Signal-to-interference ratio percolation for Cox point processes

Abstract

We study signal-to-interference plus noise ratio (SINR) percolation for Cox point processes, i.e., Poisson point processes with a random intensity measure. SINR percolation was first studied by Dousse et al. in the case of a two-dimensional Poisson point process. It is a version of continuum percolation where the connection between two points depends on the locations of all points of the point process. Continuum percolation for Cox point processes was recently studied by Hirsch, Jahnel, and Cali.We study the SINR graph model for a stationary Cox point process in two or higher dimensions, in case of a bounded and integrable path-loss function. We show that if this function has compact support or if the stationary intensity measure evaluated at a unit box has some exponential moments, then the SINR graph has an infinite connected component in case the spatial density of points is large enough and the interferences are sufficiently reduced (without vanishing). This holds if the intensity measure is asymptotically essentially connected, and also if the intensity measure is only stabilizing but the connection radius is large. We also provide estimates on the critical interference cancellation factor.A prominent example of the intensity measure is the two-dimensional Poisson-Voronoi tessellation, which is used for modelling real-world street systems. We show that its total edge length in a unit square has some exponential moments. We conclude that its SINR graph percolates for any bounded path-loss function with power-law decay of exponent larger than 2.MSC 2010. Primary 82B43, 60G55, 60K35; secondary 90B18.

SINR percolation for Cox point processes with random powers

“…More precisely, in our first main result, Theorem 2.1, we present sufficient conditions for the existence of a supercritical percolation phase, i.e., a nontrivial regime for the intensity of the underlying CPP and nonvanishing interference, such that the Cox SINR graph with random powers percolates. This substantially extends the results of [8][9][10] from the case of a homogeneous PPP in R 2 with constant powers to that of a CPP in R d , d ≥ 2, with random and possibly unbounded powers, combining the methods of [28] for the case of a CPP with constant powers and those of [22] for the case of a PPP with random powers (both in dimension 2 or higher). We will discuss the relationship of Theorem 2.1 to these results in detail in Section 4.…”

“…For the proof, we combine the approach used in [22,Theorem 4.5] for handling random radii and the approach used in [28, Theorem 2.4] for dealing with the spatial correlations of the directing measure of the CPP. To begin with, an easy coupling argument (see [27,Section 4.2.3.4]) implies that as long as the powers are bounded, all positive results of [28] about percolation in the Cox SINR graph for asymptotically essentially connected are applicable. More precisely, we have the following proposition for the Cox SINR graph with random bounded powers.…”

“…Note that we have formulated Condition 1 in Proposition 4.1 more generally than the statement in [28], and Condition 3 does not appear in [28]. However, the proof from [28] can also be adapted to these more general cases.…”

“…On the other hand, in [28], the two extensions described above were for the first time considered jointly, giving rise to the SINR graph based on CPPs, the Cox SINR graph, but without random powers. In [28] it was established that for sufficiently large intensities and sufficiently connected environments, the Cox SINR graph percolates almost surely at least for nonvanishing interference. In [27,Section 4.2.3.4] it was anticipated that the case of random but bounded powers might easily be handled via the same methods; see Section 4.1 for further details.…”

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

“…Connectivity for D2D networks on street systems using percolation theory was explored only recently in [14], building on the theoretical results from [15] regarding percolation of Cox models. Very recently, percolation of the SINR graph associated with Cox processes has been studied in [16]. Cox processes cluster their points more than Poisson point processes [17] and, in general, their percolation properties cannot be simply derived by a comparison to this latter model [18].…”