Stochastic Geometry Research Articles

Membrane-mediated dimerization potentiates PIP5K lipid kinase activity.

The phosphatidylinositol 4-phosphate 5-kinase (PIP5K) family of lipid-modifying enzymes generate the majority of phosphatidylinositol 4,5-bisphosphate [PI(4,5)P2] lipids found at the plasma membrane in eukaryotic cells. PI(4,5)P2 lipids serve a critical role in regulating receptor activation, ion channel gating, endocytosis, and actin nucleation. Here, we describe how PIP5K activity is regulated by cooperative binding to PI(4,5)P2 lipids and membrane-mediated dimerization of the kinase domain. In contrast to constitutively dimeric phosphatidylinositol 5-phosphate 4-kinase (PIP4K, type II PIPK), solution PIP5K exists in a weak monomer-dimer equilibrium. PIP5K monomers can associate with PI(4,5)P2-containing membranes and dimerize in a protein density-dependent manner. Although dispensable for cooperative PI(4,5)P2 binding, dimerization enhances the catalytic efficiency of PIP5K through a mechanism consistent with allosteric regulation. Additionally, dimerization amplifies stochastic variation in the kinase reaction velocity and strengthens effects such as the recently described stochastic geometry sensing. Overall, the mechanism of PIP5K membrane binding creates a broad dynamic range of lipid kinase activities that are coupled to the density of PI(4,5)P2 and membrane-bound kinase.

Performance Analysis of Distance-Based D2D Matching Mechanism

The traditional cellular architecture where devices connect to their service base station (BS) may cause poor performance especially for edge users. Device-to-device (D2D) communication enables nearby user as a relay to help BS forward information, thereby improving the network coverage and quality of service (QoS) of edge users. This paper proposes a distance-based D2D matching mechanism for general cellular networks, where a relay user who successfully connects to its targeted BS can transmit data to its closest user for D2D communication. A link of BS to D2D pair contains two sublinks, which occur at different time phases in each cell. Assuming a nonsynchronous system, we consider that there exists cross-layer interference for D2D links. Based on the techniques of stochastic geometry, we develop the performance of coverage probability and ergodic rate of the D2D network. A key intermediate step in this analysis is the derivation of the interference expressions for D2D links caused by BSs and cochannel D2D users. Then, we derive the meta distribution of the signal-to-interference ratio (SIR) to capture the performance changes of individual links. Simulation results demonstrate that our matching mechanism based on the appropriate time resource allocation favors the edge users with a higher probability of successful communication and transmission rate.

Interference and secrecy analysis based on randomly spacial model in clustered WSNs

Clustering and layering are widely employed in large-scale wireless sensor networks (WSNs) to alleviate data implosion, reduce transmission delay and improve energy efficiency. It is vital to conduct performance analysis in clustered WSNs for better defining and designing networks. Here, a general analytical framework based on a randomly spacial model is proposed to conduct inter-cluster interference analysis and physical-layer security analysis for clustered WSNs. It is assumed that two adjacent cluster heads exchange confidential massage with a passive eavesdropper and the non-head nodes in clusters could be selected as cooperative jamming (CJ) nodes to disturb the eavesdropper. All mentioned nodes are randomly located in their specific areas. Under the above settings, three scenarios, namely interference, eavesdropping and CJ are considered, and a stochastic geometry tool based on kinematic measure is employed for analysis. Comparing to existing stochastic geometry methods, the proposed analytical framework can handle arbitrarily shaped, disjoint and tiered networks. The results obtained from extensive simulations have verified the rationality of the framework and demonstrated the impact of different parameters on the performance metrics of interest. The comparison with PPP also shows the advantages of this proposed model.

Estimation of forest stand characteristics using individual tree detection, stochastic geometry and a sequential spatial point process model

Airborne Laser Scanning (ALS) results in point-wise measurements of canopy height, which can further be used for Individual Tree Detection (ITD). However, ITD cannot find all trees because small trees can hide below larger tree crowns. Here we discuss methods where the plot totals and means of tree-level characteristics are estimated in such context. The starting point is a previously presented Horvitz–Thompson-like (HT-like) estimator, where the detectability is based on the larger tree crowns and a tuning parameter α that models the detection condition. We propose a new method which is based on modeling the spatial pattern of hidden tree locations using a sequential spatial point process model, with a tuning parameter θ. We also explore whether the variability of the tuning parameters α and θ can be predicted using ALS features to improve the predictions. The accuracy of stand density, dominant height and mean height is used as comparison criteria in a cross-validation procedure. The HT-like estimator with empirically estimated tuning parameter α performed the best. The overall performance of the new method was comparable. The new method was computationally less demanding, which makes it attractive for practical use.

Modeling and Performance Analysis of Statistical Priority-Based Multiple Access: A Stochastic Geometry Approach

Statistical priority-based multiple access (SPMA) protocol has attracted much attention in virtue of its support for multi-priority traffic, and the guarantee of low-latency and high-reliability transmissions for high-priority. In this work, we propose an analytical framework to study the performance of SPMA from spatial perspective with tools from the stochastic geometry. We consider two kinds of priority traffic, including high-priority traffic and low-priority traffic. In SPMA, a packet is split into multiple bursts to reduce the collision probability, and the turbo coding, frequency hopping, and time hopping are employed to further decrease the packet loss rate. We first derive the analytical expressions for the medium access probability (MAP) and burst success probability of two priority users in closed form, taking into account the potential transmitters (PTs) density, ratio of different traffic users, amount of orthogonal resources, channel occupancy statistics (COS) threshold, and statistical sliding window (SSW). Based on the derived MAP and burst success probability, we further obtain the packet success probability and spatial throughput. After evaluating the effect of key parameters on the above performance metrics, we provide guidelines on optimal design of several key system parameters, such as the COS threshold and PTs density, to guarantee the high-priority user a 99% packet success probability.

3D Two-Hop Cellular Networks With Wireless Backhauled UAVs: Modeling and Fundamentals

In this paper, we characterize the performance of a three-dimensional (3D) two-hop cellular network in which terrestrial base stations (BSs) coexist with unmanned aerial vehicles (UAVs) to serve a set of ground user equipment (UE). In particular, a UE connects either directly to its serving terrestrial BS by an access link or connects first to its serving UAV which is then wirelessly backhauled to a terrestrial BS (joint access and backhaul). We consider realistic antenna radiation patterns for both BSs and UAVs using practical models developed by the third generation partnership project (3GPP). We assume a probabilistic channel model for the air-to-ground transmission, which incorporates both line-of-sight (LoS) and non-line-of-sight (NLoS) links. Assuming the max-power association policy, we study the performance of the network in both amplify-and-forward (AF) and decode-and-forward (DF) relaying protocols. Using tools from stochastic geometry, we analyze the joint distribution of distance and zenith angle of the closest (and serving) UAV to the origin in a 3D setting. Further, we identify and extensively study key mathematical constructs as the building blocks of characterizing the received signal-to-interference-plus-noise ratio (SINR) distribution. Using these results, we obtain exact mathematical expressions for the coverage probability in both AF and DF relaying protocols. Furthermore, considering the fact that backhaul links could be quite weak because of the downtilted antennas at the BSs, we propose and analyze the addition of a directional uptilted antenna at the BS that is solely used for backhaul purposes. The superiority of having directional antennas with wirelessly backhauled UAVs is further demonstrated via simulation.

STAR-RIS Aided NOMA in Multicell Networks: A General Analytical Framework With Gamma Distributed Channel Modeling

The simultaneously transmitting and reflecting reconfigurable intelligent surface (STAR-RIS) is capable of providing full-space coverage of smart radio environments. This work investigates STAR-RIS aided downlink non-orthogonal multiple access (NOMA) multi-cell networks, where the energy of incident signals at STAR-RISs is split into two portions for transmitting and reflecting. We first propose a fitting method to model the distribution of composite small-scale fading power as the tractable Gamma distribution. Then, a unified analytical framework based on stochastic geometry is provided to capture the random locations of RIS-RISs, base stations (BSs), and user equipments (UEs). Based on this framework, we derive the coverage probability and ergodic rate of both the typical UE and the connected UE. In particular, we obtain closed-form expressions of the coverage probability in interference-limited scenarios. We also deduce theoretical expressions in conventional RIS aided networks for comparison. The analytical results show that optimal energy splitting coefficients of STAR-RISs exist to simultaneously maximize the system coverage and ergodic rate. The numerical results demonstrate that: 1) STAR-RISs are able to meet different demands of UEs located on different sides; 2) STAR-RISs with appropriate energy splitting coefficients outperform conventional RISs in the coverage and the rate performance.

Density Estimation in Randomly Distributed Wireless Networks

Networks of randomly distributed nodes appear in various fields, including forestry and wireless communications, and can often be modeled, using stochastic geometry theory, as Poisson point processs (PPPs). In these contexts, estimation of nodes density is important for monitoring and optimizing the network. Originally, this problem has been addressed in forestry where the trees are the nodes and, assuming these are distributed according to an infinite two-dimensional homogeneous PPP, the spatial density can be estimated by measuring the distances from one reference tree to its neighbors. However, in many other scenarios, nodes could result invisible with some probability, for example depending on distance. In this paper, we derive the Cramér-Rao bounds and new estimators for the node spatial density, taking into account a limited capability in sensing neighbors. As an example, we provide estimators of the spatial density of transmitting devices in wireless networks with links affected by thermal noise, path loss, and shadowing.

Visible Light and Reconfigurable Intelligent Surfaces for Beyond 5G V2X Communication Networks at Road Intersections

Traffic safety at road intersections can be enhanced by improving opportunistic exchange of safety messages between vehicles. In urban intersection scenario, obstacles such as buildings and road side installations/signboards block the line-of-sight communication between vehicles. In order to enhance the reliability of communication link, the existing vehicular solution utilizes relay placement at road intersection. However, as the vehicular density increases, the existing vehicular-radio frequency (V-RF) communication with relaying tends to suffer from higher interference, lower packet reception rate, and longer communication delays due to channel congestion. In contrast to the existing solutions, we propose to use practical deployment strategies namely hybrid vehicular-visible light communication (V-VLC)/V-RF with relaying and reconfigurable intelligent surface (RIS) aided V-RF solutions to improve the communication range for urban vehicle-to-vehicle (V2V) communication. We present stochastic geometry based analytical framework to analyze the performance of proposed solutions in terms of outage probability, throughput and delay outage rate (DOR). Numerical results illustrate that the proposed solutions can achieve considerable performance improvement in outage, throughput while ensuring low latency as compared to conventional V-RF with relaying. These deployment strategies may also serve as a preferred alternative for Intelligent Transportation System (ITS) to meet ultra-high reliable and ultra-low latency communication for beyond 5 G (B5G) vehicular networks.

Performance Comparison of Wireless Aerial 3D Cellular Network Models

In this letter, stochastic geometry tools are utilized to compare the downlink performance of two wireless cellular network models with Unmanned Aerial Vehicles (UAVs), in terms of coverage probability and average achievable rate. The considered models are: i) a truncated octahedron-based cellular network, referred to as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">regular</i> in the following, and ii) a binomial-Voronoi cellular network. The locations of all UAV-user equipments (UAV-UEs) are modeled as a three-dimensional (3D) Poisson Point Process (PPP) in both cellular models. Analytical expressions on coverage probability and achievable average rate are derived, and numerical results are presented. The binomial-Voronoi network is also compared via Monte-Carlo simulations to a regular network with uniformly random UAV-UE receiver selection policy.

On the SIR Meta Distribution for Cache-Enabled Wireless Networks With Random Discontinuous Transmission: Analysis and Optimization

A fine-grained analysis of the cache-enabled networks is crucial for system design. In this paper, we focus on the meta distribution of the signal-to-interference ratio for the cache-enabled networks where the locations of the base stations are modeled as a Poisson point process. With the application of the random caching and the random discontinuous transmission schemes, we derive the moments of the conditional successful transmission probability, the exact meta distribution and its beta approximation by utilizing stochastic geometry. The closed-form expressions of the mean and variance of the local delay (i.e., the jitter) are also derived. We then consider the maximization of the mean successful transmission probability and the minimization of the average system transmission delay by jointly optimizing the caching probability and the BS active probability. Finally, the numerical results demonstrate the superiority of the proposed optimization schemes over the existing caching strategies and reveal the impacts of the key network parameters on the cache-enabled networks in terms of successful transmission probability, successful transmission probability variance, meta distribution, mean local delay and jitter.

Edge Computing and Multiple-Association in Ultra-Dense Networks: Performance Analysis

The recent advances and unprecedented ubiquity of computation-intensive applications such as virtual reality and mobile augmented reality force new approaches to handle the accompanying challenges. Ultra-Dense Network (UDN) as a leading direction in 5G and beyond offers an opportunistic degree of freedom to be exploited where a massive number of low-power and low-cost Small Cells (SCs) are deployed. In particular, integrating Edge Computing Servers (ECSs) within the SCs at the edge of the cellular network paves the way to tackle several challenges including the processing of computation-intensive tasks with low latency. To exploit the full potentials of UDNs and ECSs, we deploy multiple associations of SCs to the same user while partitioning and offloading its computation-intensive task to the integrated ECSs therein. In this regard, we formulate a mathematical framework using tools from stochastic geometry to evaluate the average processing delay per user. Extensive Monte-Carlo simulations are conducted to verify the accuracy of our analytical results under different system parameters. Results show the existence of an optimal order of multiple-association. In addition, we propose a novel offline task division approach which significantly reduces the overall delay.

The Thermodynamics of a Stochastic Geometry Model with Applications to Non-Extensive Statistics

The Thermodynamics of a Stochastic Geometry Model with Applications to Non-Extensive Statistics

The impact of mobility on physical layer security in 5G uRLLC

Due to the openness nature of wireless medium, security issues have increasingly become a bottleneck that restricts the development of ultra-reliable and low-latency communications (uRLLCs). Physical layer security (PLS) technique has been proposed to fulfill the security and confidentiality of information transmission by exploiting the characteristics of the wireless channel, which caters to the features of uRLLC. Furthermore, PLS also shows great practicality in artificial intelligence field, especially in wireless intelligent networks. However, the previous works on the study of PLS ignored the significance of mobility and limited packet length constraint required by uRLLC for satisfying low latency, in this paper, we investigate the impact of mobility on the secrecy performance of uRLLC by using the Random WayPoint (RWP) model and the Random Direction (RD) model. Specifically, with the tools of stochastic geometry, to observe the impact of key system parameters on the secrecy performance, we establish the closed-form expression of connection outage probability, secrecy outage probability, and decoding error probability-based secrecy transmission capacity (DEP-STC). Furthermore, we derive the condition that achieves a positive DEP-STC under two moving models, which can offer the network designer some greatly significant insights into achieving perfect secrecy. Simulations validate our derived theoretical results, and indicate that RWP moving receiver can obtain a higher security level than RD moving one, while RWP eavesdropper can lead to a lower security.

Generalized limit theorems for U-max statistics

Abstract $U{\hbox{-}}\textrm{max}$ statistics were introduced by Lao and Mayer in 2008. Such statistics are natural in stochastic geometry. Examples are the maximal perimeters and areas of polygons and polyhedra formed by random points on a circle, ellipse, etc. The main method to study limit theorems for $U{\hbox{-}}\textrm{max}$ statistics is via a Poisson approximation. In this paper we consider a general class of kernels defined on a circle, and we prove a universal limit theorem with the Weibull distribution as a limit. Its parameters depend on the degree of the kernel, the structure of its points of maximum, and the Hessians of the kernel at these points. Almost all limit theorems known so far may be obtained as simple special cases of our general theorem. We also consider several new examples. Moreover, we consider not only the uniform distribution of points but also almost arbitrary distribution on a circle satisfying mild additional conditions.

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.

Finite Point Processes in a Truncated Octahedron-Based 3D UAV Network

In this paper, stochastic geometry tools are utilized to analyze the performance of a wireless network with Unmanned Aerial Vehicles (UAVs). The evaluation is based on the concept of a truncated octahedron that is the fundamental three-dimensional (3D) shape that models the basic cell in an entirely aerial cellular structure with UAV nodes only. The spatial locations of the aerial users, are modeled as a 3D Binomial Point Process (BPP). Subsequently, stochastic geometry analysis in a reference truncated octahedron located at the origin is performed and exact expressions for the distances between aerial users, are derived. To this end, expressions for evaluating the volume of the regions of the 3D cell in which an aerial user may fall, are derived. Then, by defining the signal-to-interference-plus-noise (SINR) ratio at the reference UAV-base station (UAV-BS) receiver, a performance analysis in terms of coverage probability is conducted under two interference scenarios, i.e., a pure intra-cell interference and an intra- plus inter-cell interference case. As intermediate step in the coverage probability analysis, expressions for relevant distance distributions and the Laplace transform of the aggregate interference power distribution are derived. For the sake of completeness, coverage probability analysis under intra-cell interference is also performed and compared to a 3D cell modeled as a sphere. The analytical expressions are also compared to simulation results to validate the accuracy of the proposed framework and reveal valuable insights.

Performance Analysis of Multi-Antenna UAV Networks With 3D Interference Coordination

Exploiting multiple-input multiple-output (MIMO) technology in UAV networks will mine the potential of spatial multiplexing. However, considering UAVs 3D distribution and line-of-sight (LOS)/non-line-of-sight (NLOS) channel dynamics, multi-antenna transmission will exacerbate network irregularity and render interference management a complicated problem to solve. This paper proposes 3D coordination model for interference management via multi-cell beamforming and signal-level cooperation in multi-antenna UAV networks, and analyzes system performance by deriving semi-closed expression of coverage probability using stochastic geometry. Specifically, UAVs are deployed according to 3D Poisson point process (PPP) with maximum height limit <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> , and user-centric cell group selection is considered according to specific signal threshold <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\eta $ </tex-math></inline-formula> for interference coordination with the consideration of LOS/NLOS channel conditions. Zero-forcing beamforming and joint transmission is performed within the UAV cluster for interference mitigation with limited channel state information feedback taken account. In addition, considering the irregularity of UAV group, system performance is construed as three scenarios and coverage probability is derived by Gamma approximation with proposed novel coordinate system transformation. Numerical results match well with simulations and provide optimal deployment parameters for UAV deployments, where coverage probability can reach 92% when SIR threshold <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$T=0$ </tex-math></inline-formula> dB.

User-Centric Cluster Design and Analysis for Hybrid Sub-6GHz-mmWave-THz Dense Networks

The terahertz (THz) waves with enormous bandwidth can be used along with the existing sub-6GHz and millimeter wave (mmWave) bands to achieve the ever evolving ecosystem of applications that need to be supported by the modern wireless networks. This paper investigates a user-centric dynamic base station (BS) clustering design for a hybrid network where THz, mmWave, and sub-6GHz BSs coexist. Invoking the proposed clustering model, the BS cooperation within each band is made possible by considering long term channel variations and all the surrounding BSs within a cluster with tunable size. A typical user is associated with the best BS cluster, from either a sub-6GHz, mmWave or THz tier based on the maximum signal-to-interference-plus-noise-ratio (SINR) criterion or the maximum rate criterion. Using tools from stochastic geometry, we assess the performance of the proposed user-centric hybrid system in terms of SINR and rate coverage performances, while accounting for: band specific propagation models, directional beamfroming, and BSs random locations. The accuracy of the analytical results is validated with Monte-Carlo simulations. The obtained results recognize a clear coverage/rate trade-off where a high fraction of THz BSs improves the rate significantly but may degrade the coverage performance. Thus, with carefully planned networks, enabling user-centric BS cooperation for hybrid wireless systems can achieve ultra-high rates while maintaining sufficient coverage in sixth-generation (6G) networks.

Stochastic analysis of urban V2X communications: Orthogonality versus non-orthogonality

In this paper, we study the broadcasting performance of urban Vehicle-to-everything (V2X) communications at road intersections. In such environments, dense buildings obstruct the radio channel between vehicles in adjacent road segments and Non-Line-Of-Sight (NLOS) signal propagation reduces the received power level. Therefore, messages that originate from adjacent roads traveling in a non-parallel direction usually have weaker signals than those from the road segment traveling in parallel opposite direction. Besides, the high vehicular density leads to insufficient orthogonal channel resources and high level of interference. We propose to apply Non-Orthogonal Multiple Access (NOMA) to V2X communications at urban road intersections to enhance the spectrum efficiency and improve the Package Delivery Ratio (PDR) performance. With NOMA, signals from multiple transmitters are decoded together and the interference from undesired users can be canceled if these messages are retrieved. Specifically, we study two NOMA-based V2X communication schemes, namely, NOMA-V2X decoded by Successive Interference Cancellation (SIC-V2X) and NOMA-V2X decoded by Joint Decoding (JD-V2X). Based on the tools developed in stochastic geometry, we derive and compare the PDR expressions for both NOMA schemes and the Orthogonal Multiple Access (OMA) scheme. Our results indicate that 1) both NOMA schemes outperform the conventional OMA scheme and the PDR of LOS/NLOS communications with two-user access increases by 51%/369%; 2) for four-user access, the proposed NOMA scheme shows 375% goodput enhancement as compared with the OMA scheme; and 3) JD-V2X provides significant PDR enhancement compared with SIC-V2X in the high data rate regime. Finally, we also demonstrate that the random access based NOMA schemes can achieve lower transmission latency than collision-free scheduling schemes when the road is over 56% jammed.