” and “it can be shown”, which may be true only to a mature mathematician. On the whole, the authors have been generous in writing down all the steps in solving the example problems.The book comprises ten chapters. Each chapter contains several solved problems clarifying the introduced concepts. Some of the examples are taken from the recent literature and serve to illustrate the applications in various fields of engineering and science. At the end of each chapter, there are assignment problems with two levels of difficulty. A list of references is provided at the end of the book.This book is the product of a close collaboration between two mathematicians and an engineer. The engineer has been helpful in pinpointing the problems which engineering students encounter in books written by mathematicians.

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Matrix-geometric solutions in stochastic models : an algorithmic approach

Matrix-Geometric Solutions in Stochastic Models

Structural Stochastic Matrices of M/G/1 Type and Their Applications.

: A versatile class is introduced of point processes on the real line, which are closely related to finite-state Markov processes. Many relevant probability distributions, moment and correlation formulas are given in forms which are computationally tractable. Several point processes, such as renewal processes of phase type, Markov-modulated Poisson processes and certain semi-Markov point processes appear as particular cases. The treatment of a substantial number of existing probability models can be generalized in a systematic manner to arrival processes of the type discussed in this paper. Several qualitative features of point processes, such as certain types of fluctuations, grouping, interruptions and the inhibition of arrivals by bunch inputs can be modelled in a way which remains computationally tractable.

A versatile Markovian point process

We study a single-server queue in which the server takes a vacation whenever the system becomes empty. The service and vacation times and the arrival process are all assumed to be mutually independent. The successive service times and the vacation times each form independent, identically distributed sequences with general distributions. A new class of non-renewal arrival processes is introduced. As special cases, it includes the Markov-modulated Poisson process and the superposition of phase-type renewal processes. Algorithmically tractable equations for the distributions of the waiting times at an arbitrary time and at arrivals, as well as for the queue length at an arbitrary time, at arrivals, and at departures are established. Some factorizations, which are known for the case of renewal input, are generalized to this new framework and new factorizations are obtained. The algorithmic implementation of these results is discussed.

Marcel F. Neuts

Papers

Matrix-geometric solutions in stochastic models : an algorithmic approach

Matrix-Geometric Solutions in Stochastic Models

Structural Stochastic Matrices of M/G/1 Type and Their Applications.

A versatile Markovian point process

A single-server queue with server vacations and a class of non-renewal arrival processes