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This IMA Volume in Mathematics and its Applications LINEAR ALGEBRA,
MARKOV CHAINS, AND QUEUEING MODELS is based on the proceedings of a
workshop which was an integral part of the 1991-92 IMA program on
"Applied Linear Algebra." We thank Carl Meyer and R.J. Plemmons for
editing the proceedings. We also take this opportunity to thank the
National Science Founda tion, whose financial support made the
workshop possible. A vner Friedman Willard Miller, Jr. xi PREFACE
This volume contains some of the lectures given at the workshop Lin
ear Algebra, Markov Chains, and Queueing Models held January 13-17,
1992, as part of the Year of Applied Linear Algebra at the
Institute for Mathematics and its Applications. Markov chains and
queueing models play an increasingly important role in the
understanding of complex systems such as computer, communi cation,
and transportation systems. Linear algebra is an indispensable tool
in such research, and this volume collects a selection of important
papers in this area. The articles contained herein are
representative of the underlying purpose of the workshop, which was
to bring together practitioners and re searchers from the areas of
linear algebra, numerical analysis, and queueing theory who share a
common interest of analyzing and solving finite state Markov
chains. The papers in this volume are grouped into three major
categories-perturbation theory and error analysis, iterative
methods, and applications regarding queueing models."
This book concerns modern methods in scientific computing and
linear algebra, relevant to image and signal processing. For these
applications, it is important to consider ingredients such as: (1)
sophisticated mathematical models of the problems, including a
priori knowledge, (2) rigorous mathematical theories to understand
the difficulties of solving problems which are ill-posed, and (3)
fast algorithms for either real-time or data-massive computations.
Such are the topics brought into focus by these proceedings of the
Workshop on Scientific Computing (held in Hong Kong on March 10-12,
1997, the sixth in such series of Workshops held in Hong Kong since
1990), where the major themes were on numerical linear algebra,
signal processing, and image processing.
Here is a valuable text and research tool for scientists and
engineers who use or work with theory and computation associated
with practical problems relating to Markov chains and queuing
networks, economic analysis, or mathematical programming.
Originally published in 1979, this new edition adds material that
updates the subject relative to developments from 1979 to 1993.
Theory and applications of nonnegative matrices are blended here,
and extensive references are included in each area. You will be led
from the theory of positive operators via the Perron-Frobenius
theory of nonnegative matrices and the theory of inverse
positivity, to the widely used topic of M-matrices. On the way,
semigroups of nonnegative matrices and symmetric nonnegative
matrices are discussed. Later, applications of nonnegativity and
M-matrices are given; for numerical analysis the example is
convergence theory of iterative methods, for probability and
statistics the examples are finite Markov chains and queuing
network models, for mathematical economics the example is
input-output models, and for mathematical programming the example
is the linear complementarity problem. Nonnegativity constraints
arise very naturally throughout the physical world. Engineers,
applied mathematicians, and scientists who encounter nonnegativity
or generalizations of nonnegativity in their work will benefit from
topics covered here, connecting them to relevant theory.
Researchers in one area, such as queuing theory, may find useful
the techniques involving nonnegative matrices used by researchers
in another area, say, mathematical programming. Exercises and
biographical notes are included with each chapter.
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