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.

翻译：低地球轨道（LEO）卫星产业正在迅速扩张，由于轨道权遵循先到先得原则，运营商竞相发射卫星，导致形成了规模日益庞大、立体化的星座系统，其中卫星在多样化的高度范围内运行。为满足分析此类复杂网络的需求，本文利用三维泊松点过程（PPP）建立了一个新的LEO星座分析框架。具体而言，我们引入了一种随机高度模型（RHM），该模型通过对名义球壳上均匀PPP生成的点施加随机径向位移，能够捕捉多种高度分布。在此基础上，我们推导了下行链路覆盖概率的解析表达式。为验证模型的适用性，我们展示了包括星链在内的多个主要卫星星座的高度分布符合该模型的假设。随后通过蒙特卡洛模拟证明，我们的RHM模型计算出的覆盖概率与这些实际网络高度吻合。最后，通过解析表达式与模拟结果的一致性，我们验证了解析表达式的准确性。因此，本研究为理解和预测卫星高度统计分布如何影响网络性能提供了有力工具。