The $(n,k)$th Nearest Neighbors in Poisson Point Processes and Applications to Cell-Free Networks

This paper investigates the properties related to nearest neighbors of two independent homogeneous Poisson point processes (PPP) in the same Euclidean space <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathbb {R}^{d}$</tex-math></inline-formula>. The presented concepts and results help formalize fundamental aspects of cell-free networks related to the spatial distribution of transmitters and receivers. This work defines and characterizes the neighborhood degree <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(n,k)$</tex-math></inline-formula> of points in different PPPs, i.e., <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(n,k)$</tex-math></inline-formula>th neighbors means that the first point is the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula>th nearest point of its PPP to the second point, which is in turn the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula>th nearest point of its PPP to the first point. Several bounds and approximations related to the distribution of links between user equipments and access points are derived. The distribution of the link distance is obtained in closed form and includes, as a special case, the classical distribution for the nearest neighbors in a PPP.