Author | Fayolle, Guy. author |
---|---|
Title | Random Walks in the Quarter-Plane [electronic resource] : Algebraic Methods, Boundary Value Problems and Applications / by Guy Fayolle, Roudolf Iasnogorodski, Vadim Malyshev |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 1999 |
Connect to | http://dx.doi.org/10.1007/978-3-642-60001-2 |
Descript | XV, 156 p. online resource |
and History -- 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 -- 2.2 Restricting the Equation to an Algebraic Curve -- 2.3 The Algebraic Curve Q(x, y) = 0 -- 2.4 Galois Automorphisms and the Group of the Random Walk -- 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 -- 3.2 Analytic Continuation -- 3.3 More about Uniformization -- 4 The Case of a Finite Group -- 4.1On the Conditions for H to be Finite -- 4.2 Rational Solutions -- 4.3 Algebraic Solution -- 4.4 Final Form of the General Solution -- 4.5 The Problem of the Poles and Examples -- 4.6 An Example of Algebraic Solution by Flatto and Hahn -- 4.7 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 BVP in the Complex Plane -- 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 -- 6 The Genus 0 Case -- 6.1 Properties of the Branches -- 6.2 Case 1: ?01 = ??1,0 = ??1,1 = 0 -- 6.3 Case 3: ?11 = ?10 = ?01 = 0 -- 6.4 Case 4: ??1,0 = ?0,?1 = ??1,?1= 0 -- 6.5 Case 5: MZ= My= 0 -- 7 Miscellanea -- 7.1 About Explicit Solutions -- 7.2 Asymptotics -- 7.3 Generalized Problems and Analytic Continuation -- 7.4 Outside Probability -- References