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

Kim, Seyong, Jinseok Choi+3 more|arXiv (Cornell University)|Jul 7, 2023

Satellite Communication SystemsEngineering3 citations

TL;DR

This paper proposes a unified stochastic geometry model for satellite-terrestrial integrated networks (STINs) using a spherical point process with random heights to jointly analyze satellite and terrestrial base stations. By modeling satellite and ground stations as marked Poisson point processes on concentric spheres, the authors derive closed-form expressions for rate coverage probability, demonstrating STINs' dual benefits: coverage extension in rural areas and data offloading in urban hotspots.

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.

Motivation & Objective

• To address the lack of a tractable, unified analytical framework for modeling satellite-terrestrial integrated networks (STINs).
• To enable performance analysis of STINs by jointly capturing satellite and terrestrial network deployments in a single stochastic geometry model.
• To evaluate the two primary benefits of STINs: coverage extension in remote rural areas and data offloading in dense urban environments.
• To derive closed-form expressions for rate coverage probability under realistic channel conditions, including shadowed-Rician fading and path loss.
• To provide quantitative insights into system design trade-offs based on key parameters such as transmit power, density, height distributions, and biasing factors.

Proposed method

• Models satellite and terrestrial base stations as marked Poisson point processes on concentric spheres, with random heights as marks to ensure visibility to the typical user.
• Applies stochastic geometry tools to derive the signal-to-interference-plus-noise ratio (SINR) distribution by conditioning on the association of the typical user with either satellite or terrestrial network.
• Uses conditional coverage probability expressions based on the Laplace transform of interference, incorporating path loss, shadowed-Rician fading, and power biasing.
• Derives the rate coverage probability by integrating over the distribution of the nearest base station distance under both satellite and terrestrial associations.
• Employs the generalized Laplace transform technique to handle the complex interference from both satellite and terrestrial tiers, enabling analytical tractability.
• Introduces a normalized SINR framework to decouple the effects of path loss and transmit power, simplifying the derivation of coverage expressions.

Experimental results

Research questions

• RQ1 How can a unified analytical model be constructed to jointly represent satellite and terrestrial networks in a single stochastic geometry framework?
• RQ2 What is the rate coverage probability of STINs under realistic channel conditions, including shadowed-Rician fading and varying path loss exponents?
• RQ3 To what extent can STINs extend coverage in remote rural areas compared to terrestrial-only networks?
• RQ4 How effectively can STINs offload traffic in high-density urban areas, and what system parameters maximize this benefit?
• RQ5 How do transmit power, base station density, height distributions, and biasing factors influence the overall performance of STINs?

Key findings

• The proposed spherical point process with random heights enables a tractable, unified model that captures both satellite and terrestrial network geometries simultaneously.
• The derived rate coverage probability expressions are analytically tractable and validated through comparison with realistic Starlink constellation simulations.
• STINs significantly extend coverage in rural areas due to the uniform spatial distribution of satellites, even at low terrestrial base station densities.
• In urban areas, STINs achieve substantial data offloading gains, improving rate coverage by balancing load between satellite and terrestrial tiers.
• Optimal performance is achieved with appropriate biasing factors and power control; excessive biasing can degrade performance due to increased interference.
• The model reveals that path loss exponent and height distributions of satellites and terrestrial BSs are critical design parameters affecting coverage and rate performance.