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This work develops an analytical framework for downlink low earth orbit (LEO) satellite communications, leveraging tools from stochastic geometry. We propose a tractable approach to the analysis of such satellite communication systems accounting for the fact that satellites are located on circular orbits. We accurately characterize this geometric property of such LEO satellite constellations by developing a Cox point process model that jointly produces orbits and satellites on these orbits. Our work differs from existing modeling studies that have assumed satellites' locations as completely random binomial point processes. For this newly developed Cox model, we derive the outage probability of the proposed network and the distribution of the signal-to-interference-plus-noise ratio (SINR) of an arbitrarily located user in the network. By presenting various key network performance metrics as functions of key network parameters, this work allows one to assess the statistical properties of downlink LEO satellite communications and thus can be used as a system-level design tool., Submitted to IEEE Journal

submitted to IEEE Journal; While roadside units (RSUs) play an essential role in vehicle-to-everything (V2X) by communicating with users, some users in congestion areas may not be well-served due to data traffic, signal attenuation, and interference. In these cases, vehicle relays can be employed to enhance the network topology to better serve those users. This paper leverages stochastic geometry to propose a novel framework for the performance analysis of heterogeneous vehicular networks with RSUs, vehicle relays, and vehicle users. We present a two-dimensional analytical model where the spatial dependence between RSUs, vehicle relays, vehicle users, and roads is accurately taken into account through a Cox point process structure. Assuming relays are backhauled to RSUs over a reserved wireless resource and users are associated with the closest RSU or relay, we derive the probability that the typical user is associated with either an RSU or a relay. Then, we derive the signal-to-interference ratio (SIR) coverage probability of the typical user. Finally, using the derived formulas, we evaluate the average effective rate of the typical user in the network. This allows us to determine the gain of the average effective rate of users that results from the deployment of relays in the network.

This research explores the utilization of relays in vehicle-to-all (V2X) communications, where roadside units (RSUs) alone may not be sufficient to ensure network connectivity due to network congestion, signal attenuation, or interference. By employing stochastic geometry, we analyze a spatially-correlated vehicular network that incorporates both RSUs and relays to serve network users on roads. Our model considers the geometric characteristics of roads, RSUs, relays, and users using Cox point processes conditionally on the same road structure. Assuming disjoint frequency resources for serving users and for enabling RSU-connected relays, each user can associate with either an RSU or a relay whichever is closest to it. We derive the association probability and coverage probability for the typical user, enabling us to assess network performance. Additionally, we investigate user throughput by considering interactions among different links within the proposed network. This paper offers practical insights for the design of two-tier vehicular networks. Specifically, we express user association, user signal-to-interference ratio (SIR), and user throughput as functions of network variables. This information aids in determining optimal relay density and operating bandwidth to enhance network reliability and maximize user throughput in vehicular networks., Comment: submitted to IEEE Journal

This work introduces two new notions of dimension, namely the unimodular Minkowski and Hausdorff dimensions, which are inspired from the classical analogous notions. These dimensions are defined for unimodular discrete spaces, introduced in this work, which provide a common generalization to stationary point processes under their Palm version and unimodular random rooted graphs. The use of unimodularity in the definitions of dimension is novel. Also, a toolbox of results is presented for the analysis of these dimensions. In particular, analogues of Billingsley's lemma and Frostman's lemma are presented. These last lemmas are instrumental in deriving upper bounds on dimensions, whereas lower bounds are obtained from specific coverings. The notions of unimodular Hausdorff size, which is a discrete analogue of the Hausdorff measure, and unimodular dimension function are also introduced. This toolbox allows one to connect the unimodular dimensions to other notions such as volume growth rate, discrete dimension and scaling limits. It is also used to analyze the dimensions of a set of examples pertaining to point processes, branching processes, random graphs, random walks, and self-similar discrete random spaces. Further results of independent interest are also presented, like a version of the max-flow min-cut theorem for unimodular one-ended trees and a weak form of pointwise ergodic theorems for all unimodular discrete spaces., Comment: 89 pages, 1 figure. This version of the paper is a merging of the previous version with arXiv:1808.02551. Earlier versions of this paper were titled `On the Dimension of Unimodular Discrete Spaces, Part I: Definitions and Basic Properties'