Ultra-Dense Networks for 5G and Beyond: Modelling, Analysis, and Applications

By Xiaoli Chu

Offers comprehensive insight into the theory, models, and techniques of ultra-dense networks and applications in 5G and other emerging wireless networks

The need for speed—and power—in wireless communications is growing exponentially. Data rates are projected to increase by a factor of ten every five years—and with the emerging Internet of Things (IoT) predicted to wirelessly connect trillions of devices across the globe, future mobile networks (5G) will grind to a halt unless more capacity is created. This book presents new research related to the theory and practice of all aspects of ultra-dense networks, covering recent advances in ultra-dense networks for 5G networks and beyond, including cognitive radio networks, massive multiple-input multiple-output (MIMO), device-to-device (D2D) communications, millimeter-wave communications, and energy harvesting communications.

Clear and concise throughout, Ultra-Dense Networks for 5G and Beyond - Modelling, Analysis, and Applications offers a comprehensive coverage on such topics as network optimization; mobility, handoff control, and interference management; and load balancing schemes and energy saving techniques. It delves into the backhaul traffic aspects in ultra-dense networks and studies transceiver hardware impairments and power consumption models in ultra-dense networks. The book also examines new IoT, smart-grid, and smart-city applications, as well as novel modulation, coding, and waveform designs.

• One of the first books to focus solely on ultra-dense networks for 5G in a complete presentation
• Covers advanced architectures, self-organizing protocols, resource allocation, user-base station association, synchronization, and signaling
• Examines the current state of cell-free massive MIMO, distributed massive MIMO, and heterogeneous small cell architectures
• Offers network measurements, implementations, and demos
• Looks at wireless caching techniques, physical layer security, cognitive radio, energy harvesting, and D2D communications in ultra-dense networks

Ultra-Dense Networks for 5G and Beyond - Modelling, Analysis, and Applications is an ideal reference for those who want to design high-speed, high-capacity communications in advanced networks, and will appeal to postgraduate students, researchers, and engineers in the field. 

• Language: English
• Publisher: Wiley
• Release date: Jan 31, 2019
• ISBN: 9781119473718

Book preview

List of Contributors

Symeon Chatzinotas

Interdisciplinary Centre for Security

Reliability and Trust

University of Luxembourg

Luxembourg

Jie Deng

School of Electronic Engineering and Computer Science

Queen Mary University of London

UK

Ming Ding

Data61, CSIRO

Australia

Trung Q. Duong

School of Electronics, Electrical Engineering and Computer Science

Queen's University Belfast

UK

Lorenzo Galati‐Giordano

Nokia Bell Labs

Dublin

Ireland

Adrian Garcia‐Rodriguez

Nokia Bell Labs

Dublin

Ireland

Sumit Gautam

Interdisciplinary Centre for Security

Reliability and Trust

University of Luxembourg

Luxembourg

Giovanni Geraci

Department of Information and Communication Technologies

Universitat Pompeu Fabra

Spain

Weisi Guo

School of Engineering

University of Warwick

UK

Zhu Han

Department of Electrical and Computer Engineering

University of Houston

USA

Tiep M. Hoang

Queen's University Belfast

UK

Een‐Kee Hong

College of Electronics and Information

Kyung Hee University

South Korea

Won‐Joo Hwang

Department of Information and Communication Engineering

Inje University

South Korea

Amir Hossein Jafari

Samsung Research America

USA

Markku Juntti

Centre for Wireless Communications

University of Oulu

Finland

Dani Korpi

Nokia Bell Labs

Espoo

Finland

Marios Kountouris

Mathematical and Algorithmic Sciences Lab

Paris Research Center

Huawei Technologies Co. Ltd.

France

Maria Liakata

Department of Computer Science

University of Warwick

UK

David López‐Pérez

Nokia Bell Labs

Dublin

Ireland

Mbazingwa E. Mkiramweni

Xidian University

China

Guillem Mosquera

Mathematics Institute

University of Warwick

UK

Hien Quoc Ngo

School of Electronics, Electrical Engineering and Computer Science

Queen's University Belfast

UK

Huy T. Nguyen

Department of Information and Communication Engineering

Inje University

South Korea

Nam‐Phong Nguyen

School of Electronics, Electrical Engineering and Computer Science

Queen's University Belfast

UK

Van Minh Nguyen

Mathematical and Algorithmic Sciences Lab

Paris Research Center

Huawei Technologies Co. Ltd.

France

Björn Ottersten

Interdisciplinary Centre for Security, Reliability and Trust

University of Luxembourg

Luxembourg

Weijie Qi

Department of Electronic and Electrical Engineering

University of Sheffield

UK

Tony Q.S. Quek

Information Systems Technology and Design Pillar

Singapore University of Technology and Design

Singapore

Taneli Riihonen

Laboratory of Electronics and Communications Engineering

Tampere University of Technology

Finland

Le‐Nam Tran

School of Electrical and Electronic Engineering

University College Dublin

Ireland

Hoang D. Tuan

School of Electrical and Data Engineering

University of Technology Sydney

Australia

Mikko Valkama

Laboratory of Electronics and Communications Engineering

Tampere University of Technology

Finland

Nguyen‐Son Vo

Institute of Fundamental and Applied Sciences

Duy Tan University

Vietnam

Quang‐Doanh Vu

Centre for Wireless Communications

University of Oulu

Finland

Thang X. Vu

Interdisciplinary Centre for Security, Reliability and Trust

University of Luxembourg

Luxembourg

Chungang Yang

School of Telecommunications Engineering

Xidian University

China

Howard H. Yang

Information Systems Technology and Design Pillar

Singapore University of Technology and Design

Singapore

Jie Zhang

Department of Electronic and Electrical Engineering

University of Sheffield

UK

Qi Zhang

The Jiangsu Key Laboratory of Wireless Communications

Nanjing University of Posts and Telecommunications

China

Preface

We are observing an ever‐increasing number of connected devices and the rapid growth of bandwidth‐intensive wireless applications. The number of wirelessly connected devices is anticipated to exceed 11.5 billion by 2019, i.e. nearly 1.5 mobile devices per capita. In addition, it is expected that we will witness a 10 000‐fold growth in wireless data traffic by the year 2030. Such unprecedented increases in mobile data traffic and network loads are pushing contemporary wireless network infrastructures to a breaking point. These predictions have raised alarm to the wireless industry and mobile network operators who are faced with the challenges of provisioning high‐rate, low‐delay, and highly reliable connectivity anytime and anywhere without significantly increasing energy consumption at the infrastructure, such as base stations, fronthaul and backhaul networks, and core networks.

The above challenges demand a paradigm shift to the wireless network infrastructure. In this context, ultra‐dense networks, which are characterized by a very high density of low‐power radio access nodes with different transmit power levels, radio frequency coverage areas, and signal/data processing capabilities, such as small cell (e.g. pico‐ or femto‐cells) access points, remote radio heads, and relay nodes, have attracted a lot of interest from the wireless industry and research community. Compared to a high‐power macrocell, each small cell has a much smaller coverage radius (from several meters to several hundred meters) and can assign a larger amount of radio resources to each user within its coverage area than in conventional cellular networks, thus improving users' quality‐of‐service (QoS) and quality‐of‐experience (QoE). Moreover, the low transmission power of small cells makes it possible for them to operate in an unlicensed spectrum; for example, the large available bandwidth in the millimeter‐wave (mm‐Wave) frequency range from 30 GHz to 300 GHz, to mitigate the scarcity of licensed spectrum in mobile networks.

While the commercial consensus is fuzzy regarding which technologies will be used for fifth generation (5G) wireless communications, ultra‐dense networks have been widely considered as a key enabler for 5G and beyond in wireless communication network implementation. In addition to their potential in provisioning ubiquitous high‐capacity wireless connectivity, ultra‐dense networks also offer numerous opportunities and flexibility to be incorporated with other 5G candidate technologies, such as mm‐Wave communications, massive multiple‐input multiple‐output (MIMO), non‐orthogonal multiple access, in‐band full‐duplex operation, simultaneous wireless information and power transfer (SWIPT), device‐to‐device (D2D) communications, and distributed caching, to enable the realization of 5G technologies and systems' full potential.

Dense coexistence of various neighboring and/or overlapping radio access nodes, dynamic network topologies, complicated co‐channel interference scenarios, backhaul provisioning, energy consumption, QoS provisioning, and security issues bring new challenges for ultra‐dense network deployment. These technical challenges need to be addressed in order to exploit the true potential of ultra‐dense networks. Although some research attempts have been made toward understanding the theoretical and practical performance limits, a comprehensive mathematical methodology to capture the dynamic topologies, large‐scale heterogeneous interference, and the high level of randomness in ultra‐dense networks essential to the system‐level analysis and design of 5G networks is missing in the existing literature. In light of the ongoing densification of radio access nodes and wireless connected devices toward 5G wireless networks, there is an urgent need for the research community, industry, and even end users to better understand the fundamental technical details as well as the achievable performance gains of ultra‐dense networks.

Ultra‐dense Networks for 5G and Beyond: Modelling, Analysis, and Applications provides a comprehensive and systematic exposition on the state‐of‐the‐art of ultra‐dense networks and their applications in 5G cellular and other wireless networks, with a delicate balance between mathematical modeling, theoretical analysis, and practical design. It contains cutting‐edge tutorials on the theoretical and technical foundations that underpin ultra‐dense networks, as well as insightful surveys of ultra‐dense network‐related emerging technological trends that will be interesting and informative to readers of all backgrounds. The book is written by researchers currently leading the research and development of ultra‐dense networks, covering a wide spectrum of topics, ranging from system modeling and performance analysis, network performance optimization, radio resource management, wireless self‐backhauling, massive MIMO to unlicensed spectrum, energy efficiency, big data analytics, physical layer security, SWIPT, distributed caching, and cooperative video streaming, etc.

The book is organized into 12 chapters, which are grouped into three parts. In the following, we provide a brief tour through the book's parts and chapters to show how this book addresses the challenges faced by ultra‐dense networks from various aspects.

Part I, including Chapters 1 to 3, presents the background information and fundamental knowledge necessary for understanding the theoretical and technical foundations of ultra‐dense networks from three main disparate aspects.

Chapter 1 explores the fundamental performance limits of ultra‐dense networks due to physical limits of radio wave propagation. Focusing on the impact of network node density on the network performance, the authors model the spatial distribution of network nodes using Poisson point processes (PPP). As a result of the close proximity of base stations and mobile devices in ultra‐dense networks, the elevated base station height and dual‐slope line‐of‐sight (LoS) and non‐line‐of‐sight (NLoS) propagations are considered in the modeling of small‐scale and large‐scale fading. Under this system model, analytical expressions are derived for ultra‐dense network downlink performance in terms of coverage probability, throughput, and average transmission rate. Then, under general multi‐slope pathloss and channel power distribution models, an analytical framework is presented to enable the analysis of asymptotic performance limits (i.e. scaling laws) of network densification.

Chapter 2 presents a general framework for performance analysis of ultra‐dense networks, particularly focusing on the impact of pathloss and multipath fading. Under a pathloss model that incorporates both LoS and NLoS propagations and a distance‐dependent multi‐path Rician fading model with a variant Rician K‐factor, the analytical expressions for both coverage probability and area spectral efficiency are derived. Comparing the performance impact of LoS and NLoS transmissions in interference‐limited ultra‐dense networks under Rician fading with that under Rayleigh fading, the analytical and simulation results show that pathloss dominates the overall system performance of ultra‐dense networks, while the impact of multi‐path fading is quite limited and does not help to mitigate the performance loss caused by the many LoS interfering links in ultra‐dense networks, especially in single‐input single‐output systems.

Chapter 3 considers the network densifications in wireless user devices, radio access nodes, and cloud edge nodes, and presents a comprehensive introduction to mean field game theory and tools, which is useful in the design and analysis of ultra‐dense networks with spatial‐temporal dynamics. Following that, the authors present a survey of the latest applications of mean field games in the 5G era in general (including device‐to‐device communications and cloud‐edge networks) and in 5G ultra‐dense networks in particular, with a focus on resource management problems such as interference mitigation, energy management, and caching.

Chapters 4 to 8 are grouped into Part II, which exploits the synergy between ultra‐dense networks and other key 5G candidate technologies.

Chapter 4 studies different schemes of wireless self‐backhauling in ultra‐dense networks, with a focus on the recently proposed in‐band full‐duplex self‐backhauling. The deployment of wireless backhauling, in general, helps to resolve the scalability problems faced by wired backhaul connections, because wireless backhauling does not require physical cables. The use of inband wireless self‐backhauling is even more intriguing than traditional wireless backhauling, because it enables the reuse of spectral resources between radio access links and backhaul links. In this respect, no additional or dedicated frequency resources are required for the backhaul, and thus the in‐band wireless self‐backhauling will be commercially beneficial. Three different in‐band wireless self‐backhauling schemes based on full‐duplex or half‐duplex operations are analyzed and compared.

Chapter 5 aims to optimally leverage the dense deployment of small cells and massive MIMO in future mobile networks. Using analytical tools from stochastic geometry, the authors derive a tight approximation of the achievable downlink rate for a two‐tier heterogeneous cellular network and use it to compare the performance between densifying small cells and expanding base station antenna arrays. The results show that increasing the density of small cells improves the downlink rate much faster than expanding antenna arrays at base stations. However, when the small cell density exceeds a certain threshold, the network capacity may start to deteriorate. On the contrary, the network capacity keeps increasing with the expansion of base station antenna arrays until it reaches an upper bound, which is caused by pilot contamination. This upper bound surpasses the maximum network capacity achieved by the dense deployment of small cells. Moreover, the authors provide practical design insights into the tradeoff between the dense deployment of small cells and massive MIMO in future high‐capacity wireless networks.

Chapter 6 introduces a promising 5G technology termed as cell‐free massive MIMO, where radio access points distributed in an area coherently serve mobile users in the area using the same time/frequency resource. It is called cell‐free because there is no boundary among cells, unlike as seen in traditional cellular networks. Cell‐free massive MIMO networks complement ultra‐dense networks by connecting closely located access points together. The connection of access points is feasible because only a limited amount of necessary information is exchanged and the central processing unit is not required to have a high computing capability. In particular, the authors study the physical layer security in cell‐free massive MIMO networks under a pilot spoofing attack, where an active eavesdropper attacks the uplink training, and present a simple counterattack scheme based on the transmit power control at all the involved radio access points.

Chapter 7 proposes to solve the spectrum crunch in ultra‐dense networks by employing massive MIMO in unlicensed spectrum and by leveraging the spatial interference suppression capabilities of multi‐antenna systems. The authors discuss the motivation and justification of using unlicensed spectrum bands in ultra‐dense networks, describe the technical fundamentals of massive MIMO in an unlicensed spectrum, and identify the main use cases and the associated challenges. The results show that massive MIMO in an unlicensed spectrum is capable of significantly boosting the performance of ultra‐dense networks in both outdoor and indoor environments.

In Chapter 8, a set of advanced optimization tools is introduced to maximize the energy efficiency for ultra‐dense networks, where the total power consumption may become a critical concern as the number of network nodes and connected devices increases. The chapter begins with the introduction of optimization techniques that are useful for maximizing energy efficiency, including concave–convex fractional programming, non‐tractable fractional programming, and the alternating direction method of multipliers for distributed implementation. The chapter then moves on to demonstrate how these optimization techniques can be applied to maximizing the energy efficiency of spectrum‐sharing dense small cell networks.

In Part III, which includes Chapter 9 to Chapter 12, the promising applications of ultra‐dense networks in the 5G era are presented and discussed.

In Chapter 9, the authors provide an insightful discussion on how big data methods can be used to improve the deployment of and the QoS in ultra‐dense networks, covering both structured and unstructured data analytics. It is demonstrated that having access to consumer data and being able to analyse it in the wireless network context would allow effective user‐centric deployment and operations of ultra‐dense networks. In particular, the chapter outlines two big data approaches for improving ultra‐dense network deployment: (1) identify the spatial‐temporal traffic and service patterns to aid targeted deployment of dense networks; and (2) identify social community patterns to assist ultra‐dense peer‐to‐peer and D2D networking.

In Chapter 10, the authors look into physical layer security opportunities and challenges for ultra‐dense networks and show how physical layer security technologies can be applied to mitigate issues caused by unreliable wireless backhaul links in ultra‐dense networks. Given that the wireless backhaul links are not reliable at all times, secrecy outage probability for a spectrum sharing ultra‐dense network under the impact of unreliable backhaul links is evaluated.

Chapter 11 discusses the application of SWIPT in energy harvesting enabled ultra‐dense networks with a distributed caching architecture. Without loss of generality, the system model consists of a source node and a destination node, which communicate with each other with the help of multiple relay nodes. Each relay is equipped with a cache memory and energy harvesting capability. Based on the time‐splitting architecture, the authors have focused on the problem of relay selection to maximize the data throughput between the relay and the destination subject to the amount of harvested energy at the relays. A separate optimization problem is formulated to maximize the energy stored at the relays subject to given QoS constraints.

Chapter 12 investigates the application of cooperative video streaming with the support of D2D caching in ultra‐dense networks. The ever‐increasing demand for video streaming services at a high data rate and high QoE is increasingly likely to cause traffic congestion at the backhaul links in ultra‐dense networks. The authors propose a joint rate allocation and description distribution optimization algorithm, where both the cache storage and downlink resources of mobile devices are exploited, to enable densely deployed base stations in order to provide high QoE video streaming services to mobile users while saving energy at the same time.

Part I

Fundamentals of Ultra‐dense Networks

1

Fundamental Limits of Ultra‐dense Networks

Marios Kountouris and Van Minh Nguyen

Mathematical and Algorithmic Sciences Lab, Paris Research Center, Huawei Technologies Co. Ltd., France

1.1 Introduction

Mobile traffic has significantly increased over the last decade, mainly due to the stunning expansion of smart wireless devices and bandwidth‐demanding applications. This trend is forecast to be maintained, especially with the deployment of fifth generation (5G) and beyond networks and machine‐type communications. A major part of the mobile throughput growth during the past few years has been enabled by the so‐called network densification, i.e. adding more base stations (BSs) and access points and exploiting spatial reuse of the spectrum. Emerging 5G cellular network deployments are envisaged to be heterogeneous and dense, primarily through the provisioning of small cells such as picocells and femtocells. Ultra‐dense networks (UDNs) will remain among the most promising solutions to boost capacity and to enhance coverage with low‐cost and power‐efficient infrastructure in 5G networks. The underlying foundation of this expectation is the presumed linear capacity scaling with the number of small cells deployed in the network. In other words, doubling the number of BSs doubles the capacity the network supports in a given area and this can be done indefinitely. Nevertheless, in this context, several important questions arise: how close are we to fundamental limits of network densification? Can UDNs indefinitely bring higher overall data throughput gains in the network by just adding more infrastructure? If the capacity growth arrives to a plateau, what will cause this saturation and how the network should be optimized to push this saturation point further? These are the questions explored in this chapter.

The performance of wireless networks relies critically on their spatial configuration upon which inter‐node distances, fading characteristics, received signal power, and interference are dependent. Cellular networks have been traditionally modeled by placing the base stations on a regular grid (usually on a hexagonal lattice), with mobile users either randomly scattered or placed deterministically. Tractable analysis can sometimes be achieved for a fixed user location with a small number of interfering BSs and Monte Carlo simulations are usually performed for accurate performance evaluation. As cellular networks have become denser, they have also become increasingly irregular. This is particularly true for small cells, which are deployed opportunistically and in hotspots and dense heterogeneous networks (HetNets). As a result, the widely used deterministic grid model has started showing its limitations and cannot be used for general and tractable performance analysis results in UDNs. Although more real‐world deployment data are needed to make conclusive statements on which is a better model for UDNs, stochastic spatial models are often a more appropriate model versus a deterministic one. In a random spatial model, the BS locations are modeled by a two‐dimensional spatial point process, the simplest being the Poisson Point Process (PPP). This model has the advantages of being scalable to multiple classes of overlaid BSs and accurate to model location randomness, especially that of UDNs and HetNets. Additionally, powerful tools from stochastic geometry can be used to derive performance results, such as coverage, average rate, and throughput, for general dense multi‐tier networks in closed form, which was not even possible for macrocellular (single‐tier) networks using a deterministic grid model. Moreover, the suitability of this mathematical abstraction can be reinforced in particular for UDNs due to the fact that network planning of a large number of small base stations is complex, making the resulting network closer to a random distribution model. The employment of PPPs allows the capture of the spatial randomness of real‐world UDN deployments (often not fully coordinated) and, at the same time, obtains precise and tractable expressions for system‐level performance metrics [1, 2]. Furthermore, it has been shown that using random spatial models does not introduce important discrepancy to a regular/deterministic model. For instance, the PPP case has nearly the exact same signal‐to‐interference ratio (SIR) statistics as a very wide class of spatial BS distributions, including the hexagonal grid, with just a small fixed SIR shift (e.g. 1.53 dB) [3].

Recent work has also considered more general models with inhibition (e.g. cluster models) or repulsion (e.g. determinantal point process) [2, 4, 5]. Point process models like the Matern hardcore process, Ginibre process, Strauss process, Cox process, and the perturbed lattice are more realistic point process models than the PPP and the hexagonal grid models since they can capture the spatial characteristics of the actual network deployments better. However, more general point processes typically result in less tractable expressions that include integrals that must be numerically evaluated, which, however, is still much simpler than an exhaustive network simulation.

There has been noticeable divergence on the conclusions of various network studies using spatial models, according to which densification is not always beneficial to the network performance. Recent and often conflicting findings based on various modeling assumptions have identified that densification may eventually stop delivering significant throughput gains at a certain point. First, it has been shown that the coverage probability does not depend on the network density and thus the throughput can grow linearly with the BS density in the absence of background noise for both the closest [1] and the strongest BS association [6]. These results assume simple models—mostly for tractability reasons—in which (i) BSs are located according to a homogeneous PPP and are placed at the same height as the UEs, and (ii) the signal propagation is modeled using the standard single‐slope unbounded pathloss and Rayleigh distribution for the small‐scale fading. By contrast, using a dual‐slope pathloss model, Rayleigh fading, and nearest BS association, [7] shows that both coverage and capacity strongly depend on the network density. More precisely, the coverage probability, expressed in terms of signal‐to‐interference‐plus‐noise ratio (SINR), is maximized at some finite BS density and there exists a phase transition on the asymptotic potential network throughput with ultra‐densification (i.e. network density goes to infinity). If the near‐field pathloss exponent is less than one, the potential throughput goes to zero with denser network deployment, whereas if it is greater than one, the potential throughput grows unboundedly as the network becomes increasingly denser.

In [8] and [9], the performance of millimeter wave systems is analyzed using stochastic geometry and considering scenarios with line‐of‐sight (LOS) and non‐line‐of‐sight (NLOS) propagation for the pathloss attenuation. More comprehensive models can be found in [10] ‐ [12], where the pathloss exponent changes with a probability that depends on the distance between BSs and UEs. In [13], the authors consider strongest cell association with bounded pathloss and lognormal shadowing and show that the coverage attains a maximum point before starting to decay when the network becomes denser. In [14], similar conclusions are obtained under Nakagami fading for the LOS propagation and both nearest and strongest BS association. Based on multi‐slope pathloss and smallest pathloss association, [15] shows that the network coverage probability first increases with BS density, and then decreases. Moreover, the area spectral efficiency will grow almost linearly as the BS density goes asymptotically large. Optimal densification in terms of maximum SINR‐coverage probability is investigated in [16]. In [17], interference scaling limits in a Poisson field with singular power law pathloss and Rayleigh fading are derived. Moreover the authors in [18] provide spectral efficiency scaling laws with spatial interference cancellation at the receiver. It is shown that linear scaling of the spectral efficiency with network density can be obtained if the number of receive antennas increases super‐linearly with the network density (or linearly in the case of bounded pathloss).

In this chapter, we aim at providing an answer to whether there are any fundamental limits to 5G UDNs due to physical limits arising from electromagnetic propagation. To tackle this question, we follow two approaches. First, we derive analytical expressions for the downlink performance of UDNs under a system model that combines PPP‐distributed elevated BSs and dual‐slope LOS/NLOS propagation affecting both small‐scale and large‐scale fading. Second, we investigate the asymptotic performance limits (scaling laws) of network densification under general multi‐slope pathloss and channel power distribution models. Using tools from extreme value theory [19], and in particular regular variation analysis [20], we present a general framework that allows us to derive the scaling regimes of the downlink SINR, coverage probability, potential throughput, and average per‐user and system‐wide rate.

Notation

The distribution function of is denoted by and . In addition, for real functions and , we say if and if . We also use notation , , to denote the convergence in distribution, convergence in probability, and almost sure (a.s.) convergence, respectively. A positive, Lebesgue measurable function on is called regularly varying with index at if for . In particular, is called slowly varying (respectively rapidly varying) (at ) if (respectively if ). We denote by the class of regularly varying functions with index . and are respectively the probability and the expectation operators. is the Palm probability with respect to the point process (in PPP – Slivnyak's Theorem) and the expectation is taken with respect to the measure . The Gauss hypergeometric function is denoted by and denotes the gamma function.

1.2 System Model

1.2.1 Network Topology

We consider a downlink dense wireless network, in which the locations of transmitters (BSs) are modeled as a homogeneous PPP of intensity , where is the dimension of the network. Users are distributed according to some independent and stationary point process (e.g. PPP), whose intensity is sufficiently larger than in order to ensure that all BSs are active, i.e. every BS has at least one active user associated within its coverage. The case of partially loaded networks can be straightforwardly applied by considering that the network intensity is , where is the network load ratio. The analysis is performed for a typical user located at the origin ; hence the link between the origin and its associated BS is a typical link. Since the network domain is limited (i.e. not the entire is considered) and points far away from the origin generate weak interference to the typical user due to pathloss attenuation, the distance from the user to any node is upper bounded by an arbitrarily large constant . Each node transmits with a power level that is independent of the others but is not necessarily constant. Lastly, we assume that all BSs may be elevated at the same height , measured in meters, whereas the typical UE is at the ground level; alternatively, can be interpreted as the elevation difference between BSs and UEs if the latter are all placed at the same height.

1.2.2 Wireless Propagation Model

Radio links in a wireless medium are susceptible to time‐varying channel impediments, interference, and background noise. These include long‐term attenuation due to pathloss, medium‐term variation due to shadowing, and short‐term fluctuations due to multi‐path fading.

The small‐scale fading between node and the typical user is denoted by and is assumed independent across time and space. In the most general model, includes all propagation phenomena and link gains except pathloss, such as transmit power, fast fading, shadowing, antenna gains, etc. We may refer to as channel power. Fast fading is usually modeled by a Rayleigh distribution of the channel amplitude (i.e. is exponentially distributed), while shadowing is modeled by a lognormal distribution of the channel power. Given node location , the variables are assumed to be non‐zero and independently distributed according to some distribution . Incorporating the channel fading into the spatial model, , forms an independently marked PPP [5].

The large‐scale pathloss function is denoted by and is assumed to be a non‐decreasing function of . For a realistic and practically relevant pathloss model, we have that a pathloss function is bounded if and only if (iff) , and unbounded otherwise. Furthermore, the pathloss function is said to be physical iff . Let denote the horizontal distance between and the typical UE, measured in meters. We consider a distance‐dependent LOS probability function , i.e. the probability that a BS located at experiences LOS propagation depends on the distance . We use and to denote the subsets of BSs in LOS and in NLOS propagation conditions, respectively. We remark that each BS is characterized by either LOS or NLOS propagation independently from the others and regardless of its operating mode as serving or interfering BS.

We consider two different pathloss models: in the first part of this chapter where we aim at deriving exact analytical expressions for the network performance, we adopt the standard power‐law (non‐bounded) model; in the second part where we focus on asymptotic limits when , we consider