On Exact Distribution of Poisson-Voronoi Area in $K$-tier HetNets with Generalized Association Rule

TL;DR

This paper derives the exact distribution and moments of Voronoi areas in multi-tier Poisson networks under a generalized user association rule, providing closed-form expressions and numerical evaluations for different network scenarios.

Contribution

It introduces a unified framework to compute the exact distribution of Voronoi areas in K-tier HetNets with a generalized association rule, extending prior models.

Findings

Exact moments of Voronoi areas are derived using Robbins' theorem.

Closed-form expressions for mean and higher-order moments are obtained.

Numerical results validate the accuracy of the exact expressions.

Abstract

This letter characterizes the exact distribution function of a typical Voronoi area in a $K$-tier Poisson network. The users obey a generalized association (GA) rule, which is a superset of nearest base station association and maximum received power based association (with arbitrary fading) rules that are commonly adopted in the literature. Combining the Robbins' theorem and the probability generating functional of a Poisson point process, we obtain the exact moments of a typical $k$-th tier Voronoi area, $k \in \{1,...,K\}$ under the GA rule. We apply this result in several special cases. For example, we prove that in multi-tier networks with the GA rule, the mean of $k$-th tier Voronoi area can exactly be expressed in a closed-form. We also obtain simplified expressions of its higher-order moments for both average and instantaneous received power based user association. In single-tier networks with exponential fading, the later association rule provides closed-form expression of the second-order moment of a typical Voronoi area. We numerically evaluate this exact expression and compare it with an approximated result.