Stochastic geometry is a branch of probability theory that deals with the study of random spatial patterns. Random point patterns, termed point processes, are the most basic such objects that appear in numerous applications. After presenting a brief introduction to point processes, we will present our work on the stochastic modeling and analysis of wireless cellular networks, which departs from the conventional approaches based on deterministic models. In addition to results related to key wireless metrics, such as coverage and rate, we will also discuss new results related to Poisson point processes that were inspired by this work. Building on this discussion, we will introduce line processes, which along with point processes, allow us to model networks on the lines (such as transportation networks). After providing a brief historical perspective and the construction of line processes, we will define a doubly stochastic point process by placing an independent Poisson point process on each line of the Poisson line process, which we term the Poisson Line Cox Process. We will discuss key properties and applications of this point process to vehicular and transportation networks. We will also briefly discuss new contributions related to path distance distributions that were inspired by this work. In the last segment, we will discuss our ongoing work on developing a new mathematical framework to study landmark-based geolocation by modeling generic landmarks (such as trees and lampposts) as point processes. In addition to presenting our early results, we will also discuss possible connections of this approach to information and coding theory.

If you are interested in more details, please refer to the following monographs. They are accessible free of cost from most university campuses. Monograph [M1] summarizes our work on cellular networks and is written to be the first introduction to this area. Monograph [M2] summarizes our work on line processes and their applications to vehicular networks.

[M1] An Introduction to Cellular Network Analysis Using Stochastic Geometry: https://link.springer.com/book/10.1007/978-3-031-29743-4

[M2] Poisson Line Cox Process: Foundations and Applications to Vehicular Networks: https://link.springer.com/book/10.1007/978-3-031-02379-8