kth Distance Distributions of n-Dimensional Mat\'ern Cluster Process

TL;DR

This paper derives the distribution of contact and nearest neighbor distances in n-dimensional Matérn cluster processes using a novel PGF-based approach, with applications in cellular and D2D network performance analysis.

Contribution

It introduces a general method to derive distance distributions in point processes using PGF, applicable to any process with known PGFL, and validates the approach through simulations.

Findings

Derived the CDF of kth contact and nearest neighbor distances.

Validated the analysis with numerical simulations.

Provided insights into clustering effects on network performance.

Abstract

In this letter, we derive the CDF (cumulative distribution function) of $k$th contact distance (CD) and nearest neighbor distance (NND) of the $n$-dimensional ($n$-D) Mat\'ern cluster process (MCP). We present a new approach based on the probability generating function (PGF) of the random variable (RV) denoting the number of points in a ball of arbitrary radius to derive its probability mass function (PMF). The proposed method is general and can be used for any point process with known probability generating functional (PGFL). We also validate our analysis via numerical simulations and provide insights using the presented analysis. We also discuss two applications, namely: macro-diversity in cellular networks and caching in D2D networks, to study the impact of clustering on the performance.