Random Distances Associated with Hexagons

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. Index Terms Random distances; distance distribution functions; regular hexagons I. THE PROBLEM Define a “unit hexagon” as the regular hexagon with a side length of 1. Picking two points uniformly at random from the interior of a unit hexagon, or between two adjacent unit hexagons sharing a side, the goal is to obtain the explicit probability density function (PDF) of the random distances between these two endpoints. II. DISTANCE DISTRIBUTIONS ASSOCIATED WITH REGULAR HEXAGONS A. Random Distances within a Unit Hexagon A unit hexagon can be decomposed into three congruent rhombuses, as shown in Fig. 1. Define each one of these rhombuses as a “unit rhombus”, i.e., with an acute angle of π 3 and a