Random Distances Associated with Hexagons

TL;DR

This paper derives explicit probability density functions for random distances within and between regular hexagons, validated by simulations, with applications in wireless networks and other engineering fields.

Contribution

It provides the first closed-form expressions for distance distributions in regular hexagons, including moments and polynomial fits, validated through simulations and probabilistic methods.

Findings

Accurate closed-form distance distribution functions for hexagons.

Validation of functions through simulations and probabilistic sums.

Provision of statistical moments and polynomial fits for practical use.

Abstract

In this report, the explicit probability density functions of the random Euclidean distances associated with regular hexagons are given, when the two endpoints of a link are randomly distributed in the same hexagon, and two adjacent hexagons sharing a side, respectively. Simulation results show the accuracy of the obtained closed-form distance distribution functions, which are important in a wide variety of applied sciences and engineering fields. In particular, hexagons are often used in wireless communication networks such as the cellular systems. The correctness of these distance distribution functions is validated by a recursion and a probabilistic sum. The first two statistical moments of the random distances, and the polynomial fits of the density functions are also given in this report for practical uses.