Spherical Point Process with Random Heights: New Approach for Modeling and Analysis of Downlink Satellite Networks

TL;DR

This work addresses the challenge of realistically modeling large-scale LEO satellite constellations that operate across multiple altitudes. It introduces a Random Height Model (RHM) built on a Poisson point process on a sphere with IID radial height displacements, enabling analytical treatment of 3D satellite clouds. By incorporating Shadowed-Rician fading, Gamma approximation, and Alzer’s bound, the authors derive exact and tractable expressions for satellite visibility, nearest-satellite distance, interference, and downlink coverage probability, and validate these results against real-world constellations such as Starlink and OneWeb. The findings show that altitude variance critically shapes network performance and that the RHM significantly reduces model-to-real-world mismatch compared with single-altitude models, offering a practical tool for design and analysis of volumetric satellite networks.

Abstract

The Low Earth Orbit (LEO) satellite industry is undergoing rapid expansion, with operators competitively launching satellites due to the first-come, first-served principle governing orbital rights. This has led to the formation of increasingly large-scale, volumetric constellation where satellites operate across a diverse range of altitudes. To address the need for analyzing such complex networks, this paper establishes a new analytical framework for LEO constellations by leveraging a 3D Poisson point process (PPP). Specifically, we introduce a random height model (RHM) that can capture various altitude distributions by applying a random radial displacement to points generated by a homogeneous PPP on a nominal shell. Building on this, we derive an analytical expression for the downlink coverage probability. To motivate our model, we show that the altitude distributions of several leading satellite constellations, including Starlink, align with our model's assumptions. We then demonstrate through Monte Carlo simulations that the coverage probability of our RHM closely matches that of these real-world networks. Finally, we confirm the accuracy of our analytical expressions by showing their agreement with simulation results. Our work thereby provides a powerful tool for understanding and predict how the statistical distribution of satellite altitudes impacts network performance.

Figures (10)

• Figure 1: The multi-altitude distribution of four major commercial constellations: Starlink, OneWeb, Strela, and Globalstar. (a) The complete global distribution of the constellations (Source: https://satellitetracker3d.com), and (b) the resulting subset of satellites visible to a typical ground user (Source: https://N2YO.com).
• Figure 2: Illustration of satellite networks. (a) is general SPPP where the points are distributed on the surface of sphere with radius $R_{\sf S}$. (b) is the proposed RHM where each point on the surface of SPPP has random height toward radial direction.
• Figure 3: (a) Illustrates the visible region in a traditional single-altitude model, where all satellites share one altitude, forming a 2D spherical cap. (b) Shows the visible region under the RHM, where satellites at various altitudes expand the region from a 2D cap into a 3D volume. (c) Defines the geometry at altitude $h$: the spherical cap $\mathcal{A}(r,h)$ is the region within a distance $r$ of the user, and $R(h)$ is the maximum distance defining the total visible area. (d) Depicts the interference region at altitude $h$, defined as the total visible region with a central exclusion zone of radius $r$ removed.
• Figure 4: Comparison PDF of the distance to the nearest satellite $R$ where $R_{\sf S} = 550$ km and $h_{\max}$ = 100 km.
• Figure 5: Histogram of altitude of commercial satellite in operation.

Theorems & Definitions (11)

• Lemma 1
• proof
• Lemma 2
• Lemma 2
• Lemma 3
• Proposition 1
• Theorem 1
• Proposition 2
• Corollary 1
• proof