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

TL;DR

This paper derives the exact distribution of path distances from a typical intersection to the k-th nearest node in a Manhattan Poisson line Cox process, aiding network planning in intelligent transport and V2X systems.

Contribution

It provides a novel exact cumulative distribution function for path distances in a Cox process driven by Manhattan lines, useful for network analysis and planning.

Findings

Exact CDF of path distance derived and expressed via integer partition function

Numerical evaluation method for practical values of k presented

Application demonstrated using an urban map for real-world relevance

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 $p\!\left(k\right)$, 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 \ac{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 \ac{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. We corroborate the applicability of our distance distribution model using the map of an urban area.