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Random Walks in the Quarter Plane

Algebraic Methods, Boundary Value Problems, Applications to Queueing Systems and Analytic Combinatorics

  • Guy Fayolle
  • Roudolf Iasnogorodski
  • Vadim Malyshev

Part of the Probability Theory and Stochastic Modelling book series (PTSM, volume 40)

Table of contents

  1. Front Matter
    Pages i-xvii
  2. The General Theory

    1. Front Matter
      Pages 1-1
    2. Guy Fayolle, Roudolf Iasnogorodski, Vadim Malyshev
      Pages 3-8
    3. Guy Fayolle, Roudolf Iasnogorodski, Vadim Malyshev
      Pages 9-35
    4. Guy Fayolle, Roudolf Iasnogorodski, Vadim Malyshev
      Pages 37-53
    5. Guy Fayolle, Roudolf Iasnogorodski, Vadim Malyshev
      Pages 55-117
    6. Guy Fayolle, Roudolf Iasnogorodski, Vadim Malyshev
      Pages 119-154
    7. Guy Fayolle, Roudolf Iasnogorodski, Vadim Malyshev
      Pages 155-170
    8. Guy Fayolle, Roudolf Iasnogorodski, Vadim Malyshev
      Pages 171-182
    9. Guy Fayolle, Roudolf Iasnogorodski, Vadim Malyshev
      Pages 183-191
  3. Applications to Queueing Systems and Analytic Combinatorics

    1. Front Matter
      Pages 193-193
    2. Guy Fayolle, Roudolf Iasnogorodski, Vadim Malyshev
      Pages 195-200
    3. Guy Fayolle, Roudolf Iasnogorodski, Vadim Malyshev
      Pages 201-219
    4. Guy Fayolle, Roudolf Iasnogorodski, Vadim Malyshev
      Pages 221-241
  4. Back Matter
    Pages 243-248

About this book

Introduction

This monograph aims to promote original mathematical methods to determine the invariant measure of two-dimensional random walks in domains with boundaries. Such processes arise in numerous applications and are of interest in several areas of mathematical research, such as Stochastic NetworksAnalytic Combinatorics, and Quantum Physics. This second edition consists of two parts.

Part I is a revised upgrade of the first edition (1999), with additional recent results on the group of a random walk. The theoretical approach given therein has been developed by the authors since the early 1970s. By using Complex Function TheoryBoundary Value ProblemsRiemann Surfaces, and Galois Theory, completely new methods are proposed for solving functional equations of two complex variables, which can also be applied to characterize the Transient Behavior of the walks, as well as to find explicit solutions to the one-dimensional Quantum Three-Body Problem, or to tackle a new class of Integrable Systems.

Part II borrows special case-studies from queueing theory (in particular, the famous problem of Joining the Shorter of Two Queues) and enumerative combinatorics (CountingAsymptotics).

Researchers and graduate students should find this book very useful.

Keywords

60G50, 39B32, 32A26, 30D05, 46N50 algebraic methods analytic combinatorics boundary value problems functional equations random walks in the quarter plane

Authors and affiliations

  • Guy Fayolle
    • 1
  • Roudolf Iasnogorodski
    • 2
  • Vadim Malyshev
    • 3
  1. 1.INRIA Paris-­‐RocquencourtLe ChesnayFrance
  2. 2.ParisFrance
  3. 3.Faculty of Mechanics and MathematicsMoscow Lomonosov State UniversityMoscowRussia

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-50930-3
  • Copyright Information Springer International Publishing AG 2017
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-50928-0
  • Online ISBN 978-3-319-50930-3
  • Series Print ISSN 2199-3130
  • Series Online ISSN 2199-3149
  • Buy this book on publisher's site
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