Probabilistic modeling and analysis of spatial telecommunication systems have never been more important than they are today. In particular, it is an essential research area for designing and developing next-generation communication networks that are based on multihop message transmission technology. These lecture notes provide valuable insights into the underlying mathematical discipline, stochastic geometry, introducing the theory, mathematical models and basic concepts. They also discuss the latest applications of the theory to telecommunication systems.
The text covers several of the most fundamental aspects of quality of service: connectivity, coverage, interference, random environments, and propagation of malware. It especially highlights two important limiting scenarios of large spatial systems: the high-density limit and the ergodic limit. The book also features an analysis of extreme events and their probabilities based on the theory of large deviations. Lastly, it includes a large number of exercises offering ample opportunities for independent self-study.
Author(s): B. Jahnel, W.König
Series: Compact Textbooks in Mathematics
Edition: 1
Publisher: Birkhäuser
Year: 2020
Language: English
Pages: 200
Tags: Point Processes, Palm, Cox, Marked, Percolations, Large Deviations
Preface
Acknowledgments
Contents
1 Introduction
2 Device Locations: Point Processes
2.1 Point Processes
2.2 Poisson Point Processes
2.3 Campbell Moment Formulas
2.4 Marked Poisson Point Processes
2.5 Palm Calculus
3 Random Environments: Cox Point Processes
3.1 Cox Point Processes
3.2 Random Tessellations
3.3 Inversion Formulas of Palm Calculus
4 Coverage and Connectivity: Boolean Models
4.1 Boolean Models
4.2 Coverage Properties
4.3 Long-Range Connectivity in Homogeneous Boolean Models
4.4 Intermezzo: Phase Transition in Discrete Percolation
4.5 Proof of Phase Transition in Continuum Percolation
4.6 Asymptotic Connectivity and Percolation
4.7 Bounded-Hop Percolation and Coverage
4.8 Percolation for Cox Point Processes
5 Interference: Signal-to-Interference Ratios
5.1 Describing Interference
5.2 Signal-to-Interference Ratios
5.3 SINR Percolation
5.4 Medium Access Control
6 Large Systems: Convergence of Point Processes
6.1 Convergence of Random Point Measures
6.2 High-Density Limits
6.3 Intermezzo: Stationary Processes and Birkhoff's Theorem
6.4 Ergodic Theorem for Point Processes
6.5 Applications in Telecommunications
6.6 Ergodic Theorem for Marked Point Processes
6.7 Empirical Stationary and Individual Fields
6.8 Describing Connectivity with Empirical Individual Fields
7 Events of Bad Quality of Service: Large Deviations
7.1 Introductory Example
7.2 Principles of Large Deviations
7.3 LDP in the High-Density Setting
7.4 Empirical Degree Measures in a Highly-Dense System
7.5 Connectivity to a Base Station in Dense Systems
7.6 High Exceedances of Interference
7.7 LDP in the Ergodic Limit
7.8 Connectivity to Receivers in Large Boxes
8 Random Malware Propagation: Interacting Markov Processes
8.1 Markov Chains in Continuous Time
8.2 Richardson Models
8.3 Contact Processes
8.4 Chase-Escape Models
Bibliography
Index
