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

This paper develops a stochastic-geometry framework for high-altitude platform station (HAPs) networks in which platforms execute cyclic patrol trajectories anchored to designated service regions. We introduce two small-circle ring Cox process models on the spherical Earth. In the small-circle ring Poisson Cox process (SCR-PCP), platforms form one-dimensional Poisson point processes on localized patrol rings, whereas in the small-circle ring binomial Cox process (SCR-BCP), each ring contains a fixed number of uniformly distributed platforms. We establish the isotropy of both models and derive spatial statistics, including the distributions of the nearest-anchor, nearest-ring, and nearest-HAPs distances, together with the joint serving distance and serving ring angle distribution required for SCR-BCP analysis. Building on these results, we derive coverage probability expressions under nearest-HAPs association by decomposing aggregate interference into same-ring and other-ring components and characterizing their conditional Laplace transforms. To account for the flight dynamics of patrol-based HAPs, we integrate a steady circular flight propulsion model with the communication analysis and introduce a coverage energy efficiency (CEE) metric. This yields an analytical condition for the energy-optimal patrol radius that balances coverage performance against the propulsion cost of circular flight. Numerical results reveal fundamental differences between intensity-driven (SCR-PCP) and finite-fleet (SCR-BCP) deployments and demonstrate that patrol geometry, platform density, and cruising velocity should be jointly optimized to achieve energy-efficient HAPs operation.