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The study of distribution functions (with respect to areas, perimeters) for partitioning a plane (space) by a random field of straight lines (hyperplanes) and for obtaining Voronoi diagrams is a classical problem in statistical geometry. Moments for such distributions have been investigated since 1972 [1]. We give a complete solution of these problems for the plane, as well as for Voronoi diagrams. The following problems are solved: 1. A random set of straight lines is given on the plane, all shifts are equiprobable, and the distribution law has the form What is the area (perimeter) distribution of the parts of the partition? 2. A random set of points is marked on the plane. Each point A is associated with a "region of attraction," which is a set of points on the plane to which A is the closest of the marked set. The idea is to interpret a random polygon as the evolution of a segment on a moving one and construct kinetic equations. It is sufficient to take into account a limited number of parameters: the covered area (perimeter), the length of the segment, and the angles at its ends. We show how to reduce these equations to the Riccati equation using the Laplace transform. [ABSTRACT FROM AUTHOR]

We consider a point process (PP) generated by superimposing an independent Poisson point process (PPP) on each line of a 2D Poisson line process (PLP). Termed PLP-PPP, this PP is suitable for modeling networks formed on an irregular collection of lines, such as vehicles on a network of roads and sensors deployed along trails in a forest. Inspired by vehicular networks in which vehicles connect with their nearest wireless base stations (BSs), we consider a random bipartite associator graph in which each point of the PLP-PPP is associated with the nearest point of an independent PPP through an edge. This graph is equivalent to the partitioning of PLP-PPP by a Poisson Voronoi tessellation (PVT) formed by an independent PPP. We first characterize the exact distribution of the number of points of PLP-PPP falling inside the ball centered at an arbitrary location in R 2 as well as the typical point of PLP-PPP. Using these distributions, we derive cumulative distribution functions (CDFs) and probability density functions (PDFs) of kth contact distance (CD) and the nearest neighbor distance (NND) of PLP-PPP. As intermediate results, we present the empirical distribution of the perimeter and approximate distribution of the length of the typical chord of the zero-cell of this PVT. Using these results, we present two close approximations of the distribution of node degree of the random bipartite associator graph. In a vehicular network setting, this result characterizes the number of vehicles connected to each BS, which models its load. Since each BS has to distribute its limited resources across all the vehicles connected to it, a good statistical understanding of load is important for an efficient system design. Several applications of these new results to different wireless network settings are also discussed. [ABSTRACT FROM AUTHOR]

We consider a point process (PP) generated by superimposing an independent Poisson point process (PPP) on each line of a 2D Poisson line process (PLP). Termed PLP-PPP, this PP is suitable for modeling networks formed on an irregular collection of lines, such as vehicles on a network of roads and sensors deployed along trails in a forest. Inspired by vehicular networks in which vehicles connect with their nearest wireless base stations (BSs), we consider a random bipartite associator graph in which each point of the PLP-PPP is associated with the nearest point of an independent PPP through an edge. This graph is equivalent to the partitioning of PLP-PPP by a Poisson Voronoi tessellation (PVT) formed by an independent PPP. We first characterize the exact distribution of the number of points of PLP-PPP falling inside the ball centered at an arbitrary location in R2 as well as the typical point of PLP-PPP. Using these distributions, we derive cumulative distribution functions (CDFs) and probability density functions (PDFs) of kth contact distance (CD) and the nearest neighbor distance (NND) of PLP-PPP. As intermediate results, we present the empirical distribution of the perimeter and approximate distribution of the length of the typical chord of the zero-cell of this PVT. Using these results, we present two close approximations of the distribution of node degree of the random bipartite associator graph. In a vehicular network setting, this result characterizes the number of vehicles connected to each BS, which models its load. Since each BS has to distribute its limited resources across all the vehicles connected to it, a good statistical understanding of load is important for an efficient system design. Several applications of these new results to different wireless network settings are also discussed.

The automobile industry has been evolving with rapid growth over the last two decades. Conventionally, the narrow band dedicated short-range communications is used for the wireless connectivity of vehicular networks in the automobile industry. However, due to its shorter wireless range, higher latency, and lower capacity; cellular-assisted vehicular communications can be adopted to improve networks performance by disrupting the nearby infrastructure. The performance of cellular-assisted vehicular communications is typically compromised if jammers are present in their vicinity. Thus, it is important to characterize the interference in cellular-assisted vehicular communications with jammers. In this paper, we investigate jamming interference due to clustered-jammers for vehicular networks by considering that vehicles, road-side units, and pedestrians are modeled using an independent one-dimensional Poisson line process while cellular base stations are modeled using an independent two-dimensional homogeneous Poisson point process. The results show that jamming disrupts the performance of the vehicular networks and that this performance is severely compromised by increasing the density of jamming clusters and their transmission power. Therefore, the prime focus of the system designers should be to introduce anti-jamming scenario, when the transmission power and the number of jamming clusters in a network increase.

In this paper, we investigate physical-layer security of the uplink millimeter wave communications for a cellular vehicle-to-everything (C-V2X) network comprised of a large number of base stations (BSs) and different categories of V2X nodes, including vehicles, pedestrians, and road side units. Considering the dynamic change and randomness of the topology of the C-V2X network, we model the roadways, the V2X nodes on each roadway, and the BSs by a Poisson line process, a 1D Poisson point process (PPP), and a 2D PPP, respectively. We propose two uplink association schemes for a typical vehicle, namely, the smallest-distance association (SDA) scheme and the largest-power association (LPA) scheme, and we establish a tractable analytical framework to comprehensively assess the security performance of the uplink transmission, by leveraging the stochastic geometry theory. Specifically, for each association scheme, we first obtain new expressions for the association probability of the typical vehicle, and then derive the overall connection outage probability and secrecy outage probability by calculating the Laplace transform of the aggregate interference power. Numerical results are presented to validate our theoretical analysis, and we also provide interesting insights into how the security performance is influenced by various system parameters, including the densities of V2X nodes and BSs. Moreover, we show that the LPA scheme outperforms the SDA scheme in terms of secrecy throughput. [ABSTRACT FROM AUTHOR]

With the increasing demand for greener and more energy efficient transportation solutions, electric vehicles (EVs) have emerged to be the future of transportation across the globe. However, currently, one of the biggest bottlenecks of EVs is the battery. Small batteries limit the EVs driving range, while big batteries are expensive and not environmentally friendly. One potential solution to this challenge is the deployment of charging roads, i.e., dynamic wireless charging systems installed under the roads that enable EVs to be charged while driving. In this paper, we use tools from stochastic geometry to establish a framework that enables evaluating the performance of charging roads deployment in metropolitan cities. We first present the course of actions that a driver should take when driving from a random source to a random destination in order to maximize dynamic charging during the trip. Next, we analyze the distribution of the distance to the nearest charging road. This distribution is vital for studying multiple performance metrics such as the trip efficiency, which we define as the fraction of the total trip spent on charging roads. Next, we derive the probability that a given trip passes through at least one charging road. The derived probability distributions can be used to assist urban planners and policy makers in designing the deployment plans of dynamic wireless charging systems. In addition, they can also be used by drivers and automobile manufacturers in choosing the best driving routes given the road conditions and level of energy of EV battery.

A comprehensive vehicular network analysis requires modeling the street system and vehicle locations. Even when Poisson point processes (PPPs) are used to model the vehicle locations on each street, the analysis is barely tractable. That holds for even a simple average-based performance metric—the success probability, which is a special case of the fine-grained metric, the meta distribution (MD) of the signal-to-interference ratio (SIR). To address this issue, we propose the transdimensional approach as an alternative. Here, the union of 1D PPPs on the streets is simplified to the transdimensional PPP (TPPP), a superposition of 1D and 2D PPPs. The TPPP includes the 1D PPPs on the streets passing through the receiving vehicle and models the remaining vehicles as a 2D PPP ignoring their street geometry. Through the SIR MD analysis, we show that the TPPP provides good approximations to the more cumbrous models with streets characterized by Poisson line/stick processes; and we prove that the accuracy of the TPPP further improves under shadowing. Lastly, we use the MD results to control network congestion by adjusting the transmit rate while maintaining a target fraction of reliable links. A key insight is that the success probability is an inadequate measure of congestion as it does not capture the reliabilities of the individual links. [ABSTRACT FROM AUTHOR]

This paper concerns the performance of vehicle-to-everything (V2X) communications. More precisely, we analyze the broadcast of safety-related V2X communications in cellular networks where base stations and vehicles are assumed to share the same spectrum and vehicles broadcast their safety messages to neighboring users. We model the locations of vehicles as a Poisson line Cox point process and the locations of users as a planar Poisson point process. We assume that users are associated with their closest base stations when there is no vehicle within a certain distance $\rho $. On the other hand, users located within a distance $\rho $ from vehicles are associated with the vehicles to receive their safety messages. We quantify the properties of this vehicle-prioritized association using the stochastic geometry framework. We derive the fractions of users that receive safety messages from vehicles. Then, we obtain the expression for the signal-to-interference ratio of the typical user evaluated on each association type. To address the impact of vehicular broadcast on the cellular network, the paper also derives the effective rate offered to the typical user in this setting. [ABSTRACT FROM AUTHOR]

Vehicle-to-vehicle (V2V) communication-based cooperative vehicular networks are a key technology for future smarter transportation systems as they provide various applications for addressing road safety and traffic congestion. To alleviate the performance limitations of existing dedicated short-range communication (DSRC) protocols, cellular-assisted V2V communications have been proposed recently, wherein cellular base stations (BSs) relay the transmission from one vehicle to another. Such cellular-assisted communications have shown promise for more reliable V2V communications with a wider capacity and longer transmission distances. In this study, we theoretically analyze the performance of cellular-relay (CR) V2V communications, in which a vehicle first transmits a message to its nearest BS via uplink, and then, the BS forwards the message to a destination vehicle via downlink. We model the road segments through a Poisson line process and the positions of vehicles through a Poisson point process on the roads; subsequently, we derive a theoretical expression for the probability of a successful CR transmission and the mean local delay in the CR transmission. Furthermore, we propose a performance metric to evaluate the difference between the performances of CR and direct V2V communications, and it can be applied to automatic transmission mode selection (i.e., CR or direct) for V2V communications. We evaluate the analytical results using numerical examples and demonstrate the impacts of various system parameters on the performances of CR and direct V2V communications. [ABSTRACT FROM AUTHOR]

We introduce a general framework for the modeling and analysis of vehicular networks by defining street systems as random 1D subsets of $\mathbb {R}^{2}$. The street system, in turn, specifies the random intensity measure of a Cox process of vehicles, i.e., vehicles form independent 1D Poisson point processes on each street. Models in this Coxian framework can characterize streets of different lengths and orientations forming intersections or T-junctions. The lengths of the streets can be infinite or finite and mutually independent or dependent. We analyze the reliability of communication for different models, where reliability is the probability that a vehicle at an intersection, a T-junction, or a general location can receive a message successfully from a transmitter at a certain distance. Further, we introduce a notion of equivalence between vehicular models, which means that a representative model can be used as a proxy for other models in terms of reliability. Specifically, we prove that the Poisson stick process-based vehicular network is equivalent to the Poisson line process-based and Poisson lilypond model-based vehicular networks, and their rotational variants. [ABSTRACT FROM AUTHOR]

This paper develops a framework to study multi-hop relaying in a vehicular network consisting of vehicles and Road Side Units (RSUs), and the effect of this relaying on the network coverage and the communication delay. We use a stochastic geometry model that consists of a combination of Poisson Line Process (PLP) and 1D Poisson Point Process (PPP) to reliably characterize the vehicular network layout and the locations of the vehicles and the RSUs. Using this model, we analyze the effect of the different network parameters on the coverage provided by the RSUs to the vehicles. Then, we investigate how the uncovered vehicles can receive their intended packets by relaying them through multiple hops that form connected paths to the RSUs. We also analyze the delay introduced to packet delivery due to multi-hop relaying. Namely, we present results that illustrate the coverage gains achieved through multi-hop relaying and the delays induced. Such results could be used by network planners and operators to decide on the different configurations and operational parameters of the vehicular network to suit particular scenarios and objectives. [ABSTRACT FROM AUTHOR]

In this paper, we present the downlink coverage and rate analysis of a cellular vehicle-to-everything (C-V2X) communication network where the locations of vehicular nodes and road side units (RSUs) are modeled as Cox processes driven by a Poisson line process (PLP) and the locations of cellular macro base stations (MBSs) are modeled as a 2D Poisson point process (PPP). Assuming a fixed selection bias and maximum average received power based association, we compute the probability with which a typical receiver, an arbitrarily chosen receiving node, connects to a vehicular node or an RSU and a cellular MBS. For this setup, we derive the signal-to-interference ratio (SIR)-based coverage probability of the typical receiver. One of the key challenges in the computation of coverage probability stems from the inclusion of shadowing effects. As the standard procedure of interpreting the shadowing effects as random displacement of the location of nodes is not directly applicable to the Cox process, we propose an approximation of the spatial model inspired by the asymptotic behavior of the Cox process. Using this asymptotic characterization, we derive the coverage probability in terms of the Laplace transform of interference power distribution. Further, we compute the downlink rate coverage of the typical receiver by characterizing the load on the serving vehicular nodes or RSUs and serving MBSs. We also provide several key design insights by studying the trends in the coverage probability and rate coverage as a function of network parameters. We observe that the improvement in rate coverage obtained by increasing the density of MBSs can be equivalently achieved by tuning the selection bias appropriately without the need to deploy additional MBSs. [ABSTRACT FROM AUTHOR]

In designing a network to link n points in a square of area n, we might be guided by the following two desiderata. First, the total network length should not be much greater than the length of the shortest network connecting all points. Second, the average route length (taken over source-destination pairs) should not be much greater than the average straight-line distance. How small can we make these two excesses? Speaking loosely, for a nondegenerate configuration, the total network length must be at least of order n and the average straight-line distance must be at least of order n1/2, so it seems implausible that a single network might exist in which the excess over the first minimum is o(n) and the excess over the second minimum is o(n1/2). But in fact we can do better: for an arbitrary configuration, we can construct a network where the first excess is o(n) and the second excess is almost as small as O (log n). The construction is conceptually simple and uses stochastic methods: over the minimum-length connected network (Steiner tree) superimpose a sparse stationary and isotropic Poisson line process. Together with a few additions (required for technical reasons), the mean values of the excess for the resulting random network satisfy the above asymptotics; hence, a standard application of the probabilistic method guarantees the existence of deterministic networks as required (speaking constructively, such networks can be constructed using simple rejection sampling). The key ingredient is a new result about the Poisson line process. Consider two points a distance r apart, and delete from the line process all lines which separate these two points. The resulting pattern of lines partitions the plane into cells; the cell containing the two points has mean boundary length approximately equal to 2r + constant(log r). Turning to lower bounds, consider a sequence of networks in [0√n-]2 satisfying a weak equidistribution assumption. We show that if the first excess is O (n) then the second excess cannot be o(√log n). © Applied Probability Trust 2008.

In this paper, we consider a vehicular network in which the wireless nodes are located on a system of roads. We model the roadways, which are predominantly straight and randomly oriented, by a Poisson line process (PLP) and the locations of nodes on each road as a homogeneous 1D Poisson point process. Assuming that each node transmits independently, the locations of transmitting and receiving nodes are given by two Cox processes driven by the same PLP. For this setup, we derive the coverage probability of a typical receiver, which is an arbitrarily chosen receiving node, assuming independent Nakagami- $m$ fading over all wireless channels. Assuming that the typical receiver connects to its closest transmitting node in the network, we first derive the distribution of the distance between the typical receiver and the serving node to characterize the desired signal power. We then characterize coverage probability for this setup, which involves two key technical challenges. First, we need to handle several cases as the serving node can possibly be located on any line in the network and the corresponding interference experienced at the typical receiver is different in each case. Second, conditioning on the serving node imposes constraints on the spatial configuration of lines, which requires careful analysis of the conditional distribution of the lines. We address these challenges in order to characterize the interference experienced at the typical receiver. We then derive an exact expression for coverage probability in terms of the derivative of Laplace transform of interference power distribution. We analyze the trends in coverage probability as a function of the network parameters: line density and node density. We also provide some theoretical insights by studying the asymptotic characteristics of coverage probability. [ABSTRACT FROM AUTHOR]

The analysis of point patterns with linear structures is of interest in many applications. To detect anisotropy in such patterns, in particular in the case of a columnar structure, we introduce a functional summary statistic, the cylindrical K-function, which is a directional K-function whose structuring element is a cylinder. We further introduce a class of anisotropic Cox point processes, called Poisson line cluster point processes. The points of such a process are random displacements of Poisson point processes defined on the lines of a Poisson line process. Parameter estimation for this model based on moment methods or Bayesian inference is discussed in the case where the underlying Poisson line process is latent. To illustrate the proposed methods, we analyse two- and three-dimensional point pattern datasets. The three-dimensional dataset is of particular interest as it relates to the minicolumn hypothesis in neuroscience, which claims that pyramidal and other brain cells have a columnar arrangement perpendicular to the surface of the brain. [ABSTRACT FROM AUTHOR]

Consider the following random spatial network: in a large disk, construct a network using a stationary and isotropic Poisson line process of unit intensity. Connect pairs of points using the network, with initial/final segments of the connecting path formed by travelling off the network in the opposite direction to that of the destination/source. Suppose further that connections are established using 'near geodesics', constructed between pairs of points using the perimeter of the cell containing these two points and formed using only the Poisson lines not separating them. If each pair of points generates an infinitesimal amount of traffic divided equally between the two connecting near geodesics, and if the Poisson line pattern is conditioned to contain a line through the centre, then what can be said about the total flow through the centre? In Kendall (2011) it was shown that a scaled version of this flow has asymptotic distribution given by the 4-volume of a region in 4-space, constructed using an improper anisotropic Poisson line process in an infinite planar strip. Here we construct a more amenable representation in terms of two 'seminal curves' defined by the improper Poisson line process, and establish results which produce a framework for effective simulation from this distribution up to an L1 error which tends to 0 with increasing computational effort. [ABSTRACT FROM AUTHOR]

At each point of a Poisson point process of intensity λ in the hyperbolic plane, center a ball of bounded random radius. Consider the probability Pr that, from a fixed point, there is some direction in which one can reach distance r without hitting any ball. It is known (see Benjamini, Jonasson, Schramm and Tykesson (2009)) that if λ is strictly smaller than a critical intensity λgv then Pr does not go to 0 as r → ∞. The main result in this note shows that in the case λ = λgv, the probability of reaching a distance larger than r decays essentially polynomially, while if λ > λgv, the decay is exponential. We also extend these results to various related models and we finally obtain asymptotic results in several situations. [ABSTRACT FROM AUTHOR]

Vehicular networks can aid in traffic monitoring, autonomous driving, and car accidents prevention. Yet, the deployment of these networks has been delayed due to the limited spectrum, especially for the case of unlicensed operations. To handle this issue, the Federal Communications Commission (FCC) proposed to permit Wi-Fi devices to operate in the 5.9 GHz band allocated to the intelligent transportation system (ITS). In a recent work, we analyzed the impact of the coexistence of dedicated short range communications (DSRC) and Wi-Fi on future DSRC network deployments by developing a stochastic geometry analytical model that considers a dynamic medium access probability (MAP) of DSRC nodes which uses carrier sense multiple access with collision avoidance (CSMA/CA). This previous work was based on the standard 2D homogeneous Poisson Point Process (PPP) model. In this work, we model the roads using the more applicable but more complex Poisson line process (PLP) Cox point process. We generate performance metrics represented through coverage probability and area system throughput, and we compare these results to our earlier work. The importance of this work is two-fold. First, it allows a further understanding of the impact of DSRC-Wi-Fi coexistence on future DSRC network deployments, and second, it highlights the effectiveness of the PLP in modeling the distribution of vehicles in an area by producing more accurate performance results. [ABSTRACT FROM AUTHOR]

For the Poisson line process the empirical polygon is defined thanks to ergodicity and laws of large numbers for some characteristics, such as the number of edges, the perimeter, the area, etc... One also has, still thanks to the ergodicity of the Poisson line process, a canonical relation between this empirical polygon and the polygon containing a given point. The aim of this paper is to provide numerical simulations for both methods: in a previous paper (Paroux, Advances in Applied Probability, 30:640–656, ) one of the authors proved central limit theorems for some geometrical quantities associated with this empirical Poisson polygon, we provide numerical simulations for this phenomenon which generates billions of polygons at a small computational cost. We also give another direct simulation of the polygon containing the origin, which enables us to give further values for empirical moments of some geometrical quantities than the ones that were known or computed in the litterature. [ABSTRACT FROM AUTHOR]

Using results for the distribution of perimeters of random polygons arising from random lines in a plane, we obtain new analytic approximations to the distributions of areas and local line densities for random polygons and compute various limiting properties of random polygons. Using simulation, we show that the lengths of adjacent sides of polygons generated by random line processes in the plane are correlated with ρ=0.616±0.001. [ABSTRACT FROM AUTHOR]

Abstract: In order to optimize the acoustic properties of a stacked fiber non-woven, the microstructure of the non-woven is modeled by a macroscopically homogeneous random system of straight cylinders (tubes). That is, the fibers are modeled by a spatially stationary random system of lines (Poisson line process), dilated by a sphere. Pressing the non-woven causes anisotropy. In our model, this anisotropy is described by a one parametric distribution of the direction of the fibers. In the present application, the anisotropy parameter has to be estimated from 2d reflected light microscopic images of microsections of the non-woven. After fitting the model, the flow is computed in digitized realizations of the stochastic geometric model using the lattice Boltzmann method. Based on the flow resistivity, the formulas of Delany and Bazley predict the frequency-dependent acoustic absorption of the non-woven in the impedance tube. Using the geometric model, the description of a non-woven with improved acoustic absorption properties is obtained in the following way: First, the fiber thicknesses, porosity and anisotropy of the fiber system are modified. Then the flow and acoustics simulations are performed in the new sample. These two steps are repeated for various sets of parameters. Finally, the set of parameters for the geometric model leading to the best acoustic absorption is chosen. [Copyright &y& Elsevier]