Spatial stochastic network models [Elektronische Ressource] : scaling limits and Monte-Carlo methods / vorgelegt von Florian Voß

Sujets

• Mathematics

Informations

• Publié par: universitat_ulm
• Publié le: 01 janvier 2009
• Nombre de lectures: 5
• Langue: English
• Poids de l'ouvrage: 3 Mo

Extrait

Universität Ulm
Institut für Stochastik
Spatial Stochastic Network Models
Scaling Limits and Monte–Carlo Methods
Dissertation
zur Erlangung des Doktorgrades Dr. rer. nat.
der Fakultät für Mathematik und Wirtschaftswissenschaften der
Universität Ulm
vorgelegt von
Florian Voß
aus
Seeheim-Jugenheim
Dezember 2009Amtierender Dekan: Prof. Dr. Werner Kratz
1. Gutachter: Prof. Dr. Volker Schmidt
2.hter: Prof. Dr. Ulrich Stadtmüller
Tag der Promotion: 19. Februar 2010Contents
1 Introduction 5
1.1 Aims and motivation . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 The Geostoch library . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Preliminaries from stochastic geometry 13
2.1 Basic notation and definitions . . . . . . . . . . . . . . . . . . . . 13
2.2 Point processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.1 Point processes as random counting measures . . . . . . . 14
2.2.2 Fundamental properties . . . . . . . . . . . . . . . . . . . 16
2.2.3 Ergodicity and mixing . . . . . . . . . . . . . . . . . . . . 18
2.2.4 Palm distribution . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.5 Poisson point processes . . . . . . . . . . . . . . . . . . . . 20
2.3 Marked point processes . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.1 Definitions and basic properties . . . . . . . . . . . . . . . 20
2.3.2 Intensity measure and Palm distribution . . . . . . . . . . 21
2.3.3 Ergodic theorem . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.4 Jointly stationary point processes and Neveu’s formula . . 24
2.4 Random closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4.1 Definitions and basic properties . . . . . . . . . . . . . . . 25
2.4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4.3 Ergodicity and mixing . . . . . . . . . . . . . . . . . . . . 27
2.5 Random measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5.1 Definitions and basic properties . . . . . . . . . . . . . . . 29
2.5.2 Intensity measure and Palm distribution . . . . . . . . . . 30
2.5.3 Random measures associated with random closed sets . . . 30
3 Random tessellations and point processes on their edges 33
3.1 Deterministic tessellations . . . . . . . . . . . . . . . . . . . . . . 34
3.2 Random . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2.1 Random tessellations as marked point processes . . . . . . 36
3.2.2 as random sets and random measures 38
3.3 Mean value formulae . . . . . . . . . . . . . . . . . . . . . . . . . 39
12 Contents
3.4 Random tessellation models . . . . . . . . . . . . . . . . . . . . . 40
3.4.1 Poisson–Voronoi tessellation . . . . . . . . . . . . . . . . . 40
3.4.2 Poisson–Delaunay . . . . . . . . . . . . . . . . 41
3.4.3 Poisson line tessellation . . . . . . . . . . . . . . . . . . . . 42
3.4.4 Iterated tessellations . . . . . . . . . . . . . . . . . . . . . 42
3.5 Point processes on the edges of random tessellations . . . . . . . . 43
3.5.1 Cox processes . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.5.2 Cox pro on the edges of random tessellations . . . . 46
3.5.3 Thinnings of the vertices . . . . . . . . . . . . . . . . . . . 48
3.5.4 Voronoi tessellations of point processes on random tessel-
lations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4 Simulation algorithms for the typical cell 51
4.1 Typical Voronoi cells of Poisson processes and Cox processes on PLT 53
4.1.1 Radial simulation of stationary Poisson processes . . . . . 53
4.1.2 Simulation of the typical Voronoi cell of PVT . . . . . . . 54
4.1.3 SimofthetVoronoicellofCoxprocessesonPLT 55
4.2 Typical Voronoi cell of Cox processes on PVT . . . . . . . . . . . 57
4.2.1 Direct simulation algorithm . . . . . . . . . . . . . . . . . 58
4.2.2 An indirect simulation algorithm . . . . . . . . . . . . . . 61
4.3 Typical Voronoi cell of Cox processes on PDT . . . . . . . . . . . 69
4.4 T V cell of Cox pro on nested tessellations . . . 73
e4.4.1 Representation formula for the Palm version T of T . . . . 74
4.4.2 The simulation algorithm . . . . . . . . . . . . . . . . . . 76
4.5 Typical Voronoi cell for thinned vertex sets . . . . . . . . . . . . . 77
4.5.1 Vertices of PDT . . . . . . . . . . . . . . . . . . . . . . . . 78
4.5.2 V of PLT . . . . . . . . . . . . . . . . . . . . . . . . 78
4.5.3 Vertices of PVT . . . . . . . . . . . . . . . . . . . . . . . . 79
4.6 Numerical results obtained by Monte Carlo simulation . . . . . . 80
4.6.1 Scaling invariance for Cox processes and thinnings . . . . . 80
4.6.2 Comparison of direct and indirect simulation algorithms . 81
4.6.3 of the typical cell of PVT and Cox processes
on PDT, PLT and PVT . . . . . . . . . . . . . . . . . . . 81
4.6.4 ComparisonofthetypicalcellofCoxprocessesonPVT/PVT
and PVT/PLT nestings . . . . . . . . . . . . . . . . . . . 84
4.6.5 Numerical results for thinnings . . . . . . . . . . . . . . . 85
4.6.6 Implementation tests . . . . . . . . . . . . . . . . . . . . . 87
5 Euclidean and shortest path connection distances 89
5.1 Spatial stochastic models for two-level hierarchical networks . . . 90
5.1.1 Components of low and highhy levels on random
tessellations . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.1.2 Service zones and their inner structure . . . . . . . . . . . 91Contents 3
5.1.3 Cost functionals for two-level hierarchical models . . . . . 92
5.2 The typical Euclidean distance D . . . . . . . . . . . . . . . . . . 93
5.2.1 Distributional properties . . . . . . . . . . . . . . . . . . . 94
5.2.2 Example: Cox processes on PLT . . . . . . . . . . . . . . . 99
5.3 Statistical estimators . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.3.1 Estimators for distribution function and density of D . . 101
5.3.2 Almost sure convergence of the maximal error . . . . . . . 102
5.3.3 Numerical results from Monte-Carlo simulation . . . . . . 104
5.4 Shortest path connection lengths . . . . . . . . . . . . . . . . . . 108
5.4.1 Typical shortest path length C . . . . . . . . . . . . . . . 108
5.4.2 Representation formula . . . . . . . . . . . . . . . . . . . . 108
5.5 Statistical estimators . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.5.1 Estimators for the density of C . . . . . . . . . . . . . . . 112
5.5.2 Almost sure convergence of the maximal error . . . . . . . 115
5.5.3 Rates of convergence and variances . . . . . . . . . . . . . 117
5.5.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . 118
6 Scaling limits 123
6.1 Scaled typical Euclidean distances and shortest path lengths . . . 124
6.2 Asymptotic exponential distribution for ! 0 . . . . . . . . . . . 125
6.3 Weibull for !1 . . . . . . . . . . . . 127
6.4 Proof of Theorem 6.2 . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.4.1 Some auxiliary results . . . . . . . . . . . . . . . . . . . . 128
6.4.2 Typical Euclidean distance . . . . . . . . . . . . . . . . . . 129
6.4.3 Shortest path length vs. scaled Euclidean distance . . . . . 130
6.5 Proof of Lemma 6.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.6.1 Mixing tessellations . . . . . . . . . . . . . . . . . . . . . . 141
6.6.2 Second moment of perimeter of the typical cell . . . . . . . 143
6.6.3 Asymptotic Weibull distribution of shortest path lengths . 145
6.7 Numerical results and possible extensions . . . . . . . . . . . . . . 147
6.7.1 Numerical results . . . . . . . . . . . . . . . . . . . . . . . 148
6.7.2 Possible extensions . . . . . . . . . . . . . . . . . . . . . . 149
7 Empirical and parametric densities for shortest path length 151
7.1 Distributional properties of C for Cox processes and thinnings . . 152
7.1.1 Densities estimated from Monte–Carlo simulation . . . . . 152
7.1.2 Means and variances . . . . . . . . . . . . . . . . . . . . . 155
7.2 Parametric densities for the typical shortest path length . . . . . 157
7.2.1 The truncated Weibull distribution . . . . . . . . . . . . . 158
7.2.2 Mixtures of exponential and Weibull distributions . . . . . 159
7.3 Fitting of parametric densities . . . . . . . . . . . . . . . . . . . . 161
7.4 Application to real network data . . . . . . . . . . . . . . . . . . . 1624 Contents
8 Capacity distributions 167
8.1 Modeling of capacities . . . . . . . . . . . . . . . . . . . . . . . . 168
8.2 Capacities at locations with fixed distance to HLC . . . . . . . . . 171
8.2.1 Representation formula . . . . . . . . . . . . . . . . . . . . 171
8.2.2 Estimation of the distribution function . . . . . . . . . . . 173
8.3 Capacities at points of Cox processes . . . . . . . . . . . . . . . . 174
8.3.1 Representation formula . . . . . . . . . . . . . . . . . . . . 174
8.3.2 Estimation of the density . . . . . . . . . . . . . . . . . . . 176
8.4 Numerical results and possible extensions . . . . . . . . . . . . . . 177
8.4.1 Numerical results . . . . . . . . . . . . . . . . . . . . . . . 178
8.4.2 Possible extensions . . . . . . . . . . . . . . . . . . . . . . 180
9 Conclusion and outlook 183
A Mathematical background 187
A.1 Convergence concepts . . . . . . . . . . . . . . . . . . . . . . . . . 187