The meta distribution of the signal-to-interference-ratio (SIR) is an important performance indicator for wireless networks because, for ergodic point processes, it describes the fraction of ...scheduled links that achieve certain reliability, conditionally on the point process. In this paper, we calculate the moments of the meta distribution in vehicular ad hoc networks (VANETs) along high-speed motorways. Due to the high speeds, the drivers maintain large safety distances, and the Poisson point process (PPP) becomes a poor deployment model. Because of that, we model the distribution of inter-vehicle distance equal to the sum of a constant hardcore distance and an exponentially distributed random variable. We design a novel discretization model for the locations of vehicles which can be used to approximate well the meta distribution of the SIR due to the hardcore process. We validate the model against synthetic motorway traces. On the other hand, the PPP overestimates significantly the coefficient-of-variation of the meta distribution due to the hardcore process, and its predictions fail. In addition, we show that the calculation of the meta distribution becomes especially meaningful in the upper tail of the SIR distribution.
Poisson Voronoi tessellations have been used in modeling many types of systems across different sciences, from geography and astronomy to telecommunications. The existing literature on the ...statistical properties of Poisson Voronoi cells is vast, however, little is known about the properties of Voronoi cells located close to the boundaries of a compact domain. In a domain with boundaries, some Voronoi cells would be naturally clipped by the boundary, and the cell area falling inside the deployment domain would have different statistical properties as compared to those of non-clipped Voronoi cells located in the bulk of the domain. In this paper, we consider the planar Voronoi tessellation induced by a homogeneous Poisson point process of intensity
λ
>
0
in a quadrant, where the two half-axes represent boundaries. We show that the mean cell area is less than
λ
-
1
when the seed is located exactly at the boundary, and it can be larger than
λ
-
1
when the seed lies close to the boundary. In addition, we calculate the second moment of cell area at two locations for the seed: (i) at the corner of a quadrant, and (ii) at the boundary of the half-plane. We illustrate that the two-parameter Gamma distribution, with location-dependent parameters calculated using the method of moments, can be of use in approximating the distribution of cell area. As a potential application, we use the Gamma approximations to study the degree distribution for secure connectivity in wireless sensor networks deployed over a domain with boundaries.
Interference statistics in vehicular networks have long been studied using the Poisson Point Process (PPP) for the locations of vehicles. In roads with a small number of lanes and restricted ...overtaking, this model becomes unrealistic because it assumes that the vehicles can come arbitrarily close to each other. In this paper, we model the headway distance (the distance between the head of a vehicle and the head of its follower) equal to the sum of a constant hardcore distance and an exponentially distributed random variable. We study the mean, the variance, and the skewness of interference at the origin with this deployment model. Even though the pair correlation function becomes complicated, we devise a simple formulae to capture the impact of the hardcore distance on the variance of interference in comparison with a PPP model of equal intensity. In addition, we study the extreme scenario, where the interference originates from a lattice. We show how to relate the variance of interference due to a lattice to that of a PPP under Rayleigh fading.
In mobile wireless networks with blockage, different users, and/or a single user at different time slots, may be blocked by some common obstacles. Therefore, the temporal correlation of interference ...does not depend only on the user displacement law but also on the spatial correlation introduced by the obstacles. In this letter, we show that in mobile networks with a high density of users, blockage increases the temporal correlation of interference, while in sparse networks blockage has the opposite effect.
Random geometric graphs consist of randomly distributed nodes (points), with pairs of nodes within a given mutual distance linked. In the usual model the distribution of nodes is uniform on a square, ...and in the limit of infinitely many nodes and shrinking linking range, the number of isolated nodes is Poisson distributed, and the probability of no isolated nodes is equal to the probability the whole graph is connected. Here we examine these properties for several self-similar node distributions, including smooth and fractal, uniform and nonuniform, and finitely ramified or otherwise. We show that nonuniformity can break the Poisson distribution property, but it strengthens the link between isolation and connectivity. It also stretches out the connectivity transition. Finite ramification is another mechanism for lack of connectivity. The same considerations apply to fractal distributions as smooth, with some technical differences in evaluation of the integrals and analytical arguments.
In the original (1961) Gilbert model of random geometric graphs, nodes are placed according to a Poisson point process, and links formed between those within a fixed range. Motivated by wireless ad ...hoc networks "soft" or "probabilistic" connection models have recently been introduced, involving a "connection function" H(r) that gives the probability that two nodes at distance r are linked (directly connect). In many applications (not only wireless networks), it is desirable that the graph is connected; that is, every node is linked to every other node in a multihop fashion. Here the connection probability of a dense network in a convex domain in two or three dimensions is expressed in terms of contributions from boundary components for a very general class of connection functions. It turns out that only a few quantities such as moments of the connection function appear. Good agreement is found with special cases from previous studies and with numerical simulations.
The Lorentz gas is a billiard model involving a point particle diffusing deterministically in a periodic array of convex scatterers. In the two dimensional finite horizon case, in which all ...trajectories involve collisions with the scatterers, displacements scaled by the usual diffusive factor
are normally distributed, as shown by Bunimovich and Sinai in 1981. In the infinite horizon case, motion is superdiffusive, however the normal distribution is recovered when scaling by
, with an explicit formula for its variance. Here we explore the infinite horizon case in arbitrary dimensions, giving explicit formulas for the mean square displacement, arguing that it differs from the variance of the limiting distribution, making connections with the Riemann Hypothesis in the small scatterer limit, and providing evidence for a critical dimension
d
=6 beyond which correlation decay exhibits fractional powers. The results are conditional on a number of conjectures, and are corroborated by numerical simulations in up to ten dimensions.
We approximate the uniform measure on an equilateral triangle by a measure supported on n points. We find the optimal sets of points (n-means) and corresponding approximation (quantization) error for ...n≤4, give numerical optimization results for n≤21, and a bound on the quantization error for narrow right∞. The equilateral triangle has particularly efficient quantizations due to its connection with the triangular lattice. Our methods can be applied to the uniform distributions on general sets with piecewise smooth boundaries.
While the Euclidean distance characteristics of the Poisson line Cox process (PLCP) have been investigated in the literature, the analytical characterization of the path distances is still an open ...problem. In this paper, we solve this problem for the stationary Manhattan Poisson line Cox process (MPLCP), which is a variant of the PLCP. Specifically, we derive the exact cumulative distribution function (CDF) for the length of the shortest path to the nearest point of the MPLCP in the sense of path distance measured from two reference points: (i) the typical intersection of the Manhattan Poisson line process (MPLP), and (ii) the typical point of the MPLCP. We also discuss the application of these results in infrastructure planning, wireless communication, and transportation networks.
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