Meta Distribution Research Articles

Radar Detection in Vehicular Networks: Fine-Grained Analysis and Optimal Channel Access

Automotive radar is a critical feature in advanced driver-assistance systems. It is important in enhancing vehicle safety by detecting the presence of other vehicles in the vicinity. The performance of radar detection is, however, affected by the interference from radars of other vehicles as well as the variation in the target radar cross-section (RCS) due to varying physical features of the target vehicle. Considering such interference and random RCS, this work provides a fine-grained performance analysis of radar detection. Specifically, using stochastic geometry, we calculate the meta distribution of the signal-to-interference-and-noise ratio that permits the reliability analysis of radar detection at individual vehicles. We also evaluate the delay aspect of radar detection, namely, the mean local delay which is the average number of transmission attempts needed until the first successful target detection. For a given target distance, we obtain the optimal transmit probability that maximizes the density of successful detection while keeping the mean local delay below a threshold. We also provide several system design insights in terms of the fraction of reliable radar links, transmission delay, the density of vehicles, and congestion control. Finally, leveraging the derived framework, we present an online algorithm to select the transmit probability in a distributed manner, particularly when system parameters such as vehicular density and interferer locations are unknown to individual vehicles. We demonstrate the efficacy of the proposed online framework for selecting the optimal transmit probability within a deadline of 0.1 ms, which is critical for sustaining automotive safety applications and services.

Performance Analysis of Cache-Enabled User Association for Hybrid Heterogeneous Cellular Networks

Hybrid heterogeneous cellular networks (HCNets) with sub-6G and millimeter-wave (mmWave) base stations (BSs) achieves good performance-and-cost tradeoffs. Via noticing the negligible retrieving latency and the potential joint transmission (JT) gain of sub-6G BSs, we propose a novel cache-enabled user association scheme for hybrid HCNets with limited local storage, in which partial sub-6G local storage caches contents with relatively low popularity for higher overall content diversity. To sufficiently exploit the large bandwidth of mmWave BSs, they are endowed with higher association priority. Then, the success probability and meta distribution of achievable rate are studied for the proposed scheme, respectively. We further provide the optimization of caching parameters and the condition for superior performance over the MPC-based scheme. Analytical and numerical results show that the network performance can be improved for larger sub-6G BS cooperation size and the array size at mmWave BSs, while the mmWave BS density has two adverse effect for low-to-medium and high rate threshold. For the proposed scheme, the content diversity in combination with JT gain benefits the success probability. And its performance advantage increases for larger local storage and/or higher association probability with mmWave BSs.

Spatio-Temporal Analysis of Meta Distribution for Cell-Center/Cell-Edge Users

Emergence of various types of wireless applications has brought about fast growing and diversified traffic in cellular networks. To gain a comprehensive understanding of the influences caused by the differentiated and dynamic traffic is vital for the design of the next-generation wireless networks. In this paper, we develop a mathematical framework for meta-distribution analysis by utilizing queueing theory and stochastic geometry to capture the spatial (geographical location) and temporal randomness (queue status) of traffic. We derive the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${k}$ </tex-math></inline-formula> -th moment of the conditional successful transmission probability (STP) and the closed-form expressions of the meta distribution for the cell-center users (CCUs) and the cell-edge users (CEUs), respectively. The results are further extended to obtain the analytical expression of the meta distribution by taking the temporal random arrival of traffic into consideration. Moreover, the mean local delays for the CCUs and CEUs are derived. Finally, the impact of key network parameters on the meta distribution and the corresponding mean local delay is investigated.

The Transdimensional Poisson Process for Vehicular Network Analysis

A comprehensive vehicular network analysis requires modeling the street system and vehicle locations. Even when Poisson point processes (PPPs) are used to model the vehicle locations on each street, the analysis is barely tractable. That holds for even a simple average-based performance metric -- the success probability, which is a special case of the fine-grained metric, the meta distribution (MD) of the signal-to-interference ratio (SIR). To address this issue, we propose the transdimensional approach as an alternative. Here, the union of 1D PPPs on the streets is simplified to the transdimensional PPP (TPPP), a superposition of 1D and 2D PPPs. The TPPP includes the 1D PPPs on the streets passing through the receiving vehicle and models the remaining vehicles as a 2D PPP ignoring their street geometry. Through the SIR MD analysis, we show that the TPPP provides good approximations to the more cumbrous models with streets characterized by Poisson line/stick processes; and we prove that the accuracy of the TPPP further improves under shadowing. Lastly, we use the MD results to control network congestion by adjusting the transmit rate while maintaining a target fraction of reliable links. A key insight is that the success probability is an inadequate measure of congestion as it does not capture the reliabilities of the individual links.

Towards a Deep Analysis of Millimeter Wave D2D Underlaid Cellular Networks

This paper provides a deep analysis for millimeter-wave device-to-device (mm-wave D2D) underlaid cellular networks to investigate how the performance of individual users or link percentiles under constraints are affected by the interaction between D2D and cellular communication modes integrated with the salient mm-wave features. Under a general Nakagami fading model, we derive the upper bounds for the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$b$ </tex-math></inline-formula> -moments of the conditional signal-to-interference-noise ratio (SINR) distributions for the two modes given the network realization, and further calculate the upper bounds of SINR meta distributions (MDs). Moreover, we extend the MD-based analysis to evaluate the fine-grained performance for the transmission effectiveness, through deriving the bounding results for the user-perceived rate MD and spatial rate capacity (SRC), the latter of which is defined as the effective density of users satisfying a given transmission rate requirement with a certain probability. Numerical results validate the tightness of the proposed bounds and reveal the different effects of the network configuration parameter associated with mm-wave on the link and network performances from the view of individual users.

Meta Distribution of the SINR for mmWave Cellular Networks With Clusters

In order to satisfy the requirement of extremely high data rate in traffic hotspot regions, millimeter wave (mmWave) has attracted significant attention in wireless communication networks. While the coverage performance of mmWave networks based on the distribution of signal-to-interference-plus-noise ratio (SINR) has been widely studied, it provides only very limited information on the link reliability. In this paper, we provide a fine-grained performance analysis of the mmWave networks with hotspots. Specifically, we first establish a general and tractable framework to investigate the performance of mmWave networks using the Poisson cluster process integrated with several features of the mmWave band. Both open and closed association strategies are considered. To show what fraction of users in the networks achieves target reliability when the SINR is given, we derive the tier association probability and the moments of the conditional SINR distribution, based on which the exact meta distributions of SINR are given. Interestingly, in clustered mmWave networks, the widely used standard and generalized beta approximations do not work well when the blockage effect is severe. To resolve this issue, we provide a modified approximation by scaling the standard beta distribution, which is shown to be closer to the exact results. We conduct extensive simulations to study the impact of mmWave and deployment features on the performance of clustered mmWave networks. Numerical results reveal that the optimal scattering variance of mmWave base stations scales with the cluster size to maximize the number of concurrent reliable links, and increasing the antenna directivity results in more reliable communications than elevating the transmit power of mmWave base stations.

Fine-Grained Analysis on Downlink LEO Satellite-Terrestrial mmWave Relay Networks

This letter investigates the downlink integrated low earth orbit (LEO) satellite-terrestrial millimeter-wave decode-and-forward relay network, where the satellite and terrestrial relays locate in uniform distribution over the spherical cap surface of the LEO and in Poisson point process over the ground, respectively. In order to obtain fine-grained information on the coverage performance, the meta distribution (MD) of signal-to-interference-to-noise ratio (SINR) is adopted for network performance evaluation, which provides the probability that certain percentage of satellite-relays-user links are able to reach a target SINR threshold. The closed-form and analytical expression of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${b}$ </tex-math></inline-formula> -th moments of the complementary cumulative distribution functions (CCDFs) of SINR for the satellite-to-relay link and relay-to-user link are then derived, respectively. In addition, we further give the analytical MD expression via the beta approximation. Finally, numerical results are provided to validate our analytical results.

Meta Distributions—Part 1: Definition and Examples

Meta distributions (MDs) have emerged as a powerful tool in the analysis of wireless networks. Compared to standard distributions, they enable a clean separation of the different sources of randomness, resulting in sharper, more refined results. In particular, they capture the disparity of the performances of individual links or users. In this first part of a two-letter series, we start from first principles and give the formal definition of MDs and present several simple yet illustrative examples. Part 2 explores the properties of the MD in more depth and offers multiple interpretations and applications.

On the Performance of Downlink Non-Orthogonal Multiple Access Wireless Networks With Directional Beamforming and Limit of the User Number

Non-orthogonal multiple access (NOMA) is an enabling and key wireless technology for the next generation of mobile communications. Generally, previous works focus on the performance in NOMA networks, based on some assumptions. For instance, the number of users in a NOMA group, is two-category: 1) only two users; 2) any number of users without restriction. And just a certain factor of channel gain, e.g., the propagation path loss or the small scale fading, is subjectively considered to be dominant. Besides, there is lack of the closed-form expression for the probability density function (PDF) of ordered channel gain. In this paper, we investigate a downlink cellular NOMA network with the directional beamforming and the upper limit for user number (ULFUN) in a NOMA group. Meanwhile, the path loss and the Rayleigh fading are taken into account as a whole for ordering channel gains. Based on the derived closed-form PDF of ordered channel gain, we analyze the performance for an arbitrary user in the network and derive closed-form expressions for the outage probability, the approximate meta distribution of conditional success probability (CSP) and the ergodic data rate. The provided simulations demonstrate that the proposed analytical results are accurate and reasonable. Moreover, it illustrates that the existence of ULFUN can have a positive impact on the performance gain in low SINR environments, while in terms of the outage probability, ULFUN would become a non-negligible and negative factor in the improvement of performance for high SINR environments.

Meta Distribution of Downlink SIR for Binomial Point Processes

The meta distribution (MD) of the signal to interference ratio (SIR) extends stochastic geometry analysis from spatial averages to reveals find-grained information about the network performance. There have been several efforts to establish the MD framework for the Poisson point process (PPP) and other ergodic point processes. However, the MD analysis for finite point processes is overlooked. In this letter, we develop the MD of the binomial point process (BPP), which is practical for cases with a priori knowledge about the number of devices as well as their geographical spatial existence. For such finite models, we define the MD as a location-dependent likelihood of a receiver to achieve a required SIR with a probability more than a predefined threshold. This letter also extends the MD of the BPP to find the MD of finite PPP and verifies the convergence of the newly derived MD to the ergodic PPP's MD. The obtained analytical derivations are validated using Monte-Carlo simulations.

Meta Distributions—Part 2: Properties and Interpretations

In the companion letter (Part 1), we have defined and exemplified meta distributions (MDs) as a natural extension of the concepts of the mean and distribution of a random variable. Here we provide an in-depth discussion of the properties and interpretations of MDs. It includes original results on the calculation of MDs in the monotone case and two applications to simple Poisson wireless networks models.

The SINR Meta Distribution in Poisson Cellular Networks

This work studies the signal-to-interference-plus-noise ratio (SINR) meta distribution (MD) in cellular networks with a focus on the Poisson model. Firstly, we show that for stationary base station point processes, arbitrary fading, and power-law path loss with exponent α, the base station density λ and the noise power σ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> impact the SINR MD only through η\pmb \triangleq λ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α/2</sup> /σ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> , termed the network signal-to-noise ratio (NSNR). Next, we show that for Poisson cellular networks, the SINR MD can be written as g(x)θ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-2/α</sup> when the target SINR θ and the target reliability x jointly satisfy a constraint. We derive this constraint and the integral of g(x). Lastly, we discuss several extensions of the results to more general models and architectures.

The Meta Distribution of the Signal-to-Interference Ratio for Long Range Wide Area Networks With Power Control

To reduce energy consumption, a device in a long range wide area network (LoRaWAN) needs to adjust its transmit power according to the distance from its tagged gateway. It is important to measure the performance of the LoRaWANs uplink with power control. In this article, we focus on the analysis of the coverage probability and the meta distribution of the signal-to-interference ratio (SIR) for a LoRaWAN uplink with fractional power control (FPC). The LoRaWAN uplink is analyzed based on the Poisson point process. We present the possible reductions in transmit power of devices whilst ensuring that the received signal power is greater than the receiver sensitivity. We derive the coverage probability of a LoRaWAN uplink and show how power control influences it. Finally, utilizing the meta distribution of SIR, the fine-grained information of the LoRaWAN is revealed. The results show that the power control greatly increases the successful probability of edge-devices with little effect on the probability of inner-devices if an appropriate FPC coefficient is chosen. This is because the LoRa signal can be demodulated at a very low required SIR threshold.

SIR Meta Distribution in the Heterogeneous and Hybrid Networks

With the development of the technology, the wireless systems are becoming more heterogeneous with the introduction of various power nodes including femtocells, relays, or distributed antennas. Among the research of wireless network performance, the meta distribution of the signal-to-interference ratio (SIR) has attracted significant attention. Compared to the standard success (coverage) probability, the meta distribution provides much more fine-grained information about the network performance. In this paper, we analyze the meta distribution of the SIR in the multi-tier heterogeneous and hybrid networks, where each tier is based on a homogeneous independent Poisson point process model. For the open tiers (the users can associate with any tier) and the closed tiers (the users can only associate with a certain tier), we study the b th moment of the conditional success probability for the typical user and give the beta approximation of the meta distribution from analysis and simulations. Furthermore, we analyze the per-link rate control for open tiers and closed tiers, which answers the question: “how to set the SIR threshold to meet a target reliability?”. We give the approximate value of the SIR threshold to meet a target reliability and show how the value is related to the path loss exponent and densities.

Meta Distribution of Downlink SIR in a Poisson Cluster Process-Based HetNet Model

The performance analysis of heterogeneous cellular networks (HetNets), that relied mostly on the homogeneous Poisson point process (PPP) for the spatial distribution of the users and base stations (BSs), has seen a major transition with the emergence of the Poisson cluster process (PCP)-based models. With the combination of PPP and PCP, it is possible to construct a general HetNet model which can capture formation of hotspots and spatial coupling between the users and the BSs. While the downlink coverage analysis of this model in terms of the distribution of the received downlink signal-to-interference ratio ( SIR ) is well understood by now, more fine grained analysis in terms of the meta distribution of SIR is an open problem. In this letter, we solve this problem by deriving the meta distribution of the downlink SIR assuming that the typical user connects to the BS providing the maximum received power.

On the Uplink SIR Meta Distribution for AF Relay Networks

A fine-grained performance analysis of uplink transmission is provided in dual-hop amplify-and-forward (AF) relay networks. In order to improve the cell-edge coverage, multiple relay nodes are deployed around the base station and AF relay transmission is established via the aid of the scheduled relay node. The locations of the cell-edge users and relay nodes are modeled by correlated stochastic geometry models. In contrast to the conventional performance analysis based on the coverage probability, the distribution of the conditional coverage probability (CCP), which is called signal-to-interference ratio (SIR) meta distribution, is investigated by considering the random location of the nodes as well as the underlying channel. The moments of the CCP are analytically derived and the approximation of uplink SIR meta distribution is presented in semi-closed-form expression. Numerical results verify the accuracy of the analytical expressions and show the impact of the system parameters on the network performance.

Stochastic geometry based analysis for heterogeneous networks: a perspective on meta distribution

The meta distribution as a new performance metric can provide much more fine-grained information about the individual link reliability, and is of great value for the analysis and design of the future cellular networks. In this paper, we investigate the stochastic geometry based analysis for heterogeneous networks from the perspective on the meta distribution. The comprehensive overview for the fundamental framework of the meta distribution is provided, which involves the concepts of the meta distribution and its related performance metric (e.g., mean local delay and spatial outage capacity) and the efficient calculation methods of the meta distribution. The insights of the meta distribution are also stated by the comparison with standard success probability. The various applications of the meta distribution to heterogeneous networks are summarized and categorized by different types of technologies. Furthermore, some open issues and future work are discussed to promote the development and application of the meta distribution.

The Meta Distributions of Secrecy Rate for the Downlink NOMA Systems

In this letter, the Meta distributions (MDs) of the secrecy rate in downlink non-orthogonal multiple access (NOMA) systems with randomly located eavesdroppers are investigated. Due to the complexity of calculation, the MDs for NOMA users are approximated by Beta approximation and the analytical expressions for the first two moments of conditional secrecy connection probability of NOMA users are derived, while perfect/imperfect successive interference cancellations on the near user are considered. Finally, the accuracy of the result is verified by Monte Carlo simulations. The results show that the MD can capture the statistics of the secrecy rate in a given secrecy connection probability.

A Unified Framework for SINR Analysis in Poisson Networks With Traffic Dynamics

We study the performance of wireless links for a class of Poisson networks, in which packets arrive at the transmitters following Bernoulli processes. By combining stochastic geometry with queueing theory, two fundamental measures are analyzed, namely the transmission success probability and the meta distribution of signal-to-interference-plus-noise ratio (SINR). Different from the conventional approaches that assume independent active states across the nodes and use homogeneous point processes to model the locations of interferers, our analysis accounts for the interdependency amongst active states of the transmitters in space and arrives at a non-homogeneous point process for the modeling of interferers’ positions, which leads to a more accurate characterization of the SINR. The accuracy of the theoretical results is verified by simulations, and the developed framework is then used to devise design guidelines for the deployment strategies of wireless networks.

The Joint and Product Meta Distributions of the SIR and Their Applications to Secrecy and Cooperation

The meta distribution (MD) of the signal-to-interference ratio (SIR) provides more fine-grained information about the link performance than the standard success probability. This paper focuses on a fundamental extension of the SIR MD—the joint MD of the SIR at different locations, and studies its applications to physical layer security and cooperative reception. The concept of the joint MD of the SIR is formally introduced for two or more users (locations), as well as the joint conditional success probability and the $n$ -th order product MD, which is a simpler version of the joint MD. The joint MD is first applied to physical layer security. The network reliability of the secrecy transmission based on the MD is studied, taking into account the signal and interference correlations between the legitimate user and the eavesdropper. Next the joint MD is applied to cooperative reception, where we consider the scenario that the base station sends a message to a group of users, and the goal is that at least one user successfully receives the message. The moments of the conditional probability of the event that the transmission succeeds at at least one of the two users are derived. The results demonstrate the practical relevance of the joint MD.