Dynamical System Characterization of Heterogeneous Walker Satellite Networks: An Orbit-Aware Stochastic Geometry Perspective

Heterogeneous and in particular multi-altitude low Earth orbit (LEO) satellite constellations exhibit complex spatial and temporal structures, which require new modeling tools for their performance analysis. In this paper, we develop an orbit-aware stochastic geometry framework modeling today's LEO satellites on various orbits and various altitudes. In particular, we characterize such a system as the superposition of multiple Walker point processes and formulate it as a dynamical system determined by an initial condition and the rotation speeds of satellites and Earth. We show that when the speeds are rationally commensurable, the proposed satellite system is periodic. Then, we show that the system is ergodic when the speeds are rationally independent, establishing a theoretical link between time averages of the system and the expectation of it under the invariant measure. We derive the nearest-satellite distance distribution of a typical receiver at a given latitude and analyze the signal to interference-plus-noise ratio (SINR) coverage probability of the typical receiver. We then derive the ergodic throughput of the downlink communication to the typical receiver. Overall, the proposed framework offers a rigorous and tractable tool for analyzing downlink performance in Walker-type heterogeneous LEO satellite networks.

Joint Visibility Analysis of RIS in Non-Terrestrial Networks through Stochastic Geometry

Non-Terrestrial networks (NTNs) are a key theme in upcoming 6G communications, especially for ubiquitous coverage. Urban environments, comprising of high rise buildings often result in blocking the line of sight (LoS) path between the user equipment (UE) and the NTN base station (NTN-BS). In this paper we investigate the situation where reconfigurable intelligent surfaces (RIS) are deployed on the building roof-tops to ensure multi-hop connectivity between the UE and the NTN-BS. In such a scenario, it becomes crucial to statistically study the LoS visibility of the RIS from the UE as well as from the NTN-BS, hence termed as joint visibility. In this work, accounting for the dual stochasticity arising from the locations of the RIS deployed buildings and the respective random building heights, we statistically study the probability of joint RIS visibility in a two-dimensional (2D) scenario considering a deterministic location of the NTN-BS. Further, we study the joint RIS visibility statistics conditional on the UE-NTN link being LoS or non-LoS. For the RISs deployed as a point point process (PPP) having exponentially distributed heights, the expected RISs jointly visible under the unconditional and conditional geometric settings are derived in closed form. Interestingly, in the 2D setting, the maximum expected RISs jointly visible, unconditionally, is twice the Basel number $(π^2/ 6)$. The simulated results are analyzed over building density, average building height, the altitude and position of the NTN-BS. We also illustrate probability heatmaps, demonstrating the strongest chance to have a RIS used conditioned on the system geometry. This study is expected to be useful in planning the deployment of RIS in urban areas, improving the signal and for assessing economic aspects.