![]() the density of relays and users) allowing for good connectivity of the network. Performing numerical simulations with original path-finding algorithms, we estimate critical parameters (e.g. Using renormalisation techniques, we prove the existence of phase transitions between different connectivity regimes, in particular those where percolation can be solely ensured by the relays or, on the contrary, where a sufficient density of users is essential. Percolation of this random graph (existence of an infinite connected component with positive probability) is interpreted as good connectivity of the network. The network is then modelled by a connectivity graph as follows: vertices are the atoms of both these processes and fixed-range connections between them possible only along the PVT edges or between network nodes located on adjacent PVT edges. Network users are given by a Cox process supported by the edges of the PVT while additional network relays are given by a Bernoulli process on the vertices of the PVT. We see the street system of a city as a planar Poisson-Voronoi tessellation (PVT). In this thesis, we study new mathematical models of D2D networks in urban environments. An application of significant economic interest for operators is the one of the uberisation of networks, where an operator having no (or very few) network infrastructure could build a mobile network relying only on its end-devices (users). One of the main paradigms investigated to address this challenge, called Device-to-Device (D2D) communication, consists in allowing for short-range direct communications between network devices. The fifth generation of cellular networks is expected to provide coverage for an unprecedented number of devices over large areas. TheyĪlso provide a guideline for achieving connectivity using minimal Our resultsĬan be used to decide on the goodness of any channel rendezvousĪlgorithm by computing the expected resultant connectivity. Probability which results in a connected network. Moreover, weįind the trade-off between deployment-density versus rendezvous Of using either type of rendezvous techniques. Regions: channel abundance, optimal, and channel deprivation.įor each region we show the requirement and the outcome Present, we characterize and analyze the connectivity for all the We derive the critical number of channels which maintains supercriticality Secondary network in terms of the available number of channels,ĭeployment densities, number of simultaneous transmissions per MLG, we study both cases of primaries’ absence and presence.įor both cases, we define and characterize connectivity of the We show how percolation occurs in the MLG by coupling it withĪ typical discrete percolation. Multiple channels as parallel edges in a graph and build a multilayered Following the disk graph model, we represent the Show that invisibility is even more pronounced with channelĪbundance. We model this invisibility as a Poisson thinning process and Protocols, it becomes difficult for two nodes to agree on a commonĬhannel, thereby potentially remaining invisible to each other. We show that, with more channelĪbundance, even with the use of either type of rendezvous Protocols: naive ones which do not guarantee a common channelĪnd advanced ones which do. The existence of multi-channel, we use two types of rendezvous Phenomenon which we define as channel abundance. Secondary nodes have plenty of vacant channels to choose from–a In particular, we focus on the scenario where the Of Dynamic Spectrum Access networks using percolation This indicates that connectivity claims made in the literature using the geometric disc abstraction in general hold also for the more irregular shapes found in practice.Įffects of multi-channel and rendezvous protocols on the connectivity In other words, anisotropic radiation patterns and spotty coverage allow an unbounded connected component to appear at lower ENC levels than perfect circular coverage allows. We compare networks of geometric discs to other simple shapes and/or probabilistic connections, and we find that when transmission range and node density are normalized across experiments so as to preserve the expected number of connections (ENC) enjoyed by each node, discs are the “hardest ” shape to connect together. In this paper we examine how these issues affect network connectivity. In reality communication channels are unreliable and communication range is generally not rotationally symmetric. Puffco support warranty.Models of wireless ad-hoc and sensor networks are often based on the geometric disc abstraction: transmission is assumed to be isotropic, and reliable communication channels are assumed to exist (apart from interference) between nodes closer than a given distance.
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