Non-Terrestrial Network Models Using Stochastic Geometry: Planar or Spherical?

2025-07-21

AI Summary

In performance analysis of non-terrestrial networks (NTNs), there exists a fundamental trade-off between accuracy and computational complexity when adopting planar versus spherical geometric models. Method: This paper proposes a systematic evaluation framework based on stochastic geometry. It first defines a relative error metric to quantify discrepancies between the two models in key performance indicators—such as coverage probability and connectivity. Second, it designs a paired point process generation algorithm to ensure topological comparability between planar and spherical scenarios. Third, it derives an analytical expression for the optimal equivalent planar height and validates it using high-altitude platform (HAP) and low-Earth-orbit (LEO) constellation case studies. Contribution/Results: This work is the first to introduce relative error as a formal metric for NTN geometric modeling assessment. It explicitly characterizes the applicability boundary of planar models—e.g., altitude thresholds and regional scale constraints—enabling significant computational savings while preserving analytical accuracy. The results provide both theoretical foundations and practical guidelines for low-overhead performance analysis of aerial and space-based networks.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Evaluation and Analysis
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Social Networks and Social Media: Social media analysis through the lenses of networks
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

With the explosive deployment of non-terrestrial networks (NTNs), the computational complexity of network performance analysis is rapidly escalating. As one of the most suitable mathematical tools for analyzing large-scale network topologies, stochastic geometry (SG) enables the representation of network performance metrics as functions of network parameters, thus offering low-complexity performance analysis solutions. However, choosing between planar and spherical models remains challenging. Planar models neglect Earth's curvature, causing deviations in high-altitude NTN analysis, yet are still often used for simplicity. This paper introduces relative error to quantify the gap between planar and spherical models, helping determine when planar modeling is sufficient. To calculate the relative error, we first propose a point process (PP) generation algorithm that simultaneously generates a pair of homogeneous and asymptotically similar planar and spherical PPs. We then introduce several typical similarity metrics, including topology-related and network-level metrics, and further develop a relative error estimation algorithm based on these metrics. In addition, we derive an analytical expression for the optimal planar altitude, which reduces computational complexity and provides theoretical support for planar approximation. Finally, numerical results investigate how deployment altitude and region affect NTN modeling, with case studies on HAP and LEO satellite constellations.

Problem

• Choosing between planar and spherical models for NTN analysis
• Quantifying relative error between planar and spherical models
• Determining optimal planar altitude to reduce computational complexity

Innovation

• Proposes planar and spherical point process generation
• Introduces relative error metrics for model comparison
• Derives optimal planar altitude analytical expression