On a Class of Dynamical Poisson-Voronoi Tessellations

TL;DR

This paper analyzes the properties of handover events in a dynamic Poisson-Voronoi tessellation model with mobile stations, focusing on the statistical characteristics of user connection changes over time.

Contribution

It introduces a novel analysis of the handover point process in a dynamic Poisson-Voronoi tessellation, including its stationarity, intensity, and joint distribution of inter-event times.

Findings

The handover point process is stationary.

Explicit expressions for the intensity and inter-event time distributions are derived.

The analysis extends to multi-speed scenarios and identifies key Markovian state variables.

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, a 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.