Abstract:In this work, we use percolation theory to study the feasibility of large-scale connectivity of relay-augmented deviceto-device (D2D) networks in an urban scenario featuring a haphazard system of streets and canyon shadowing allowing only for line-of-sight (LOS) communications in a finite range. We use a homogeneous Poisson-Voronoi tessellation (PVT) model of streets with homogeneous Poisson users (devices) on its edges and independent Bernoulli relays on the vertices. Using this model, we demonstrate the exis… Show more
“… Figure 3: Phase transition diagram of with hypothetical ranges of ( p , H ). Estimated values for , , and are obtained in the planar case via Monte Carlo simulations; see Subsection 2.4 and the earlier publication [30]. …”
Section: Model and Main Resultsmentioning
confidence: 99%
“…We prove that the critical range , numerically estimated in [30], is nontrivial in the sense that for large enough, including some .…”
Section: Introductionmentioning
confidence: 91%
“… Critical connectivity range: For , there exists a critical connectivity range separating the following two connectivity regimes: Permanently supercritical : For the graph percolates with positive probability for all . User critical : For the graph exhibits a nontrivial phase transition in ; i.e. it does not percolate for small , and it percolates with positive probability for large enough . We prove that the critical range , numerically estimated in [30], is nontrivial in the sense that for large enough, including some . As a corollary we obtain the existence of a nontrivial phase transition in the PVT hard-geometric bond percolation model . …”
Section: Introductionmentioning
confidence: 93%
“…In a recent work [30], a new mathematical model for the so-called device-to-device wireless networks in obstructive urban environments was proposed and studied numerically. This model is meant to represent direct wireless connections between users (and possibly some relays) taking place in an urban environment, with limited-range connections possible only within the line of sight along the streets.…”
Section: Introductionmentioning
confidence: 99%
“…It is in light of these recent developments that we introduced in our previous work [30] a new percolation model (originally in the planar case) for Cox processes supported by PVTs.…”
In this work, we study a new model for continuum line-of-sight percolation in a random environment driven by the Poisson–Voronoi tessellation in the d-dimensional Euclidean space. The edges (one-dimensional facets, or simply 1-facets) of this tessellation are the support of a Cox point process, while the vertices (zero-dimensional facets or simply 0-facets) are the support of a Bernoulli point process. Taking the superposition Z of these two processes, two points of Z are linked by an edge if and only if they are sufficiently close and located on the same edge (1-facet) of the supporting tessellation. We study the percolation of the random graph arising from this construction and prove that a 0–1 law, a subcritical phase, and a supercritical phase exist under general assumptions. Our proofs are based on a coarse-graining argument with some notion of stabilization and asymptotic essential connectedness to investigate continuum percolation for Cox point processes. We also give numerical estimates of the critical parameters of the model in the planar case, where our model is intended to represent telecommunications networks in a random environment with obstructive conditions for signal propagation.
“… Figure 3: Phase transition diagram of with hypothetical ranges of ( p , H ). Estimated values for , , and are obtained in the planar case via Monte Carlo simulations; see Subsection 2.4 and the earlier publication [30]. …”
Section: Model and Main Resultsmentioning
confidence: 99%
“…We prove that the critical range , numerically estimated in [30], is nontrivial in the sense that for large enough, including some .…”
Section: Introductionmentioning
confidence: 91%
“… Critical connectivity range: For , there exists a critical connectivity range separating the following two connectivity regimes: Permanently supercritical : For the graph percolates with positive probability for all . User critical : For the graph exhibits a nontrivial phase transition in ; i.e. it does not percolate for small , and it percolates with positive probability for large enough . We prove that the critical range , numerically estimated in [30], is nontrivial in the sense that for large enough, including some . As a corollary we obtain the existence of a nontrivial phase transition in the PVT hard-geometric bond percolation model . …”
Section: Introductionmentioning
confidence: 93%
“…In a recent work [30], a new mathematical model for the so-called device-to-device wireless networks in obstructive urban environments was proposed and studied numerically. This model is meant to represent direct wireless connections between users (and possibly some relays) taking place in an urban environment, with limited-range connections possible only within the line of sight along the streets.…”
Section: Introductionmentioning
confidence: 99%
“…It is in light of these recent developments that we introduced in our previous work [30] a new percolation model (originally in the planar case) for Cox processes supported by PVTs.…”
In this work, we study a new model for continuum line-of-sight percolation in a random environment driven by the Poisson–Voronoi tessellation in the d-dimensional Euclidean space. The edges (one-dimensional facets, or simply 1-facets) of this tessellation are the support of a Cox point process, while the vertices (zero-dimensional facets or simply 0-facets) are the support of a Bernoulli point process. Taking the superposition Z of these two processes, two points of Z are linked by an edge if and only if they are sufficiently close and located on the same edge (1-facet) of the supporting tessellation. We study the percolation of the random graph arising from this construction and prove that a 0–1 law, a subcritical phase, and a supercritical phase exist under general assumptions. Our proofs are based on a coarse-graining argument with some notion of stabilization and asymptotic essential connectedness to investigate continuum percolation for Cox point processes. We also give numerical estimates of the critical parameters of the model in the planar case, where our model is intended to represent telecommunications networks in a random environment with obstructive conditions for signal propagation.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
