Kth Distance Distributions of n-Dimensional Matérn Cluster Process

Abstract

In this letter, we derive the CDF (cumulative distribution function) of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> th contact distance (CD) and nearest neighbor distance (NND) of the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> n</i> -dimensional ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -D) Matérn cluster process (MCP). We present a new approach based on relationship between the probability mass function (PMF) and the probability generating function (PGF) of the random variable (RV) denoting the number of points in a ball of arbitrary radius to derive these CDFs. 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.