The Localization Problem in Networks of Uniformly Deployed Nodes

Abstract

Consider a bidimensional domain S sube Rfr <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> and throw two statistically independent uniform poisson point processes with constant densities equal to <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">PL</sub> and <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">PNL</sub> , respectively. The first point process identifies the spatial distribution of a set of nodes which has information about their position, hereafter denoted as L-nodes, while the other one is used to model the spatial distribution of nodes which need to localize themselves, hereafter denoted as NL-nodes. Both kind of nodes are equipped by the same kind of transceiver, and communicate over a channel affected by shadow fading. The goal of this paper is to derive the probability that a randomly chosen NL-node over the domain S gets localized as a function of a variety of transmission parameters. As many random graph properties, the localization probability is a monotone graph property presenting thresholds. We derive finite thresholds for the localization probability. The envisaged scenario refers to the case in which the number of deployed nodes of both point processes is finite. Simulation results closely match the theoretical derivations confirming the effectiveness of the employed probabilistic model.