Stochastic Geometry Research Articles

A Tractable Approach to Coverage Analysis in Downlink Satellite Networks

Satellite networks are promising to provide ubiquitous and high-capacity global wireless connectivity. Traditionally, satellite networks are modeled by placing satellites on a grid of multiple circular orbit geometries. Such a network model, however, requires intricate system-level simulations to evaluate coverage performance, and analytical understanding of the satellite network is limited. Continuing the success of stochastic geometry in a tractable analysis for terrestrial networks, in this paper, we develop novel models that are tractable for the coverage analysis of satellite networks using stochastic geometry. By modeling the locations of satellites and users using Poisson point processes on the surfaces of concentric spheres, we characterize analytical expressions for the coverage probability of a typical downlink user as a function of relevant parameters, including path-loss exponent, satellite height, density, and Nakagami fading parameter. Then, we also derive a tight lower bound of the coverage probability in tractable expression while keeping full generality. Leveraging the derived expression, we identify the optimal density of satellites in terms of the height and the path-loss exponent. Our key finding is that the optimal average number of satellites decreases logarithmically with the satellite height to maximize the coverage performance. Simulation results verify the exactness of the derived expressions.

The Meta Distribution of SINR in UAV-Assisted Cellular Networks

Mounting compact and lightweight base stations on unmanned aerial vehicles (UAVs) is a cost-effective and flexible solution to provide seamless coverage on the existing terrestrial networks. While the coverage probability in UAV-assisted cellular networks has been widely investigated, it provides only the first-order statistic of signal-to-interference-plus-noise ratio (SINR). In this paper, to analyze high-order statistics of SINR and characterize the disparity among individual links, we provide a meta distribution (MD)-based analytical framework for UAV-assisted cellular networks, in which the probabilistic line-of-sight channel and realistic antenna pattern are taken into account for air-to-ground transmissions. To accurately characterize the interference from UAVs, we relax the widely applied uniform off-boresight angle (OBA) assumption and derive the exact distribution of OBA. Using stochastic geometry, for both steerable and vertical antenna scenarios, we obtain mathematical expressions for the moments of condition success probability, the SINR MD, and the mean local delay. Moreover, we study the asymptotic behavior of the moments as network density approaches infinity. Numerical results validate the tightness of the theoretical results and show that the uniform OBA assumption underestimates the network performance, especially in the regime of moderate altitude of UAV. We also show that when UAVs are equipped with steerable antennas, the network coverage and user fairness can be optimized simultaneously by carefully adjusting the UAV parameters.

Fluid Antenna With Linear MMSE Channel Estimation for Large-Scale Cellular Networks

The concept of reconfigurable fluid antennas (FA) is a potential and promising solution to enhance the spectral efficiency of wireless communication networks. Despite their many advantages, FA-enabled communications have limitations as they require an enormous amount of spectral resources in order to select the most desirable position of the radiating element from a large number of prescribed locations. In this paper, we present an analytical framework for the outage performance of large-scale FA-enabled communications, where all user equipments (UEs) employ circular multi-FA array. In contrast to existing studies, which assume perfect channel state information, the developed framework accurately captures the channel estimation errors on the performance of the considered network deployments. In particular, we focus on the limited coherence interval scenario, where a novel sequential linear minimum mean-squared error (LMMSE)-based channel estimation method is performed for only a very small number of FA ports. Next, for the communication of each BS with its associated UE, a low-complexity port-selection technique is employed, where the port that provides the highest signal-to-interference-plus-noise-ratio is selected among the ports that are estimated to provide the strongest channel from each FA. By using stochastic geometry tools, we derive both analytical and closed-form expressions for the outage probability, highlighting the impact of channel estimation on the performance of FA-based UEs. Our results reveal the trade-off imposed between improving the network’s performance and reducing the channel estimation quality, indicating new insights for the design of FA-enabled communications.

Terahertz-band Non-orthogonal Multiple Access: System- and Link-level Considerations

Non-orthogonal multiple access (NOMA) communications promise high spectral efficiency and massive connectivity, serving multiple users over the same time-frequency-code resources. Higher data rates and massive connectivity are also achieved by leveraging wider bandwidths at higher frequencies, especially in the terahertz (THz) band. This work investigates the prospects and challenges of combining these algorithmic and spectrum enablers in THz-band NOMA communications. We consider power-domain NOMA coupled with successive interference cancellation at the receiver, focusing on multipleinput multiple-output (MIMO) systems as antenna arrays are crucial for THz communications. On the system level, we study the scalability of THz-NOMA beamforming, clustering, and spectrum/power allocation algorithms. We also motivate stochastic geometry techniques for performance analysis and system modeling and discuss the interplay role between the reconfigurable intelligent surfaces and NOMA. On the link level, we highlight the challenges in channel estimation and data detection and the constraints on computational complexity. We further illustrate future research directions. When properly configured and given sufficient densification, THz-band NOMA communications can significantly improve the performance and capacity of future wireless networks.

A Game-Theoretic Framework for Coexistence of WiFi and Cellular Networks in the 6-GHz Unlicensed Spectrum

The recently unlocked 6-GHz spectrum is accessible to cellular and WiFi networks for unlicensed use while they conform to the constraints imposed by the incumbent nodes. We allow only a fraction of the cellular base stations (BSs) and a fraction of WiFi access points (APs) to use the 6-GHz band so that sources of interference are spatially segregated and made sparse, thereby decreasing the overall interference to each other. Through our proposed framework, we control this fraction as we group portions of cellular and WiFi network elements into entities competing with the other entities for the spectrum resources. These entities interact to satisfy their Quality of Service demands by playing a non-cooperative game. The action of an entity corresponds to the fraction of its network elements (WiFi APs and cellular BSs) operating in the 6-GHz band. We use tools from stochastic geometry to derive the theoretical performance metrics for users of each radio access technology, which helps us capture the aggregate behaviour of the network in a snapshot. Due to the decentralized nature of the game, we find the solution using distributed Best Response Algorithm ( <monospace xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D-BRA</monospace> ), which improves the average datarate by 11.37% and 18.59% for cellular and WiFi networks, respectively, with random strategy as the baseline. The results demonstrate how the system parameters affect the performance of a network at equilibrium and highlight the throughput gains of the networks as a result of using the 6-GHz bands, which offer considerably larger bandwidths. We tested our framework using real-world data, which shows that practical implementation of multi-entity spectrum sharing is feasible even when the spatial distribution of the network elements and population are non-homogeneous.

Performance of Offloading Strategies in Collocated Deployments of Millimeter Wave NR-U Technology

5G New Radio (NR) technology operating in millimeter wave (mmWave) band is expected to be utilized in areas with high and fluctuating traffic demands such as city squares, shopping malls, etc. The latter may result in quality of service (QoS) violations. To deal with this challenge, 3GPP has recently proposed NR unlicensed (NR-U) technology that may utilize 60 GHz frequency band. In this paper, we investigate the deployment of NR-U base stations (BS) simultaneously operating in licensed and unlicensed mmWave bands in presence of competing WiGig traffic, where NR-U users may use unlicensed band as long as session rate requirements are met. To this aim, we utilize the tools of stochastic geometry, Markov chains, and queuing systems with random resource requirements to simultaneously capture NR-U/WiGig coexistence mechanism and session service dynamics in the presence of mmWave-specific channel impairments. We then proceed comparing performance of different offloading strategies by utilizing the eventual session loss probability as the main metric of interest. Our results show non-trivial behaviour of the collision probability in the unlicensed band as compared to lower frequency systems. The baseline strategy, where a session is offloaded onto unlicensed band only when there are no resources available in the licensed one, leads to the best performance. The offloading strategy, where sessions with heavier-than-average requirements are immediately directed onto unlicensed band results in just <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$2\!-\!5\%$</tex-math></inline-formula> performance loss. The worst performance is observed when sessions with smaller-than-average requirements are offloaded onto unlicensed band.

Wireless-powered backscatter communications for smart sustainable cities: A secrecy analysis

Due to its ability to enable self-sustainable communications, wirelessly powered backscatter communication (WP-BackCom) has been deemed a promising technology for supporting smart sustainable cities. We note that the existing works on analyzing the secrecy performance of WP-BackComs have largely ignored the spatial randomness of nodes that is used to model a large-scale network. To fill this gap, this paper evaluates the secrecy outage probability of the large-scale WP-BackComs based on a tractable framework of stochastic geometry (SG). In particular, the imperfect successive interference cancellation (SIC) at each gateway (GW) and eavesdropper (Eve), and the non-linear energy harvesting (EH) model and energy-causality constraint at each backscatter user (BU) are considered. The energy outage probability and secrecy outage probability expressions are derived in the general case at first, then the closed-form expressions are obtained under the special scenario, i.e., the no-noise and the typical propagation loss scenario. Numerical simulations validate the analytical results and examine the impacts of various parameters on the secrecy performance. In particular, there exists an optimal reflection coefficient for secrecy outage probability minimization under certain conditions, and the optimal reflection coefficient increases with the increasing SIC coefficient at GW.

Fluctuations of cell geometry and their nonequilibrium thermodynamics in living epithelial tissue.

We measure different contributions to entropy production in a living functional epithelial tissue. We do this by extracting the functional dynamics of development while at the same time quantifying fluctuations. Using the translucent Drosophila melanogaster pupal epithelium as an ideal tissue for high-resolution live imaging, we measure the entropy associated with the stochastic geometry of cells in the epithelium. This is done using a detailed analysis of the dynamics of the shape and orientation of individual cells which enables separation of local and global aspects of the tissue behavior. Intriguingly, we find that we can observe irreversible dynamics in the cell geometries but without a change in the entropy associated with those degrees of freedom, showing that there is a flow of energy into those degrees of freedom. Hence, the living system is controlling how the entropy is being produced and partitioned into its different parts.

Performance analysis of flexible indoor and outdoor user distribution in urban multi-tier heterogeneous network

The next generation heterogeneous networks (hetnet) refer to a complex eco system of radio cells with varying coverage ranges and technologies that interoperate to deliver the required capacity and coverage. This paper addresses the impact of large scale transmissions and analyses the performance of a typical user distribution in an urban multi-tier hetnet. A Boolean scheme of building blockage has been used to abstract the urban environment, which affects the architecture of hetnet with millimetre wave (mmW) access. A distribution mechanism is proposed to enable the indoor users to operate in mmW band whereas the outdoor users to operate in microwave (μWave) band. An indoor user can be connected with mmW band if the line-of-sight (LOS) link exists. Otherwise, the exploitation is provided with μWave band to enhance the coverage. By employing tools from stochastic geometry, the mathematical expressions are derived for coverage probability, rate and spatial average channel capacity of a user in indoor and outdoor environments. The exact mathematical results show that an impact of small cell occupation ratio results in maximised spatial average channel capacity of an outdoor hetnet. The significant enhancement is provided with mmW small cell in terms of rate and coverage.

Performance Analysis of Power Line Communication System for Massive Indoor IoT Applications

This study explores the utilization of low-voltage indoor power line communication (PLC) technology within the framework of a split centralized radio access network (C-RAN) architecture, aimed at enhancing 5G indoor mobile coverage for the Internet of Things (IoT). A primary focus of this investigation is the comprehensive evaluation of the system’s performance in terms of radio and access latency. Several critical variables come into play, including available bandwidth, the chosen transmission technology, and the number of wireless IoT devices within the network. An analytical model, employing queuing theory, Markovian models, and stochastic geometry, is formulated to assess the radio-access delay spanning the air interface and the innovative PLC-based front-hauling system proposed in our previous study. The precision of this model is validated through simulations, ensuring that the proposed system effectively meets the stringent latency constraints required by eCPRI, thus contributing to the improvement of 5G indoor mobile coverage for the IoT network.

Non-Orthogonal Multiple Access for UAV-Aided Heterogeneous Networks: A Stochastic Geometry Model

In this work, we explore the potential benefits of deploying unmanned aerial vehicles (UAVs) as aerial base stations (ABSs) with sub-6GHz bands and small cells terrestrial base stations (TBSs) with millimeter wave (mmWave) bands in a hybrid heterogeneous networks (HetNets). A flexible non-orthogonal multiple access (NOMA) based user association policy is proposed. By using the tools from stochastic geometry, new analytical expressions for association probability, coverage probability and spectrum efficiency are derived for characterizing the performance of UAV-aided HetNets under an appropriate Air-to-Ground (A2G) channel model and the Ground-to-Ground (G2G) channels with a LoS ball blockage model. Finally, we provide insights on the proposed hybrid HetNets by numerical results. We confirm that i) the proposed NOMA enabled HetNets is capable of achieving superior performance compared with the OMA enabled ABSs by setting power allocation factors and targeted signal-to-interference-plus-noise ratio (SINR) threshold properly; ii) there is a tradeoff between the association probabilities and the spectrum efficiency in the NOMA enabled ABSs tier; iii) the coverage probability and spectrum efficiency of the NOMA enabled ABSs tier is largely affected by the imperfect successive interference cancellation (ipSIC) coefficient, power allocation factors and SINR threshold; iv) compared with only sub-6GHz ABSs, mmWave enabled TBSs are capable of enhancing the spectrum efficiency of the HetNets when the mmWave line-of-sight (LoS) link is available.

Analysis of the Delay Distribution in Cellular Networks by Using Stochastic Geometry

Analysis of the Delay Distribution in Cellular Networks by Using Stochastic Geometry

A Spatio-Temporal Analytical Model for Statistical Priority-Based Multiple Access Network

In this letter, we propose a spatio-temporal analytical framework to comprehensively assess the performance of a statistical priority-based multiple access (SPMA) network. By combining tools from stochastic geometry with queueing theory, we characterize both the macroscopic interference distribution of the spatially interacting priority queues, and the microscopic temporal evolution of each priority queue. The proposed analytical framework allows to derive the analytical expressions of burst success probability, medium access probability (MAP), and packet success probability, and evaluate the impacts of key system parameters on a stable SPMA network. The advantage of SPMA is further highlighted by compared with the traditional CSMA scheme.

Differential Framework for Submeter-Accurate Vehicular Navigation With Cellular Signals

A framework that could achieve submeter-level-accurate horizontal navigation with carrier phase differential measurements from cellular signals is developed. This framework, termed CD-cellular, is composed of a base and a rover in a cellular environment, both making carrier phase measurements to the same cellular base transceiver stations (BTSs). The base shares its carrier phase measurements with the mobile rover, which in turn employs an extended Kalman filter to obtain a coarse estimate of its states, followed by a batch weighted nonlinear least squares (B-WNLS) estimator to solve for the integer ambiguities, and finally a point-solution WNLS to estimate its own states. The framework is designed to guarantee that after some time, the rover's position error remains below a pre-defined threshold with a desired probability. This is achieved by leveraging models of the BTS positions from stochastic geometry. Experimental results on an unmanned aerial vehicle (UAV) in an open semi-urban environment with multipath-free, line-of-sight (LOS) conditions are presented, showing that the developed framework achieves a 70.48 cm position root mean-squared error (RMSE) over a trajectory of 2.24 km, measured with respect to the UAV's navigation solution from its onboard GPS-inertial navigation system (INS).

Semantic Sensing Performance Analysis: Assessing Keyword Coverage in Text Data

In this paper, to provide the theoretical reference for semantic sensing performance, the stochastic geometry tool is used to make the quantitative coverage performance analysis of the semantic keywords. This method can help understand semantic network performance from the macroscopic view rather than the connection-oriented perspective. Firstly, through the semantic mapping operation, the context space is transformed into the semantic space in the semantic sensing process. Secondly, given the text data with different statement styles, e.g., literary and scientific texts, two typical semantic reflection models are established with the help of stochastic geometry theory, where the keyword coverage expressions are derived. Thirdly, under various parameter settings, the correctness and effectiveness of the proposed models have been evaluated through the simulation results and analysis.

Modeling and Analysis of UAV-Assisted Mobile Network with Imperfect Beam Alignment

With the rapid development of emerging 5G and beyond (B5G), Unmanned Aerial Vehicles (UAVs) are increasingly important to improve the performance of dense cellular networks. As a conventional metric, coverage probability has been widely studied in communication systems due to the increasing density of users and complexity of the heterogeneous environment. In recent years, stochastic geometry has attracted more attention as a mathematical tool for modeling mobile network systems. In this paper, an analytical approach to the coverage probability analysis of UAV-assisted cellular networks with imperfect beam alignment has been proposed. An assumption was considered that all users are distributed according to Poisson Cluster Process (PCP) around base stations, in particular, Thomas Cluster Process (TCP). Using this model, the impact of beam alignment errors on the coverage probability was investigated. Initially, the Probability Density Function (PDF) of directional antenna gain between the user and its serving base station was obtained. Then, association probability with each tier was achieved. A tractable expression was derived for coverage probability in both Line-of-Sight (LoS) and Non-Line-of-Sight (NLoS) condition links. Numerical results demonstrated that at low UAVs altitude, beam alignment errors significantly degrade coverage performance. Moreover, for a small cluster size, alignment errors do not necessarily affect the coverage performance.

A New 5G Cellular Broadcasting Approach: Hybrid LDM-NOMA and FeMBMS-OMA Analysis Based on Stochastic Geometry

A New 5G Cellular Broadcasting Approach: Hybrid LDM-NOMA and FeMBMS-OMA Analysis Based on Stochastic Geometry

Decentralized Federated Learning Over Slotted ALOHA Wireless Mesh Networking

Federated Learning (FL) presents a mechanism to allow decentralized training for machine learning (ML) models inherently enabling privacy preservation. The classical FL is implemented as a client-server system, which is known as Centralised Federated Learning (CFL). There are challenges inherent in CFL since all participants need to interact with a central server resulting in a potential communication bottleneck and a single point of failure. In addition, it is difficult to have a central server in some scenarios due to the implementation cost and complexity. This study aims to use Decentralised Federated learning (DFL) without a central server through one-hop neighbors. Such collaboration depends on the dynamics of communication networks, e.g., the topology of the network, the MAC protocol, and both large-scale and small-scale fading on links. In this paper, we employ stochastic geometry to model these dynamics explicitly, allowing us to quantify the performance of the DFL. The core objective is to achieve better classification without sacrificing privacy while accommodating for networking dynamics. In this paper, we are interested in how such topologies impact the performance of ML when deployed in practice. The proposed system is trained on a well-known MINST dataset for benchmarking, which contains labelled data samples of 60K images each with a size 28×28 pixels, and 1000 random samples of this MNIST dataset are assigned for each participant’ device. The participants' devices implement a CNN model as a classifier model. To evaluate the performance of the model, a number of participants are randomly selected from the network. Due to randomness in the communication process, these participants interact with the random number of nodes in the neighborhood to exchange model parameters which are subsequently used to update the participants’ individual models. These participants connected successfully with a varying number of neighbours to exchange parameters and update their global models. The results show that the classification prediction system can achieve 95% accuracy using the three different model optimizers in the training settings (i.e., SGD, ADAM, and RMSprop optimizers). Consequently, the DFL over mesh networking shows more flexibility in IoT systems, which reduces the communication cost and increases the convergence speed which can outperform CFL.

Distributed Communication and Sensing System Co-Design for Improved UAV Network Resilience

The progress in wireless technology over the past decade led to the rapid adoption of unmanned aerial vehicles (UAVs) for various applications. As the interest in UAVs is accelerating, increased attention is paid to the reliability and resilience of UAV-based systems with respect to the collision avoidance. One of the ways to improve this aspect is to utilize the RADAR functionality. In this work, we consider a cellular network employed for communication jointly with RADAR operation. The critical parameter that affects the RADAR algorithm is the radar cross-section (RCS). Since the task of obtaining the RCS of a complex-shaped object is extremely challenging, we first propose a novel, accurate, and fast method of scattered field assessment. We further perform radio network planning for the cellular deployment, as well as link budget estimations for the RADAR system that co-exists with it. Under this system model, we carry out detailed Monte-Carlo simulations of the RADAR detection process to obtain reliable statistical results and answer the question of how the actual bistatic RCS model affects the detection algorithm. We then apply mathematical modeling based on stochastic geometry to estimate the collision probability without the need to simulate an extensive number of flight-hours. Our numerical results confirm the robustness to RCS pattern nulls, which is crucial for safety-centric applications such as collision avoidance.

Energy Efficiency Analysis in Heterogeneous Networks: A Stochastic Geometry Perspective

Energy Efficiency Analysis in Heterogeneous Networks: A Stochastic Geometry Perspective