Self-Similarity of Human Society Geometry. Self-Similarity of Wireless Networks

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State of the Art of Stochastic Modeling of the Wireless Networks Topologies

In the past decades, the research community has successfully modeled network topologies by extensively using Poisson Point Process (PPP). The seminal work of [29] has enriched the community knowledge on the achievable limits of capacity, routing, etc. Further works, e.g., [18, 30, 31], have studied in detail the routing and communication properties of these topologies.
In [32] the authors overview results driven by stochastic geometry and random graphs on connectivity, capacity, outage probability and other fundamental limits of wireless networks.
The authors start by giving the mathematical preliminaries, basic notions, notations and properties of point processes, boolean models and random geometric graphs. Then, using stochastic geometric tools they characterize interference, outage and throughput. Basics of percolation and connectivity are reviewed in other to further pass to the analysis of capacity. Other applications are also shortly reviewed such as routing, information propagation and secrecy. Dierent spatial stochastic models like Poisson point process, Poisson hard-core process, Strauss process and the perturbed triangular lattice are used in [33] to t the locations of base-stations in cellular networks. The data was obtained from a public database. The authors introduce a metric called deployment gain in order to estimate the coverage performance achieved by a data set.
A Matern hard core process is used in [34] in order to model concurrent transmissions in carrier-sense multiple access (CSMA) networks. As the eld of multi-tier and cognitive wireless networks received increased interested, the need for characterizing the interference in these new type of scenarios was answered in numerous works [35]. In [36], the authors provide and extensive survey on the literature related to stochastic geometry models for single-tier as well as multi-tier and cognitive cellular wireless networks. The authors overview the models and methods used for multi-tier networks together with the relevant metrics. Information dissemination within a network modeled by a Poisson point process has been studied (e.g. [37], [38]).
As the study of cellular networks with tools of stochastic geometry became more intense, special attention has been given to enhancing the accuracy of the estimation of signal-tointerference- plus-noise-ration (SINR). In this sense, the authors of [39] have shown that the SINR values experienced by a user with respect to dierent base stations are related to an instance of the two-parameter Poisson-Dirichlet process.

Mobility Estimation in Cellular Networks by Means of Stochastic Geometry

A preliminary study we have performed during the rst part of the Ph.D period consisted in the estimation of mobile user equipment (UE) speed by exploiting network information and stochastic geometry. This study came as a continuation of the work on mobility estimation by means of signal processing with the intention to exploit network provided information and stochastic geometry. The lessons learned from this study were fundamental for the creation of the hyperfractal model.
UE’s speed information is useful for optimizing handover (HO), reduce call drop or networking signaling ow, optimizing the UE-to-evolved NodeB (eNB) attachment and radio resource utilization eciency, for example during channel allocation decision and in multicarrier deployment scenarios. For instance, one may favor the HO of high-speed UEs to large coverage cells (e.g., macro cells) or to higher coverage carriers in multi-carrier scenarios. On the other hand, for macro cell trac ooading purpose, low or medium speed UEs are preferred to HO to small cells, see Figure 2.3. It is also known that UE’s speed information can be benecial to the optimal conguration of HO parameters, for example in setting the ltering coecients of power measurements, as well as for determining most suitable transmission scheduling strategy [53].

Stochastic Geometry and User History based Mobility Estimation: STRAIGHT

In [64], the authors propose a mobility estimator for cellular networks that is based on the following dependency between the UE velocity and average cell residence time: E[Th] = R 2E[V ] (2.3). where: Th is the HO call cell residence time, dened as the time spent by a mobile in a given cell to which the call was handed over from a neighboring cell before crossing to another cell, R is the cell radius, V is the user speed. This expression assumes uniformly distributed mobiles in a cellular network and, secondly, the mobiles moving in straight lines with direction uniformly distributed in [0; 2). Equation (2.3) can be re-written as: E[V ] = R 2E[Th] (2.4).
for showing that the user speed is proportional to the ratio of cell radius over the cell residence time. The assumption of linear trajectory of UEs is clearly unrealistic. One can easily nd that UEs have erratic trajectories and with many direction changes which will thus lengthen the cell residence time and lead to incorrect estimation of the UE’s speed. As the time a UE stays in a cell depends not only on the UE’s speed but also on its trajectory, we can interpret that the erratic trajectory can be translated into an increase in its travel distance.
From (2.4), one may also interpret R 2 as the average distance traveled by an UE in the cell and thus the increase in traveled distance would mean an increase to a notion of the cell radius, such that a \virtual » or equivalent cell radius should be used to replace the simple R in (2.4) for taking into account erratic trajectory or similar factor.

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Self-Similarity of Human Society Geometry. Self-Similarity of Wireless Networks

From rapid prototyping to tissue-engineering, fractals have been extensively used in biology and medicine. In fact, as nature is seen as having a fractal \nature », it is easy to understand why fractals have been used from the modeling of the universe to the modeling of the behavior of atoms. One may argue that the universe itself is fractal as a whole, the debate on this topic having led to the birth of fractal cosmology [72].
Cities yield some of the best examples of fractals (see Figure 3.3) as we shall further argue throughout this section.

Table of contents :

List of Figures
List of Tables
Les Hyperfractales pour la Modelisation des Reseaux sans Fil 
0.1 Les Modeles d’ Hyperfractales
0.1.1 Un Modele de Propagation qui Genere la Necessite d’Elargir le
Modele Topologique. L’Eet Canyon
0.1.2 Le Modele Hyperfractale pour les Relais
0.2 Des Applications pour les Reseaux sans Fil
0.2.1 Procedure de Validation avec des Donnees
0.2.2 Etude de la Vitesse de Propagation de l’Information dans un Reseau Urbain Tolerant aux Retards
0.2.3 Le Routage de Bout en Bout dans un Reseau Hyperfractale avec Relais
0.2.4 Goulot d’Etranglement
1 Introduction and Motivation 
1.1 Technological Context
1.2 Motivation
1.3 Contributions and Structure of the Thesis
1.4 Publications
2 Classic Stochastic Geometry 
2.1 Notions of Classic Stochastic Geometry
2.1.1 Model fundamentals
2.1.2 Poisson Point Process
2.1.3 Poisson Line Process
2.1.4 Voronoi Tessellations
2.2 State of the Art of Stochastic Modeling of the Wireless Networks Topologies
2.3 Mobility Estimation in Cellular Networks by Means of Stochastic Geometry
2.3.1 Mobility State Estimation
2.3.2 System Model
2.3.3 Stochastic Geometry and User History based Mobility Estimation: STRAIGHT
2.3.4 Stretch Parameter Computation
2.3.5 Performance Evaluation
2.3.6 Lessons Learned
3 Self-Similar Geometry. The Hyperfractal Model 
3.1 Self-similar Geometry. Fractals
3.1.1 Fractal Dimension
3.1.2 The Sierpinski Triangle
3.2 Self-Similarity of Human Society Geometry. Self-Similarity of Wireless Networks
3.3 Poisson-Shots on Fractal Maps as Precursors of the Hyperfractal Model .
3.3.1 Denition of Poisson-shots on Fractal Maps
3.3.2 Towards the Hyperfractal
3.4 The Hyperfractal Model
3.4.1 Propagation Model as Feature of the Topological Model. Urban Canyon Model
3.4.2 The Support
3.4.3 The Hyperfractal Model for Mobile Users
3.4.4 Hyperfractal Model for Relays
3.5 Stochastic Geometry of the Hyperfractal Model
3.5.1 Typical Points of , r and
3.5.2 Fundamental Properties of the Typical Points
3.5.3 An Alternative Method for Computing the Number of Relays in the Map
3.6 Concluding Remarks
4 Model Fitting with Traces. Computation of Fractal Dimension 
4.1 Theoretical Foundation
4.1.1 Density-to-Length Criteria and the Computation of the Fractal Dimension
4.1.2 The Spatial Intersection Density Criterion
4.1.3 The Time Interval Intersection Criterion
4.2 Data Fitting Examples
4.3 Concluding Remarks
5 Application to Ad-Hoc Networks. End-to-End Energy versus Delay 
5.1 Introduction and Motivation
5.2 System Model
5.2.1 Preliminary Study on Connectivity with no Energy Constraints
5.3 Main Results
5.3.1 Path Cumulated Energy
5.3.2 Path Maximum Energy
5.3.3 Remarks on the Network Throughput Capacity
5.3.4 Simulations
5.4 Short Study on Load and Bottleneck
5.4.1 System Model
5.4.2 Main Results
5.5 Simulations
5.6 Concluding Remarks
6 Application to Ad-Hoc networks. Delay-Tolerant Networks 
6.1 Introduction and Motivation
6.2 System Model
6.2.1 Canyon Eect
6.2.2 Broadcast Algorithm
6.3 Main Results
6.3.1 Upper Bound
6.3.2 Lower Bound
6.3.3 Asymptotic to Poisson Uniform
6.3.4 Extension with Limited Radio Range
6.3.5 Information Teleportation
6.4 Simulations in a System Level Simulator
6.4.1 QualNet Network Simulator Conguration
6.4.2 Urban Vehicular Environment Modeling and Scenario Description
6.4.3 Validation of Upper and Lower Bounds: Constant Speed
6.4.4 Validation of Bounds Under Speed Variation
6.5 Simulations in a Self-Developped Discrete Time Event-Based Simulator .
6.5.1 Information Spread Under Hyperfractal Model and Teleportation Phenomenon
6.5.2 Validation of Upper and Lower Bounds on the Average Broadcast Time in the Entire Network
6.5.3 Validation of Bounds on Average Broadcast Time Under Speed Variation
6.6 Concluding Remarks
7 Conclusion and Future Work 
7.1 Conclusion
7.2 Future Work
7.2.1 Generalization of the Model for Nodes
7.2.2 Generalization to Poisson Points on Poisson Lines
7.2.3 Generalization to Poisson Voronoi Tessellations
7.2.4 In-Depth Percolation for a Finite Window
A Proofs 


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