Random walks in the quarter plane: algebraic methods, boundary value problems, applications to queueing systems and analytic combinatorics
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
[Cham]
Springer
[2017]
|
| Ausgabe: | Second edition |
| Schriftenreihe: | Probability theory and stochastic modelling
volume 40 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | xvii, 248 Seiten Diagramme |
| ISBN: | 9783319509280 |
Internformat
MARC
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| 100 | 1 | |a Fayolle, Guy |d 1943- |0 (DE-588)120682036 |4 aut | |
| 245 | 1 | 0 | |a Random walks in the quarter plane |b algebraic methods, boundary value problems, applications to queueing systems and analytic combinatorics |c Guy Fayolle, Roudolf Iasnogorodski, Vadim Malyshev |
| 250 | |a Second edition | ||
| 264 | 1 | |a [Cham] |b Springer |c [2017] | |
| 264 | 4 | |c © 2017 | |
| 300 | |a xvii, 248 Seiten |b Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
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| 650 | 7 | |a Promenades aléatoires (Mathématiques) |2 ram | |
| 650 | 7 | |a Random walks (statistiek) |2 gtt | |
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Datensatz im Suchindex
| _version_ | 1804177388026724352 |
|---|---|
| adam_text | Titel: Random walks in the quarter plane
Autor: Fayolle, Guy
Jahr: 2017
Contents
Part I The General Theory
1 Probabilistic Background....................................................................3
1.1 Markov Chains............................................................................3
1.2 Random Walks in a Quarter Plane..............................................4
1.3 Functional Equations for the Invariant Measure..........................6
2 Foundations of the Analytic Approach..............................................9
2.1 Fundamental Notions and Definitions..........................................9
2.1.1 Covering Manifolds........................................................10
2.1.2 Algebraic Functions........................................................12
2.1.3 Elements of Galois Theory............................................13
2.1.4 Universal Covering and Uniformization........................15
2.1.5 Abelian Differentials and Divisors..................................15
2.2 Restricting the Equation to an Algebraic Curve..........................16
2.2.1 First Insight (Algebraic Functions)................................16
2.2.2 Second Insight (Algebraic Curve)..................................17
2.2.3 Third Insight (Factorization)..........................................18
2.2.4 Fourth Insight (Riemann Surfaces)................................19
2.3 The Algebraic Curve Q(x,y) =0................................................20
2.3.1 Branches of the Algebraic Functions on the Unit
Circle..............................................................................23
2.3.2 Branch Points................................................................25
2.4 Galois Automorphisms and the Group of the Random Walk ... 29
2.4.1 Construction of the Automorphisms | and rj on S..........31
2.5 Reduction of the Main Equation to the Riemann Torus..............32
vii
viii Contents
3 Analytic Continuation of the Unknown Functions
in the Genus 1 Case............................................................................37
3.1 Lifting the Fundamental Equation
onto the Universal Covering........................................................37
3.1.1 Lifting of the Branch Points..........................................39
3.1.2 Lifting of the Automorphisms on the Universal
Covering........................................................................39
3.2 Analytic Continuation..................................................................41
3.3 More About Uniformization........................................................45
4 The Case of a Finite Group................................................................55
4.1 Conditions for Tí to be Finite......................................................55
4.1.1 Criterion for Groups of Order 4....................................61
4.1.2 Criterion for Groups of Order 6....................................62
4.1.3 Criterion for Groups of Order Am..................................65
4.1.4 Criterion for Groups of Order Am - 2............................71
4.2 Further General Results..............................................................72
4.2.1 A Theorem About 6s......................................................72
4.2.2 Form of the General Criterion........................................73
4.3 On Some Symmetric Quantities of the p Function......................75
4.4 Examples....................................................................................77
4.4.1 H of Order 4..................................................................77
4.4.2 H of Order 6..................................................................78
4.5 Various Comments......................................................................78
4.6 Rational Solutions........................................................................79
4.6.1 The Case N(f) ^ 1........................................................81
4.6.2 The Case Nif) = 1........................................................83
4.7 Algebraic Solutions......................................................................91
4.7.1 The Case N(f) = 1........................................................91
4.7.2 The Case Nif ) ¿I........................................................96
4.8 Final Form of the General Solution............................................99
4.9 The Problem of the Poles and Examples....................................103
4.9.1 Rational Solutions..........................................................104
4.10 An Example of an Algebraic Solution by Flatto and Hahn..........114
4.11 Two Queues in Tandem..............................................................117
5 Solution in the Case of an Arbitrary Group......................................119
5.1 Informal Reduction to a Riemann-Hilbert-Carleman BVP..........119
5.2 Introduction to BVPs in the Complex Plane................................121
5.2.1 A Bit of History............................................................121
5.2.2 The Sokhotski-Plemelj Formulae..................................122
5.2.3 The Riemann Boundary Value Problem
for a Closed Contour......................................................123
Contents ix
5.2.4 The Riemann BVP for an Open Contour........................126
5.2.5 The Riemann-Carleman Problem with a Shift................128
5.3 Further Properties of the Branches Defined by Q(x,y) = 0 ... . 135
5.4 Index and Solution of the BVP (5.1.5)........................................145
5.5 Complements..............................................................................151
5.5.1 Analytic Continuation....................................................151
5.5.2 Computation of w..........................................................151
6 The Genus 0 Case................................................................................155
6.1 Properties of the Branches..........................................................155
6.2 Case 1: p0i = P-i,o = P-1,1 =0..................................................157
6.3 Case 3: pu = pw = poi = 0........................................................158
6.4 Case 4: p_ito = po-i = P-1,-1 =0............................................161
6.4.1 Integral Equation............................................................161
6.4.2 Series Representation......................................................162
6.4.3 Uniformization................................................................162
6.4.4 Setting a Boundary Value Problem (BVP)....................164
6.5 Case 5: Mx = My = 0..................................................................164
6.5.1 Playing with the Uniformization....................................169
7 Criterion for the Finiteness of the Group in the Genus 0 Case .... 171
7.1 The Main Theorem......................................................................171
7.2 Proof of Part (a) of Theorem 7.1.1..............................................172
7.2.1 Limit Conformai Gluing When Passing
from Genus 1 to Genus 0..............................................173
7.2.2 Limit of the Uniformization When Passing from
Genus 1 to Genus 0........................................................176
7.2.3 Second Form of the Criterion
in the Zero Drift Case....................................................179
7.2.4 Another Proof of the Criterion........................................180
7.3 Proof of Part (b) of Theorem 7.1.1..............................................181
7.3.1 Miscellaneous Remarks..................................................181
8 Miscellanea..........................................................................................183
8.1 On Explicit Solutions..................................................................183
8.2 Asymptotics................................................................................184
8.2.1 Large Deviations............................................................184
8.2.2 Martin Boundary............................................................186
8.2.3 Random Walks Absorbed on the Axes..........................187
8.3 Generalized Problems and Analytic Continuation........................187
8.3.1 Arbitrary Finite Jumps....................................................187
8.3.2 Space Inhomogeneity......................................................188
8.3.3 Dimension 3..............................................................189
X Contents
8.4 Transient Behavior and Laplace Transforms................................189
8.4.1 Sojourn Time in a Jackson Network
with Overtaking..............................................................189
8.4.2 Diffusion Process............................................................191
8.5 Outside Probability......................................................................191
Part II Applications to Queueing Systems and Analytic
Combinatorics
9 A Two-Coupled Processor Model............ ................................195
9.1 Model and Equations..................................................................195
9.2 Reduction to a Boundary Value Problem on a Circle..................197
9.3 The Case pq = /x¡ /12 . Solutions with Elliptic Integrals................198
9.4 The Case pq ^ p p 2....................................................................199
9.4.1 p + q = 0: Rational Solutions........................................199
10 Joining the Shorter of Two Queues: Reduction
to a Generalized BVP..........................................................................201
10.1 Model and Historical Remarks....................................................201
10.2 Equations....................................................................................202
10.2.1 Reduction of the Number of Unknown Functions..........203
10.3 Meromorphic Continuation to the Complex Plane......................204
10.4 A Fredholm Integral Equation for Q(x) When a / ß..................208
10.4.1 Complete Solution of System (10.2.4)............................216
10.4.2 Explicit Integral Forms for Equal Service Rates
(a = ß)..........................................................................216
10.5 A Functional Equation for Gì (y)................................................217
10.5.1 A Non-standard BVP for Gì (y)......................................218
10.5.2 Poles and Residues of Gi(y): Miscellaneous Issues. . . . 218
11 Counting Lattice Walks in the Quarter Plane....................................221
11.1 A Functional Equation for a Tri-Variate CGF............................222
11.2 Group Classification of the 79 Main Random Walks..................224
11.3 Holonomy and Algebraicity of the Generating Functions............225
11.4 The Nature of the Counting Generating Functions......................227
11.5 Some Exact Asymptotics............................................................229
11.5.1 The Simple Random Walk............................................230
11.6 A General Approach for Walks with Small Steps and Eight
Possible Neighbors......................................................................234
11.6.1 Outline of the Arguments..............................................234
11.6.2 Reduction to a BVP......................................................235
11.6.3 Solution of the BVP......................................................236
Contents xi
11.6.4 Computation of F(0,0,z), F(l,0,z), F(0, l,z),
F(l, l,z)........................................................................236
11.6.5 Groups of Order 4..........................................................237
11.6.6 On the Singularities of the Generating Functions..........238
References....................................................................................................243
Index............................................................................................................247
|
| any_adam_object | 1 |
| author | Fayolle, Guy 1943- Iasnogorodski, Roudolf 1938- Malyshev, Vadim 1938- |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.2/82 |
| dewey-search | 519.2/82 |
| dewey-sort | 3519.2 282 |
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| discipline | Mathematik |
| edition | Second edition |
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| id | DE-604.BV044232660 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T07:47:17Z |
| institution | BVB |
| isbn | 9783319509280 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-029638254 |
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| series | Probability theory and stochastic modelling |
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| spelling | Fayolle, Guy 1943- (DE-588)120682036 aut Random walks in the quarter plane algebraic methods, boundary value problems, applications to queueing systems and analytic combinatorics Guy Fayolle, Roudolf Iasnogorodski, Vadim Malyshev Second edition [Cham] Springer [2017] © 2017 xvii, 248 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Probability theory and stochastic modelling volume 40 Promenades aléatoires (Mathématiques) ram Random walks (statistiek) gtt Random walks (Mathematics) Viertelebene (DE-588)4553601-6 gnd rswk-swf Irrfahrtsproblem (DE-588)4162442-7 gnd rswk-swf Irrfahrtsproblem (DE-588)4162442-7 s Viertelebene (DE-588)4553601-6 s DE-604 Iasnogorodski, Roudolf 1938- (DE-588)120682060 aut Malyshev, Vadim 1938- (DE-588)120682079 aut Erscheint auch als Online-Ausgabe 978-3-319-50930-3 Probability theory and stochastic modelling volume 40 (DE-604)BV042008213 40 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029638254&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Fayolle, Guy 1943- Iasnogorodski, Roudolf 1938- Malyshev, Vadim 1938- Random walks in the quarter plane algebraic methods, boundary value problems, applications to queueing systems and analytic combinatorics Probability theory and stochastic modelling Promenades aléatoires (Mathématiques) ram Random walks (statistiek) gtt Random walks (Mathematics) Viertelebene (DE-588)4553601-6 gnd Irrfahrtsproblem (DE-588)4162442-7 gnd |
| subject_GND | (DE-588)4553601-6 (DE-588)4162442-7 |
| title | Random walks in the quarter plane algebraic methods, boundary value problems, applications to queueing systems and analytic combinatorics |
| title_auth | Random walks in the quarter plane algebraic methods, boundary value problems, applications to queueing systems and analytic combinatorics |
| title_exact_search | Random walks in the quarter plane algebraic methods, boundary value problems, applications to queueing systems and analytic combinatorics |
| title_full | Random walks in the quarter plane algebraic methods, boundary value problems, applications to queueing systems and analytic combinatorics Guy Fayolle, Roudolf Iasnogorodski, Vadim Malyshev |
| title_fullStr | Random walks in the quarter plane algebraic methods, boundary value problems, applications to queueing systems and analytic combinatorics Guy Fayolle, Roudolf Iasnogorodski, Vadim Malyshev |
| title_full_unstemmed | Random walks in the quarter plane algebraic methods, boundary value problems, applications to queueing systems and analytic combinatorics Guy Fayolle, Roudolf Iasnogorodski, Vadim Malyshev |
| title_short | Random walks in the quarter plane |
| title_sort | random walks in the quarter plane algebraic methods boundary value problems applications to queueing systems and analytic combinatorics |
| title_sub | algebraic methods, boundary value problems, applications to queueing systems and analytic combinatorics |
| topic | Promenades aléatoires (Mathématiques) ram Random walks (statistiek) gtt Random walks (Mathematics) Viertelebene (DE-588)4553601-6 gnd Irrfahrtsproblem (DE-588)4162442-7 gnd |
| topic_facet | Promenades aléatoires (Mathématiques) Random walks (statistiek) Random walks (Mathematics) Viertelebene Irrfahrtsproblem |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029638254&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV042008213 |
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