A Survey of Typical-Cell Volume Distributions in Poisson--Voronoi and Poisson--Delaunay Tessellations: Analytical Theory, High-Dimensional Limits, and Wireless Applications

Abstract

Random spatial tessellations generated by point processes provide fundamental models for proximity, space partitioning, and local geometry in stochastic systems. Poisson--Voronoi and Poisson--Delaunay tessellations induced by homogeneous Poisson point processes form a canonical dual pair used in stochastic geometry, computational geometry, spatial statistics, and wireless-network analysis. Their typical-cell volume distributions provide important geometric inputs for modeling coverage, traffic load, clustering, connectivity, and other system characteristics. Despite extensive study, the literature remains analytically asymmetric. For Poisson--Voronoi cell volumes, exact integral representations exist in certain planar settings, while a recent scale--shape factorization provides an exact general-dimensional representation with conditional Gamma structure. However, the normalized shape laws and unbounded facet-count mixture remain implicit, and tractable unconditional closed-form distributions are unavailable. Practical modeling therefore relies largely on simulation, moment characterizations, and empirical approximations. By contrast, Poisson--Delaunay simplex volumes admit dimension-explicit PDFs, CDFs, and moment formulas derived through Mellin-transform analysis and Meijer's \(G\)-function representations. Motivated by this contrast, this paper surveys typical-cell volume distributions in Poisson--Voronoi and Poisson--Delaunay tessellations. We review the main analytical methods, synthesize exact and approximate results, summarize emerging high-dimensional limits, and discuss wireless-network applications, including load modeling, cooperative transmission, and three-dimensional architectures. We also identify open problems concerning unconditional Poisson--Voronoi distributions, non-Poisson spatial models, data-driven geometric inference, and dimension-aware network modeling.