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.

翻译：本文为高空平台站（HAPs）网络建立了随机几何框架，其中平台沿指定服务区域执行循环巡逻轨迹。我们在球形地球表面引入两种小圆环考克斯过程模型：在小圆环泊松考克斯过程（SCR-PCP）中，平台在局部巡逻环上形成一维泊松点过程；而在小圆环二项考克斯过程（SCR-BCP）中，每个环包含固定数量的均匀分布平台。我们证明了两种模型的各向同性，并推导了空间统计特性，包括最近锚点、最近环和最近HAPs距离的分布，以及用于SCR-BCP分析所需的联合服务距离和服务环角度分布。基于这些结果，我们通过将聚合干扰分解为同环干扰和异环干扰并刻画其条件拉普拉斯变换，推导了最近HAPs关联下的覆盖概率表达式。为考虑巡逻式HAPs的飞行动力学，我们将稳态圆形飞行推进模型与通信分析相结合，并引入覆盖能效（CEE）指标。这产生了能量最优巡逻半径的分析条件，该条件平衡了覆盖性能与圆形飞行的推进成本。数值结果揭示了强度驱动型（SCR-PCP）部署与有限机群型（SCR-BCP）部署之间的根本差异，并表明应联合优化巡逻几何、平台密度和巡航速度以实现高能效的HAPs运行。