The Influence of Canyon Shadowing on Device-to-Device Connectivity in Urban Scenario

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 existence of a minimal threshold for relays below which large-scale connectivity of the network is not possible, regardless of all other network parameters. Through simulations, we estimate this threshold to 71.3%. Moreover, if the mean street length is not larger than some threshold (predicted to 74.3% of the communication range; which might be the case in a typical urban scenario) then any (whatever small) density of users can be compensated by equipping more crossroads with relays. Above this latter threshold, good connectivity requires some minimal density of users, compensated by the relays in a way we make explicit. The existence of the above regimes brings interesting qualitative arguments to the discussion on the possible D2D deployment scenarios.

Continuum line-of-sight percolation on Poisson–Voronoi tessellations

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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].

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“…We prove that the critical range , numerically estimated in [30], is nontrivial in the sense that for large enough, including some .…”

“… 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 . …”

“…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.…”

“…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.…”