Problem
Research questions and friction points this paper is trying to address.
Poisson-Voronoi tessellation
Poisson-Delaunay tessellation
typical-cell volume distribution
stochastic geometry
wireless network applications
A Survey of Typical-Cell Volume Distributions in Poisson--Voronoi and Poisson--Delaunay Tessellations: Analytical Theory, High-Dimensional Limits, and Wireless Applications
🤖 AI Summary
This study addresses the absence of general closed-form solutions for volume distributions in Poisson-Voronoi and Delaunay tessellations, a key limitation in wireless network modeling. By integrating Mellin transforms, Meijer G-functions, and stochastic geometry theory, this work systematically compares the analytical properties of these two tessellation types. Through a synthesis of exact solutions and approximation methods, it unveils scale-shape decomposition mechanisms and high-dimensional limiting behaviors. The research establishes a comprehensive analytical framework for volume distributions, offering novel perspectives for wireless network performance analysis. Furthermore, it identifies critical theoretical challenges and future research directions, thereby advancing the theoretical foundations of stochastic geometry in communication system modeling.
📝 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Poisson-Voronoi tessellation
Poisson-Delaunay tessellation
typical-cell volume distribution
high-dimensional limits
wireless network applications
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