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Stochastic Geometry of Cylinders: Characterizing Inter-Nodal Distances for 3D UAV Networks

Yunfeng Jiang, Zhiming Huang, Jianping Pan

TL;DR

This paper addresses the challenge of obtaining an exact analytical characterization of coverage probability in finite 3D networks where nodes are uniformly distributed in a cylinder. It introduces a four-step geometric–probabilistic framework that combines an exact distance distribution between cylinder-bound points, serving-distance via order statistics, interference modeling through conditional Laplace transforms, and a tower-law integration to yield $P_c$ without relying on asymptotic or PPP assumptions. The authors derive the exact distance PDF for two random points in a finite cylinder and assemble a closed-form integral that captures finite-node and boundary effects, with extensive Monte Carlo validation showing near-perfect accuracy and clear gains over PPP models. The results generalize to any confined 3D wireless system and have practical impact for UAV, underwater, and robotic networks by enabling boundary-aware reliability analyses and design insights.

Abstract

The analytical characterization of coverage probability in finite three-dimensional wireless networks has long remained an open problem, hindered by the loss of spatial independence in finite-node settings and the coupling between link distances and interference in bounded geometries. This paper closes this gap by presenting the first exact analytical framework for coverage probability in finite 3D networks modeled by a binomial point process within a cylindrical region. To bypass the intractability that has long hindered such analyses, we leverage the independence structure, convolution geometry, and derivative properties of Laplace transforms, yielding a formulation that is both mathematically exact and computationally efficient. Extensive Monte Carlo simulations verify the analysis and demonstrate significant accuracy gains over conventional Poisson-based models. The results generalize to any confined 3D wireless system, including aerial, underwater, and robotic networks.

Stochastic Geometry of Cylinders: Characterizing Inter-Nodal Distances for 3D UAV Networks

TL;DR

This paper addresses the challenge of obtaining an exact analytical characterization of coverage probability in finite 3D networks where nodes are uniformly distributed in a cylinder. It introduces a four-step geometric–probabilistic framework that combines an exact distance distribution between cylinder-bound points, serving-distance via order statistics, interference modeling through conditional Laplace transforms, and a tower-law integration to yield without relying on asymptotic or PPP assumptions. The authors derive the exact distance PDF for two random points in a finite cylinder and assemble a closed-form integral that captures finite-node and boundary effects, with extensive Monte Carlo validation showing near-perfect accuracy and clear gains over PPP models. The results generalize to any confined 3D wireless system and have practical impact for UAV, underwater, and robotic networks by enabling boundary-aware reliability analyses and design insights.

Abstract

The analytical characterization of coverage probability in finite three-dimensional wireless networks has long remained an open problem, hindered by the loss of spatial independence in finite-node settings and the coupling between link distances and interference in bounded geometries. This paper closes this gap by presenting the first exact analytical framework for coverage probability in finite 3D networks modeled by a binomial point process within a cylindrical region. To bypass the intractability that has long hindered such analyses, we leverage the independence structure, convolution geometry, and derivative properties of Laplace transforms, yielding a formulation that is both mathematically exact and computationally efficient. Extensive Monte Carlo simulations verify the analysis and demonstrate significant accuracy gains over conventional Poisson-based models. The results generalize to any confined 3D wireless system, including aerial, underwater, and robotic networks.

Paper Structure

This paper contains 15 sections, 8 theorems, 31 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Related works
  3. System Model and Problem Formulation
  4. System Model
  5. Problem Formulation
  6. Coverage Probability Analysis
  7. Distance Distribution between Any Two Nodes
  8. Distance Distribution from the Serving Node
  9. Modeling the Aggregated Interference
  10. Deriving the Coverage Probability
  11. Simulation Results
  12. Validation of the Distance Distribution
  13. Coverage Probability Analysis
  14. Comparative Analysis with the PPP Model
  15. CONCLUSION

Key Result

Theorem 1

Consider two nodes drawn independently and uniformly from a cylinder with base radius $R$ and height $H$. Let $L$ denote their Euclidean distance, and let $l$ be a realization of $L$. Denote by $D_{xy}$ and $D_z$ their planar and vertical separations, respectively, and define the squared variables $ where $f_{XY}(r)$ and $f_Z(z)$ represent the PDFs of the squared horizontal and vertical distance c

Figures (5)

  • Figure 1: A UAV network in a cylindrical volume $\mathcal{C}$ of height $H$ and radius $R$.
  • Figure 2: Validation of the theoretical PDF of the inter-node distance against Monte Carlo simulation results for two representative cylindrical geometries. The analytical curves show near-perfect agreement with simulation data.
  • Figure 3: Coverage probability as a function of the number of UAVs for cylinders with varying heights.
  • Figure 4: The impact of the Nakagami-$m$ fading parameter on network coverage probability.
  • Figure 5: Comparison of coverage probability: Proposed BPP model vs. PPP approximation vs. Monte Carlo simulation (ground truth).

Theorems & Definitions (13)

  • Theorem 1
  • proof : Proof Sketch
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 2
  • proof
  • Lemma 3: Theorem 2.3.13 in 8
  • Lemma 4: Theorem 2.2.2 in 8
  • ...and 3 more