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

Abstract

In this paper 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 point process driven by the Manhattan Poisson line process. The CDF is expressed as a sum over the integer partition function <formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex>$p\!\left(k\right)$</tex></formula> , which allows us to numerically evaluate the CDF in a simple manner. The distance distributions can be used to study the k-coverage of broadcast signals in intelligent transportation systems (ITS) transmitted from a \ac{RSU} that is located at an intersection. They can also be insightful for network dimensioning in urban vehicle-to-everything (V2X) systems, because they can yield the exact distribution of network load within a cell, provided that the \ac{RSU} is located 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 the distance distribution model using the map of an urban area.