AI 中文总结
本文综述泊松-沃罗诺伊与泊松-德劳内镶嵌的典型单元体积分布，梳理分析方法、相关结果与高维极限，探讨其在无线网络的应用并指出开放问题。
AI 中文摘要
由点过程生成的随机空间镶嵌为随机系统中的邻近性、空间划分及局部几何提供了基础模型。齐次泊松点过程诱导的泊松-沃罗诺伊（Poisson--Voronoi）与泊松-德劳内（Poisson--Delaunay）镶嵌构成了随机几何、计算几何、空间统计及无线网络分析中常用的正则对偶对，其典型单元体积分布是建模覆盖范围、业务负载、聚类、连通性及其他系统特性的重要几何输入。尽管已有大量研究，但相关文献仍存在分析不对称性：平面场景中泊松-沃罗诺伊单元体积存在精确积分表示，近期的尺度-形状分解则提供了含条件伽马结构的通用维精确表示，但归一化形状律与无界面计数混合仍为隐式，且无易于处理的无条件闭式分布，实际建模因此主要依赖模拟、矩表征及经验近似；相比之下，泊松-德劳内单纯形体积则存在通过梅林变换分析与梅耶G函数表示推导的显式维概率密度函数（PDF）、累积分布函数（CDF）及矩公式。受该对比启发，本文对泊松-沃罗诺伊与泊松-德劳内镶嵌中的典型单元体积分布展开综述，回顾主要分析方法，综合精确与近似结果，总结新兴的高维极限，并讨论其在无线网络中的应用，包括负载建模、协作传输及三维架构；同时指出关于无条件泊松-沃罗诺伊分布、非泊松空间模型、数据驱动几何推断及感知维度的网络建模等开放问题。
英文摘要
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.
Comments23 pages, 3 figures, 3 tables, Accepted for publication in IEEE Access
