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

Xinlei Yu, Qimei Cui, Yuanjie Wang, Na Li, Xiaofeng Tao, Mikko Valkama
2020 Science China Information Sciences  
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
more » ... tribution 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. et al. Stochastic geometry based analysis for heterogeneous networks: a perspective on meta distribution. Sci China Inf Sci, 2020, 63 (12) : 223301, https://doi.Macro BS WLAN Relay Micro BS Pico/Family BS User (a) (b) Figure 1 (Color online) Heterogeneous cellular network and its stochastic geometry based model. (a) Illustration of an HCN and (b) a realization of 3-tier PPP model [8] @Copyright 2012 IEEE. required [7] . Network densification naturally leads to the heterogeneity of cellular network, i.e., at the same time of increasing the density of the macro base stations (BSs), various low power nodes such as micro BSs, pico BSs, and relays are placed throughout the macro cell network, especially for the hotspot area of data traffic, as shown in Figure 1 (a) [8] . Besides, technologies such as massive multi-input multi-output (MIMO), device to device (D2D) networking and offloading, cloud radio access networks (C-RANs), BS cooperation, unmanned aerial vehicles (UAVs), and vehicle ad hoc networks (VANETs) have the potential to apply in 5G/B5G networks as the heterogeneous and flexible multiple physical layer technologies and radio access technologies (RATs) [2, 9] . With the network densification and heterogeneity, the cellular network becomes more complex and the interference has been one of the biggest obstacle compared with the noise [10, 11] . Meanwhile, various network node deployment makes the network no longer follow the traditional hexagonal lattice model, which challenges the network performance analysis of dense heterogeneous network. Coverage, reliability, QoE and energy still remain a major concern in the 6G path. This decade of the dense heterogeneous network development is also the decade of stochastic geometry, which is one of the most powerful mathematical tools in the modeling and performance analysis of wireless networks with randomly distributed nodes. A stochastic geometry based analysis approach has its unique advantage to evaluate the interference of wireless networks and capture the average performance of entire networks by characterizing the performance of a typical node [11, 12] . Stochastic geometry and its related techniques were first focused on ad hoc networks [13] [14] [15] [16] [17] [18] , because the interference has to be deal with carefully under the random deployment of transmitters and receivers and the relatively freedom and uncontrollable channel access protocol, such as Aloha, and carrier sense multiple access/collision detect (CSMA/CD). Subsequently, a lot of work was attempted to summarize the existing studies and promote the study of the modeling and analysis of cellular networks based on stochastic geometry. Monograph [10] provided an overview of the interference as well as the signal-to-interference ratio (SIR) analysis in large wireless networks, which were modeled by regular lattice networks and homogeneous Poisson point process (PPP). The Poisson cluster process (PCP) model and general motioninvariant model are also introduced preliminarily, where the Palm theory was used for the analysis of the more general model. Ref. [19] surveyed various techniques based on stochastic geometry and random geometric graphs comprehensively, discussed the applications of point process and percolation theory, and presented related results including connectivity, capacity, throughput, and outage probability in appeared literatures. Ref. [12] stated the potential research interests on some general spatial models, such as binomial point process (BPP), PCP, Matern hard core point process, and determinantal point process, which captures the temporal and spatial correlations of node locations compared with the most popular Yu X L, et al. Sci China Inf Sci December 2020 Vol. 63 223301:3 PPP model. These researches jointly motivated the original studies of the modeling and analysis of cellular networks using stochastic geometry, especially for heterogenous cellular networks (HCNs) [8, [20] [21] [22] [23] . Ref. [20] modeled the location of BSs in downlink cellular networks as a homogeneous PPP, and presented a tractable approach to analyze the key metrics of coverage probability (i.e., the distribution of the signalto-interference-plus-noise ratio) and rate firstly. This study shows the signal-to-interference-plus-noise ratio (SINR) performance simulated by actual BS deployment is lower bounded by the proposed model, and upper bounded by grid model. The accuracy of both bound is equal, but the lower bound from the proposed model is much better because stochastic geometry provides a tractable approach for analysis. Ref. [8] developed a K-tier downlink HCN model which is modeled by K tiers independent homogeneous PPP with different BS density, transmit power, and SINR threshold, and analyzed the coverage probability and average data rate, where the coverage probability was derived under the assumption that SINR is greater than 1. A realization of 3-tier HCN model is shown in Figure 1(b) , where the user association scheme of the maximum averaged received power is adopted, and the first tier is the macro BSs denoted by red circle markers, and the 2, 3-tiers are the pico BSs and the femto BSs denoted by green triangle and black square markers, respectively. And Ref. [21] analyzed the SINR in downlink multi-tier HCN with flexible cell association. Refs. [8, 21] both show that under the interference-limited assumption of typical HCNs, the noise has limited effect on coverage probability, and the coverage probability is independent of the BS tiers K, and the parameters of each tier, such as BS transmit power, density. Ref. [22] used various point processes including PPP, Poisson hard-core process, Strauss process, and the perturbed triangular lattice to model the BSs' deployment in actual cellular networks, and confirmed the accuracy of the SINR performance obtained by stochastic geometry model as a tight lower bound of the actual cellular networks. Ref. [23] discussed the BS cooperation in downlink K-tier HCNs which are modeled by a non-homogeneous PPP superimposed by the independent homogeneous PPP with different parameters of each tier based on the mapping theorem [24, Theorem 2.34] and the superposition property [24, Section 2.5] of PPP. Subsequently, stochastic geometry was widely used in the performance analysis of cellular networks, in which PPP served as an important role to model the real network distribution because it owns some superior properties like tractable probability generating functional (PGFL) [24, Theorem 4.9] and Slivnyak's theorem [24, Theorem 8.10]. Some detailed research findings for the stochastic geometry based analysis on cellular networks were surveyed and summarized by [25, 26] , which include HCNs, MIMO, BS cooperation, relay, physical layer security, mobility, Internet of Things (IoT), and D2D. Besides, monographs [24, 27, 28] and the recently occurred new preprint [29] are the most complete, detailed and comprehensive documents for the stochastic geometry theory and its applications on wireless communications, which are of great significance for researchers. Monograph [30] provided a stochastic geometry based analytical framework for multi-antenna wireless networks. Most of the above studies focused on the performance metric-success (coverage) probability (or outage probability, i.e., the complement of success probability), which is the complementary cumulative distribution function (CCDF) of the SINR or SIR. In addition to the success probability, the meta distribution as a new performance metric with more fine-gained information of SIR was presented firstly in [31] in 2016. The meta distribution is a new perspective for the performance analysis based on the stochastic geometry framework, which is the CCDF of the conditional success probability given the transmitter point process. The meta distribution has been focused on by researchers. Up to now, there have been over 50 literatures, as shown in References, that study the theory of meta distribution and its related applications, and the number of papers trends to increase rapidly. This survey focuses on the theory of meta distribution and its related applications, and hopes to motivate the future work on meta distribution. In the following sections of this paper, Section 2 reviews the fundamentals of meta distribution including the definitions, efficient calculation methods, and the superiority of meta distribution compared with the success probability. Section 3 introduces various applications on meta distribution classed by different technologies. Section 4 presents the challenges and open issues of meta distribution. And Section 5 concludes and remarks this paper.
doi:10.1007/s11432-020-2875-7 fatcat:sc7xl37bf5dejmeishd7igohjm