Distance Distributions in Regular Polygons

This paper derives the exact cumulative density function of the distance between a randomly located node and any arbitrary reference point inside a regular \el-sided polygon. Using this result, we obtain the closed-form probability density function (PDF) of the Euclidean distance between any arbitrary reference point and its $n$-th neighbour node, when $N$ nodes are uniformly and independently distributed inside a regular $\ell$-sided polygon. First, we exploit the rotational symmetry of the regular polygons and quantify the effect of polygon sides and vertices on the distance distributions. Then we propose an algorithm to determine the distance distributions given any arbitrary location of the reference point inside the polygon. For the special case when the arbitrary reference point is located at the center of the polygon, our framework reproduces the existing result in the literature.Comment: 13 pages, 5 figure

On the Distribution of Random Geometric Graphs

Random geometric graphs (RGGs) are commonly used to model networked systems that depend on the underlying spatial embedding. We concern ourselves with the probability distribution of an RGG, which is crucial for studying its random topology, properties (e.g., connectedness), or Shannon entropy as a measure of the graph's topological uncertainty (or information content). Moreover, the distribution is also relevant for determining average network performance or designing protocols. However, a major impediment in deducing the graph distribution is that it requires the joint probability distribution of the $n(n-1)/2$ distances between $n$ nodes randomly distributed in a bounded domain. As no such result exists in the literature, we make progress by obtaining the joint distribution of the distances between three nodes confined in a disk in $\mathbb{R}^2$. This enables the calculation of the probability distribution and entropy of a three-node graph. For arbitrary $n$, we derive a series of upper bounds on the graph entropy; in particular, the bound involving the entropy of a three-node graph is tighter than the existing bound which assumes distances are independent. Finally, we provide numerical results on graph connectedness and the tightness of the derived entropy bounds.Comment: submitted to the IEEE International Symposium on Information Theory 201

Downlink Coverage and Rate Analysis of Low Earth Orbit Satellite Constellations Using Stochastic Geometry

As low Earth orbit (LEO) satellite communication systems are gaining increasing popularity, new theoretical methodologies are required to investigate such networks' performance at large. This is because deterministic and location-based models that have previously been applied to analyze satellite systems are typically restricted to support simulations only. In this paper, we derive analytical expressions for the downlink coverage probability and average data rate of generic LEO networks, regardless of the actual satellites' locality and their service area geometry. Our solution stems from stochastic geometry, which abstracts the generic networks into uniform binomial point processes. Applying the proposed model, we then study the performance of the networks as a function of key constellation design parameters. Finally, to fit the theoretical modeling more precisely to real deterministic constellations, we introduce the effective number of satellites as a parameter to compensate for the practical uneven distribution of satellites on different latitudes. In addition to deriving exact network performance metrics, the study reveals several guidelines for selecting the design parameters for future massive LEO constellations, e.g., the number of frequency channels and altitude.Comment: Accepted for publication in the IEEE Transactions on Communications in April 202

Performance Analysis of Arbitrarily-Shaped Underlay Cognitive Networks: Effects of Secondary User Activity Protocols

This paper analyzes the performance of the primary and secondary users (SUs) in an arbitrarily-shaped underlay cognitive network. In order to meet the interference threshold requirement for a primary receiver (PU-Rx) at an arbitrary location, we consider different SU activity protocols which limit the number of active SUs. We propose a framework, based on the moment generating function (MGF) of the interference due to a random SU, to analytically compute the outage probability in the primary network, as well as the average number of active SUs in the secondary network. We also propose a cooperation-based SU activity protocol in the underlay cognitive network which includes the existing threshold-based protocol as a special case. We study the average number of active SUs for the different SU activity protocols, subject to a given outage probability constraint at the PU and we employ it as an analytical approach to compare the effect of different SU activity protocols on the performance of the primary and secondary networks.Comment: submitted to possible IEEE Transactions publicatio

Stochastic Geometry Modeling and Analysis of Single- and Multi-Cluster Wireless Networks

This paper develops a stochastic geometry-based approach for the modeling and analysis of single- and multi-cluster wireless networks. We first define finite homogeneous Poisson point processes to model the number and locations of the transmitters in a confined region as a single-cluster wireless network. We study the coverage probability for a reference receiver for two strategies; closest-selection, where the receiver is served by the closest transmitter among all transmitters, and uniform-selection, where the serving transmitter is selected randomly with uniform distribution. Second, using Matern cluster processes, we extend our model and analysis to multi-cluster wireless networks. Here, the receivers are modeled in two types, namely, closed- and open-access. Closed-access receivers are distributed around the cluster centers of the transmitters according to a symmetric normal distribution and can be served only by the transmitters of their corresponding clusters. Open-access receivers, on the other hand, are placed independently of the transmitters and can be served by all transmitters. In all cases, the link distance distribution and the Laplace transform (LT) of the interference are derived. We also derive closed-form lower bounds on the LT of the interference for single-cluster wireless networks. The impact of different parameters on the performance is also investigated