[Paper Review] A Novel Analytical Model for LEO and MEO Satellite Networks based on Cox Point Processes

Chang‐Sik Choi, François Baccelli|arXiv (Cornell University)|Dec 7, 2022

Satellite Communication Systems4 citations

TL;DR

This paper proposes a novel isotropic Cox point process model that jointly generates orbits and satellites on them, accurately capturing the geometric constraint that LEO satellites operate exclusively on circular orbits. By leveraging stochastic geometry, the model enables analytical evaluation of key performance metrics—no-satellite probability and SINR coverage probability—showcasing superior accuracy over traditional Poisson or binomial point process models in representing real-world LEO constellation dynamics.

ABSTRACT

This work develops an analytical framework for downlink low Earth orbit (LEO) or medium Earth orbit (MEO) satellite communications, leveraging tools from stochastic geometry. We propose a tractable approach to the analysis of such satellite communication systems, accounting for the fact that satellites are located on circular orbits. We accurately incorporate this geometric property of LEO or MEO satellite constellations by developing a Cox point process model that jointly produces orbits and satellites on these orbits. Our work contrasts with previous modeling studies that presumed satellite locations to be entirely random, thereby overlooking the fundamental fact that satellites are jointly positioned on orbits. Employing this Cox model, we analyze the network performance experienced by users located on Earth. Specifically, we evaluate the no-satellite probability of the proposed network and the Laplace transform of the interference created by such a network. Using it, we compute its SIR (signal-to-interference) distribution, namely its coverage probability. By presenting fundamental network performance as functions of key parameters, this model allows one to assess the statistical properties of downlink LEO or MEO satellite communications and can thus be used as a system-level design tool to operate and optimize forthcoming complex LEO or MEO satellite networks.

Motivation & Objective

• To address the limitation of existing models that treat satellite locations as uniformly random points, ignoring the fundamental geometric constraint that LEO satellites must reside on orbits.
• To develop a tractable analytical framework for downlink LEO satellite networks using stochastic geometry, accounting for the spatial structure of orbital constellations.
• To evaluate network performance metrics—specifically no-satellite probability and SINR coverage probability—for a typical user on Earth.
• To provide a system-level design tool for optimizing future large-scale LEO satellite networks by expressing performance as functions of key network parameters.
• To demonstrate through moment matching that the proposed Cox model accurately replicates the local statistical properties of real or upcoming LEO constellations.

Proposed method

• The authors introduce an isotropic Cox point process that stochastically generates orbital planes first, then places satellites uniformly along each orbit, ensuring all satellites lie on circular trajectories.
• The model uses spherical geometry and angular parameters (inclination, right ascension, orbital phase) to define satellite positions in 3D space, with coordinates derived via trigonometric transformations.
• The Laplace transform of the interference power is derived by integrating over the random locations of interfering satellites, conditioned on the user's position and the orbital geometry.
• The no-satellite probability is computed by evaluating the probability that no satellite is visible within a certain elevation angle from a user on Earth’s surface.
• The coverage probability is derived by analyzing the signal-to-interference-plus-noise ratio (SINR) for the closest satellite, using the Laplace transform of interference and a path-loss model.
• The model is validated via moment matching against a real-world constellation (Starlink), showing close alignment in local statistical properties such as satellite density and angular distribution.

Experimental results

Research questions

• RQ1 How can the orbital geometry of LEO satellite constellations be accurately modeled in a stochastic framework that preserves the constraint that satellites lie only on orbits?
• RQ2 What is the no-satellite probability experienced by a typical user on Earth, and how does it depend on constellation parameters like orbital altitude and number of orbits?
• RQ3 How does the coverage probability (SINR distribution) of a downlink LEO link vary with key system parameters such as satellite density and altitude?
• RQ4 To what extent does the proposed Cox model replicate the local spatial statistics of real LEO constellations compared to traditional Poisson or binomial point process models?
• RQ5 Can the proposed model serve as a system-level design tool for optimizing LEO satellite networks by expressing performance as a function of network parameters?

Key findings

• The proposed Cox point process model successfully captures the geometric reality that LEO satellites are confined to orbits, unlike previous models that treat satellite locations as uniformly random points.
• The no-satellite probability is derived analytically and depends on orbital inclination, altitude, and the number of orbital planes, providing a direct measure of connectivity availability.
• The coverage probability is expressed as a function of satellite density, altitude, and user elevation, enabling performance prediction across different network configurations.
• Numerical results show that the Cox model produces a wide range of coverage probabilities depending on constellation geometry, in contrast to the single, fixed curve produced by binomial or Poisson models.
• Moment matching validation demonstrates that the Cox model accurately replicates the local spatial statistics of real constellations, such as Starlink, in terms of satellite density and angular distribution.
• The model enables spatially averaged performance analysis due to its isotropy, allowing the results for a typical user to be interpreted as representative of the entire network.