There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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

In this paper, we demonstrate the existence of fair end-to-end window-based congestion control protocols for packet-switched networks with first come-first served routers. Our definition of fairness generalizes proportional fairness and includes arbitrarily close approximations of max-min fairness. The protocols use only information that is available to end hosts and are designed to converge reasonably fast. Our study is based on a multiclass fluid model of the network. The convergence of the protocols is proved using a Lyapunov function. The technical challenge is in the practical implementation of the protocols.

https://www-inst.eecs.berkeley.edu/~ee228a/fa03/228A03/papers/Mo.pdf

Fair end-to-end window-based congestion control

Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)

Basic Tools of Probabilistic Modelling The Queue with Server of Walking Type and Its Applications to Computer System Modelling Queueing Network Models Queueing Networks with Multiple Classes of Positive and Negative Customers and Product Form Solution Markov-Modulated Queues Diffusion Approximation Methods for General Queueing Networks Approximate Decomposition and Iterative Techniques for Closed Model Solution Synthesis Problems in Single-Resource Systems: Characterisation and Control of Achievable Performance Control of Performance in Mutliple-Resource Systems A Queue with Server of Walking Type.

Analysis and synthesis of computer systems

Queuing networks are studied at selected points in the steady state, namely, at the moments when jobs of a given class arrive into a given node (either from the outside or from other nodes) and at the moments when jobs of a given class leave a given node (either for the outside or for other nodes). The processes defined by these points are known to be, in general, non-Potsson, interdependent, and serially correlated; therefore the relation between the distribution of the system state embedded at those moments and the steady-state (or random point) distribution is not obvious a priori. For a large class of networks having product-form equihbrium distribnttons it is shown that (a) if the given job class belongs to an open subchain, the state distributions at input pomts, output points, and random points are identical, and (b) if the job class belongs to a closed subchain, the distribution at input and output points ts the same as the steady-state distribution of a network with one less job in that subchain.

The Distribution of Queuing Network States at Input and Output Instants

. A single-server processor-sharing system with M job classes is analyzed in the steady state. The scheduling strategy considered divides the total processor capacity in unequal fractions among the different job classes. More precisely, if there are N~jobs of classj in the system, j = 1, 2 ..... M, each class k job receives a fraction gh/(~M.~ giN~) of the processor capacity. Earlier analyses of this system are shown to be incorrect and new expressions for the conditional expected response times Wk(t) of class k jobs with required service time t are obtained (for general required service time distributions). These yield the asymptotic behavior of W~(t) as t ~ oo and rather simple formulas in the exponential case. The unconditional average response times are also obtained. KEY WORDS AND PHRASES: processor sharing, strategy, egalitarian, discriminatory job classes, Laplace transform, Fourier-Stiehjes transform CR CATEGORIES: 4.35.4.6, 5.5, 6.2

Isi Mitrani

Papers

Analysis and synthesis of computer systems

The Distribution of Queuing Network States at Input and Output Instants

Sharing a Processor Among Many Job Classes

Probabilistic Modelling

Spectral expansion solution for a class of Markov models: application and comparison with the matrix-geometric method