Stochastic Geometry Research Articles

A SIC-Based BS Coordination Scheme for Full Duplex Cellular Networks

Full Duplex (FD) in cellular networks is expected to increase the cell spectral efficiency. However, while the downlink (DL) spectral efficiency (SE) increases with FD, the uplink (UL) SE decreases because of the Base Station to Base Station (BS) interference. In this paper, assuming a three-node model, we propose a method based on Successive Interference Cancellation (SIC) to reduce the BS-to-BS interference present in FD cellular networks. The approach consists in coordinating BSs to enable the decoding and the suppression of undesired signals that impair uplink transmissions. We analyze both distributed and Centralized Radio Access Networks (CRAN) architectures. Stochastic geometry is used to derive the coverage probability and mean data rate of the proposed scheme. In the distributed scenario, the FD UL average data rate is increased by 25% with our solution compared to a classical FD network, while our FD scheme still outperforms Half-Duplex (HD) on the DL. In the centralized scenario, our solution outperforms HD by 10% and classical FD by 78% on the UL, while preserving classical FD gains on the DL.

Stochastic Geometry Analysis of IRS-Assisted Downlink Cellular Networks

Using stochastic geometry tools, we develop a comprehensive framework to analyze the downlink performance of various types of users (e.g., users served by direct base station (BS) transmissions and indirect intelligent reflecting surface (IRS)-assisted transmissions) in a cellular network with multiple BSs and IRSs. For the proposed users, we provide the approximate expressions for the performance in terms of coverage probability, ergodic capacity, and energy efficiency (EE). The proposed stochastic geometry framework can capture the impact of channel fading, locations of BSs and IRSs, arbitrary phase-shifts and interference experienced by a typical user supported by direct transmission and/or IRS-assisted transmission. For IRS-assisted transmissions, we first model approximate the desired signal power from the nearest IRS as a sum of scaled generalized gamma (GG) random variables whose parameters are functions of the IRS phase shifts. Then, we derive the Laplace Transform (LT) of the received signal power in a closed form. Also, we approximate the aggregate interference from multiple IRSs as the sum of normal random variables. Then, we derive the LT of the aggregate interference from all IRSs and BSs. The derived LT expressions are used to calculate coverage probability, ergodic capacity, and EE for users served by direct BS transmissions as well as users served by IRS-assisted transmissions. Finally, we derive the overall network coverage probability, ergodic capacity, and EE based on the fraction of direct and IRS-assisted users, which is defined as a function of the deployment density of IRSs, as well as blockage probability of direct transmission links. Numerical results validate the derived analytical expressions and extract useful insights related to the number of IRS elements, large-scale deployment of IRSs and BSs, and the impact of IRS interference on direct transmissions.

Stochastic Geometry-Based Analysis of Cache-Enabled Hybrid Satellite-Aerial-Terrestrial Networks With Non-Orthogonal Multiple Access

Due to the emergence of non-terrestrial platforms with extensive coverage, flexible deployment, and reconfigurable characteristics, the hybrid satellite-aerial-terrestrial networks (HSATNs) can accommodate a great variety of wireless access services in different applications. To effectively reduce the transmission latency and facilitate the frequent update of files with improved spectrum efficiency, we investigate the performance of cache-enabled HSATN, where the user retrieves the required content files from the cache-enabled aerial node (AN) or the satellite with the non-orthogonal multiple access (NOMA) scheme. If the required content files of the user are cached in the AN, the cache-enabled node would serve directly. Otherwise, the user would retrieve the content file from the satellite system, where the satellite system seeks opportunities for proactive content pushing to ANs during the user content delivery phase. Specifically, taking into account the uncertainty of the number and location of ANs, along with the channel fading of terrestrial users, the outage probability and hit probability of the considered network are, respectively, derived based on stochastic geometry. Numerical results unveil the effectiveness of the cache-enabled HSATN with the NOMA scheme and proclaim the influence of key factors on the system performance. The realistic, tractable, and expandable framework, as well as associated methodology, provide both useful guidance and a solid foundation for evolved networks with advanced configurations in the performance of cache-enabled HSATN.

Performance Analysis of Statistical Priority-Based Multiple Access Network With Directional Antennas

This letter proposes an analytical framework to assess the network performance of a wireless network adopting statistical priority-based multiple access (SPMA) scheme with nodes equipping with directional antennas. By taking into account the spatial randomness of nodes and the impact of directional antenna, we first derive the analytical expressions of medium access probability (MAP), burst success probability, and packet success probability with tools from stochastic geometry. We then verify the correctness of the analytical framework by adopting three typical directional antenna models. By comparing with the scenario where nodes are equipped with omnidirectional antennas, we show the superiority of the directional antennas in limiting the aggregated network interference and enhancing the packet success probability.

Reliability Benefit of Location-Based Relay Selection for Cognitive Relay Networks

In this article, we develop an analytical framework to study the impact of location-based relay selection strategy on the reliability of cognitive relay networks. By utilizing the tool of stochastic geometry, we first derive a closed-form expression for the reliability-enhanced region (RER), where relaying transmission can achieve higher transmission reliability than direct transmission. Then, we adopt the normalized reliability gain (NRG) to quantify the reliability benefit obtained by using relaying transmission compared to direct transmission, and we obtain the spatial distribution of NRG in the RER. Subsequently, by taking the spatial random nature of relays’ distribution into account, we investigate the reliability benefit obtained by secondary networks with the optimal location-based relay selection (OLB-RS) strategy. To reduce the feedback overhead during relay selection, we propose a region-aware relay selection (RA-RS) strategy and obtain the achievable reliability benefit. The results indicate that the reliability is highly dependent on the location of relay, and the OLB-RS strategy is to select the relay closest to the midpoint between the corresponding secondary source and destination.

Binomial Line Processes: Distance Distributions

We introduce the binomial line process (BLP), a novel spatial stochastic model for the characterization of streets in the statistical evaluation of wireless and vehicular networks. Existing stochastic geometry models for streets, e.g., Poisson line processes (PLP) and Manhattan line processes (MLP) lack an important aspect of city-wide street networks: streets are denser in the city center and sparse near the suburbs. Contrary to these models, the BLP restricts the generating points of the streets to a fixed radius centered at the origin of the Euclidian plane, thereby capturing the inhomogeneity of the streets with respect to the distance from the center. We derive a closed-form expression for the contact distribution of the BLP from a random location on the plane. Leveraging this, we introduce the novel Binomial line Cox process (BLCP) to emulate points on individual lines of the BLP and derive the distribution to the nearest BLCP point from an arbitrary location. Using numerical results, we highlight that the spatial configuration of the streets is remarkably distinct from the perspective of a city center user to that of a suburban user. The framework developed in this paper can be integrated with the existing models of line processes for more accurate characterization of streets in urban and suburban environments.

Semi-Grant-Free NOMA: A Stochastic Geometry Model

Grant-free (GF) transmission holds promise in terms of low latency communication by directly transmitting messages without waiting for any permissions. However, collision situations may frequently happen when limited spectrum is occupied by numerous GF users. The non-orthogonal multiple access (NOMA) technique can be a promising solution to achieve massive connectivity and fewer collisions for GF transmission by multiplexing users in power domain. We utilize a semi-grant-free (semi-GF) NOMA scheme for enhancing network connectivity and spectral efficiency by enabling grant-based (GB) and GF users to share the same spectrum resources. With the aid of semi-GF protocols, uplink NOMA networks are investigated by invoking stochastic geometry techniques. We propose a novel <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dynamic protocol</i> to interpret which part of the GF users are allocated in NOMA transmissions via transmitting various channel quality thresholds by an added handshake. We utilize open-loop protocol with a fixed average threshold as the benchmark to investigate performance improvement. It is observed that dynamic protocol provides more accurate channel quality thresholds than open-loop protocol, thereby the interference from the GF users is reduced to a large extent. We analyze the outage performance and diversity gains under two protocols. Numerical results demonstrate that dynamic protocol is capable of enhancing the outage performance than open-loop protocol.

Design and Analysis of Coded Caching Schemes in Stochastic Wireless Networks

Coded caching is a technique that promises significant reductions in network traffic by exploiting the multicast opportunities among multiple cache-enabled users. Most works in this area investigate the fundamental performance limits of coded caching from an information-theoretic perspective. In this paper, from a practical perspective, we focus on the design and analysis of the coded caching scheme in large-scale stochastic networks with dynamic traffic. Aiming at the challenges of the large number of users, dynamic packet requests, and distance-dependent interference, the packet caching scheme and packet delivery scheme are jointly designed to ensure the availability of the coded caching gain. To characterize the coded caching performance, the queue dynamic of the packet requests and channel state are first analyzed. Then by combining the successive group decoding theory and stochastic geometry approach, the network interference is characterized and the successful transmission probability of the multicast coded signal is further derived. Moreover, the coded caching scheme is optimized to minimize the average delay of the packet request, which is proved to ensure service fairness and reduce the complexities of network performance analysis. The analytical and numerical results provide useful insights for the provisioning and planning of the coded caching network.

A stochastic confocal ellipsoid channel model for high speed railway MIMO communication systems

This paper investigates the channel model for multiple-input multiple-output (MIMO) communication systems in high speed railway (HSR) networks. Specially, a three-dimensional (3D) confocal stochastic geometry ellipsoid model is proposed for modeling the MIMO channels. Normalized space–time correlation function(ST CF), Spatial cross correlation function (CCF) and level crossing rate (LCR) of both theoretical and simulation model are have been derived and analyzed. As a 3D GBSM, the proposed confocal ellipsoid model has lower correlation in spatial cross-correlation function (CCF) compared with that of the corresponding two-dimensional (2D) ellipse model. Measurement data of different HSR scenarios verify the applicability of the confocal ellipsoid model.

A Stochastic Geometry Analysis for Energy-Harvesting-Based Device-to-Device Communication

The rapidly developing energy harvesting (EH) technology is a promising solution to the durability issue in the battery-powered Internet of Things (IoT) systems. In this article, underlaid device-to-device (D2D) transmission powered by radio signals harvested from cellular systems is studied. By considering the dilemmas among EH, D2D transmission opportunity, and interference management, we propose two transmission policies: 1) Policy 1 requires that the available power in the battery should be no less than the D2D transmission power and 2) Policy 2 not only sets the constraint on available power but also introduces the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">guard zone</i> rule to protect D2D transmissions from severe interference. The employment of a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">guard zone</i> in this situation is technically challenging since the original distribution of energy arrival will thereby be changed. We derive expressions in closed or semiclosed forms for the considered D2D transmission performance metrics with the stochastic geometry framework and Poisson hole process. With numerical simulation results, the influences of varying network parameters on D2D performances are illustrated. The results show that by introducing a guard zone, the D2D successful transmission rate can be increased by 41.2%. All the developed D2D frameworks and the summarized useful remarks are used to provide meaningful design insights and guidelines for the deployment strategies of EH-based D2D wireless networks.

An analytical model for quantifying the efficiency of traffic-data collection using instrumented vehicles

Emerging instrumented vehicles (IVs), when equipped with high-precision positioning devices (e.g., DGPS) and ranging sensors (e.g., radar, LIDAR, cameras), are capable of generating high-quality traffic data. This study evaluates the efficiency of such data-collection procedures; this remains largely unknown because variable penetration rates of IVs rarely arise in the real world. We propose an analytical model that establishes a quantitative relationship between the ratio of collected trajectory points to total traffic trajectory points (RCT) and the IV penetration rate, according to stochastic geometry theory. With this, the data-collection efficiency (DCE) of IVs can be effectively evaluated. A simulation approach is developed to generate IV data and thereby validate the proposed analytical model, using a comprehensive set of traffic scenarios; these data consist of eight micro-trajectory datasets from the next-generation simulation (NGSIM) program and four typical sensor IV deployments. The numerical analysis demonstrates that the model perfectly reflects the simulated IV data for all traffic scenarios. In addition, an analytical comparison of the DCEs for fixed sensors, probe vehicles, and IVs reveals that IVs are the most efficient method for collecting traffic data in road networks. This study proposes a theory to predict the percentage of trajectory points that can be collected by a certain percentage of IVs within the entire traffic flow.

An Evolutionary Game for Mobile User Access Mode Selection in sub-6 GHz mmWave Cellular Networks

By utilizing the combination of two powerful tools i.e., stochastic geometry (SG) and evolutionary game theory (EGT), in this paper, we study the problem of mobile user (MU) mode selection in heterogeneous sub-$6$ GHz/millimeter wave (mmWave) cellular networks. Particularly, by using SG tools, we first propose an analytical framework to assess the performance of the considered networks in terms of average signal-to-interference-plus-noise (SINR) ratio, average rate, and mobility-induced time overhead, for scenarios with user mobility{.} According to the SG-based framework, an EGT-based approach is presented to solve the problem of access mode selection. Specifically, two EGT-based models are considered, where for each MU its utility function depends on the average SINR and the average rate, respectively, while the time overhead is considered as a penalty term. A distributed algorithm is proposed to reach the evolutionary equilibrium, where the existence and stability of the equilibrium is theoretically analyzed and proved. Moreover, we extend the formulation by considering information delay exchange and evaluate its impact on the convergence of the proposed algorithm. Our results reveal that the proposed technique can offer better spectral efficiency and connectivity in heterogeneous sub-$6$ GHz/mmWave cellular networks with mobility, compared with the conventional access mode selection techniques.

A Spatiotemporal Model for Hard-Deadline Multistream Traffic in Uplink IoT Networks

Ultrareliable and delay-intolerant message delivery is one of the key components for Internet of Things (IoT) and cyber–physical systems to support real-time control and real-time interaction. In this article, we provide a spatiotemporal framework that captures the packet loss rate (PLR) for large-scale grant-free uplink IoT networks with multiple types of hard-deadline traffic. An independent and a prioritized packet scheduling schemes are proposed and investigated for efficiently realizing frequency diversity. Tools from stochastic geometry and queuing theory are utilized to account for the macroscopic coverage probability and microscopic PLR. The expressions of devices’ coverage, steady state distribution, and PLR of services are derived for both scheduling schemes. Detailed system-level simulations are used to identify network design guidelines. The results show that the independent scheme provides better network scalability and consumes lower average transmit power at the devices, while the prioritized scheme enhances the PLR performance of high priority service and requires lower peak transmit power of devices.

UAV Networks Through a Spatial Repulsion Model

In UAV networks, to avoid the collision and strong co-frequency interference caused by UAVs’ high agility and strong LOS links, spatial repulsion exists between UAVs, which has not been modeled in the current work. In this letter, by setting an adjustable repulsion zone for each UAV, a tractable model of UAV networks with spatial repulsion is considered and the impact of repulsion to cell selection and interference partitions under the air-to-ground channel is theoretically characterized, based on which the coverage performance is analyzed via stochastic geometry. Specifically, due to the serving potential of the closer NLOS UAV, cell selection is based on the average signal-to-interference ratio instead of distance, thus the distribution of the distance between serving UAV and the typical user with spatial repulsion needs to rederive. In addition, we obtain more accurate interference Laplace functions by quantifying the irregular LOS/NLOS interference partitioning with the repulsion. The results show that the coverage probability can be improved by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2.2\times $ </tex-math></inline-formula> with repulsion when the signal to interference ratio threshold equals −5dB.

Performance Analysis for AOA-Based Localization Under Millimeter-Wave Wireless Networks

Millimeter-Wave (mmWave) communication is a promising solution for achieving high data rate and low latency in 5G wireless networks. Since directional beamforming and antenna arrays are exploited in the mmWave networks, accurate angle-of-arrival (AOA) measurements can be obtained and utilized for localization purposes. In this work, we consider the AOA-based positioning for the mmWave networks using stochastic geometry and analyze how the Cramér-Rao lower bound (CRLB) is affected by the spatial distribution of nodes, including the target and participating anchors. In order to apply the CRLB on a network setting with random node locations, we propose an accurate approximation of the CRLB using the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\lceil L/4 \rceil $ </tex-math></inline-formula> -th value of the ordered distances where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> is the number of participating anchors. These findings provide us deep insight into optimum network design that meets specified localization requirements.

Caching and Multicasting for Fog Radio Access Networks

Cache-enabled Fog radio access network (F-RAN) is a promising technology to alleviate the traffic congestion and boost the contents delivery success rate. The efficiency of disseminating the cached contents in F-RAN can be boosted by enabling multicast service at the fog access points (F-APs). This paper proposes a joint random caching and multicasting optimization scheme for wireless backhauled F-RAN. Using tools from stochastic geometry, the expression of the successful transmission probability (STP) is derived by carefully analyzing the different types of serving F-APs and interferers. Then, a closed-form expression of the asymptotic STP in the high SNR region is derived to reduce the complexity. The joint caching and multicasting optimization problem is formulated to maximize the STP. The optimization problem is complex and non-convex in general. A novel projected cuckoo search algorithm (PCSA) is proposed to obtain the optimal content placement that maximizes the STP. The numerical simulation results show that PCSA outperforms the original cuckoo search algorithm (CSA) and the proposed asymptotically joint caching and multicasting scheme outperforms the benchmark caching schemes by up to 15% higher STPs.

Stochastic Geometry-Based Analysis of the Impact of Underlying Uncorrelated IoT Networks on LoRa Coverage

IoT networks are more and more present nowadays. Some IoT protocols share the same bandwidth leading to interference on neighboring networks and decrease of overall coverage. To contribute to this problem, an analytical study of the coverage of a LoRa network with underlying uncoordinated IoT networks for uplink transmissions is presented in this paper. Using stochastic geometry, closed form analytical expressions are proposed allowing to analyze the success and coverage probabilities for a LoRa network. An appropriate model of the path loss including real-life values is used to characterize the log-distance propagation parameters. The interference comes from both the LoRa network itself and the underlying IoT networks, modeled with an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> -stable distribution based on recent measurements. It is shown that for an environment with a huge amount of surrounding uncoordinated IoT networks, the gateways deployment should be doubled to reach a decent coverage probability, compared to an environment where the underlying interfering networks are not considered.

Downlink Performance Analysis in MIMO UAV-Cellular Communication With LOS/NLOS Propagation Under 3D Beamforming

Integrating unmanned aerial vehicles (UAVs) as the base station is a recent progress and a promising solution to assist cellular networks to improve coverage. In this paper, an analytical framework is proposed to analyze coverage probability and capacity, using stochastic geometry. We incorporate the air-to-ground (A2G), and two-piece path loss model that undergoes millimeter wave (mmWave) channel fading in both line-of-site (LOS) and non-line-of-site (NLOS). Furthermore, we assume that all users’ equipment’s (UEs) are clustered around Terrestrial Base Stations (TBS) and UAVs, according to Poisson Cluster Process (PCP), i.e., Matern cluster process. We also propose Multiple-Input-Multiple-Output (MIMO) transmission where multiple single-antenna UAVs simultaneously communicate with TBS. Initially, we obtain expressions for association probability with each tier. After that, we derive downlink coverage expression based on signal-to-interference-plus-noise-ratio (SINR), by assuming 3D directional beamforming with UAV height being not fixed. Numerical results demonstrate the efficiency and accuracy of the proposed approach.

CoMP-JT Scheme for D2D Communication in Industrial LiFi Networks

This paper focuses on the combined technique of device-to-device (D2D) and coordinated multipoint joint transmission (CoMP-JT) to bridge the indoor industrial wireless gap with LiFi-based networks. It also considers the mobility challange of industrial internet-of-things (IIoT) within a realistic scenario. Using the semiangle at half illuminance of the LiFi access point and D2D transmitting IIoT, we derive a coverage model for the communication range. Using stochastic geometry, the analytical expressions for CoMP-JT probability and coverage performance are derived in terms of different parameters, such as transmitter densities and bias factor. The analytical approximations match quite well with the Monte Carlo method-based simulation results.

MIMO Assisted Networks Relying on Intelligent Reflective Surfaces: A Stochastic Geometry Based Analysis

Intelligent reflective surfaces (IRSs) are invoked for improving both the spectral efficiency (SE) and energy efficiency (EE). Specifically, an IRS-aided multiple-input multiple-output network is considered, where the performance of randomly roaming users is analyzed by utilizing stochastic geometry tools. As such, to distinguish the superposed signals at each user, the passive beamforming weight at the IRSs and detection weight vectors at the users are jointly designed. As a benefit, by adopting a zero-forcing-based design, the intra-cell interference imposed by the IRS can be suppressed. In order to evaluate the performance of the proposed network, we first derive the approximated channel statistics in the high signal-to-noise-ratio (SNR) regime. Then, we derive the closed-form expressions both for the outage probability and for the ergodic rate of users. Both the high-SNR slopes of ergodic rate and the diversity orders of outage probability are derived for gleaning further insights. The network's SE and EE are also derived. Our numerical results are provided to confirm that: i) the high-SNR slope of the proposed network is one; ii) the SE and EE can be significantly enhanced by increasing the number of IRS elements.