On a Class of Dynamical Poisson-Voronoi Tessellations

TL;DR

This work develops a rigorous stochastic-geometry framework for dynamical Poisson-Voronoi tessellations generated by mobile stations moving in the plane. By constructing the head-point process on the upper-half plane and the radial-bird process, the authors derive a stationary handover point process and provide explicit formulas for handover intensity, inter-handover times, and Palm distributions of distances at handover epochs. They establish a Markovian structure for the handover dynamics in the single-speed case and extend the analysis to two-speed and multi-speed mobility, including pure and mixed handover types and their intensities. The results have direct implications for performance metrics in dynamic wireless networks, including satellite and terrestrial constellations, and lay groundwork for future extensions to spherical geometry and more complex mobility models.

Abstract

Consider a dynamical network model featuring mobile stations on the Euclidean plane. The initial locations of the stations are given by a homogeneous Poisson point process. The stations are all moving at a constant speed and in a random direction. Consider fixed users located in the Euclidean plane, which are served by the mobile stations. Each user stays connected to the nearest station at any given point of time. Since the stations are moving, an user disconnects and connects with different stations over time, by always selecting which ever station is the closest. This gives rise to a dynamical version of the Poisson-Voronoi tessellation. The focus of this paper is on the sequence of ``handover'' events of a typical user, which are the epochs when its association changes. This defines a point process on the time-axis, the ``handover point process''. We show that this point process is stationary and we determine its main properties, in particular its intensity and the joint distribution of its inter-event times. We also analyze the handover Palm distributions of several variables of practical interest. This includes the distance to the closest mobile stations and the point process of all other mobile stations at handover epochs. The analysis is conducted both in the single-speed and in the multi-speed scenarios. It leads to the identification of the three dimensional state variables that ``Markovize'' the association dynamics. The analysis is based on a specific system of non-compact particles. The motivations are in the modeling of low or medium orbit satellite wireless communication networks. The model studied here is a planar ``caricature'' of this problem, which is initially defined on the sphere.

Figures (47)

• Figure 3.1: Summary for the handover process in the single-speed case.
• Figure 3.2: Summary for the handover process in the two-speed case.
• Figure 4.2: The birds together with their heads form the radial bird particle process ${\mathcal{P} }_c\equiv {\mathcal{H} }_c$.
• Figure 4.3: The blue $\textcolor{blue}{\times}$'s forms the handover point process on the time-axis. The blue nodes ($\textcolor{blue}{\bullet}$), block nodes ($\bullet$) and red nodes ($\textcolor{red}{\bullet}$), respectively, on individual vertical blue, black and red lines, respectively, forms the point process corresponding to the distance of all the mobile base stations at handover times, at a time corresponding to a head and at a typical time, respectively.
• Figure 4.4: Scenario in Observation \ref{['observation:in-out']}. All birds with head point inside and outside, respectively, of the half-ball of height $h'$, intersect the vertical line below and above the level $h'$, respectively.

Theorems & Definitions (168)

• Remark 4.1
• Definition 4.2: Head point process
• Lemma 4.3: Mapping Lemma
• proof : Proof Lemma \ref{['lemma:tips_density']}
• Remark 4.4
• Remark 4.5
• Remark 4.7: Distance function as a branch of a hyperbola on $\mathbb H^+$
• Definition 4.8: Radial bird closed set
• Remark 4.9
• Definition 4.10: Radial bird Particle Process (RBPP)