Hard-core Point Process Research Articles

Analysis of Coverage and Capacity for UAV-Aided Networks with Directional mmWave Communications

Millimeter wave (mmWave) unmanned aerial vehicle (UAV)-aided networks have enormous application potential due to their large bandwidth and ultra-high speed, being regarded as an effective technology for improving the reliability of military and civilian fields. However, due to their complex electromagnetic spectrum environment and the sensitivity of mmWaves to blocking effects, its performance analysis faces certain difficulties. This article investigates the coverage and network capacity of mmWave UAV-aided networks under significant blocking effects and complex electromagnetic environments; for this purpose, we equipped each UAV with mmWave antennas featuring adjustable beamwidth and direction. A Matérn hard-core point process (MHCPP) with repulsion constraints was also employed to reflect the minimum distance constraints to isolate the mutual interference between UAVs. Then, using a stochastic geometric analysis, we derived the coverage and capacity characteristics and further obtained a closed-form expression for the network coverage probability. Finally, the simulation results showed that the network throughput could reach 86% when the density of UAVs was half of that of ground base stations (GBSs) in the city center, validating the efficiency and accuracy of our theoretical derivations.

Multi-RIS Aided Multi-User Mmwave MIMO System Modelled by Matérn Hard-Core Point Processes

Multi-RIS Aided Multi-User Mmwave MIMO System Modelled by Matérn Hard-Core Point Processes

On a special class of gibbs hard-core point processes modeling random patterns of non-overlapping grains

Inspired by issues of formal kinetics in materials science, we consider a class of processes with density with respect to an inhomogeneous finite Poisson point process, which may be regarded as a generalization of the classical Strauss hard-core process. We prove expressions for the intensity measure and the void probabilities, together with upper and lower bounds. A discussion on some special cases of interest, links with literature and a comparison between Matérn I and Strauss hard-core process are also provided. We apply such a special class of point processes in modeling patterns of non-overlapping grains and in the study of the mean volume density of particular birth-and-growth processes.

Satellite-Aerial Communications With Multi-Aircraft Interference

Satellite-aerial communication (SAC) is envisioned as a fundamental component of the sixth-generation (6G) wireless networks. Motivated by its importance, we investigate a SAC system including a geostationary satellite (S), a target aircraft (TA), and a set of interfering aircraft (IA). Specifically, TA sends signals to S in the presence of IA interference. Considering the trajectory, hierarchy, and safety distance of the aircraft’s flight routes, we propose a novel three-dimensional stacked Poisson line hardcore point process. That is, we introduce safety distances to the stacked Poisson line Cox process in order to describe the locations of IA in the sky. We also propose two approximations, namely, the equi-dense model and the discretization model, to maintain the tractability of the analysis. To this end, the uplink coverage probability is studied by using the two proposed mathematical models. Moreover, we investigate the coverage probability of the aviation use case with predefined flight altitudes. Finally, numerical results and Monte Carlo simulations are presented to validate the accuracy of the proposed analysis.

Cooperative Satellite-Aerial-Terrestrial Systems: A Stochastic Geometry Model

Nowadays, satellite and aerial platforms are playing an important role in realizing global seamless wireless coverage. In this paper, a cooperative satellite-aerial-terrestrial network (SATN) is considered, in which two kinds of relaying links, satellite and aerial relaying links, are used to assist a group of aerial terminals to forward their information to a remote terrestrial destination (D). Specifically, we model these aerial platforms sharing the same frequency band as a Matérn hard-core point process type-II. Also, a group of aerial jammers at D’s side is modeled as a Poisson point process. To demonstrate the end-to-end (e2e) performance of the two relaying links, the statistical characteristics of the received signal-to-interference are characterized and then a closed-form expression for the outage probability (OP) over the uplink from the aerial source to the satellite/the aerial relay, the downlink from the satellite/the aerial relay to D, and the inter-aerial relay link are derived. Numerical results are presented to verify the proposed analysis models and compare the outage performance of the considered cooperative SATN with the two relay links under numerous scenarios.

Stochastic Geometry Analysis of Sojourn Time in RF/VLC Hybrid Networks

The spectrum scarcity in the radio frequency (RF) communication in indoor environments motivates the integration of an alternative technology like visible light communication (VLC) with the existing RF architecture that results in a hybrid RF/VLC network. While VLC helps offloading the congested RF spectrum by offering capacity-per-area improvements, the resulting heterogeneity and narrow coverage areas of optical base stations (BSs) impose several challenges for user mobility such as unnecessary handovers. To help addressing these challenges, in this paper, we derive the mean and the distribution of sojourn time in RF/VLC hybrid networks. The mathematical analysis conducted in this paper makes use of the tools from stochastic geometry and abstracting the BSs’ locations via two independent homogeneous Poisson point processes (PPPs). Since PPP modeling is yet to be well established for RF/VLC hybrid networks, we compare the PPP based analytical results to those obtained for an actual deployment, a Matérn hard-core point process (MHCPP) based deployment, and a deterministic square lattice deployment of VLC luminaries. Furthermore, we utilize the sojourn time distribution to calculate the unnecessary handover probability. Our numerical results show the interplay between the sojourn time and the receiver field of view as a function of BS density and they highlight the cost of BS densification in terms of unnecessary handovers.

Multi-UAV Enabled Aerial-Ground Integrated Networks: A Stochastic Geometry Analysis

Multiple unmanned aerial vehicles (UAVs) can function as aerial base stations to provide flexible and reliable communication services for massive ground devices (GDs). It is quite a challenging task to analyze such multi-UAV networks when considering practical mutually exclusive relationships among UAVs. Based on the tools of stochastic geometry, we in this paper develop a theoretical framework for modeling and analyzing aerial networks with UAVs following Matérn hard-core point process (MHCPP). As the tractable probability generating functional (PGFL) of repulsive point processes is unavailable, we employ an approximate approach based on the Poisson point process to analyze the cumulative interference and the signal-to-interference ratio (SIR) of a typical GD. By considering both line-of-sight (LOS) and none-line-of-sight (NLOS) communications, we obtain the approximation expressions of the network coverage probability and average rate. Finally, extensive simulation results are presented to validate the efficiency and accuracy of our proposed framework.

Spectral efficiency analysis of distributed millimeter wave massive MIMO system using spatial point processes

This paper considers a downlink distributed millimeter wave massive multi-input multi-output (D-mmMIMO) system, consisting of multiple randomly distributed radio access units (RAUs), the distribution of which is modeled as either a Poisson point process (PPP) or a Matérn hard-core point process (MHCPP). Compared with the PPP model where RAUs are randomly distributed without restriction, the MHCPP model constructed from a PPP by dependently thinning is more realistic, since it imposes a minimum distance between two RAUs so that they are not too close to each other. When a hybrid precoding algorithm based on antenna array response vectors is adopted, the upper bounds on average spectral efficiencies of the UEs for both the PPP D-mmMIMO system and MHCPP D-mmMIMO system are derived. Finally, simulation results are provided to demonstrate the validity of the developed analysis. The presented results also show that, when the MHCPP is generated from the PPP used in the PPP D-mmMIMO system, the MHCPP D-mmMIMO system with a low hard-core distance achieves a quite close performance to the PPP D-mmMIMO system, and increasing hard-core distance has a negative impact on spectral efficiency. In contrast, when the MHCPP is constructed from a different PPP, which system performs better depends on the intensities of two PPPs as well as hard core distance.

Stochastic Analysis of Cooperative Satellite-UAV Communications

This paper considers a dual-hop cooperative satellite-unmanned aerial vehicle (UAV) communication system including a satellite (S) and a group of cluster headers (CHs) which are respectively with a group of uniformly distributed UAVs. Specifically, the CHs serve as aerial decode-and-forward relays to forward the information transmitted by the S to the UAVs. Moreover, free-space optical (FSO) and radio frequency (RF) technologies are respectively adopted over S-CH and CH-UAV links to exploit the high directivity of FSO over long-distance transmission and the omnidirectional coverage ability of RF. The positions of the CHs in the three-dimensional space follow the Matérn hard-core point processes type-II, in which each CH cannot be closer to another by less than a predefined distance. Three different cases over CH-UAV links are considered during the performance modeling: interference-free, interference-dominated, and interference-and-noise. Then, the coverage performance of the S-CH link and the CH-UAV link in the aforementioned three cases is studied and the closed-form analytical expressions of the coverage probability (CP) over both two links are derived. Additionally, the asymptotic expressions for the CP over S-CH link and CH-UAV link in the interference-free case are derived. Finally, numerical results are provided to validate our proposed analytical models and thus some meaningful conclusions are achieved.

Stochastic Geometry Analysis of User Mobility in RF/VLC Hybrid Networks

The integration of visible light communication (VLC) with existing radio frequency (RF) networks has emerged as a new network architecture to meet the rapidly growing traffic demand. The resulting RF/VLC hybrid network structures offer capacity-per-area improvements due to the use of two technologies operating at different frequency bands and the relatively higher base station (BS) density. However, the reduced BS coverage footprints and the heterogeneous BS types result in challenges for user mobility such as frequent handovers and the need for suitable BS association policies. To help addressing these challenges, in this paper, we conduct a user mobility analysis for RF/VLC hybrid networks by deriving the user-to-BS association probabilities and handover rates. The analysis makes use of stochastic geometry and modeling BSs’ locations via a Poisson point process (PPP). Since PPP modeling has not yet been well established for hybrid RF/VLC networks, we support the applicability of our approach by comparing the user mobility performance to those obtained for an actual deployment, a Matérn hard-core point process (MHCPP) based deployment, and a deterministic square lattice deployment of VLC luminaries. Furthermore, since the handover rates directly depend on the association policies, we consider two popular association policies. Our numerical results show that the PPP, the MHCPP, the square lattice, and the actual deployments have comparable performance in terms of handover rates regardless of the association policy, and they highlight the tradeoff between balancing network load and handover rates achieved by the association policies.

Outage in Motorway Multi-Lane VANETs With Hardcore Headway Distance Using Synthetic Traces

In this paper we analyze synthetic mobility traces generated for three-lane unidirectional motorway traffic to find that the locations of vehicles along a lane are better modeled by a hardcore point process instead of the widely-accepted Poisson point process (PPP). In order to capture the repulsion between successive vehicles while maintaining a level of analytical tractability, we make a simple extension to PPP: We model the inter-vehicle distance along a lane equal to the sum of a constant hardcore distance and an exponentially distributed random variable. We calculate the J-function and the Ripley's K-function for this hardcore point process. We fit its parameters to the available traces, and we illustrate that the higher the average speed along a lane, the more prominent the hardcore component becomes. In addition, we consider a transmitter-receiver link on the same lane, and we generate simple formulae for the moments of interference under reduced Palm measure for that lane, and without conditioning for other lanes. We illustrate that under Rayleigh fading a shifted-gamma approximation for the distribution of interference per lane provides a very good fit to the simulated outage probability using the synthetic traces, while the fit using the PPP is poor.

Secure UAV Random Networks With Minimum Safety Distance

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.

A Scalar Version of the Caflisch‐Luke Paradox

AbstractConsider an infinite cloud of hard spheres sedimenting in a Stokes flow in the whole space ℝd. Despite many contributions in fluid mechanics and applied mathematics, there is so far no rigorous definition of the associated effective sedimentation velocity. Calculations by Caflisch and Luke in dimension d = 3 suggest that the effective velocity is well‐defined for hard spheres distributed according to a weakly correlated and dilute point process, and that the variance of the sedimentation speed is infinite. This constitutes the Caflisch‐Luke paradox. In this contribution, we consider a scalar version of this problem that displays the same difficulties in terms of interaction between the differential operator and the randomness, but is simpler in terms of PDE analysis. For a class of hardcore point processes we rigorously prove that the effective velocity is well‐defined in dimensions d > 2 and that the variance is finite in dimensions d > 4, confirming the formal calculations by Caflisch and Luke, and opening a way to the systematic study of such problems . © 2020 Wiley Periodicals LLC

Performance of Indoor VLC System Under Random Placement of LEDs With Nonimaging and Imaging Receiver

This article analyzes the performance of multiple-input-multiple-output indoor visible light communication (VLC) system by randomly deploying the light-emitting-diodes (LEDs) using Matern hardcore point process (MHCP). Furthermore, photodetectors (PDs) with two different field of view (FOVs) have been utilized for an imaging as well as nonimaging receiver structures. It is a widely known fact that in a conventional VLC system, signal-to-noise ratio (SNR) profile inside the room varies with respect to the LED placement and the PDs position. Consequently, the proposed work attempts to achieve uniform SNR across the room, by utilizing MHCP based LED placement at the transmitter, and a nonimaging receiver with four PDs using 1-FOV, 2-FOV, and imaging receiver configurations. Simulation results show that random deployment of LED using MHCP configuration results in a more uniform SNR profile inside the room as compared to regular LED deployment schemes. Furthermore, three different power allocation schemes for LEDs namely equal power, distance-based power, and an optimal power allocation are proposed. For the different power allocation schemes, the average SNR and the variance of the received optical power (OP) inside the room are derived and compared. In addition, the closed-form expression for the bit-error-rate (BER) probability is derived for the proposed MHCP configuration using <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">on</small> - <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">off</small> keying as a modulation scheme. Results show improvement in BER performance with the OP allocation in comparison to other power allocation schemes both for imaging and nonimaging receiver configuration.

Energy Efficiency Analysis and Optimization of Multiantenna Heterogeneous Cellular Networks Modeled by Matérn Hard-core Point Process

Energy Efficiency Analysis and Optimization of Multiantenna Heterogeneous Cellular Networks Modeled by Matérn Hard-core Point Process

Energy efficiency optimization for heterogeneous cellular networks modeled by Matérn hard-core point process

The Poisson point process (PPP) has been widely used in wireless network modeling and performance analysis due to the independence between its nodes. Therefore, it may not be a suitable model for many of the exclusive networks between the nodes. This paper analyzes the energy efficiency (EE) and optimizes the two-tier heterogeneous cellular networks (HetNets). Considering the mutual exclusion between macro base stations (MBSs) distribution, the deployment of MBSs is modeled by the Matern hard-core point process (MHCPP), and the deployment of pico base stations (PBSs) is modeled by the PPP. We adopt a simple approximation method to study the signal to interference ratio (SIR) distribution in two-tier MHCPP-PPP networks and then derive the coverage probabilities, the average data rates and the energy efficiency of HetNets. Finally, an optimization algorithm is proposed to improve the EE of HetNets by controlling the transmit power of PBSs. The simulation results show that the EE of a system can be effectively improved by selecting the appropriate transmit power for the PBSs. In addition, two-tier MHCPP-PPP HetNets have higher energy efficiency than two-tier PPP-PPP HetNets.

On Interference Dynamics in Matérn Networks

A thorough understanding of the temporal dynamics of interference in wireless networks is crucial for the design of communication protocols, scheduling, and interference management. This article applies stochastic geometry to investigate the interference dynamics in a network of nodes that use carrier sense multiple access (CSMA). This type of network is approximated by a Matérn hard-core point process of type II with Nakagami fading. We derive and analyze expressions for the variance, covariance, and correlation of the interference power at an arbitrary location. Results show that even though the commonly used Poisson approximation to CSMA may have the same average interference as the Matérn model, the interference dynamics behaves significantly different. In this way, this study reveals a pitfall in the modeling of wireless systems and contributes to the theory of interference calculus.

Performance Analysis of C-RAN with Different Deployment Distributions

In cloud radio access networks (C-RAN) for 5G and beyond, the user-centric system model has gained popularity during downlink performance analysis for various reasons like tractability of stochastic geometric approach and distributed beamforming methods. In this paper, the coverage analysis of C-RAN with different distributions of BSs, e.g. uniform, PPP, Matern hard-core point process (MHCPP) of types I, II, and III are studied. We present the coverage analysis with the uniform distribution of BSs and compare it with other deployment distributions of BSs. The effect of threshold distance on the performance of MHCPP distributions is explored. The Monte Carlo simulation of heterogeneous C-RAN with different cluster point processes is observed and compared with the PPP model. The effect of the range of child clusters on the performance analysis is observed in the cluster point processes. To avoid the complexity in the study, interference is not considered in the analysis, and its counterpart simulations are also not considered while performing the comparison.

Performance Analysis of Multi-Cell Full-Duplex Cellular Networks

We analyze the performance of a cellular network, where Poisson point process distributed half-duplex (HD) downlink (DL) and uplink (UL) users are served by multiple full-duplex (FD) base stations (BSs). To address the surge in interference in the network due to the simultaneous operation in time and frequency of the FD BSs, we (a) adopt a self-interference cancellation scheme at each BS, and (b) apply linear interference alignment in each cell to cancel the intra-cell interference. Further, to better capture the distribution of the FD BSs, we model the BSs as a Matérn hard-core point process, in which a minimum distance is imposed between points. The performance of both UL and DL users is analyzed by deriving general expressions and closed-form approximations for the outage probability and throughput. Next, simulations are carried out for both macro and micro cell environments under both FD and HD operations with respect to various network parameters. Our results reveal several fundamental characteristics and the necessary conditions required for the successful deployment of such networks.

Research on Resource Efficiency Optimization Model of TDMA-Based Distributed Wireless Ad Hoc Networks

Because of networking flexibility, strong robustness and expansibility, and low cost of operation and maintenance, distributed wireless ad hoc networks are widely used in ultra-density 5th-Generation (5G) networks, Device to Device (D2D) communications and industrial wireless sensor networks. However, compared with the centralized ones, the Time Division Multiple Access(TDMA)-based distributed network lacks centralized coordination by a control center, while its resource scheduling only depends on partial information, and the high receiving failure probability caused by neighbors’ cumulative interference outside the scope of maintenance may reduce the resource efficiency and restrict its usability according to the existing protocols severely. The simulation results show that the transmission failure probability can be up to 20% when maintaining two-hops neighbors in TDMA-based coordinated distributed scheduling. But the research on the cumulative interference outside the scope of maintenance and the optimal neighbor maintenance scope is still unclear at present. In this paper, we establish a resource efficiency model and an interference model of control messages and data messages under different neighbor maintenance by hardcore point process (HCPP) considering cumulative interference, cost of Media Access Control (MAC) layer and channel multiplex. Furtherly, we build an optimization model to maximize the resource efficiency under scheduling delay and receiving success probability constraints, thus we can obtain the optimal scope of maintenance in different network conditions. We conduct extensive theoretical analysis and simulations to evaluate the correctness of models. Simulation results show that the resource efficiency optimization model can provide guidance for optimal protocol parameters design.