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3D Stochastic Geometry Model for Aerial Vehicle-Relayed Ground-Air-Satellite Connectivity

Yulei Wang, Yalin Liu, Yaru Fu, Yujie Qin, Zhongjie Li

TL;DR

This work addresses the challenge of enabling global-scale connectivity via AV relays by modeling AV deployments with a 3D Matérn hard-core process on spherical surfaces and deriving an analytical two-hop GASS connectivity framework. The approach combines 3D spherical geometry with stochastic geometry to obtain distributions of link distances, ASPs, and overall connectivity, accounting for interference and Nakagami-$m$ fading. Key contributions include a 3D MHCPP-based node distribution model, closed-form distance PDFs on spherical caps, and a tractable expression for the overall two-hop connectivity. Numerical validation via Monte Carlo simulations demonstrates the model's accuracy and provides deployment guidelines for AV densities and minimum inter-AV distances to optimize performance.

Abstract

Due to their flexibility, aerial vehicles (AVs), such as unmanned aerial vehicles and airships, are widely employed as relays to assist communications between massive ground users (GUs) and satellites, forming an AV-relayed ground-air-satellite solution (GASS). In GASS, the deployment of AVs is crucial to ensure overall performance from GUs to satellites. This paper develops a stochastic geometry-based analytical model for GASS under Matern hard-core point process (MHCPP) distributed AVs. The 3D distributions of AVs and GUs are modeled by considering their locations on spherical surfaces in the presence of high-altitude satellites. Accordingly, we derive an overall connectivity analytical model for GASS, which includes the average performance of AV-relayed two-hop transmissions. Extensive numerical results validate the accuracy of the connectivity model and provide essential insights for configuring AV deployments.

3D Stochastic Geometry Model for Aerial Vehicle-Relayed Ground-Air-Satellite Connectivity

TL;DR

This work addresses the challenge of enabling global-scale connectivity via AV relays by modeling AV deployments with a 3D Matérn hard-core process on spherical surfaces and deriving an analytical two-hop GASS connectivity framework. The approach combines 3D spherical geometry with stochastic geometry to obtain distributions of link distances, ASPs, and overall connectivity, accounting for interference and Nakagami- fading. Key contributions include a 3D MHCPP-based node distribution model, closed-form distance PDFs on spherical caps, and a tractable expression for the overall two-hop connectivity. Numerical validation via Monte Carlo simulations demonstrates the model's accuracy and provides deployment guidelines for AV densities and minimum inter-AV distances to optimize performance.

Abstract

Due to their flexibility, aerial vehicles (AVs), such as unmanned aerial vehicles and airships, are widely employed as relays to assist communications between massive ground users (GUs) and satellites, forming an AV-relayed ground-air-satellite solution (GASS). In GASS, the deployment of AVs is crucial to ensure overall performance from GUs to satellites. This paper develops a stochastic geometry-based analytical model for GASS under Matern hard-core point process (MHCPP) distributed AVs. The 3D distributions of AVs and GUs are modeled by considering their locations on spherical surfaces in the presence of high-altitude satellites. Accordingly, we derive an overall connectivity analytical model for GASS, which includes the average performance of AV-relayed two-hop transmissions. Extensive numerical results validate the accuracy of the connectivity model and provide essential insights for configuring AV deployments.

Paper Structure

This paper contains 13 sections, 1 theorem, 14 equations, 2 figures.

Table of Contents

  1. Introduction
  2. System Model
  3. Network Model
  4. Transmission Model
  5. Interference Model
  6. 3D Stochastic Geometry Model
  7. 3D Distribution Model
  8. $\mathcal{A}_1$ and $r_1(\texttt{t})$
  9. $\mathcal{A}_2$ and $r_2(\texttt{t})$
  10. Overall Connectivity Model
  11. Numerical Results
  12. Conclusion
  13. Acknowledgment

Key Result

Proposition 1

$\forall i\in \{1,2\}$, $\mathcal{P}_i$ is given by and $\hat{R}_i =\{ \sqrt{H_a^2 + 2 H_1 \left(R_{\mathrm{e}} + H_a\right)},(R_{\mathrm{e}} + H_s)\cos{\vartheta_s}-\sqrt{(R_{\mathrm{e}} + H_a)^2 -(R_{\mathrm{e}} + H_s)^2\sin^2{\vartheta_s}}\}$.

Figures (2)

  • Figure 1: A typical AV-relayed two-hop transmission scenario in GASS. Herein $R_{\mathrm{e}}$ is the earth radius, $H_s$ and $H_a$ are the height of satellites and AVs over the ground, respectively.
  • Figure 2: Three metrics of $\overline{\mathcal{P}}_{success}:\overline{\mathcal{P}}_1,\overline{\mathcal{P}}_2,\overline{\mathcal{P}}_{\mathrm{overall}}$ versus different parameters $\{\check{d},\lambda_a^0,\lambda_u,\theta_1,\theta_2\}$, where $H_s$ = 600Km, $H_a$ = 1Km, $\{m_1,m_2\}=\{3,3\}$, and $\{\Omega_1,\Omega_2\}=\{1,1\}$. The "solid", "dash", and "dot" lines show the theoretical results under $\check{d}=\{0,100,200\}$m respectively. The simulation results under $\check{d} = \{100,200\}$m are not shown to avoid overlapping.

Theorems & Definitions (1)

  • Proposition 1