Distance Order Statistics for Hierarchical NTNs via Stochastic Geometry

The distance between a terrestrial terminal and an endpoint in a non-terrestrial network (NTN) greatly impacts the performance of a 6G wireless application. Considering the multitude of critical metrics influenced by this distance – such as channel degradation due to path-loss, latency incurred from propagation delay, and coverage probability dependent on node’s footprint – an analytical description of the terrestrial-NTN distance is imperative for network design. To this end, many researchers have employed the use of stochastic geometry to probabilistically describe the distance from a terrestrial terminal to the nearest node of an NTN. Although the nearest node statistics are a powerful analytical tool, this paper extends on these techniques by enabling the stochastic description of the ordered distance to any node in a hierarchical multi-layer NTN. Explicitly, we utilize stochastic geometry to derive and verify the distribution of the distance from a terrestrial terminal to i) thenth-nearest node on a specific layerk, and ii) the distance to thenth-nearest node across all layers of the NTN. An analytical understanding of these distances enable better evaluations of different hierarchical NTNs architectures and support the analysis of complex 6G applications which may not always choose the nearest node as the serving node.