Random walks in the quarter plane : algebraic methods, boundary value problems, applications to queueing systems and analytic combinatorics / Guy Fayolle, Roudolf Iasnogorodski, Vadim Malyshev.
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 Networks, Analyt...
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Full Text (via Springer) |
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| Main Authors: | , , |
| Format: | eBook |
| Language: | English |
| Published: |
Cham, Switzerland :
Springer,
2017.
|
| Edition: | Second edition. |
| Series: | Probability theory and stochastic modelling ;
v. 40. |
| Subjects: |
Table of Contents:
- Preface; Backbone of the Book; Acknowledgements; Contents; Introduction and History; Historical Comments; Contents of the Book; Part I The General Theory; 1 Probabilistic Background; 1.1 Markov Chains; 1.2 Random Walks in a Quarter Plane; 1.3 Functional Equations for the Invariant Measure; 2 Foundations of the Analytic Approach ; 2.1 Fundamental Notions and Definitions; 2.1.1 Covering Manifolds; 2.1.2 Algebraic Functions; 2.1.3 Elements of Galois Theory; 2.1.4 Universal Covering and Uniformization; 2.1.5 Abelian Differentials and Divisors; 2.2 Restricting the Equation to an Algebraic Curve.
- 2.2.1 First Insight (Algebraic Functions)2.2.2 Second Insight (Algebraic Curve); 2.2.3 Third Insight (Factorization); 2.2.4 Fourth Insight (Riemann Surfaces); 2.3 The Algebraic Curve Q(x,y) = 0; 2.3.1 Branches of the Algebraic Functions on the Unit Circle; 2.3.2 Branch Points; 2.4 Galois Automorphisms and the Group of the Random Walk; 2.4.1 Construction of the Automorphisms and on S; 2.5 Reduction of the Main Equation to the Riemann Torus; 3 Analytic Continuation of the Unknown Functions in the Genus 1 Case ; 3.1 Lifting the Fundamental Equation onto the Universal Covering.
- 4.5 Various Comments4.6 Rational Solutions; 4.6.1 The Case N(f) neq1; 4.6.2 The Case N(f) = 1; 4.7 Algebraic Solutions; 4.7.1 The Case N(f)=1; 4.7.2 The Case N(f) neq1 ; 4.8 Final Form of the General Solution; 4.9 The Problem of the Poles and Examples ; 4.9.1 Rational Solutions; 4.10 An Example of an Algebraic Solution by Flatto and Hahn; 4.11 Two Queues in Tandem; 5 Solution in the Case of an Arbitrary Group; 5.1 Informal Reduction to a Riemann
- Hilbert
- Carleman BVP; 5.2 Introduction to BVPs in the Complex Plane; 5.2.1 A Bit of History; 5.2.2 The Sokhotski
- Plemelj Formulae.
- 5.2.3 The Riemann Boundary Value Problem for a Closed Contour5.2.4 The Riemann BVP for an Open Contour; 5.2.5 The Riemann
- Carleman Problem with a Shift; 5.3 Further Properties of the Branches Defined by Q(x,y) = 0; 5.4 Index and Solution of the BVP (5.1.5); 5.5 Complements; 5.5.1 Analytic Continuation; 5.5.2 Computation of w; 6 The Genus 0 Case; 6.1 Properties of the Branches; 6.2 Case 1: p01 = p-1,0 = p-1,1 = 0; 6.3 Case 3: p11 = p10 = p01 = 0; 6.4 Case 4: p-1,0 = p0,-1 = p-1,-1 = 0; 6.4.1 Integral Equation; 6.4.2 Series Representation; 6.4.3 Uniformization.
- 3.1.1 Lifting of the Branch Points3.1.2 Lifting of the Automorphisms on the Universal Covering ; 3.2 Analytic Continuation; 3.3 More About Uniformization; 4 The Case of a Finite Group; 4.1 Conditions for calH to be Finite; 4.1.1 Criterion for Groups of Order 4 ; 4.1.2 Criterion for Groups of Order 6 ; 4.1.3 Criterion for Groups of Order 4m ; 4.1.4 Criterion for Groups of Order 4m-2; 4.2 Further General Results; 4.2.1 A Theorem About δs; 4.2.2 Form of the General Criterion; 4.3 On Some Symmetric Quantities of the Function; 4.4 Examples; 4.4.1 calH of Order 4; 4.4.2 calH of Order 6.