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A Spherical Stochastic Geometry Framework for Patrol-Based HAPs Network: Coverage and Energy Efficiency Analysis

This paper proposes a spherical stochastic geometry framework using two Cox process models to analyze the coverage probability and energy efficiency of high-altitude platform station networks with cyclic patrol trajectories, ultimately deriving an analytical condition for the energy-optimal patrol radius that balances communication performance against propulsion costs.

Original authors: Mohammad Taha Shah, Mohamed-Slim Alouini

Published 2026-06-05

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Mohammad Taha Shah, Mohamed-Slim Alouini

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the sky above our cities and oceans is about to get very busy. Instead of just satellites zooming high above or cell towers stuck on the ground, we are introducing High-Altitude Platform Stations (HAPs). Think of these as giant, solar-powered "flying cell towers" that hover in the stratosphere (about 12 to 30 miles up).

Unlike satellites that fly in fixed, long loops around the Earth, or cell towers that never move, these HAPs have a unique job: they patrol. They fly in small, circular patterns over specific neighborhoods or islands, constantly circling to provide internet and communication services.

This paper is a mathematical guidebook for designing a network of these flying patrol stations. The authors, Mohammad Taha Shah and Mohamed-Slim Alouini, created a new way to predict how well this network will work and how much energy it will use.

Here is the breakdown of their work using simple analogies:

1. The Two Ways to Organize the Patrols

The paper proposes two different "rules" for how these flying stations are arranged. Imagine you are organizing a security team patrolling a park.

Model A: The "Crowd" Approach (SCR-PCP)
Imagine a park where security guards are assigned to patrol paths, but the number of guards on each path changes randomly. Sometimes there are 2 guards, sometimes 5, sometimes 0, depending on who is available or how busy the park is.
- In the paper: This is the Poisson Cox Process. The number of HAPs on any given patrol ring is random and fluctuates. This models a flexible, dynamic network where the fleet size changes based on demand or maintenance.
Model B: The "Fixed Squad" Approach (SCR-BCP)
Now imagine a different park where every patrol path has a strict rule: there must be exactly 3 guards on that path, no more, no less. They are spread out evenly.
- In the paper: This is the Binomial Cox Process. Each patrol ring has a fixed number of HAPs ( $n_H$ ). This models a carefully planned, "engineered" fleet where you know exactly how many planes are in the sky at all times.

2. The Challenge: The Earth is Round

Most math models for networks assume the world is flat (like a piece of paper). But the Earth is a sphere.

The Analogy: If you draw a circle on a flat table, it's easy. If you draw a circle on a basketball, the geometry gets tricky. The paper insists on doing the math on a sphere because at high altitudes, the curvature of the Earth matters. Ignoring this would be like trying to navigate a ship using a flat map of a round ocean—it leads to errors.

3. The "Traffic Jam" Problem (Interference)

When you have many HAPs flying, they all transmit radio signals. If two HAPs are too close or too loud, their signals crash into each other, creating "noise" (interference) that ruins the internet connection.

The Paper's Insight:
- In the "Crowd" model, the interference is unpredictable. You might get lucky with no neighbors, or unlucky with a traffic jam.
- In the "Fixed Squad" model, the interference is predictable but constant. If you have 10 planes on a ring, you always have 9 neighbors interfering with your signal. The paper found that in this fixed model, adding more planes to the same ring actually hurts performance significantly because they are all stuck in the same small circle.

4. The "Fuel vs. Speed" Trade-off (Energy Efficiency)

This is the most practical part of the paper. Flying in a circle is hard work for an airplane.

The Analogy: Think of a car taking a sharp turn. The tighter the turn, the more you have to lean (bank) and the more fuel you burn to stay on the path.
- Small Patrol Ring: The HAP has to turn sharply. It burns a lot of energy (high propulsion cost), but it stays close to the people it serves (good signal).
- Large Patrol Ring: The HAP turns gently. It saves energy, but it might fly further away from the people it serves (weaker signal).

The authors created a metric called Coverage Energy Efficiency (CEE). They asked: "What is the perfect size for the patrol circle?"

The Answer: It depends on how fast the plane is flying and how many other planes are around.
- If the plane flies fast, it needs a wider circle to avoid burning too much fuel turning sharply.
- If there are many planes (high density), the circle needs to be smaller to keep the signal strong, even if it costs more fuel.

Summary of Findings

Geometry Matters: You cannot treat these flying networks like ground cell towers. You must account for the round Earth and the circular patrol paths.
Planning vs. Chaos: If you have a fixed number of planes (Fixed Squad), adding more planes to the same ring makes the internet worse due to interference. If you have a random crowd (Crowd model), the network is more flexible.
The Sweet Spot: There is a "Goldilocks" patrol radius. It's not too small (saves fuel but weakens signal) and not too big (strong signal but wastes fuel). The perfect size changes based on how fast the planes fly and how many of them are in the sky.

In short: The paper provides the mathematical "recipe" to build a network of flying internet stations that balances getting a strong signal with saving fuel, ensuring that these future sky-towers are both effective and energy-efficient.

Problem Statement

High-altitude platform stations (HAPs) are emerging as a critical component of non-terrestrial networks (NTNs), offering wide coverage and line-of-sight conditions with lower latency than satellites. However, existing stochastic geometry (SG) frameworks for network analysis fail to capture three defining characteristics of realistic HAP deployments simultaneously:

Geographically anchored service regions: HAPs loiter over specific areas (e.g., cities, maritime corridors) rather than traversing global orbits like satellites.
Bounded patrol trajectories: Platforms execute cyclic, local patrol paths (small circles) rather than static point placements or great-circle orbits.
Spherical Earth geometry: At stratospheric altitudes (20–50 km), Earth's curvature and horizon effects are non-negligible, rendering planar approximations inaccurate.

Current models either simplify the network to static point processes (ignoring patrol dynamics), focus on single-node physics (ignoring network interference), or apply satellite-specific orbit models (ignoring local anchoring). There is a lack of a tractable, network-scale framework that integrates spherical geometry with anchor-based cyclic patrol motion.

Methodology

The authors propose a Spherical Stochastic Geometry framework based on two novel Small-Circle Ring Cox Process models defined on a spherical shell of radius $R_H = R_\oplus + H$ (Earth radius plus altitude).

1. System Model:

Anchors: Service regions are modeled as a homogeneous Poisson Point Process (PPP) $\Phi_u$ on the spherical shell.
Patrol Rings: Each anchor $u$ generates a deterministic patrol ring $C(u, \gamma)$ in its local tangent plane with radius $a = R_H \tan \gamma$ .
Two Deployment Regimes:
- SCR-PCP (Poisson Cox Process): Platforms on each ring form a 1D PPP with linear intensity $\nu$ . This models intensity-driven, dynamic fleets where the number of active platforms fluctuates.
- SCR-BCP (Binomial Cox Process): Each ring contains a fixed number $n_H$ of platforms uniformly distributed along the trajectory. This models dimensioned, finite fleets with explicit redundancy or capacity planning.

2. Analytical Derivations:

Spatial Statistics: The authors prove the isotropy of both models and derive exact distributions for:
- Distance to the nearest anchor ( $D_u$ ).
- Distance to the nearest patrol ring ( $D_R$ ).
- Distance to the nearest HAP ( $d_0$ ).
- For SCR-BCP, a joint distribution of the serving distance and the serving anchor's central angle is derived to handle finite-population dependencies.
Coverage Probability: Under a nearest-HAP association rule, the Signal-to-Interference Ratio (SIR) is analyzed. Aggregate interference is decomposed into same-ring and other-ring components. The authors derive conditional Laplace transforms for these interference terms, accounting for the distinct intra-ring structures (Poisson vs. Binomial).
Energy Efficiency: A steady-circular-flight (SCF) propulsion model is integrated. The Coverage Energy Efficiency (CEE) metric is defined as the ratio of coverage probability to the propulsion power required for circular flight. This allows for the derivation of an energy-optimal patrol radius.

Key Contributions

Novel Stochastic Models: Introduction of SCR-PCP and SCR-BCP, which are the first Cox process models to rigorously couple global spherical topology with local, anchor-based cyclic patrol structures.
Exact Spatial Statistics: Derivation of geometry-exact distance distributions and joint statistics without resorting to planar or small-angle approximations.
Coverage Analysis: Closed-form expressions for coverage probability that explicitly handle the decomposition of interference into same-ring and inter-ring components, revealing fundamental differences between intensity-driven and finite-fleet deployments.
Energy-Aware Optimization: Integration of aerodynamic flight mechanics with communication performance to establish analytical conditions for the energy-optimal patrol radius, balancing coverage gains against propulsion costs.

Key Results

Model Distinction:
- SCR-PCP: Performance is highly sensitive to patrol radius and platform density. In sparse deployments, a non-zero optimal patrol radius exists to balance the probability of finding a serving HAP against interference. In dense deployments, smaller radii are preferred to minimize interference.
- SCR-BCP: Performance is primarily governed by the fixed per-ring population ( $n_H$ ). Increasing $n_H$ significantly degrades coverage due to strong same-ring interference. The coverage probability is relatively insensitive to patrol radius compared to SCR-PCP.
Energy Efficiency:
- The optimal patrol radius is a function of cruising velocity, platform density, and fleet size.
- Higher cruising velocities increase the propulsion penalty for tight turns, shifting the optimal patrol radius to larger values.
- In SCR-BCP, for small fleets, the optimal radius is large (to minimize propulsion cost); for large fleets, the optimal radius shrinks to mitigate same-ring interference.
Validation: Analytical results for distance distributions and coverage probabilities were validated against 3D Monte Carlo simulations, showing indistinguishable curves across various regimes.

Significance and Claims

The paper claims to fill a critical gap in NTN analysis by providing the first tractable framework that simultaneously respects spherical geometry, anchor-based topology, and cyclic patrol dynamics.

Theoretical Impact: The work establishes that patrol geometry plays a fundamentally different role depending on the deployment regime. For intensity-driven fleets (SCR-PCP), patrol radius is a key optimization variable for both coverage and energy. For dimensioned fleets (SCR-BCP), the finite population constraint dominates performance, making the system less sensitive to geometric variations once the fleet size is fixed.
Practical Implication: The derived analytical conditions for energy-optimal patrol radius provide a design guideline for operators. The paper asserts that patrol geometry, platform density, and cruising velocity must be jointly optimized to achieve energy-efficient HAP operations, rather than optimizing communication and propulsion independently.
Future Direction: The authors suggest that this framework can serve as a foundation for integrating patrol-based HAPs into multi-layer terrestrial-aerial-satellite architectures (SAGIN), enabling joint optimization of mobility and spectrum coexistence.

The paper maintains a modest tone, noting that while it offers a geometry-exact framework, it assumes idealized conditions (e.g., steady circular flight, Rayleigh fading) and does not account for complex atmospheric propagation or dynamic handover protocols, which remain areas for future work.