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In this paper, we present the success probability analysis of cooperative cellular vehicle-to-everything (C-V2X) communications, i.e., cellular-relay V2X communications. We model the spatial layout of macro base stations (MBSs) as a 2D Poisson point process (PPP) and roads as a Poisson line process (PLP), with road wireless nodes (including vehicles and roadside units) modeled as a 1D PPP on each road. For a typical source node, we calculate the signal-to-interference ratio (SIR)-based success probability of transmitting a packet to its closest destination node with the assist of its nearest MBS. We take into account three cooperative transmission schemes and derive their expressions of joint success probability in two consecutive phases considering the correlation of road topology and nodes' locations, respectively. We verify the accuracy of our analytical results through Monte-Carlo simulations. In addition, we explore the impacts of several key parameters on the success probability and discuss the selection of transmission schemes.

In the classical form, the Poisson Multi-Bernoulli Mixture (PMBM) filter uses a PMBM density to describe target birth, surviving, and death, which does not model the appearance of spawned targets. Although such a model can handle target birth, surviving, and death well, its performance may degrade when target spawning arises. The reason for this is that the original PMBM filter treats the spawned targets as birth targets, ignoring the surviving targets’ information. In this paper, we propose a Kullback-Leibler Divergence (KLD) minimization based derivation for the PMBM prediction step, including target spawning, in which the spawned targets are modeled using a Poisson Point Process (PPP). Furthermore, to improve the computational efficiency, three approximations are used to implement the proposed algorithm, such as the Variational Multi-Bernoulli (VMB) filter, the Measurement-Oriented marginal MeMBer/Poisson (MOMB/P) filter, and the Track-Oriented marginal MeMBer/Poisson (TOMB/P) filter. Finally, simulation results demonstrate the validity of the proposed filter by using the spawning model in these three approximations.

We consider random graphs G built on a homogeneous Poisson point process on R d , d ≥ 2 , with points x marked by i.i.d. random variables E x . Fixed a symmetric function h ( ⋅ , ⋅ ) , the vertexes of G are given by points of the Poisson point process, while the edges are given by pairs { x , y } with x ≠ y and | x − y | ≤ h ( E x , E y ) . We call G Poisson h -generalized Boolean model, as one recovers the standard Poisson Boolean model by taking h ( a , b ) ≔ a + b and E x ≥ 0 . Under general conditions, we show that in the supercritical phase the maximal number of vertex-disjoint left–right crossings in a box of size n is lower bounded by C n d − 1 apart from an event of exponentially small probability. As special applications, when the marks are non-negative, we consider the Poisson Boolean model and its generalization to h ( a , b ) = ( a + b ) γ with γ > 0 , the weight-dependent random connection models with max-kernel and with min-kernel and the graph obtained from the Miller–Abrahams random resistor network in which only filaments with conductivity lower bounded by a fixed positive constant are kept.

In this article, we present a new approach for managing the interference on cellular users through optimum mode selection between half-duplex (HD) and in-band full-duplex (IBFD) in such a way that device-to-device (D2D) throughput is maximized, while the quality of service (QoS) of the cellular users is guaranteed in terms of delay. To present a comprehensive view and analyse of the proposed approach in a large network, we use Poisson point process that enables us to model a large network with random parameters in such a way that is highly compatible with reality. Also, to model the cellular users’ delay, we use the queuing theory and Markov processes. Unlike other related works, the mode selection between HD and IBFD is considered as a decision variable and its optimal value is obtained. The results show that in comparison with the related works, our proposed approach leads to improvements in the D2D throughput, while through proper interference management, the impact on the cellular users’ throughput is negligible. Moreover, the QoS of the cellular users is guaranteed by keeping the delay below a certain threshold.

Persistent homology captures the appearances and disappearances of topological features such as loops and cavities when growing disks centered at a Poisson point process. We study extreme values for the lifetimes of features dying in bounded components and with birth resp. death time bounded away from the threshold for continuum percolation and the coexistence region. First, we describe the scaling of the minimal lifetimes for general feature dimensions, and of the maximal lifetimes for cavities in the Čech filtration. Then, we proceed to a more refined analysis and establish Poisson approximation for large lifetimes of cavities and for small lifetimes of loops. Finally, we also study the scaling of minimal lifetimes in the Vietoris-Rips setting and point to a surprising difference to the Čech filtration.

The Poisson multi-Bernoulli mixture (PMBM) filter is conjugate prior composed of the union of a Poisson point process (PPP) and a multi-Bernoulli mixture (MBM). Considering that the single model is not enough to guarantee stable tracking performance for maneuvering targets, in this article, a multiple model PMBM (MM-PMBM) filter is proposed to cope with this problem. The proposed MM-PMBM filter extends the single-model PMBM filter recursion to multiple motion models by exploiting the jump Markov system (JMS). The performance of the proposed algorithm is examined from two scenarios with different detection probabilities. Moreover, the robustness of Markovian model transition probability matrices (TPMs) for the proposed MM-PMBM filter is also explored. The simulation results demonstrate that the proposed MM-PMBM filter performs well in terms of the tracking accuracy, including the target states and cardinality estimates, and also has good tolerance with respect to different TPMs.

For a powered Poisson process, we define Signal-to-Interference-plus-Noise Ratio(SINR) and thesinr network as a Telecommunication Network. We define the Empirical Measures (empirical powered measur...

As one of the key technologies in 5th generation mobile networks, device-to-device (D2D) communication technology can significantly improve spectrum utilization, throughput and energy efficiency as well as reduce congestion in the networks. However, due to the spectrum sharing between D2D users (DUs) and cellular users (CUs), it may also cause serious interferences. In this paper, we adopt appropriate stochastic geometric tools to model and analyze the mutual interferences, based on which the coverage probabilities of different users are derived to evaluate the network performance. Specifically, we model the locations of CUs as a Poisson point process (PPP), while the locations of DUs are modelled as a Poisson hole process (PHP), which is driven by the PPP of CUs’ locations due to the exclusion between the positions of CUs and DUs. Furthermore, the approximated expressions of user coverage probabilities are obtained by two methods, which are consistent with the simulation results. The first method is to dissolve the hole, and the second method is to approximate the PHP through a thinned PPP. In addition, the user ergodic data rates and the sum data rate of the networks are also derived through the second approach. The impacts of key parameters (such as the density of DUs and the size of the warning radius) on the ergodic rates and sum data rate are analyzed.

This letter characterizes the exact distribution function of a typical Voronoi area in a $K$ -tier Poisson network. The users obey a generalized association (GA) rule, which is a superset of nearest base station association and maximum received power based association (with arbitrary fading) rules that are commonly adopted in the literature. Combining the Robbins’ theorem and the probability generating functional of a Poisson point process, we obtain the exact moments of a typical $k$ -th tier Voronoi area, $k\in \{1, {\dots },K\}$ under the GA rule. We apply this result in several special cases. For example, we prove that in multi-tier networks with the GA rule, the mean of $k$ th tier Voronoi area can exactly be expressed in a closed-form. We also obtain simplified expressions of its higher-order moments for both average and instantaneous received power based user association. In single-tier networks with exponential fading, the later association rule provides closed-form expression of the second-order moment of a typical Voronoi area. We numerically evaluate this exact expression and compare it with an approximated result.

In this paper, a novel Poisson hole process (PHP) modeling of wireless networks is proposed. Contrary to the prior PHP models with circular-shaped holes, we utilized circular sector holes in a random direction to capture the spatial separation between tiers in a millimeter wave (mmWave) heterogeneous cellular network (HCN). In this case, small cell base stations and macrocell base stations are distributed as a PHP and Poisson point process (PPP). Using tools from stochastic geometry, we derive approximate analytical expressions by regarding the effect of one or several holes on coverage probability. Simulation results reveal that compared to conventional PPP-modeling of HCNs, the proposed approaches can provide about 2–3 dB more accurate analysis in terms of signal-to-interference-and-noise ratio coverage probability. Moreover, some interesting insights about the effect of holes on coverage probability and the relation between proposed hole configuration and prior models with circular holes is discovered. It turns out that the analysis based on the proposed PHP model can provide a more general study than prior works.

The spatial correlation of the RF-powered nodes that harvest enough energy plays a critical role in the performance evaluation of large-scale wirelessly powered communication networks (WPCNs). However, such correlation is mostly ignored in the literature for analytical tractability. In this paper, we explore the correlation and propose a new point process, named energized point process (EPP), as a model for the RF-powered nodes that succeed in harvesting energy. Specifically, we focus on the spatial correlation of the energy harvested from a Poisson field of RF power sources. Two energy harvesting models with different degrees of practicality are introduced to concretize the EPP with the aim of visualizing the point process of the energized nodes and characterizing its density and pair correlation function theoretically. It turns out that for both models the resulting process exhibits positive correlation. With the first-and second-order statistics of the EPP, we use a fitted Poisson cluster process to provide good approximations of the success probability and area spectral efficiency in the information transmission phase. Numerical results investigate the impacts of the key parameters related to the energy transfer on the spatial correlation of the EPP and the communication performance. An important conclusion is that although the Poisson point process has been so far the most widely used model for wirelessly powered networks, it is inadequate due to the positive energy correlation.

The primary approach to increase coverage and capacity in infrastructure-based wireless networks is densification. Densification, however, presents a major challenge when serving mobile users, which is the overhead associated with the increased rate of base station handovers. Assuming a prior knowledge of a mobile’s trajectory and base stations’ locations, we formulate the problem of determining the sequence of handovers that optimize the trade-off between the mobile user’s perceived throughput and handover overheads in noise-limited environments. Under appropriate conditions, we show that the problem reduces to determining a maximum weight path in a directed acyclic graph induced by the mobile user’s trajectory. In practice, knowledge of a mobiles’ trajectory may be limited and one may also want to limit the handover complexity, whence we propose a new class of mobility-driven greedy association policies. The greedy policies are based on defining a handover support set, which constrains both the possible handovers and the complexity/information requirements. In a setting where base station locations follow a Poisson point process, we show that the performance of such handover processes follows a continuous-time Markov process which can be analyzed using complex variable techniques. This enables one to explore the optimization size/shape of the handover support set for mobility-driven greedy handover strategies and their relative performance compared to traditional association policies.

Primary Health Centres (PHC) are the first referral health facilities for Indonesian people to seek treatment. The varying distribution of PHC location in central Surabaya, therefore, causes its process to follow the Non-homogeneous Poisson Point Process (NHPP). We use Bayesian analysis coupled with Markov Chain Monte Carlo (MCMC) to model the mixture Poisson regression on NHPP intensity estimation. The result shows that two mixture components are significantly involved in the model along with four variables; i.e., the total population, the number of clean households, Accessibility Index, and the length of road; that produce the smallest Deviance Information Criteria (DIC). Keywords: Bayesian Analysis; Mixture Poisson Regression; NHPP Intensity; Primary Health Centres

Owing to its unparalleled tractability, the Poisson point process (PPP) has emerged as a popular model for the analysis of cellular networks. Considering a stationary point process of users, which is independent of the base station (BS) point process, it is well known that the typical user does not lie in the typical cell and thus it may not truly represent the typical cell performance. Inspired by this observation, we present a construction that allows a direct characterization of the downlink performance of the typical cell. For this, we present an exact downlink analysis for the 1-D case and a remarkably accurate approximation for the 2-D case. Several useful insights about the differences and similarities in the two viewpoints (typical user vs. typical cell) are also provided.

We consider transmission of packets over queue-length sensitive unreliable links, where packets are randomly corrupted through a noisy channel whose transition probabilities are modulated by the queue-length. The goal is to characterize the capacity of this channel. We particularly consider multiple-access systems, where transmitters dispatch encoded symbols over a system that is a superposition of continuous-time $GI_k/GI/1$ queues. A server receives and processes symbols in order of arrivals with queue-length dependent noise. We first determine the capacity of single-user queue-length dependent channels. Further, we characterize the best and worst dispatch processes for $GI/M/1$ queues and the best and worst service processes for $M/GI/1$ queues. Then, the multiple-access channel capacity is obtained using point processes. When the number of transmitters is large and each arrival process is sparse, the superposition of arrivals approaches a Poisson point process. In characterizing the Poisson approximation, we show that the capacity of the multiple-access system converges to that of a single-user $M/GI/1$ queue-length dependent system, and an upper bound on the convergence rate is obtained. This implies that the best and worst server behaviors of single-user $M/GI/1$ queues are preserved in the sparse multiple-access case., Comment: final version, to appear in IEEE Open Journal of the Communications Society

In this article, we analyze the signal-to-interference ratio (SIR) coverage probability of a multi-cell downlink system with random traffic arrivals. Based on a theoretical model, in which the base stations (BSs) and devices are deployed as independent Poisson point processes (PPPs), we first present the sufficient and necessary conditions of the ε-stable region for the queueing system. Then, by taking into account the impact of spatially queueing interactions among BSs, we focus on the steady status of the queues and introduce a two-queue-length approximated model for the system. Specifically, in the studied new model, the BSs are separated into two sets in the steady state: the long-queue BS set and the short-queue BS set. The locations of the former are modeled by a Neyman-Scott process and those of the latter are modeled as a residual hole process. By applying the first-order statistic approximation, we further approximate the hole process by a homogeneous PPP with the same density. To model the deployment of the BSs precisely, the related parameters in the approximated model are fitted. Using tools from stochastic geometry and queueing theory, we derive the SIR coverage probability in the steady state. An iterative algorithm is proposed to calculate the active probability of the BSs. To reduce the computational complexity, Beta distribution is applied to approximate the probability density function of the service rate in each iteration of the algorithm. Finally, the effect of the fitting parameters and the accuracy of our analysis are presented via Monte Carlo simulations.

In this paper, we provide degree distributions for $AB$ random geometric graphs, in which points of type $A$ connect to the closest $k$ points of type $B$. The motivating example to derive such degree distributions is in 5G wireless networks with multi-connectivity, where users connect to their closest $k$ base stations. It is important to know how many users a particular base station serves, which gives the degree of that base station. To obtain these degree distributions, we investigate the distribution of area sizes of the $k-$th order Voronoi cells of $B$-points. Assuming that the $A$-points are Poisson distributed, we investigate the amount of users connected to a certain $B$-point, which is equal to the degree of this point. In the simple case where the $B$-points are placed in an hexagonal grid, we show that all $k$-th order Voronoi areas are equal and thus all degrees follow a Poisson distribution. However, this observation does not hold for Poisson distributed $B$-points, for which we show that the degree distribution follows a compound Poisson-Erlang distribution in the 1-dimensional case. We then approximate the degree distribution in the 2-dimensional case with a compound Poisson-Gamma degree distribution and show that this one-parameter fit performs well for different values of $k$. Moreover, we show that for increasing $k$, these degree distributions become more concentrated around the mean. This means that $k$-connected $AB$ random graphs balance the loads of $B$-type nodes more evenly as $k$ increases. Finally, we provide a case study on real data of base stations. We show that with little shadowing in the distances between users and base stations, the Poisson distribution does not capture the degree distribution of these data, especially for $k>1$. However, under strong shadowing, our degree approximations perform quite good even for these non-Poissonian location data., 23 pages, 13 figures

The noise in daily infection counts of an epidemic should be super-Poissonian due to intrinsic epidemiological and administrative clustering. Here, we use this clustering to classify the official national SARS-CoV-2 daily infection counts and check for infection counts that are unusually anti-clustered. We adopt a one-parameter model of $\phi'_i$ infections per cluster, dividing any daily count $n_i$ into $n_i/\phi'_i$ 'clusters', for 'country' $i$. We assume that $n_i/\phi'_i$ on a given day $j$ is drawn from a Poisson distribution whose mean is robustly estimated from the four neighbouring days, and calculate the inferred Poisson probability $P'_{ij}$ of the observation. The $P'_{ij}$ values should be uniformly distributed. We find the value $\phi_i$ that minimises the Kolmogorov-Smirnov distance from a uniform distribution. We investigate the $(\phi_i, N_i)$ distribution, for total infection count $N_i$. We find that most of the daily infection count sequences are inconsistent with a Poissonian model. Most are found to be consistent with the $\phi_i$ model. The 28-, 14- and 7-day least noisy sequences for several countries are best modelled as sub-Poissonian, suggesting a distinct epidemiological family. The 28-day least noisy sequence of Algeria has a preferred model that is strongly sub-Poissonian, with $\phi_i^{28} < 0.1$. TJ, TR, RU, BY, AL, AE, and NI have preferred models that are also sub-Poissonian, with $\phi_i^{28} < 0.5$. A statistically significant ($P^{\tau} < 0.05$) correlation was found between the lack of media freedom in a country, as represented by a high Reporters sans frontieres Press Freedom Index (PFI$^{2020}$), and the lack of statistical noise in the country's daily counts. The $\phi_i$ model appears to be an effective detector of suspiciously low statistical noise in the national SARS-CoV-2 daily infection counts., Comment: 30 pages, 17 figures, 12 tables; zenodo.5262698 at https://zenodo.org/record/5262698, live git at https://codeberg.org/boud/subpoisson, archived git at https://archive.softwareheritage.org/swh:1:rev:086b14906401f8484258cead5bc7152439a686ee ; v2: comparison with alternative methods and correlation with PFI, data updated to May 2021; v3: refs added, version accepted in PeerJ

We generalise disagreement percolation to Gibbs point processes of balls with varying radii. This allows to establish the uniqueness of the Gibbs measure and exponential decay of pair correlations in the low activity regime by comparison with a sub-critical Boolean model. Applications to the Continuum Random Cluster model and the Quermass-interaction model are presented. At the core of our proof lies an explicit dependent thinning from a Poisson point process to a dominated Gibbs point process., 23 pages, 0 figure Correction, from the published version, of the proof of Section 4

Owing to its flexibility in modeling real-world spatial configurations of users and base stations (BSs), the Poisson cluster process (PCP) has recently emerged as an appealing way to model and analyze heterogeneous cellular networks (HetNets). Despite its undisputed relevance to HetNets—corroborated by the models used in the industry—the PCP’s use in performance analysis has been limited. This is primarily because of the lack of analytical tools to characterize the performance metrics, such as the coverage probability of a user connected to the strongest BS. In this paper, we develop an analytical framework for the evaluation of the coverage probability, or equivalently the complementary cumulative density function (CCDF) of signal-to-interference-and-noise ratio ( SINR ), of a typical user in a $K$ -tier HetNet under a $\max $ power-based association strategy, where the BS locations of each tier follow either a Poisson point process (PPP) or a PCP. The key enabling step involves conditioning on the parent PPPs of all the PCPs, which allows us to express the coverage probability as a product of sum-product and probability generating functionals (PGFLs) of the parent PPPs. In addition to several useful insights, our analysis provides a rigorous way to study the impact of the cluster size on the ${\it SINR}$ distribution, which was not possible using the existing PPP-based models.

International audience; The employment of stochastic geometry for the analysis and design of ultra dense networks (UDNs) has provided significant insights into network densification. In addition to the characterization of the network performance and behavior, these tools can also be exploited toward solving complex optimization problems that could maximize the capacity benefits arising in UDNs. However, this is preconditioned on the existence of tractable closed form expressions for the considered figures of merit. In this course, the present paper introduces an accurate approximation for the moment generating function (MGF) of the aggregate other-cell interference created by base stations whose positions follow a Poisson point process of given spatial density. Given the pivotal role of the MGF of the aggregate interference in stochastic geometry and the tractability of the derived MGF, the latter can be employed to substantially simplify ensuing stochastic geometry analyses. Subsequently, the present paper employs the introduced MGF to provide closed form expressions for the downlink ergodic capacity for the interference limited case, and validates the accuracy of these expressions by the use of extensive Monte Carlo simulations. The derived expressions depend on the density of users and base stations, setting out a densification road map for network operators and designers of significant value.

Compared to the standard success (coverage) probability, the meta distribution of the signal-to-interference ratio (SIR) provides much more fine-grained information about the network performance. We consider general heterogeneous cellular networks (HCNs) with base station tiers modeled by arbitrary stationary and ergodic non-Poisson point processes. The exact analysis of non-Poisson network models is notoriously difficult, even in terms of the standard success probability, let alone the meta distribution. Hence we propose a simple approach to approximate the SIR meta distribution for non-Poisson networks based on the ASAPPP ("approximate SIR analysis based on the Poisson point process") method. We prove that the asymptotic horizontal gap $G_0$ between its standard success probability and that for the Poisson point process exactly characterizes the gap between the $b$th moment of the conditional success probability, as the SIR threshold goes to $0$. The gap $G_0$ allows two simple approximations of the meta distribution for general HCNs: 1) the per-tier approximation by applying the shift $G_0$ to each tier and 2) the effective gain approximation by directly shifting the meta distribution for the homogeneous independent Poisson network. Given the generality of the model considered and the fine-grained nature of the meta distribution, these approximations work surprisingly well., This paper has been accepted in the IEEE Transactions on Communications. 14 pages, 13 figures

Let $U_1,U_2,\ldots$ be random points sampled uniformly and independently from the $d$-dimensional upper half-sphere. We show that, as $n\to\infty$, the $f$-vector of the $(d+1)$-dimensional convex cone $C_n$ generated by $U_1,\ldots,U_n$ weakly converges to a certain limiting random vector, without any normalization. We also show convergence of all moments of the $f$-vector of $C_n$ and identify the limiting constants for the expectations. We prove that the expected Grassmann angles of $C_n$ can be expressed through the expected $f$-vector. This yields convergence of expected Grassmann angles and conic intrinsic volumes and answers thereby a question of B\'ar\'any, Hug, Reitzner and Schneider [Random points in halfspheres, Rand. Struct. Alg., 2017]. Our approach is based on the observation that the random cone $C_n$ weakly converges, after a suitable rescaling, to a random cone whose intersection with the tangent hyperplane of the half-sphere at its north pole is the convex hull of the Poisson point process with power-law intensity function proportional to $\|x\|^{-(d+\gamma)}$, where $\gamma=1$. We compute the expected number of facets, the expected intrinsic volumes and the expected $T$-functional of this random convex hull for arbitrary $\gamma>0$., Comment: 31 pages, 2 figures

In this paper, we propose a multi-tier mmWave cellular framework where sub-6 GHz macro BSs (MBSs) are assumed as a Poisson point process (PPP) and small-cell BSs (SBSs), operating on either mmWave or sub-6 GHz, follows non-uniform Poisson cluster point (PCP) model. This paper proposes both centralized and distributed user association algorithms. For the centralized two-step algorithm, we aim to maximize the sum rate while satisfying quality of service (QoS) and power consumption constraints based on eigenvalue analysis. Then, we derive the association probability, the coverage probability, and the average achievable rate, cosidering directivity and blockage effect, by stochastic geometry. On this basis, a distributed user association algorithm is proposed. The simulation results demonstrate the accuracy of our theoretical analysis and also reveal the effect of some parameters on the network performance. In addition, the proposed centralized algorithm can achieve near-optimal sum rate with a low complexity.

This paper focuses on the coverage probability and ergodic capacity for millimeter wave (mmWave) user-centric dense networks, where multiple access points (APs) consist of a virtual cell for each user equipment and transmit data with mmWave antennas cooperatively. All APs are distributed according to a homogeneous Poisson point process. Different from the low-frequency band (below 3 GHz), blockages have a non-negligible effect on mmWave band. To illustrate the effect, we utilize a line-of-sight probability function, which is dependent on the link-length. Then, via stochastic geometry, the expressions for coverage probability and ergodic capacity are derived, which accounts for: blockages, different small-scale fading distributions (Nakagami, Rayleigh, and no fading), and AP cooperation. In addition, we deduce the approximate expressions for coverage probability and ergodic capacity by using the noise-limited approximation. The numerical results validate our analytical expressions and show that the AP cooperation can provide high coverage performance and distinct capacity gain in a lower-AP-density region.

Due to recent developments in the cellular communication system, stochastic process implementation is necessary. The cellular communication system exhibits random patterns in various domains, thereby compelling the utilization of stochastic processes to achieve an optimal output. User behaviors with respect to the variable geographical pattern, population density, architecture, data usage, and mobility over various cells are random in nature. Therefore, the stochastic-geometry-based Poisson point process (PPP) technique can be implemented to accurately analyze these random processes in device-to-device (D2D)-based cooperative cellular networks. The stochastic modeling entails the consideration of transmitters and receivers as the elements of stochastic point processes. The hexagonal method is not applicable for the implementation of heterogeneous network topologies, as it is not suitable for topologies, in which the cell size is not fixed. Therefore, a randomly designed heterogeneous network uses stochastic geometry as a viable solution for predicting the probabilistic parameters, including the cell interference, load distribution, coverage probability, base station (BS) mapping, and signal-to-interference-plus-noise ratio (SINR). Moreover, as a network architecture that is based on relay nodes (RNs), cellular and D2D users can be utilized in the domain of homogeneous random models. The associated phenomenon can be considered independent and Poisson. In this paper, the stochastic-geometric-based PPP approach is introduced for modeling the SINR, success probability, ergodic capacity, and outage probability for the D2D-enabled cooperative cellular network. The proposed PPP realistic model utilizes BS, RN, the cellular user (CU), and D2D user positioning method to design an interference-free network. The success probability, ergodic capacity, and outage probability for cellular and D2D users are used as metrics for evaluating the results with respect to various SINR threshold values and node densities. Moreover, the total success probability, ergodic capacity, and outage probability have been calculated for various multiple-input-multiple-output (MIMO) antenna configurations to validate the results. The results confirmed that the proposed PPP model approach outperforms the grid model and conventional multi-antenna ultra-dense network (UDN) approaches.

This paper focuses on a comprehensive framework design and performance analysis for a large-scale hotspot communication scenario where there exist large number of clustered pico base stations (P-BSs) and femto BSs (F-BSs) working on millimeter-Wave (mm-Wave). For such realistic hotspots’ deployment, to exploit the coupling between user equipments (UEs) and base stations (BSs), we model the geographical centers of UE hotspots as independent PPP around which UEs, P-BSs and F-BSs are scattered and form independent and non-homogeneous Poisson cluster processes (PCPs). However, with clustered P-BSs and F-BSs, the UEs would experience the severer intra-cluster interference so that the performance is degraded greatly. To overcome this problem in clustered networks, an effective interference management scheme is proposed by using the ratio of the first and second nearest distances of a typical UE from P-BSs so that a typical UE can be classified as either cluster-center UEs (CCUEs) or cluster-edge UEs (CEUEs). Correspondingly, the total available spectrum band is divided as CCUE-band and CEUE-band, which are occupied by P-BSs associated with CCUEs and CEUEs, respectively. Unlike P-BSs, the F-BSs randomly access the CEUE-band and CCUE-band with a given access portion factor. By using the method from stochastic geometry and the distinguishing feature of mm-Wave, such as directional beamforming with sectored antenna model and modified line of sight ball model for blockage, the UE association probability and Laplace transforms of interference are derived. This yields the derivations of downlink transmission rates. The numerical results show that the proposed clustered heterogeneous scheme with the cluster-UE classification outperforms the traditional ones without the UE classification. This approach not only captures the UE-BS coupling and the non-homogeneous nature of multi-tier networks, but also restricts the severe intra-cluster interference due to clustered BSs.

Requirements and technological advancements towards 6th generation (6G) wireless networks lead to enabling and development of massive low Earth orbit (LEO) satellite constellations to provide ubiquitous and high-capacity connectivity, particularly for maritime and airborne platforms. Consequently, new methodologies to study the performance of LEO networks are of great importance. In this paper, we derive both downlink and uplink analytical expressions for coverage probability and data rate of a massive inclined LEO constellation under general shadowing and fading. We model the LEO satellite network as a nonhomogeneous Poisson point process with general intensity in order to take into account uneven distribution of satellites along the latitudes. The results provided in this study facilitate the stochastic evaluation and design of the future massive LEO networks, regardless of satellites' exact trajectories in orbits. acceptedVersion

In UAV-enabled mmWave networks, the locations of UAVs are usually modeled by a Poisson point process or a Poisson cluster process in an infinite area. However, some typical scenarios merely deploy a fixed number of UAVs in a finite area such as hotspot areas and disaster response. To capture these characteristics, we propose a 3D UAV-enabled mmWave network model in a finite area where UAVs follow a Binomial point process (BPP). Under this setup, we derive the expression of the coverage probability for the target user at an arbitrary location rather than the region center, and further provide simplified analysis in three special cases including interference-limited case, noise-limited case and the case with the target user located at the region center. Numerical results show that the location of the target user has a significant impact on coverage probability, and a maximum coverage probability can be obtained by adjusting the height and number of UAVs.

Multi-object tracking and multi-sensor fusion algorithms are used in many areas of science and technology. For example, in an advanced field such as autonomous vehicles, multi-object tracking and multi-sensor fusion are indispensable technologies. The purpose of these technologies is to processing noisy measurements from different sensors to determine the number and state of objects with high accuracy. Some of the algorithms frequently used in multi-object tracking applications are Global Nearest Neighbor Filter (GNN), Joint Probabilistic Data Association Filter (JPDA), Multiple Hypothesis Tracking (MHT), Probability Hypothesis Density Filter (PHD), and other algorithms. These algorithms are very effective for estimating the state of objects and can combine measurements from multiple sensors to improve estimation accuracy [1,2]. But an even more powerful tool for multi-object tracking is the Poisson multi-Bernoulli mixture filter (PMBM Filter) [1]. In this paper, we will introduce the PMBM filter and simulate an algorithm using a PMBM filter that combines measurements from multiple automotive sensors to estimates object’s states. At the same time, we also compare the performance of the aforementioned algorithms in some specific simulation scenarios.

In this correspondence, we study the physical layer security in a stochastic unmanned aerial vehicles (UAVs) network from a network-wide perspective, where the locations of UAVs are modeled as a Mat$\acute{\text{e}}$rn hard-core point process (MHCPP) to characterize the minimum safety distance between UAVs, and the locations of users and eavesdroppers are modeled as a Poisson cluster process and a Poisson point process, respectively. UAVs adopt zero-forcing precoding to serve multiple ground users and emit artificial noise to combat eavesdropping. We derive the approximations for the coverage probability and secrecy probability of a typical user, with which we derive the secrecy throughput of the whole network. Numerical results show the analytical results can well approximate the simulation results. Impacts of parameters on the secrecy performance are shown., 6 pages, 5 figures

The models of random sets introduced in this chapter are based on the Poisson point process, used as seeds for the Boolean RACS, which gives good simulations of two phase connected media, for which a percolation threshold can be estimated. Properties of this model and examples of application with proper identification of its parameters are given. Generalizations include Poisson varieties (which are Poisson point processes in appropriate spaces), to build fibre or strata Boolean models, a multi-scale version based on the Cox point process, and a multi component version.

Models of stochastic point processes are basic models of random sets with a rather simple geometry. The archetype model is the Poisson point process, which simulates a population of particles without any spatial interaction. It can be used as seeds for other models developed in other chapters, called grain models. Variants of the Poisson point process use constraints like repulsion, as for the hard-core process. Multi scale models are obtained by means of Cox processes. Iterated Poisson varieties provide models with clusters like alignments on lines or in planes. A short introduction to Gibbs and to Determinantal point processes is given. Finally models of random trees are obtained by branching processes.

Many spatial statistics methods require neighbourhood structures such as the one determined by a Voronoi tessellation, so understanding statistical properties of Voronoi cells is crucial. While distributions of cell properties when data locations follow an unbounded homogeneous Poisson process have been studied, little attention has been given to how these properties change when a boundary is imposed. This is important when geographical data are gathered within a restricted study area, such as a national boundary or a coastline. We study the effects of imposing a boundary on the cell properties of a Poisson Voronoi tessellation. The area, perimeter and number of edges of individual cells with and without boundary conditions are investigated by simulation. Distributions of these properties differ substantially when boundaries are imposed, and these differences are affected by proximity to the boundary. We also investigate how changes in such properties when boundaries are imposed vary over two-dimensional space.

The Dilution RF (DRF) combines primary RF located on Poisson points by addition. The Dilution RF is infinite divisible according to the addition operation. The scalar RF built on a Poisson point process and on Poisson varieties, as well as the multivariate case to simulate multispectral data are presented.

We construct unbiased estimators for the distribution of the number of points inside random stopping sets based on a Poisson point process. Our approach is based on moment identities for stopping sets, showing that the random count of points inside the complement S¯ of a stopping set S has a Poisson distribution conditionally to S. The proofs do not require the use of set-indexed martingales, and our estimators have a lower variance when compared to standard sampling. Numerical simulations are presented for examples such as the convex hull and the Voronoi flower of a Poisson point process, and their complements. Ministry of Education (MOE) Submitted/Accepted version This research is supported by the Ministry of Education, Singapore, under its Tier 1 Grant MOE2018-T1-001-201 RG25/18.

We define a random model for the moments of the new eigenfunctions of a point scatterer on a two-dimensional rectangular flat torus. In the deterministic setting, Seba conjectured these moments to be asymptotically Gaussian, in the semi-classical limit. This conjecture was disproved by Kurlberg–Ueberschar on Diophantine tori. In our model, we describe the accumulation points in distribution of the randomized moments, in the semi-classical limit. We prove that asymptotic Gaussianity holds if and only ifs some function, modeling the multiplicities of the Laplace eigenfunctions, diverges to $$+\infty $$ .

We study a class of reinforcement models involving a Poisson process on the vertices of certain infinite graphs G. When a vertex fires, one of the edges incident to that vertex is selected. The edge selection is biased towards edges that have been selected many times previously, and a parameter α governs the strength of this bias. We show that for various graphs (including all graphs of bounded degree), if α 1 (the very strong reinforcement regime) then the random subgraph consisting of edges that are ever selected by this process does not percolate (all connected components are finite). Combined with results appearing in a companion paper, this proves that on these graphs, with α sufficiently large, all connected components are in fact trees. If the Poisson firing rates are constant over the vertices, then these trees are of diameter at most 3. The proof of nonpercolation relies on coupling with a percolation-type model that may be of interest in its own right.

Small-cell networks (SCNs), especially those operating in millimeter-wave bands, are sensitive to blockages. In this letter, we develop a three-dimensional (3D) SCN model considering blockages to investigate the impact of base-station (BS) height, BS density and blockage density on the downlink coverage probability. More specifically, we model the blockages as cylinders whose locations follow a Poisson point process and model the locations of BSs as a Poisson hole process. We assume that all the BSs are of the same height and the blockage height follows an exponential distribution. Based on the 3D SCN model, we derive the exact integral expression of coverage probability for general SCNs and the closed-form expression of coverage probability for ultra-dense SCNs. Our analytical results are verified to be reliable through simulations. The numerical results quantify the impact of the blockage density and the BS height on the coverage probability. For a small blockage density, elevated BSs always degrade the coverage probability, while the coverage probability first increases and then decreases with the BS height when the blockage density becomes sufficiently large.

Caching popular contents in advance is an important technique to achieve low latency and reduced backhaul congestion in future wireless communication systems. In this article, a multi-cell massive multi-input-multi-output system is considered, where locations of base stations are distributed as a Poisson point process. Assuming probabilistic caching, average success probability (ASP) of the system is derived for a known content popularity (CP) profile, which in practice is time-varying and unknown in advance. Further, modeling CP variations across time as a Markov process, reinforcement $Q$ -learning is employed to learn the optimal content placement strategy to optimize the long-term-discounted ASP and average cache refresh rate. In the $Q$ -learning, the number of $Q$ -updates are large and proportional to the number of states and actions. To reduce the space complexity and update requirements towards scalable $Q$ -learning, two novel (linear and non-linear) function approximations-based $Q$ -learning approaches are proposed, where only a constant (4 and 3 respectively) number of variables need updation, irrespective of the number of states and actions. Convergence of these approximation-based approaches are analyzed. Simulations verify that these approaches converge and successfully learn the similar best content placement, which shows the successful applicability and scalability of the proposed approximated $Q$ -learning schemes.

In this paper, we consider a realistic K tier heterogeneous cellular network (HetNet) consisting of two types of coexisting tiers: Poisson cluster process (PCP) based tiers with base-stations (BSs) and users (UE) distributed as PCP and are coupled, and Poisson point process (PPP) based tier with BSs and UEs distributed as mutually-independent PPPs. We derive the spectral efficiency (SE) for a typical user belonging to one of the tier under max-power association and open access. An important intermediate step involves deriving expressions for the Laplace transforms (LT) of total received power and interference which also opens new avenues for computation of other performance metrics. We also investigate the impact of clustering over the performance of both type of users via numerical simulation and provide important design insights.

Given data from a Poisson point process with intensity $(x,y) \mapsto n \mathbf{1}(f(x)\leq y),$ frequentist properties for the Bayesian reconstruction of the support boundary function $f$ are derived. We mainly study compound Poisson process priors with fixed intensity proving that the posterior contracts with nearly optimal rate for monotone and piecewise constant support boundaries and adapts to H\"older smooth boundaries with smoothness index at most one. We then derive a non-standard Bernstein-von Mises result for a compound Poisson process prior and a function space with increasing parameter dimension. As an intermediate result the limiting shape of the posterior for random histogram type priors is obtained. In both settings, it is shown that the marginal posterior of the functional $\vartheta =\int f$ performs an automatic bias correction and contracts with a faster rate than the MLE. In this case, $(1-\alpha)$-credible sets are also asymptotic $(1-\alpha)$-confidence intervals. As a negative result, it is shown that the frequentist coverage of credible sets is lost for linear functions indicating that credible sets only have frequentist coverage for priors that are specifically constructed to match properties of the underlying true function., Comment: The first version of arXiv:1703.08358 has been expanded and rewritten. We decided to split it in two separate papers, a new version of arXiv:1703.08358 and this article

Under the standard multiple target tracking models and a Poisson point process birth model, the Poisson multi-Bernoulli mixture (PMBM) filter provides the closed-form recursion to computing the posterior density over the set of targets. Without approximations, the PMBM computational complexity rapidly rises in time due to the increasing number of data association hypotheses. This paper presents innovative strategies for merging Bernoulli components for the same potential target reducing the number of single-target hypotheses in the PMBM filter, aiming to lower its computational complexity while keeping its performance high. We use several measures to compute the similarity between different Bernoulli components. Simulation results show that the proposed algorithms show performance close to the PMBM filter without Bernoulli merging, as measured by the generalized optimal sub-pattern assignment (GOSPA) metric, with a significantly reduced execution time.

International audience; In this paper we study splittings of a Poisson point process which are equivariant under a conservative transformation. We show that, if the Cartesian powers of this transformation are all ergodic, the only ergodic splitting is the obvious one, that is, a collection of independent Poisson processes. We apply this result to the case of a marked Poisson process: under the same hypothesis, the marks are necessarily independent of the point process and i.i.d. Under additional assumptions on the transformation, a further application is derived, giving a full description of the structure of a random measure invariant under the action of the transformation.

We investigate the maximal degree in a Poisson–Delaunay graph in $\mathbf{R}^{d}$, $d\geq 2$, over all nodes in the window $\mathbf{W}_{\rho }:=\rho^{1/d}[0,1]^{d}$ as $\rho $ goes to infinity. The exact order of this maximum is provided in any dimension. In the particular setting $d=2$, we show that this quantity is concentrated on two consecutive integers with high probability. A weaker version of this result is discussed when $d\geq 3$.

This article develops models and algorithms for continuous-discrete multiple target filtering, in which the multi-target system is modelled in continuous time and measurements are available at discrete time steps. In order to do so, this paper first proposes a statistical model for multi-target appearance, dynamics and disappearance in continuous time, based on continuous time birth/death processes and stochastic differential equations. The multitarget state is observed at known time instants based on the standard measurement model, and the objective is to compute the distribution of the multi-target state at these time steps. For the Wiener velocity model, we derive a closed-form formula to obtain the best Gaussian Poisson point process fit to the birth density based on Kullback-Leibler minimisation. The resulting discretised model gives rise to the continuous-discrete Gaussian Poisson multi-Bernoulli mixture (PMBM) filter, the continuous-discrete Gaussian mixture probability hypothesis density (PHD) filter and the continuous-discrete Gaussian mixture cardinality PHD (CPHD) filter. The proposed filters are specially useful for multi-target estimation when the time interval between measurements is non-uniform.

To overcome the shortcomings of Dedicated Short Range Communications (DSRC), cellular vehicle-to-everything (C-V2X) communications have been proposed recently, which has a variety of advantages over traditional DSRC, including longer communication range, broader coverage, greater reliability, and smooth evolution path towards 5G. In this paper, we consider an LTE-based C-V2X communications network in irregular Manhattan grids. We model the macrobase stations (MBSs) as a 2D Poisson point process (PPP) and model the roads as a Manhattan Poisson line process (MPLP), with the roadside units (RSUs) modeled as a 1D PPP on each road. As an enhancement architecture to DSRC, C-V2X communications include vehicle-to-vehicle (V2V) communication, vehicle-to-infrastructure (V2I) communication, vehicle-to-pedestrian (V2P) communication, and vehicle-to-network (V2N) communication. Since the spectrum for PC5 interface in 5.9 GHz is quite limited, cellular networks could share some channels to V2I links to improve spectral efficiency. Thus, according to Maximum Power-based Scheme, we adopt the stochastic geometry approach to compute the signal-to-interference ratio- (SIR-) based success probability of a typical vehicle that connects to an RSU or an MBS and the area spectral efficiency of the whole network over shared V2I and V2N downlink channels. In addition, we study the asymptotic characteristics of success probability and provide some design insights according to the impact of several key parameters on success probability.

We study an inhomogeneous random connection model in the connectivity regime. The vertex set of the graph is a homogeneous Poisson point process $\mathcal{P}_s$ of intensity $s>0$ on the unit cube $S=\left(-\frac{1}{2},\frac{1}{2}\right]^{d},$ $d \geq 2$ . Each vertex is endowed with an independent random weight distributed as $W$, where $P(W>w)=w^{-\beta}1_{[1,\infty)}(w)$, $\beta>0$. Given the vertex set and the weights an edge exists between $x,y\in \mathcal{P}_s$ with probability $\left(1 - \exp\left( - \frac{\eta W_xW_y}{\left(d(x,y)/r\right)^{\alpha}} \right)\right),$ independent of everything else, where $\eta, \alpha > 0$, $d(\cdot, \cdot)$ is the toroidal metric on $S$ and $r > 0$ is a scaling parameter. We derive conditions on $\alpha, \beta$ such that under the scaling $r_s(\xi)^d= \frac{1}{c_0 s} \left( \log s +(k-1) \log\log s +\xi+\log\left(\frac{\alpha\beta}{k!d} \right)\right),$ $\xi \in \mathbb{R}$, the number of vertices of degree $k$ converges in total variation distance to a Poisson random variable with mean $e^{-\xi}$ as $s \to \infty$, where $c_0$ is an explicitly specified constant that depends on $\alpha, \beta, d$ and $\eta$ but not on $k$. In particular, for $k=0$ we obtain the regime in which the number of isolated nodes stabilizes, a precursor to establishing a threshold for connectivity. We also derive a sufficient condition for the graph to be connected with high probability for large $s$. The Poisson approximation result is derived using the Stein's method., Comment: 21 pages, calculations are simplified significantly and results are proved under much weaker conditions

We derive transport-entropy inequalities for mixed binomial point processes, and for Poisson point processes. We show that when the finite intensity measure satisfies a Talagrand transport inequality, the law of the point process also satisfies a Talagrand type transport inequality. We also show that a Poisson point process (with arbitrary ${\sigma}$-finite intensity measure) always satisfies a universal transport-entropy inequality \`a la Marton. We explore the consequences of these inequalities in terms of concentration of measure and modified logarithmic Sobolev inequalities. In particular, our results allow one to extend a deviation inequality by Reitzner [31], originally proved for Poisson random measures with finite mass., Comment: 33 pages, comments are welcome