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

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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 time 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]. In this preliminary study, we address the problem of UE mobility estimation for the enhancement of HO procedure in mobile cellular networks. We propose a novel scheme to estimate the UE mobility class based on the behavior of the UEs in the network, the correlation between the mobility and the characteristics of the environment, and the UE history information available at the eNBs. We use stochastic geometry for network modeling.
The method is following the Long Term Evolution (LTE) Mobility State Estimation (MSE) legacy model and has negligible additional computational complexity to the eNB and the UE.

Stochastic Geometry and User History based Mobility Estima- tion: 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.
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. Figure 2.6 depicts the concept of equivalent cell radius. That is, a UE that has an erratic path through a cell i of radius R is similar to that the UE has a straight path through the cell with radius Req. In short, an increase in a UE’s drift can be considered as contributing to an increase of the equivalent cell radius.
To be precise, let be the « stretch » parameter dened as = Req R , where R is the actual radius of a cell and Req is the equivalent cell radius of the cell. It expresses a dilatation or stretch of the actual cell radius induced by the typical trajectories of the users in the cell.

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.
From Plato onwards, there has been an eort into demonstrating cities as examples of Euclidean geometry and proofs of the man’s power over nature. This has led to the separation of art from science as if the human society development is purely articial. However this simplistic point of view has always been contradicted to some extent and more strongly in the last 50 years. When it has been realized that the physical form of cities is generated by social and economic constraint, the idea that the organically growth of cities is optimal has received more credibility. The view about the shape and form of cities has become that their irregularity and messiness is simply a supercial manifestation of a deeper order. In his remarkable work, [1, 2], Michael Batty argues that \cities are fractal in form » and that much of the pre-existing urban theory is a theory of the fractal city. As the organization of economic activities in cities displays self-similarity properties, it comes as a natural consequence that urban road networks, as location of human activities, inherits self-similarity. This has been shown in [73]. Furthermore, it is immediate to notice that vehicular trac inherits the same self-similarity. The self-similarity of urban trac in time has been proven by data tting in [74, 75]. These works support the adequacy of self-similar processes in modeling the vehicular trac time series over various time scales.
Furthermore, [76] shows that the requests for cars in platforms such as Uber display a self-similar pattern. Passing now to networks, one does not lack references to the self-similarity of trac in networks [77, 78]. In [79], the authors use self-similarity to simulate the network trac and in [80] the authors show that the trac in Ethernet is self-similar. [81] proposes a multi-fractal model for high-speed networks and the authors of [82] propose self-similarity for the simulation of IP trac. In which regards wireless networks, fractal geometry has not been exploited extensively. In [83], the authors use the properties of fractals in order to miniaturize an antenna. Yet the subject where the self-similarity has been used the most intensively in the area of wireless networks is for the modeling of the coverage and coverage border [84, 85]. In [86], the authors claim that the placement of base-station is self-similar, which seems rather intuitive as the base-stations, themselves, follow social agglomerations where the cellular trac is more intense.
A pioneering work has been done by using Poisson shots on Cantor maps for representing sensor networks. This model is, in fact, the precursor of the hyperfractal model and will further be debated in more details due to the importance it for the topic of this thesis.

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Poisson-Shots on Fractal Maps as Precursors of the Hyperfractal Model

The precursor model of the hyperfractal model is the model called Poisson shots on fractal maps, and more precisely, the Poisson shots on Cantor maps. This model was generated by the motivation to model a network of transmitters and receivers in a setup that resembles the Virtual Multiple-Input-Single-Output (Virtual-MISO) communication scenario yet other communication scenarios can be envisaged. We shall brie y remind this model that was developed in [87] due to the importance it had in the process of generation of the hyperfractal model.

Denition of Poisson-shots on Fractal Maps

The Cantor maps are the support of the population of transmitters and receivers in the following model from [88]. The sensor networks inherit the property of self-similarity from the Cantor map. As the communication scenario is not relevant in this part of the manuscript, we shall not elaborate on this but only focus on the mathematical model that served as inspiration for the main contribution of this thesis.

Propagation Model as Feature of the Topological Model. Urban Canyon Model

The main goal is to develop a model for communications of vehicles in urban settings. As cars move on the streets of the city, the consequences of the radio propagation in this environment cannot be overlooked for an accurate modeling. Buildings are made of concrete, glass and steel which generate a formidable obstacle for radio wave propagation.
This is the so-called canyon eect (see Figure 3.6) that implies that the signal emitted by a mobile node propagates only on the axis where it stands on [89, 90] and cannot penetrate the barrier created by the building walls.
This eect is further exacerbated when deploying millimeter wave (mmWave) technology. Measurements for mmWave have shown that the buildings materials (tinted glass) are highly attenuative and very re ective [91]. Communication in millimeter wave is directive and possible with good quality when vehicles are in line-of-sight [92, 93]. This features and the urban architecture characteristics lead to the existence of dead zones of coverage and decrease drastically the possibility of routing a packet through intersections. We therefore decide that the canyon eect is a fundamental characteristic of the communication scenarios addressed by the hyperfractal model and we include it in the design of the model, from the very beginning.

Table of contents :

List of Figures
List of Tables
Acknowledgements
Abstract
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 
Bibliography

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