On the $k$ Nearest-Neighbor Path Distance from the Typical Intersection in the Manhattan Poisson Line Cox Process

Abstract

In this paper, we consider a Cox point process driven by the Manhattan Poisson line process. We calculate the exact cumulative distribution function (CDF) of the path distance (L1 norm) between a randomly selected intersection and the k-th nearest node of the Cox process. The CDF is expressed as a sum over the integer partition function p(k), which allows us to numerically evaluate the CDF in a simple manner for practical values of k. These distance distributions can be used to study the k-coverage of broadcast signals transmitted from a road side unit (RSU) located at an intersection in intelligent transport systems (ITS). Also, they can be insightful for network dimensioning in vehicle-to-everything (V2X) systems, because they can yield the exact distribution of network load within a cell, provided that the RSU is placed at an intersection. Finally, they can find useful applications in other branches of science like spatial databases, emergency response planning, and districting.