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 relations, space partitioning, and local geometry in stochastic systems. Among them, Poisson–Voronoi and Poisson–Delaunay tessellations induced by homogeneous Poisson point processes form a canonical dual pair widely used in stochastic geometry, computational geometry, spatial statistics, and wireless-network analysis. In many applications, the typical-cell volume distribution provides an important geometric input for modeling coverage regions, traffic load, clustering structure, connectivity, and other system-level characteristics. Despite extensive study, the literature remains fragmented and analytically asymmetric. For Poisson–Voronoi cell volumes, exact integral representations are available 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 still unavailable. Practical modeling therefore continues to rely largely on simulations, moment-based characterizations, and empirical approximations. By contrast, Poisson–Delaunay simplex volumes admit directly evaluable, dimension-explicit probability density functions, cumulative distribution functions, and moment formulas derived through Mellin-transform analysis and Meijer’s $G$ -function representations. Motivated by this contrast, this paper presents a structured survey of typical-cell volume distributions in Poisson–Voronoi and Poisson–Delaunay tessellations. We review the principal analytical methodologies, synthesize established exact and approximate results, summarize emerging high-dimensional limit behavior, and discuss representative wireless-network applications, including load modeling, cooperative transmission, and three-dimensional network architectures. We also identify open problems involving unconditional Poisson–Voronoi distributions, non-Poisson spatial models, data-driven geometric inference, and dimension-aware network modeling.