M T LTE-A S C N G

BUDAPEST UNIVERSITY OF TECHNOLOGY AND ECONOMICS

Faculty of Electrical Engineering and Informatics

Department of Networked Systems and ServicesMobile Communications and Quantum Technologies Laboratory

MODELLING TWO-TIER LTE-ADVANCED

SMALL CELL NETWORKSWITH STOCHASTIC GEOMETRY

Ph. D. Thesis ofof

Zoltán Jakó

Scientific supervisor:

Gábor Jeney, PhD.

Budapest, 2017

c© 2017, All rights reserved to the author

This document was typeset in LATEX 2ε applying Adobe R©Times font family

i

Nyilatkozat Alulírott Jakó Zoltán kijelentem, hogy ezt a doktori értekezést magam készí-tettem és abban csak a megadott forrásokat használtam fel. Minden olyan részt, amelyet szószerint, vagy azonos tartalomban, de átfogalmazva más forrásból átvettem, egyértelmuen, aforrás megadásával megjelöltem.

Declaration I, undersigned Zoltán Jakó hereby declare that this Ph.D. dissertation wasmade by myself, and I only used the sources given at the end. Every part that was quotedword-for-word, or was taken over with the same content, I noted explicitly by giving thereference to the source.

Budapest, Hungary, January 30, 2017

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Jakó Zoltán

Abstract

The amount of speech and data communication proceeded through mobile network is in-creasing rapidly and it is foreseen that this growing tendency will be continuous in thefuture. In LTE-Advanced (LTE-A) the conventional access network (formed by EvolvedNode-B or shorter eNBs), due to increased traffic, can not guarantee the Quality of Service(QoS). Therefore, in order to reduce the load (and fulfil the QoS requirements) the macrocell structure should be augmented with smaller cells. The literature applies the term “Smallcells” for smaller cells underlaid to an umbrella macro cell. These small cells operate onthe same spectra, as the macro eNBs, however with a reduced transmission power. Due tothe reduced transmission power, small cells can cover a smaller area (compared to macroeNBs). The potential deployment places of small cells are indoor (e.g. buildings) or denselypopulated areas.

The goal of this dissertation is to investigate the effect of the small cell layer in an LTE-A access network. The investigation is proceeded with the mathematical tools providedby Stochastic geometry. The usage of small cells can increase the coverage areas and thesystem throughput of the mobile access network. Nevertheless, the deployment of smallcells modifies the interference pattern of the given area significantly, which also modifies theprobability of coverage (or the probability of service outage). The dissertation representshow to create models for these multi-layer networks and calculate mathematically (with theaid of Stochastic geometry) important and essential system parameters such as the probabil-ity of outage/coverage and the overall system throughput. Simulations are provided in thedissertation in order to validate the correctness and accuracy of the proposed mathematicsforms.

iii

Kivonat

A mobil hálózaton keresztül lebonyolított beszéd és adatforgalom is egyre jobban növekszikés az elorejelzések szerint ez a növekedési tendencia folytatódik a közeljövoben is. Az LTE-Advanced-ban (LTE-A) a „hagyományos” makró bázisállomások (az ún. Evolved Node-B,vagy röviden eNB) alkotta hozzáférési hálózat önmagában már nem tudja garantálni a szol-gáltatás minoségét, ilyen fokozott adatforgalom mellett. Ezért a makró cellák tehermentesí-tésére kisebb cellákat is üzembeállítanak, hogy a megfelelo minoségu szolgáltatás biztosítottlegyen. Ezek a kisebb cellák ugyanazon a frekvenciasávon üzemelnek, mint a makrócellák.Ugyanakkor jóval kisebb teljesítménnyel sugároznak, így pedig sokkal kisebb területet fed-nek le. Tipikusan olyan helyszínekre telepítenek típusú cellákat, ahol a már meglévo makróhozzáférési hálózat nem elegendo: pl. épületekbe (femtocellákat), surun lakott városrészek-re (pikó- és mikró cellákat). A szakirodalom „kis cellák (small cells)” gyujtonévvel illetiezeket a kis bázisállomásokat.

Jelen Ph. D. disszertáció a LTE-Advanced hozzáférési hálózat „kis cellákkal” történo ki-bovítésének lehetoségét és annak hatásait vizsgálja. A vizsgálatokhoz felhasználom a szto-chasztikus geometria átal biztosított matematikai eszköztárat. A kis cellák alkalmazása je-lentosen megnöveli a mobil hálózat lefedettségét és ezáltal a mobil felhasználók adatátvitelisebességét is. Ugyanakkor módosíthatja az adott terület interferencia térképét, ami pedigbefolyásolja a szolgáltatás kiesés- és a lefedettség valószínuséget is.

A disszertációban bemutatom, hogy lehet a sztochasztikus geometria által biztosítottmatematika apparátussal, modellezni kétrétegu LTE-A kis cellás hálózatokat és kiszámolniolyan paramétereket (pl. szolgáltatás kiesés/ lefedettség valószínusége, rendszerszintu kapa-citás), amelyek felhasználhatóak a mobil szolgáltatóknál hálózattervezés során. A disszertá-cióban bemutatott sztochasztikus geometrián alapuló vizsgálatok során kapott eredeménye-ket (ahol csak lehetett) szimulációkkal vetettem össze, hogy igazoljam a bemutatott formulákhelyességét és érvényességét.

v

Köszönetnyilvánítás

Mindenekelott szeretnék köszönetet mondani konzulensemnek Dr. Jeney Gábornak. Hasz-nos tanácsai, ötletei és kritikái nélkülözhetetlen segítséget nyújtott Ph. D. hallgatóként vég-zett kutatómunkámban és disszertációm elkészítése során egyaránt. Neki tartozom azértis köszönettel, hogy színvonalas nemzetközi projektben foglalkozhattam izgalmas kutatá-si feladatokkal, és szakmai fejlodésemhez minden feltételt megteremtett. Különösen hálásvagyok szerzotársaimnak és kollégáimnak a közös munkáért és publikációkért. Végül, denem utolsósorban szeretném megköszönni a Mobil Kommunikáció és KvantumtechnológiákLaboratórium (MCL) tagjainak a kiváló légkört, amelyet biztosítottak számomra az eredmé-nyes munkához.

vii

Contents

Abbreviations xvii

Variables and Symbols xviii

1 Introduction 11.1 Background of the Research . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.1 Brief History of Stochastic Geometry and Related works . . . . . . 61.2 Motivation and Organization of the Document . . . . . . . . . . . . . . . . 8

1.2.1 Goal of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2.2 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . 9

2 System model 112.1 Location of Small Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1.1 Poisson Point Process . . . . . . . . . . . . . . . . . . . . . . . . 142.1.2 Application of PPP model . . . . . . . . . . . . . . . . . . . . . . 152.1.3 Poisson cluster Process . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.1 Error function and complementary Error function . . . . . . . . . . 192.2.2 Gamma function and incomplete Gamma function . . . . . . . . . 192.2.3 Jensen’s inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.4 Relationship Between Random Variables and Moments . . . . . . . 20

2.2.4.1 Exponential Distribution . . . . . . . . . . . . . . . . . . 202.2.4.2 Gamma Distribution . . . . . . . . . . . . . . . . . . . . 212.2.4.3 Non-central Chi-squared Distribution . . . . . . . . . . . 222.2.4.4 Weibull Distribution . . . . . . . . . . . . . . . . . . . . 232.2.4.5 Lognormal Distribution . . . . . . . . . . . . . . . . . . 23

2.2.5 Fundamentals of Stochastic geometry . . . . . . . . . . . . . . . . 232.2.5.1 Probability Generating Functional . . . . . . . . . . . . 24

ix

x CONTENTS

2.3 General Interference Characterization . . . . . . . . . . . . . . . . . . . . 252.3.1 Small cell interference in the macrocell . . . . . . . . . . . . . . . 262.3.2 Femtocell interference at another femtocell . . . . . . . . . . . . . 27

3 Small Cell modelling with PPP 293.1 Interference Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 Outage and Coverage Probability in PPP . . . . . . . . . . . . . . . . . . . 35

3.2.1 Lognormal fading . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2.2 Rayleigh and Nakagami-m fading . . . . . . . . . . . . . . . . . . 36

3.2.2.1 Rayleigh and Nakagami-m fading using Lévy distribution 373.2.2.2 Rayleigh and Nakagami-m fading using PGFL . . . . . . 38

3.2.3 Rice fading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.3 Results for Outage/Coverage Probability . . . . . . . . . . . . . . . . . . . 44

3.3.1 Results for Lognormal Fading . . . . . . . . . . . . . . . . . . . . 443.3.2 Results for Rayleigh and Nakagami-m Fading . . . . . . . . . . . . 453.3.3 Results for Rician Fading . . . . . . . . . . . . . . . . . . . . . . . 49

3.4 Average System Throughput . . . . . . . . . . . . . . . . . . . . . . . . . 513.4.1 Probability distribution of SIR . . . . . . . . . . . . . . . . . . . . 523.4.2 Overall Throughput of a Two-tier System . . . . . . . . . . . . . . 53

3.5 Results of Throughput enhancement with Small Cells . . . . . . . . . . . . 55

4 Small Cell modelling with PCP 594.1 Interference Characterization . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.1.1 Interference Characterization with Monte Carlo Simulations . . . . 614.1.2 Service Outage Probability in cluster based Small cells . . . . . . . 62

4.1.2.1 Approximated form for coverage probability . . . . . . . 634.2 Results for Outage/Coverage Probability . . . . . . . . . . . . . . . . . . . 66

4.2.1 Results for Thomas cluster Process based model . . . . . . . . . . 674.2.2 Results for Matérn cluster Process based model . . . . . . . . . . . 75

4.3 Results of Throughput enhancement . . . . . . . . . . . . . . . . . . . . . 79

5 Conclusive remarks 815.1 Further Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

A Appendix 85A.1 Transforming Random Variables . . . . . . . . . . . . . . . . . . . . . . . 85

A.1.1 Exponential distribution . . . . . . . . . . . . . . . . . . . . . . . 85A.1.2 Gamma distribution . . . . . . . . . . . . . . . . . . . . . . . . . 85A.1.3 Rice distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

CONTENTS xi

A.2 Lévy distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86A.3 Calculation of Raw moments . . . . . . . . . . . . . . . . . . . . . . . . . 88

A.3.1 Normal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 88A.3.2 Uniform distribution . . . . . . . . . . . . . . . . . . . . . . . . . 90

A.4 Relative errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Bibliography 93

List of Publications 99

xii CONTENTS

List of Figures

1.1 Downtown LTE-A network extended with Small cells . . . . . . . . . . . . 31.2 3GPP LTE-Advanced Two-tier Small cell architecture [1] . . . . . . . . . . 4

2.1 An LTE-A Two-tier small cell network . . . . . . . . . . . . . . . . . . . . 122.2 Illustration of Homogeneous Poisson point process . . . . . . . . . . . . . 152.3 PPP based Random graph model . . . . . . . . . . . . . . . . . . . . . . . 162.4 cluster processes as Small cell deployment modelling . . . . . . . . . . . . 172.5 Illustration of Cross-tier and Co-tier interference and PRB assignment . . . 26

3.1 Probability density functions for different fast fadings . . . . . . . . . . . . 323.2 Validation of the proposed forms with Monte-Carlo simulations . . . . . . 343.3 The outage probabilities for Lognormal faded channel . . . . . . . . . . . . 453.4 Probability of outage in Rayleigh, Nakagami-m fading . . . . . . . . . . . 473.5 Probability of Coverage with Rayleigh, Nakagami-m fading types and path

loss exponents (α) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.6 Probability of outage in Rician fading . . . . . . . . . . . . . . . . . . . . 493.7 Probability of Coverage with Rician fading with various path loss exponents

(α) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.8 Probability of coverage for macro UE vs. Pc

Ps . . . . . . . . . . . . . . . . . 513.9 Throughput enhancement with Small Cells . . . . . . . . . . . . . . . . . . 56

4.1 Empirical c.d.f. results from Monte-Carlo simulations . . . . . . . . . . . . 614.2 Outage probability in Poisson Point process and Poisson cluster Process

cases with different nakagami-m fading . . . . . . . . . . . . . . . . . . . 684.3 Result for outage probabilities vs. different γ and c values, ‖z‖= 100 m . . 694.4 Result for outage probabilities with different T values, ‖z‖= 100 m, m = 4 704.5 Comparing PPP and PCP node deployment . . . . . . . . . . . . . . . . . 704.6 Outage probability for different thresholds and average number of Small

cells, ‖z‖= 100 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

xiii

xiv LIST OF FIGURES

4.7 Outage probability for different β and average number of small cells, ‖z‖=100 m, T = 0dB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.8 Outage probability based on cluster size . . . . . . . . . . . . . . . . . . . 744.9 Probability of service outage in case on Matérn cluster process (T = 1) . . . 764.10 Validating results with Monte Carlo simulations . . . . . . . . . . . . . . . 774.11 Results for coverage probabilities for different distances (‖z‖) and power

fraction(

Pc

Ps

). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.12 Throughput enhancement vs. Ns (Thomas- and Matérn cluster processes) . 80

A.1 Validating Poisson point process interference and outage probability resultswith simulations (Relative error) . . . . . . . . . . . . . . . . . . . . . . . 91

A.2 Validating Poisson cluster process outage probability results with simula-tions (Relative error) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

List of Tables

1 Common Variables and Symbols . . . . . . . . . . . . . . . . . . . . . . . xix

3.1 E√

h Fading moments . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Throughputs used for Average System capacity calculation in MIMO 2× 2

configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

xv

xvi LIST OF TABLES

Abbreviations

The abbreviations used in the thesis are summarized here in alphabetical order.

3GPP 3rd Generation Partnership Project

AMC Adaptive Modulation and Coding

BS Base station

c.d.f. Cumulative distribution function

CDMA Code division multiple access

CSG Closed Subscriber Group

DSL Digital subscriber line

eNB Evolved Node-B

E-UTRA Evolved Universal Terrestrial Radio Access

EPC Evolved Packet Core

FDD Frequency Division Duplexing

GSM Global System for Mobile Communications

HeNB Home Evolved Node-B

HO Handover or Handoff

IP Internet protocol

ISM Industrial, scientific and medical radio bands

xvii

xviii LIST OF TABLES

LoS Line-of-sight

LTE Long Term Evolution

LTE-A Long Term Evolution Advanced

MGF Moment Generating Function

MIMO Multiple-input and multiple-output

NLoS Non-Line-of-sight

OFDM(A) Orthogonal Frequency-Division Multiple (Access)

UE User equipment

PCP Poisson cluster process

p.d.f. Probability density function

PGFL Probability Generating Functional

PPP Poisson point process

PRB Physical Resource Block

QoS Quality of Service

QoE Quality of Experience

SUI Stanford University Interm

SaS Symmetric alpha stable distribution

SIR Signal-to-interference ratio

TDD Time Division Duplexing

UE User Equipment

UMTS Universal Mobile Telecommunications System

LIST OF TABLES xix

Table 1: Common Variables and Symbols

Name DescriptionPc Emitted power of Macro cell eNB (in Watt)Ps Emitted power of small cell eNB (in Watt)z,‖z‖ vector of the designated UE, distance of the designated UE from origin (in

meter)x, ‖x− z‖ vector of the small cell, distance between designated user and small cellh, hc, hx, hx+y fast fading, independent and identically distributed random variableg(z) path loss at the given location zΨ

[dB]log shadowing (log-normal fading) component in dB

Ki Constant parameters from path loss modelλ density parameter of Poisson Point processNs Mean value of small cellsNc Actual number of usersI(z) Interference at zUi number of users in the ith small cellλp density of parent points in Poisson cluster processΦp, Φ set of parent points in Poisson cluster process, set of small cellsc mean value of the daughter points in Poisson cluster processδ 2 Variance of symmetrical normal distributionR Radius of the cluster in Matérn cluster processRs Radius of small cell coverageK Parameter of Rician fadingP!

0(·) reduced Palm distributionP0(·) Palm distributionGN(·) Probability Generating FunctionalGs(·) Poisson distribution’s moment-generating functionf (·) probability density functionΘc PRBs assigned by macro eNBΘi the set of PRBs assigned by the ith small cell base stationE· Operator of expected valueL · Operator of Laplace transformerf (·) Gauss error functionerfc (·) complementary Gauss error functionIP· Operator of probabilityIPcov Probability of coverageIPout Probability of outageT Threshold valueΓ(·) (complete) Gamma functionΓ(·, ·) Incomplete lower Gamma functionγ(·) Incomplete upper Gamma function

xx LIST OF TABLES

1 Introduction

Nowadays, under the definition of modern mobile communication system we usually meanLTE (Long-Term Evolution). The LTE system was standardized by 3GPP (3rd GenerationPartnership Project) [1] around 2005 and a few years later became a primary mobile systemin the world, that offers high data rate communication.

Nevertheless, evolution and research process are incessantly runs in background, in or-der to make faster and better networks (based on principals of LTE). The updated versionof LTE is the so-called LTE-Advanced (LTE-A) system. According to the literature LTE-Ais the first mobile network that really fulfils the requirements given for 4G mobile system.The standardization of LTE-A is running under the coordination of 3GPP, however standard-ization process is not closed yet. Besides, state-of-art research tasks focus on creating andspecifying a new system, that fulfils the rigorous requirements (i.e. for throughput, latencyetc.), that are already given for the next generation (so-called 5G) networks. The appearanceof the first 5G capable networks are expected around 2020 [2].

One the mainstream research direction focuses on throughput enhancement (based onincreasing the level of coverage) [3]. The importance of coverage becomes a rather impor-tant issue, relying on the fact that the carrier frequency (and the frequency band also) usedfor communication shifts for higher and higher values. For example the carrier frequencyin the currently available 2G mobile network (GSM – Global System for Mobile Commu-nications) is around 900 MHz, meanwhile LTE’s carrier frequency can be around 2 GHz.However the value of path loss highly depends on the applied carrier frequency and the ob-jects (e.g. buildings) located between the base station and mobile terminal. Furthermore,in densely populated areas, due to population, the number of (mobile) users are increasingrapidly. Therefore, the demand for mobile network services (especially high data rate or lowlatency services – such as data transfer, media stream) increase extremely. Due to the car-rier frequency shifting phenomenon and the rapidly increasing user population, the size of

1

2 CHAPTER 1. INTRODUCTION

the cells, (which is the access point for mobile devices) are reducing. In second generationGSM system the typical cell radius was around ∼30 km, nowadays in a densely populatedcity cores the average cell radius for a LTE cell is around (or less than) 500 m.

In the network planning phase, the mobile operators should take into account these as-pects, in order to serve as many users as possible, with a strictly defined service quality level(e.g. [4]). In other words the goal is to fulfil the Quality of service (QoS) requirements. Theportion of indoor voice and data communication is increasing day by day. Nowadays 60%of voice and 70% of data traffic is generated indoors [5]. Ubiquitous multimedia servicesrequire sufficient throughput and low latency supported by the network. Therefore, major re-search efforts in next-generation wireless networks are focusing on enhancement of spectralefficiency and throughput.

An essential solution to enhance the coverage and throughput is to extend the conven-tional, “one-tier” macro cell structure to a multi-tier architecture. The macrocell layer canbe augmented with second (underlaid) tier. This second tier is formed by the combination ofseveral micro-, pico- and femtocells. Micro- and picocells are already widely used in denselypopulated areas, such as shopping malls, crowded public transport stations etc. This structureis called as multi-tier network. It is anticipated that in the next-generation wireless systems– such as LTE-Advanced – the conventional macrocell structure will be ameliorated withsmaller domestic and/or customer premises cells in order to satisfy user demand in denselypopulated areas [6]. Therefore the access network becomes heterogeneous, constituted byat least two overlapping layers, namely an over-sailing macrocellular and an underlaid smallcell layer. Small cells can be created either indoors (e.g. femtocells) or outdoors (e.g. pico- ormicrocells) and in their most radical incarnation they operate within the same spectral bandas the conventional macrocellular base station (macro eNB). However their coverage areais limited. This heterogeneous network (HetNet) structure offers benefits for both the usersand for the mobile operators. The ultimate goal is to bring the eNBs/radio ports closer to thepotential users for the sake of improving the attainable reception quality. This potentiallyfacilitates supporting higher data rates. Further benefit is that – provided the majority of thehigh-rate users is off-loaded to the small cells – the macrocellular eNB becomes capable ofsupporting additional users. However, using small cells in high-velocity outdoor scenarioswould result in excessive hand-over rates and hence a potentially high call-dropping rate.

The concept of femtocell brings a paradigm shift, since these “plug and play” devices areinstalled by users, similarly to Wi-fi hotspots [7]. Femtocells are a promising cost efficientsolution for improving indoor coverage and satisfy QoS (Quality of Service) requirements.A femtocell (or Home eNB) is a small base station, that capable to carry limited 4–10 usersdata. They are usually deployed in a flat or an office in order to enhance indoor coverageand provide better QoE (Quality of Experience) to local users, by bringing the base stationcloser to the user. Home eNB concept differs from the previously used micro- or picocell

3

concept because the data collected by a HeNB is delivered to the mobile operator via publicInternet connection. The number of simultaneous connections to a femtocell is limited,hence implementation of access control is essential. The femtocell concept distinguishesthree historically evolved methods in access control:

• Open: In open access mode every potential user equipment (UE) is allowed to connectto the femtocell (if the femtocell has available resource).

• Closed: In closed access mode only a group of UEs can have access. In the 3GPPconcept [8], closed subscriber list/group (CSG white list) contains the ID’s of UEsallowed to connect to the potential femtocell. The UE also stores the allowed femtocellCSG IDs.

• Hybrid: The third hybrid access method is a combination of the previous two accessmodes, that enables UEs to camp on a cell as non-CSG users, however, the offeredbandwidth to non-CSG users is limited.

The data is brought to the mobile operators core network via wired techniques, suchas copper cable- (e.g. DSL – Digital Subscriber Line) or fiber etc. In this scenario a hugeamount data of indoor users can be served by femtocells. Therefore, this solution offloadsmacrocells [6]. At the mobile operator a femtocell Gateway collects the incoming data andmerges with the data stream collected from macro eNBs. Femtocells are low power emission

Figure 1.1: Downtown LTE-A network extended with Small cells


S1

S1

S1 S

1

X2

X2X

2

HeNB HeNB

S1

S1

S1

HeNB

S1

S1

S1

S1

X2

X2

X2

X2

MME / S-GW

HeNB GW

MME / S-GW

S1MME / S-GW

TIER 1 TIER 2

E-UTRAN

eNB

eNB eNB

Figure 1.2: 3GPP LTE-Advanced Two-tier Small cell architecture [1]

base stations, thus the emitted power is typically between 10–100 mW and the radius of thecovered area scales up to a few 10 meters.

It is foreseen that these outdoor cells are deployed to for example lamp- or utility postssimilarly given in Figure 1.1. In this case the data can be delivered to mobile operator forexample via power line communication technologies. Due to the lower distance to potentialusers utility post installed cells can offer better coverage (and QoS based services) for pedes-trian users. In both cases (outdoor and indoor solutions) goal is to bring the base station asclose as possible to users, that can guarantee the good coverage (and high data rate). Theaugmented, two-tier LTE-A access network is illustrated in Figure 1.2. The literature usesthe collecting term: “Small cells” for these low power emitting, underlaid base stations,therefore I refer them as small cells also hereafter.

Background of the Research

The importance and benefits of small cells from the previous part are clearly visible. There-fore, it is not surprising that in LTE-A and 5G systems small cells are denoted as primaryaccess points [6, 9]. According to [10] small cell installation is driven by the followingmotivations:

1.1. BACKGROUND OF THE RESEARCH 5

• increasing capacity,

• improving depth of coverage, especially inside buildings,

• improving user experience, especially the typical available data rates,

• delivering value added services, especially those enabled by high-precision locationinformation.

By the contrast Wi-fi hotspots are operating in ISM-bands (industrial, scientific and medi-cal), meanwhile small cells are operating on licensed spectra. This is exactly the same spectrathat the mobile service provider operates the macro e-NodeBs (in case of the frequency reuseis one). Thanks to the commonly used frequency band(s) the small cell users (users servedby small cells), suffer from interference generated by the macro base stations and vice versa,from the point of view of macro users (the amount of users served by macrocells) the smallcells are interference sources. Due to the ad-hoc nature of femtocells and public Internetbackhaul the central interference management (via femtocell gateway) is difficult to apply.Femtocells use public Internet to deliver data, which might suffer from significant latency(due to Internet’s best-effort nature). Furthermore, femtocells are deployed and operated bythe users, thus the location and the “uptime” is also non deterministic and mobile operatorsdo not have influence on it. One possible solution is to “upgrade” femtocells capabilities. Inother words, femtocells have to have some kind of interference detection and mitigation tech-niques. For example the interference detector monitors the available (E-UTRA) frequencybands, that can be allocated to potential users (for communication purpose), and in case ofhigh level interference on the investigated band, the scheduler do not use this band [12], [13].Another solution had been defined in the 3GPP standard [11], where the macro base stationdoes not use an amount of resources for communication. These resources are reserved forthe small cells, thus for a small cell user the effect of macrocells interference can be re-duced. However this dissertation investigates a worst case scenario, a heavily loaded system,where all resources are allocated continuously. Therefore all resources have been allocatedindependently from the interference.

Due to the specific nature of small cell concept several new technical aspects, challengesand issues emerged. The following considerations are provided:

1. Small coverage area: Small cells are low power emitting base stations, hence thecovered area is around 10 – 30 meters. Therefore to reduce the significant numberof redundant and unnecessary handovers, the speed of the mobile terminal should betaken into account during the handover decision. Large-scale superfluous handovergenerates signalling overhead and remarkably reduces QoS and QoE at UE.


2. Highly dense deployment: with high dense of small cells (e.g. hundreds of Small cellsin a macrocell),

• the interference level increases remarkably, thus interference management re-quired.

• traditional neighbour cell discovery mechanism in LTE becomes unsustainable.Therefore a dynamic neighbour cell list required to reduce measurement time.

• it is possible that multiple small cell uses the same PCI (Physical Cell Identifier)in a macrocell. The number of PCIs is limited in LTE (exactly 504), hence iden-tifier collision generates problem in handover phase i.e. during hand-in processthe source eNB can not identify the target small cell correctly.

3. Variant access control: the owner of the femtocell may modify the CSG list indepen-dently from the mobile operator, this makes the handover process more complex.

4. Backhaul route: hard handover generates a short interrupt in the communication, whilethe handover process to the target base station is completed. This interruption timemostly depends on Round Trip Time (RTT) of the message exchange between thesmall cell and the gateway or the EPC (Evolved Packet Core), which is generallyvaries according to the backhaul route latency.

Brief History of Stochastic Geometry and Related works

The history of Stochastic geometry goes back to Georges-Louis Leclerc1. He calculatedthe probability of a randomly thrown coin hits an edge of a regular mosaic paving on thefloor (1733) [14]. Another phenomenon related to Stochastic geometry is the so-called shotnoise. The effect of shot noise firstly was analysed at the early twentieth century. Schottky2

investigated the electric noise in vacuum tubes in 1918 [15]. He discovered that the cumu-lative noise in a vacuum tube has similar properties, then the effect of multiple gun shootsfired in different time periods. Campbell3 studied shot noise process and characterized itsmean and variance. In his work, Campbell presents the moments and generating functionsof the random sum of a Poisson process on the real line, but remarks that the main mathe-matical argument was due to G. H. Hardy4. Therefore the literature often refers the theoremas Campbell-Hardy theorem [16]. This theorem is the often used in probability theory and

1French mathematician, later known as Comtr de Buffon2Walter Hermann Schottky (23 July 1886–4 March 1976): German physicist3Norman Robert Campbell (1880–1949), English physicist4Godfrey Harold Hardy (1877–1947), English mathematician

1.1. BACKGROUND OF THE RESEARCH 7

Stochastic geometry. Rice in his work in [18] investigated the effect of random noise basedon Campbells’ theorem. Later, Gilbert and Pollak [19] investigated the amplitude distribu-tion of shot-noise process. Lowen and Teich [20] showed that power law base shot noisedistribution does not converge to a normal distribution as we might expect. Sousa and Sil-vester [21], using power-law shot noise, without considering fading/shadowing. Ilow andHatzinakos considered effect of fading/shadowing on interference characteristics [22].

In the last decade Stochastic geometry gain higher attention in communication networkanalysis. The modelling of wireless systems (relying on Stochastic geometry) has been lav-ishly documented [23]–[49]. Stochastic geometry is a popular investigation tool for inter-ference characterization and outage probability analysis in mobile Ad hoc networks. Sev-eral books had been written dealing with Stochastic geometry. Among these books standsout [50], where the pioneer work of Stoyan, Kendall and Mecke have been collected.

In [16] Baccelli have collected the mathematical essentials of stochastic geometry. Fur-thermore in [17] Baccelli and Błaszczyszyn provided a model for a general mobile ad hocand calculated several network performance parameters with the aid of Stochastic geometry.

Kim et al. in [23] investigate the outage probability in third generation femtocell net-works with Poisson point process (PPP) node deployment assuming lognormal fading chan-nels. Chen et al. in [24] gives an outage analysis for two-tier small cell networks comparingthe grid structure and PPP (random) deployment also with the assumption of lognormal fad-ing channels. Wang et al. in [25] models a PPP based two-tier femtocell connection orienteddeployment.

Ganti et al. in [26]–[28] investigated general wireless networks relying on the results of[16, 17]. The main contributions of Ganti had been collected into [29]. In [30] Andrews etal. investigate service outage and transmission capacity for mobile cellular systems assum-ing spatial homogeneous PPP structure. In [30] an elegant formula is given for calculatingthe coverage probability in a single-tier macrocellular structure using Stochastic geometry.This is achieved by modelling the location of the macrocellular eNBs by a 2D homogeneousPoisson Point Process. In [31] the results of [30] are extended to multi-tier networks subjectto Rayleigh fading, where the interference power decays exponentially and the eNBs/nodesof each tier obey the PPP. Interference and service analysis for third generation mobile net-works are found in [35] and [36]. Lee et al. in [42] modelled a cognitive radio networkwith Poisson cluster based transmitters. Nonetheless, this cluster based node deploymentis suitable for modelling small cell networks in urban environment, where the base stationinstallation depends on building and road structure. This dissertation dedicates a whole chap-ter for Poisson cluster modelling. Author of [43] deduces a form that allows to calculate theprobability of an UE can camp on at least one base station in HetNets, modelling the basestations locations as spatial Poission point process and assuming Rayleigh fading. Anotherpaper provides an outage analysis for relay femtocells in [44].


Hoydis et al. in [45] investigates the outage probability and the ergodic mutual informa-tion in small cell networks assuming Rician fading multiple-input multiple-output (MIMO)channels. Against this backdrop, to the best of our knowledge the coverage analysis ofmacrocellular users subjected to the interference imposed by same-frequency LTE small cellHetNets and experiencing Rician fading has not been proposed with the aid of stochasticgeometry.

Motivation and Organization of the DocumentThe dissertation analyses a two-tier LTE-A network with the aid of Stochastic geometry [50].Stochastic geometry combines the theory of probability and vector geometry. With the math-ematical tools offered by Stochastic geometry we can analyse the average behaviour of anetwork, meanwhile the several input parameters are handled as random variable. In thisdissertation for example the actual location of small cells is a random input variable. There-fore the location of the a small cell base station is modelled with two dimensional randompoint processes.

Goal of the ThesisThe goal of this dissertation (relaying on the state-of-art literature) is to propose a mathe-matical model for two-tier small cell network modelling. Some known values (e.g. the meanvalue of the user population) are given as input parameter, meanwhile other variables areremain random. Thanks to Stochastic geometry, despite of several random parameters aregiven, the model is traceable. At the first view the combination two-tier LTE-AdvancedSmall cell networks and Stochastic geometry might seems unusual to the reader. Since mo-bile telecommunication is mostly a “practical oriented” and engineer based field, meanwhilein contrast Stochastic geometry is the study of random geometric structures, focusing ontheoretical issues in mostly related to mathematical topics. Therefore the mixture of twoindependent areas, requires a proper organization of the document, in order to keep it read-able. The evaluation of the proposed formulas (given in this thesis) provide results faster,than running tediously slow Monte-Carlo simulations. Nevertheless, in this thesis the resultsof the proposed formulas are compared with Monte-Carlo simulation results for validationpurpose and in order to ensure the precision the proposed forms. The simulations are madewith MathWorks MATLABr. As far as I know there is no such detailed downlink inter-ference model for LTE-A small cells in the literature, that is proposed in this thesis. Theproposed interference model includes the effect of widely used fast fading types (Rayleigh,Nakagami-m, Rice and Weibull fading). Fast fading models are also applied for the calcula-tion of the probability service outage. The dissertation introduces two new two-dimensional

1.2. MOTIVATION AND ORGANIZATION OF THE DOCUMENT 9

point processes for modelling LTE-Advanced networks. This two introduced point processare belong to the family of Poisson cluster processes, namely: Matérn- and Thomas clusterprocess. Since small cell installation in a chosen area might not be homogeneous, thereforethese cluster based models can model these scenarios in more accurate way compared tohomogeneous point processes. The properties of Matérn- and Thomas cluster process arelavishly documented in Chapter 2.

Structure of the Thesis

The rest of the thesis is organized to the following chapters as follows.In Chapter 2, the fundamentals of this thesis are highlighted. The two-tier based system

model is introduced in this chapter in details. This chapter introduces some two-dimensionalrandom processes that can be used for small cell modelling. Firstly, the planar Poisson Pointprocess is presented in details, afterwards an adaptation (mapping) of PPP proposed, thatcan be used for modelling small cell deployment taking into consideration the road structureof the area. Secondly, the thesis introduces two Poisson cluster processes (namely Thomascluster and Matérn cluster process). This two cluster process give a precise model for a realenvironment, where the intensity of small cells in a given area is not homogeneous. After-wards, this chapter provides some required mathematical preliminaries (i.e. complete andincomplete Gamma functions, complementary Error function etc.), surveys the fundamentalaspects of Stochastic geometry and highlights the connection between random variables andfading types. Finally, the path loss model and the possible interference types are explainedhere, which is related to the system model, evidently.

Chapter 3 provides the downlink interference analysis for a Poisson Point Process (PPP)modelled two-tier small cell network. The chapter focuses on the interference caused bythe small cells. The chapter shows that the cumulative interference (from the small cells),has a probability density function (p.d.f.) and cumulative distribution function (c.d.f.), fur-thermore it follows alpha-stable distribution, more precisely Lévy distribution. Next, theoutage probability for a macrocell user is investigated for various fading types such as: log-normal fading, Rayleigh fading, Nakagami-m fading and Rice fading. The proposed formsare closed and evaluable. In the Rice fading case only a lower band for coverage probability(or upper bound for outage probability) is given. Finally, at the end of this chapter the authorhas calculated the overall system capacity with the aid of the Signal-to-Interference Ratio(SIR) distribution.

In Chapter 4, the thesis investigates the interference in case of cluster based small cellmodelling. Afterwards the thesis gives formulas to calculate the outage probability (or thecomplement event – the coverage probability) for a user served by macrocell. Due to the


complexity of the forms, they are approximated. However, according to the simulation re-sults, these approximations are still accurate.

Finally, Chapter 5 concludes and summarizes the document. Some possible further re-search topics are also given here.

2 System model

This chapter introduces the system model used for our Stochastic geometry based investiga-tions. The research provided in the thesis is strictly restricted to the access network, thus thesmall cell effects related to core network of an LTE-A system is not detailed in this disserta-tion. In this model the access network is separated into two tiers. The first tier is the macrotier and second tier is the so-called small cell tier. The model contains a macro base station(e-NodeB). Small cells form the second tier of the network. The system model is limited to afinite field denoted by R. The area of this field is represented by |R| and this parameter de-notes the coverage area of a macro eNB. The macro eNB is located at the center of R. Thispoint is denoted as the origin point of the coordinate system. In other words, the macrocellis located at the origin of the coordinate system and covers the area |R|. The small cells aredeployed by the users on this finite area |R|. An illustration of the system model is given inFigure 2.1.

Both eNB type (macro- and small eNB) operates on a fix, constant power. The emittedpower of a macro eNB is denoted by Pc, meanwhile the emitted power of the small cells aregiven by Ps. There are no power control applied for small cells. The emitted power of thesmall cells is given as an average value of the small cells market product data sheet [51].The power of the received signals at UEs receiver are denoted by Pr

c and Prs , respectively.

It is assumed that the macro eNB has an omni directional antenna. The applied schedulingalgorithm is round robin (RR). The data transmission is based on ON/OFF model; how-ever, continuous transmission is assumed in this thesis. Small cells are operating in licensed”downlink” spectra with a bandwidth of 20 MHz (E-UTRA Band 23) [52] and the carrierfrequency of this band is 2190 MHz.

In LTE advanced the air interface is based on Orthogonal Frequency Division Multiplex-ing (OFDM) together with advanced antenna techniques (i.e. MIMO) and applies adaptivemodulation and coding (AMC) in order to achieve significant throughput and spectral ef-

11

12 CHAPTER 2. SYSTEM MODEL

Figure 2.1: An LTE-A Two-tier small cell network

ficiency improvements. Higher spectral efficiency enables operators to transfer more dataper MHz of spectrum, resulting in a lower cost-per-bit. The OFDM-based air interface alsoprovides much greater deployment flexibility than 3G UMTS/HSPA with support for mul-tiple channel bandwidths as well as time and frequency duplexing modes (TDD & FDD).A 20 MHz FDD channel (3GPP Release 8) supports peak rates of at least 100 Mbps in thedownlink and 50 Mbps in the uplink. As in the context of LTE, we also consider elementarytime/frequency Physical Resource Blocks (PRB). The PRB is defined as a block of physicallayer resources that spans over one slot (typically 0.5 ms) in time and over a few adjacentOFDM sub-carriers (typically 12) in the frequency domain. In this two-tier LTE system weuse the maximum number of physical resource blocks with normal cyclic prefix [53, 54].

The small- and macrocell eNBs are assigning radio resources (PRBs) independently totheir users according to the demand of capacity requested by their users without any col-laboration. It is assumed that there is no cooperation between small cells, thus they operateindependently from each other. Moreover, there is no central coordination between smallcells and macrocell(s). On the other hand small cells operate on the same licensed spectra asthe macrocells. It is assumed that both macro and small cell base stations have MIMO andomni-directional antennas. The location of the macrocell user (macro UE) is given by vectorz. The distance between the macro UE and the macro eNB (from the origin) is the absolutevalue of the vector ‖z‖ (in meter). The effect of the fast fading is also included in the system

13

model. We include the fading effect in the received power. Parameter h denotes the fastfading, and h is an independent and identically distributed random variable. The distributionof h depends on the modelled fading type. The index of h refers to the type base station e.g.hc denotes the fast fading for a macro eNB. Further information about h is lavishly detailedin Section 2.2. In our system model the small cells are deployed lamp- and utility posts,therefore the effect of wall penetration loss is not considered. Furthermore, this allows touse outdoor path loss model for radio signal propagation modelling. The modelled two-tiersystem is interference limited, thus the effect of the thermal noise at the receiver is ignoredfor the sake of simplicity. The mobility of users are not considered in this thesis, it can beviewed as an actual snapshot of a loaded two-tier network. Note that, the snapshot naturedoes not infect the results, because random spatial processes guarantees the random locationof small cells in every scenario. Furthermore, the current position of UE (z) is unknown,only the distance between the macro eNB (located at the origin) and the (non-group) UE isfix (‖z‖).

The applied path loss for small cells is based on Stanford University Interim (SUI) chan-nel model [55], however some parameters should be modified due to system specific require-ments e.g. carrier frequency:

g[dB](z) = 12+39 · log10(‖z‖)+Ψ[dB]log (2.1)

where g(z) represents the path loss (attenuation) at the given location. Path loss due to carrierfrequency and other parameters are included to the model via the constant values (e.g. wallpenetration loss [56]). Ψ

[dB]log denotes the shadowing (log-normal fading) component in dB.

Its is given as Gaussian random variable as follows: Ψ[dB]log = 10 · log(Ψlog)∼N (0,10). Note

that Ψlog follows lognormal distribution, however Ψ[dB]log follows normal distribution.

Note that, choosing SUI model for path loss modelling does not affect the theoremsproposed by the thesis, thus other path loss models can be used also for example proposedby [57]. It is assumed that the shadowing components are i.i.d. (independent, identicallydistributed) for every base station and UE and MIMO channels are perfectly orthogonal,therefore the paths are independent from each other. The path loss does not change by theparticular transmitter/receiver antenna. The non-logarithmic version of (2.1) is given as again value:

g(z) =1Ki· 1

Ψlog· ‖z‖−α , (2.2)

where the constant parts (e.g. propagation loss due to carrier frequency) are merged into Ki.The outdoor path loss exponent is denoted by α .


An example is given here. Let us assume that the transmitter emits on Pc and the receiveris located at z. Therefore the received power at this location is:

Prc (z)

[dB] = Pc[dB]−g[dB](z), (2.3)

Prc (z) = Pc ·hc ·g(z) =

Pchc‖z‖−α

KiΨlog. (2.4)

Location of Small Cells

The dissertation introduces three random processes, that can be used for two-tier small cellmodelling. These random processes models the planar locations of the small cells on finitearea R. The first one is the widely used two dimensional homogeneous Poisson point process(PPP).

Poisson Point Process

According to the homogeneous PPP model the small cells are scatter in |R| uniformly. Theactual number of small cells follows Poisson distribution with density λ . The mean valueof the small cells, therefore Ns = λ · |R|. In this model we assume that the process is ho-mogeneous, thus the points are scattered uniformly in the planar. In this case the intensityparameter λ is constant in every part of R. The actual location of the small cells is givenby vector x. The distance between a small cell and the macro eNB (located at the origin) isgiven by the absolute value of the vector ‖x‖. Furthermore the distance between a small celland the user is given by ‖x−z‖. Note that, if the process is not homogeneous, then the inten-sity parameter should be given as λ (x). The homogeneous version of Poisson point processwidely used, accepted and popular model. Our model uses homogeneous point process, thusfor the sake of simplicity we drop the word homogeneous and refer the process simply PPPhereafter.

From our point of view the most important properties process is that the process isisotropic, stationary [26, 50]. Due to these pleasant properties of the process the mathe-matical tractability is guaranteed. One illustration of the PPP based small cell locations isgiven in Figure 2.2. A small example (for calculating the mean value of small cells) is givenin the example:

Example 2.1.1. Let us assume that the system model area is a finite square with |R| =500 m× 500 m and the intensity of small cells is λ = 4 · 10−4. Therefore the mean numberof small cells is Ns = λ · |R|= 100.

2.1. LOCATION OF SMALL CELLS 15

−500 −400 −300 −200 −100 0 100 200 300 400 500−500

−400

−300

−200

−100

0

100

200

300

400

500

x (m)

y(m

)

x

z

x−z

Macro UE

Small cell

Macro Base Station

Figure 2.2: Illustration of Homogeneous Poisson point process

Note that, the actual number of small cells is still random, the result of the upper exampleis only the mean value.

Application of PPP model

In the average system throughput calculation (given in Section 3.4) we apply PPP model inorder to calculate the system level throughput in down-town. Road structure can be repre-sented with a graph, where the vertices are crossings and the edges are roads. In this graph,shortest paths exist and can be found between two nodes, if the graph is connected. Onepossible graph structure represented in Figure 2.3, where the macrocell and small cells aredenoted by blue and green rectangles, respectively. In this case the model should take intoaccount the road structure of a district, therefore the following restrictions are given extend-ing the general PPP model:

• The location of the users is modelled with PPP. The actual number of user is given byNc.

• In general case it is expected that small cells are covering a circle areas with radius Rs(thus the area covered by one small cell is R2

s π). The whole system area is |R|.


Figure 2.3: PPP based Random graph model

• On the other hand in random graph model (city road structure) small cells are cov-ering only road segments with distance 2Rs. The roads segments are denoted bye1,e2, · · · ,en. The total length of the road segments is given by ∑i |ei|. Figure 2.3gives an illustration of the proposed random graph model. Small cells are installedalong the edges to provide extended coverage for the users i.e. small cells are locatedon the edges of the graph (e.g. installed on a street light). The second tier base stations(i.e. small cells) are assumed to be uniformly distributed along the edges. The distri-bution of the small cells thus follow a Poisson Point Process. However the area wheresmall cells possibly occur is restricted to the road structure, following [16], PPP makespossible to analyse this wireless network. The number of users in the ith small cell isdenoted by Ui. Ui’s are independent, identically distributed random variables whichfollow Poisson distribution with parameter λ .

Poisson cluster ProcessThe Poisson cluster process (PCP) based model breaks the homogeneous nature and groupsthe small cells into clusters. If one takes a photo about a realization, in some parts of thefield R the density of small cell is higher than others. Furthermore in some fraction of Ris free from small cells. This cluster based process provides an accurate model, rather than

2.1. LOCATION OF SMALL CELLS 17

−250 −200 −150 −100 −50 0 50 100 150 200 250−250

−200

−150

−100

−50

0

50

100

150

200

250

x (m)

y(m

)

Parent point Small cell

Cluster

(a) Illustration of Thomas cluster process

−250 −200 −150 −100 −50 0 50 100 150 200 250−250

−200

−150

−100

−50

0

50

100

150

200

250

x (m)

y (

m)

Parent point

Cluster

Small cell

(b) Illustration of Matérn cluster process

Figure 2.4: cluster processes as Small cell deployment modelling

PPP. Since in a real environment some parts the small cell intensity is higher (e.g. block offlats), than others (e.g. public parks). Nevertheless, the introduced cluster models remainstractable due to Stochastic geometry.

The dissertation introduces two Poisson cluster processes that can be used for small cellmodelling, namely the Thomas cluster process and Matérn cluster process [50]. Bothprocesses are belong to the family of Neyman-Scott point process [50] and composed withsuperposition of simple Poisson point processes. This provides the mathematical tractability.The base for Thomas and Matérn cluster process is a Poisson point process with parameterλp. The literature calls this points as parent points. The set of parent points is denoted by(Φp = x1,x2, . . .). In our model the actual location of the parent points is given by vectorx. Around the parent points scattered the daughter points (or the name offspring points isalso used by the literature). The daughter points are independent from each other and fromthe other parent points. Furthermore, they are identically distributed around the parent point.

The set of cluster is represented by Nxi = Ni,xi, where Ni = y1,y2, . . . denotes thei.i.d. set of daughter points. Note that Ni is independent from the parent process, in otherwords, the parent points are not members of the cluster. The whole cluster therefore can bewritten as follows:

Φ =⋃

i

(Ni∪xi) =⋃

x∈Φp

Nx.

The number of daughter points is random, and follows Poisson distribution. The meanvalue of the daughter points in a cluster is represented by c. The distance between daughter


points and the cluster centres (parent points) is given by vector y. In PCP the daughter pointsrepresents the small cells. Their actual position is given by vector xi +y. The density of thecluster process can be calculated with λ = λp · c, where c represents the average number ofsmall cells in a cluster.

The main difference between Thomas and Matérn claster is the scattering of the daughterpoints. In case of Thomas cluster process the daughter points scatter around the parentpoints according to a symmetrical normal distribution (zero mean and δ 2 variance) with thefollowing density function:

f (y) =1

2πδ 2 exp(−‖y‖

2

2δ 2

). (2.5)

However, in Matérn cluster process they are scatter around the parent points uniformlyin a circle with radius R. The density function is given by [26]:

f (y) =

1

πR2 , if ‖y‖ ≤ R0, oherwise

(2.6)

An illustration Thomas and Matérn claster is given in Figure 2.4a and Figure 2.4b, respec-tively. To compare the processes the following parameter sets are applied. In order to intro-duce these processes a short example is given:

Example 2.1.2. In both figures the system model area |R| is a finite square with |R| =500 m×500 m. The mean number of the daughter points (c) equals 50 and the mean valueof the parent points (λp · |R|) equals 4. In case of Matérn cluster process we set the radius Rto 50 meter. Therefore, the centre of the circle represents the parent point and the small cellsare scattered in a circle with diameter 100 m. In Thomas cluster the small cells are scatteredaccording to a symmetrical normal distribution, therefore 99.7% of the observations fall intothe range of 6δ . In order to compare the two processes we set the variance to δ = 16.67.Thus, the cluster (diameter) size for both processes is 100 m.

Note that in this case the actual number of small cells in a cluster is still a random vari-able, only the mean value is known (c). Furthermore, the actual number of the clusters is alsoa random variable. However, thanks to Stochastic geometry tools the model can be handledmathematically.

Mathematical PreliminariesThis section collects the essential definitions (i.e. complete and incomplete Gamma func-tions, complementary Error function etc.) and connections between random variables, that

2.2. MATHEMATICAL PRELIMINARIES 19

will be used in the following sections. The moment-generating function (MGF) and the mo-ments of the distributions are introduced in this section. The distribution of a random vari-able can be characterized with the moment-generating function. The Moment-generatingfunctions have great practical relevance not only because they can be used to easily derivemoments, but also because a probability distribution is uniquely determined by its MGF. It isimportant to mention that, all random variables have a characteristic function, however notall random variables have a moment generating function. This section also includes the defi-nition of the so-called Probability Generating Functional (PGFL), which will be an essentialinput for further calculations.

Error function and complementary Error functionThe Error function is defined by the integral [60]:

erf (s) =2√π

∫ s

0e−t2

dt. (2.7)

Evidently, the complementary Error function is defined by the integral:

erfc (s) = 1− erf (s) =2√π

∫∞

se−t2

dt. (2.8)

Gamma function and incomplete Gamma functionThe complete Gamma function is defined with the following improper integral [61]:

Γ(s) =∞∫

0

ts−1e−tdt. (2.9)

If s is a positive integer, thenΓ(s) = (s−1)!.

The upper incomplete gamma function – Γ(s,x) and lower incomplete gamma function –γ(s,x) are defined similarly to the Gamma function, only the integral limits are different:

Γ(s,x) =∞∫

x

ts−1e−tdt, (2.10)

γ (s,x) =x∫

0

ts−1e−tdt. (2.11)


The sum of lower- and upper incomplete gamma functions yields the complete Gamma func-tion:

Γ(s) = γ (s,x)+Γ(s,x) . (2.12)

If s is a positive integer, then the incomplete (upper) Gamma function can be written as [61]:

Γ(s,x) = (s−1)!e−xs−1

∑k=0

xk

k!. (2.13)

Two special values for incomplete (upper) Gamma function can be defined as follows:

Γ(s,0) = Γ(s) if Res> 0, (2.14)Γ(s,∞) = 0. (2.15)

Jensen’s inequalityJensen’s inequality is an important theory proposed by Johan Jensen a Danish mathematicianin 1906. The brief version of the theorem is given as follows [61]:

Definition 2.1. if a finite function f is given with an X random variable then:

E f (X) ≥ f (EX) if f is convex, (2.16)E f (X) ≤ f (EX) if f is concave. (2.17)

Relationship Between Random Variables and Moments

Exponential Distribution

In wireless environment the Rayleigh distribution is frequently used to model multi-path fad-ing with no direct line-of-sight (LOS) path [58, 59]. In this case the channel vary randomlythe emitted signals amplitude according to Rayleigh distribution. When we introduced thesystem model (beginning of Chapter 2.) we defined that the received power and the effectof the channel is modelled with parameter h. Instead of calculating the amplitude we cal-culate on the power domain. It is known that the square of a random variable that followsRayleigh(ω) distribution1 is an exponential(ω) distributed random variable. A brief proof isgiven in Appendix A.1.1. Let us assume that a random variable X follows Rayleigh distribu-tion with the following probability density function, denoted by fX(x,ω) (where x > 0 andω > 0, evidently):

fX(x,ω) =2xω

e(−x2/ω). (2.18)

1Note that, ω is the “scale” parameter of the distribution.


The Moment-generating function (MGF) for an exponential distributed random variableh [58]:

Eh

e−shA

=

1(1+ sA)

, (2.19)

where A is a non-negative parameter. For the sake of simplicity we choose ω = 1 hereafter.ω and A could be transformed into one other. With equation (2.19) we can calculate the frac-tional moment2 for h in Rayleigh faded channel. This will be used in Section 3.1. However,this result has been already calculated in [29], thus we only introduce the final result:

E√

h= Γ(3/2) =√

π

2, (2.20)

where Γ(x) is the Gamma function defined in (2.9).

Gamma Distribution

In wireless communication Nakagami distribution is used to model the effect of scatteredsignals that reach a receiver by multiple paths. Similarly to the previous case we investigatethe power domain, thus we need the square of the (Nakagami distributed) random variable.It is known that the Nakagami distribution is related to the Gamma distribution. Let usassume, that a given a random variable X follows Nakagami distribution with the shapeparameter m and spread parameter Ω. In case of Y is a Gamma distributed random variablewith Y ∼Gamma(m, Ω/m), then X2 ∼Y is valid between X and Y . For the sake of simplicitywe calculate with Ω= 1 hereafter. The probability density function for an Gamma distributedrandom variable is given by:

f (h,m) =hm−1e−hm

m−mΓ(m), (2.21)

meanwhile the Moment-generating function (MGF) [58]:

Eh

e−shA

=

1(1+ s A

m

)m . (2.22)

Later, to evaluate the interference distribution (Section 3.1) for a rather complex fading typesuch as Nakagami-m, we have to calculate the fractional moment of h. Thus the fractionalmoment is introduced in this section (a detailed proof is given in Appendix A.1.2):

E√

h= (2m−1)!!2m(m−1)!

√m√

π, (2.23)

2fractional moment is non-negative, but not integer moment order. For example in (2.20) the order is 1/2.


where m!! denotes the double factorial of a positive integer m. The double factorial of apositive integer m is a generalization of the usual factorial m! defined by [62]

m!! =

m · (m−2) . . .5 ·3 ·1 m > 0 odd;m · (m−2) . . .6 ·4 ·2 m > 0 even;1 m =−1,0.

Note that −1!! = 0!! = 1.It is visible that the actual value of the fractional moment depends only from the fading

parameter m. In case of Rayleigh fading (m = 1) (2.23) reduces to√

π

2 , which confirms theresult of (2.20).

Non-central Chi-squared Distribution

Now let us investigate a Rician faded channel using the method above. Since our furtherinvestigations rely on the received power instead of the amplitude, we should transform therandom variable. According to [63] the square of a Rician distributed random variable obeysthe non-central Chi-squared distribution.Therefore the p.d.f is given by:

f (h,K) = (K +1)e(−K−(K+1)h) · I0

(2√

K(K +1)h). (2.24)

A brief proof given in Appendix A.1.3. Parameter K denotes the power-ratio of the direct LOSsignal and of the scattered paths of the Rician distribution. The K = 0 scenario represents theNon-Line-of-Sight (NLoS) i.e. Rayleigh fading case. I0 (x) denotes the modified zero-orderBessel function. The Moment Generating Function (MGF) of the non-central Chi-squareddistribution is given by [64]:

Eh

e(−shA)

=

1+K1+K + sA

· exp(− sKA

1+K + sA

)(2.25)

As expected, in the Rayleigh fading scenario substituting K = 0 into (2.25) corresponds withthe MGF of the exponential distribution as given in (2.19).

The square root of the received interference power√

h follows Rician distribution, thuswe invoke the definition of the expected value to evaluate the fractional moment of h. Wewill use this form in Section 3.1. Substituting fading distribution yields,

E√

h=√

11+K

· e−K2

[(1+K) I0

(K2

)+K · I1 (K/2)

]√π

2, (2.26)


where the Modified Bessel Function of the Zero- and First order is given by I0(x) and I1(x),respectively. To validate the form we substitute K = 0 into (2.26). This corresponds toRayleigh fading and yields

√π/2 (and confirms the result of (2.20)), as expected. Again, the

actual value of the factional moment of h depends on the Rice fading parameter K.

Weibull Distribution

Due to the pleasant property of Weibull distribution if channel’s amplitude obeys Weibull dis-tribution with shape (n) and scale parameter (γ), then the kth power of h is also Weibull distri-bution with n/k and γ , which implies that the received power is also Weibull distributed [65].Using [66] to calculate the fractional moment, that will be required in Section 3.1:

E√

h=√

γ ·Γ(

1+1

2n

). (2.27)

Once again, substituting n = 1 and γ = 1 (Rayleigh fading case) yields Γ(3/2), as expected.

Lognormal Distribution

In wireless communication Lognormal distribution is used to model the effect of shadow-ing. It is also called slow-fading. In further investigation we should calculate the value ofE√

1/Ψ, where Ψ follows lognormal distribution. It is known from [67], that for anyreal or complex number s, the sth moment of a log-normally distributed variable X can becalculated with the following form:

EX s= esν+ 12 s2δ 2

, (2.28)

where µ and σ are the distribution parameters for location and scale, respectively. Therefore,reshaping E

√1/Ψ as EΨ−1/2, where s =−1/2 yields,

E√

1/Ψ= e(−1/2·µ+σ2·1/8). (2.29)

Fundamentals of Stochastic geometryIn this subsection mention some essential formulas and definitions that we will refer in thethesis later.

Definition 2.2 (Palm distribution [50]). (a heuristic approach) Let us consider a Poissonpoint process denoted by Φ. The Palm distribution probabilities are conditional probabilities


of point process events given that a point (the typical point) has been observed at a spe-cific location. Palm characteristics of homogeneous Poisson processes can be calculated asstationary characteristics, if one of the points in the set is shifted to the origin:

P0(Φ ∈R) = P(Φ+xk ∈R),

where xk is one the points. Then the gained point process distribution is the Palm distribution,denoted by P0(·). Note that, the new point process with distribution P0(·). is not stationarysince, due to the fact that there is an added point at the origin with probability 1. If the pointat the origin (o) is not counted, then we get the so-called reduced Palm distribution denotedby P!

0(·).

Definition 2.3 (Slivnyak’s Theorem [68]). Let Φ be a two-dimension homogeneous Poissonpoint process with rate λ in R, then

PΦ ∈R= P!0(Φ ∈R);

that is, the reduced Palm distribution of the point process is equal to its (original) distribution.

Probability Generating Functional

Before starting the investigation of interference and outage probability we should introducethe Probability Generating Functional (PGFL) of a process [69], [70]. For a given pointprocess denoted by N, the definition of probability generating functional can be defined asfollows,

GN(ν) = E

[∏x∈Φ

ν(x)

](2.30)

where vector x denotes the locations of the events, and the function v belongs to a suitablefamily – we require 0 ≤ ν(x) ≤ 1 for all x ∈ R, and ν to be identically 1 outside somebounded Borel set (so that there are only a finite number of non unit terms in the productabove, which converges) [50].

Relying on the results of [29], the PGFL of a (homogeneous) Poisson point process is

GN(v) = exp−λ

∫R[1−ν(x)] dx

. (2.31)

The base process of the Thomas- and Matérn cluster processes is a simple PPP (parent pointscattering) and the number daughter points in a cluster follows Poisson distribution, thereforeusing the moment-generating function of a Poisson distribution, finally gives us the PGFL ofThomas and Matérn cluster processes [29, 50]:

2.3. GENERAL INTERFERENCE CHARACTERIZATION 25

GN(ν) = exp−λp

∫R[1−Gs(ν)] dx

, (2.32)

where

Gs(ν) = exp

c[∫

Rν(x+y) f (y)dy−1

],

where f (y) is given in (2.5) and (2.6). The Poisson distribution’s moment-generating func-tion definition equals Gs(ν) = exp(c(ν−1)).

General Interference Characterization

In this section we introduce the interference types in a two-tier LTE two-tier access networks.The interference types can be grouped as follows. In case of downlink interference, theinterference is caused by base stations (macro- and small cells) to mobile users (UEs). Inuplink interference case the interference is caused by the UEs and base stations suffer fromtheir interference. This thesis focuses on the downlink interference type. Since this thesisbase model assumes a two-tier network the interference should be categorized according tothe source. When a small cell transmits it causes interference to neighbour small cell users(so-called co-tier interference) and to macrocell users (so-called cross-tier interference).

Note that, due to page limitations, this section describes just the small cell generateddownlink interferences. Therefore this subsection does not discusses the complete set ofdownlink interference scenarios.

An interference characterization is given for a code division multiplexing based systemsin [35]. However the access network part of UMTS is quite different from LTE-A accessnetwork. In LTE the radio interface is not based on code division multiplexing principle likein UMTS. Hence the investigation of LTE small cells requires a different aspect. In an LTEsystem the resources were adaptively allocated by the base stations, this way the interfer-ence level might be lower compared to UMTS. LTE-Advanced system relies on OrthogonalFrequency-Division Multiple Access (OFDMA), where the total available frequency bandis partitioned into sub-bands of sub-carriers. These sub-carriers are collected into so-calledPhysical-layer Resource Blocks (PRBs). The construction of the PRBs are briefly intro-duced here: In LTE system the transmitted signal in each slot is described by a resource gridof RB ·Mc sub-carriers and Ms OFDM symbols. Mc represents the number of sub-carriersdefined in a Physical Resource block (PRB), and Mc equals 12. The quantity RB dependson the uplink/downlink transmission bandwidth. The number of OFDM symbols in a slotdepends on the cyclic prefix length and sub-carrier spacing configuration. If the bandwidth


equals 20 MHz, then the maximum number of physical resource blocks RB=100 with Ms=7defined in [53, 54]. The small- and macrocell eNBs assigns PRBs independently to theirusers according to the demand of capacity requested by their users. The set of PRB indicesassigned by the macro base station to macrocell user c is represented by Θc. Θi denotesthe set of PRBs assigned by the ith small cell base station to their small cell user(s). If themacrocell base station and the small cell base station assign the same PRB(s) to their users,then interference level on a designated PRB increases. Interference grows at the small celluser generated by the macrocell base station or another small cell base station. To show afull description of downlink interferences we separate the small- and macrocell generatedinterferences.

Small cell interference in the macrocell

The small cell base station generated interference at a macrocell user, that uses the jth re-source block can be calculated from the following formula:

I jm(z) = ∑

x∈Φ

δx, jPshxg(‖x− z‖), (2.33)

Figure 2.5: Illustration of Cross-tier and Co-tier interference and PRB assignment

2.3. GENERAL INTERFERENCE CHARACTERIZATION 27

where the vector x represents the random location of the interference sources in the set Φ andvector z gives the actual location of the macrocell user. The path-loss g(‖x− z‖) betweenthe macro-user’s receiver and the interference source as given by (2.2). The effect of thefast fading is included to the model by hx. Parameter δx, j is an indicator, which means ifthe macrocell and the small cell base station (located at coordinate x) assigns the same jth

resource block interference occurs, otherwise the interference is 0:

δx, j =

1 if j ∈Θc and j ∈Θx0 otherwise. (2.34)

Example 2.3.1. An illustrative example is given in Figure 2.5. Assume that the macrocellbase station assigns the resource blocks with index Θc = 34,54,57. The id#1 small cellassigns resource blocks with index Θx = 34,47,52 to an UE (connected to this small cell).In this case the interference level is increasing at resource blocks with index 34: σx,34 = 1and σx, j = 0 otherwise. Therefore, this small cell does not affect the other macro assignedresource blocks.

Femtocell interference at another femtocellThe small cell base station generated interference at another small cell user can be calculatedfrom the following formula:

I js (z) = ∑

x∈Φ\kδx, jPshxg(‖x− z‖), (2.35)

where Is denotes the interference at the jth resource block. The small cell user located at zand it is assumed that the user is served by the small cell denoted by index k. The value ofδx, j indicator can be calculated from the following formula:

δx, j =

1 if j ∈Θk and j ∈Θx,0 otherwise (2.36)

where Θk is the set of PRBs assigned by the kth small cell base station and Θx is the set ofPRBs assigned by the small cell base station (in set Φ).

Example 2.3.2. An illustrative example is given in Figure 2.5. Assume that the small cellid#k assigns the resource blocks with index Θk = 47,48,49. The id#1 small cell assignsresource blocks with index Θx = 34,47,52 to an UE (connected to this small cell). Inthis case the interference level is increasing at resource blocks with index 47: σx,47 = 1 andσx, j = 0 otherwise. Therefore, this small cell does not affect the small cell id#1 assignedresource blocks.


Extending the conventional interference characterization with model of PRB allocationleads us to the following model. The actual load in the small cells can be considered as arandom variable parameter e.g. p, then the mean number of (interference source) small cellsin the given PRB equals pλ |R|, which is also a PPP. Since an independent thinning of PPPyields another PPP [16]. Therefore we can chose a subset Φt , which is also a PPP and thedensity is λt = p · λ . For example if the value of p equals one, then all small cells usesthe investigated jth PRB (δx, j = 1 for every small cell), thus Φ = Φt . On the other hand, ifp = 0.6 then only 60% of the small cell (in Φ) use the jth PRB. In the rest of the documentδx, j = 1 is assumed for the sake of simplicity.

3 Small Cell modelling with PPP

This chapter provides the downlink interference analysis for a Poisson Point Process (PPP)modelled two-tier small cell network. The chapter focuses on the interference caused bythe small cells. The thesis shows that the cumulative interference (from the small cells),can be characterized with a probability density function (p.d.f.) and cumulative distribu-tion function (c.d.f.), furthermore it follows alpha-stable distribution, more precisely Lévydistribution. Afterwards the thesis investigates the outage probability for a macrocell user,applying various fading types such as: log-normal fading, Rayleigh fading, Nakagami-mfading and Rice fading. The proposed forms are closed and evaluable. In the Rice fadingcase only a lower bound for coverage probability (or upper bound for outage probability) isverified with simulations. Finally, the thesis calculates the overall system capacity with theaid of the Signal-to-Interference Ratio (SIR) distribution.

Interference Distribution

The cumulative downlink (DL) interference power imposed by a small cells (on a chosenPRB) can be calculated with (2.33) and (2.35) for a macro cell or a small cell user, respec-tively. Although the PRB allocation process to users depends on the actual scheduler imple-mentation of the eNB, according to our assumptions this assignment process is independentfrom the other eNBs.

The PPP process is stationary and isotropic [50] and due to the pleasant property the ac-tual location of the receiver (UE in our case) does not affect the result given for the interfer-ence distribution. This phenomenon is related to Slivnyak’s Theorem given in Section 2.2.5.Therefore we can chose an arbitrary location where the receiver should settle e.g. the originof the coordinate system.

29

30 CHAPTER 3. SMALL CELL MODELLING WITH PPP

For the sake of simplicity we leave the index from the interference, hence both interfer-ence types ((2.33) and (2.35)) can be substituted. We now focus on the fast fading, thereforethe effect of shadowing is ignored, thus Ψlog=1. Furthermore we assume a heavily loadedsystem, where all resources are allocated, thus δx, j = 1 for every PRB, yields Φt = Φ.

The Laplace transform of interference (2.33) (or (2.35)) is given by:

LI(s) = EΦ,hx

exp

(−s ∑

x∈Φ

Pshxg(x)

), (3.1)

The Laplace form of the interference is related to the PPPs’ PGFL given as follows.Rewriting the sum in (3.1) exponent as a product we get

LI(s) = EΦ,hx

∏x∈Φ

exp(−sPshxg(x))

, (3.2)

which is the definition of the probability generating functional given in (2.30) with ν(x) =exp(−sPshxg(x)). Since the actual value of hx is independent from Φ itself and i.i.d, there-fore the joint expected value calculation can be separated:

LI(s) = GN (ν(x))

∣∣∣∣∣ν(x)=Ehx [exp(−sPshxg(x))]

. (3.3)

Using the probability generating functional of the PPP (2.31) and some algebraic manipula-tion given in [29], we can express this Laplace transform to the following convenient form:

LI(s) = exp−λπ(K−1

i Ps)2α E

h2α

Γ

(1− 2

α

)s

2α

, (3.4)

where E

h2α

denotes the fractional moment of h. The values for this fractional moment

depend on the fading type and already given in Section 2.2. Ki is defined in the path lossmodel (2.2).

A symmetric alpha-stable distribution (SaS) can be characterized with four parameters:characteristic exponent, skewness, scale and location parameters [71]. In case of Lévy dis-tribution the values of characteristic exponent and the skewness are restricted to 1/2 and1, respectively. The remaining two parameters (scale c and location µ) are used as inputvariables that characterizes the distribution.

Relying on the results of [29, Section 3.2.3] and [41], in case of the path loss exponentα = 4, we obtain an SaS distribution from (3.4). In [29] the authors investigated a general

3.1. INTERFERENCE DISTRIBUTION 31

Fading Type Fading moment

Without fading: E√

h= 1,

Rayleigh fading: E√

h= Γ(1+ 2

α

)= Γ

(32

)=√

π

2 (2.20),

Nakagami-m fading: E√

h= (2m−1)!!2m(m−1)!

√m

√π (2.22),

Rice fading: E√

h=√

11+K · e

−K2

[(1+K) I0

(K2

)+K · I1

(K2

)]√π

2 (2.26),

Weibull fading: E√

h=√γ ·Γ(1+ 1

2n

)(2.27),

Lognormal fading: E√

1/Ψ= e(−1/2·µ+σ2·1/8) (2.29).

Table 3.1: E√

h Fading moments

(mobile ad-hoc) wireless network. In this thesis we apply (and extend) the results to an LTE-Advanced system augmented with small cells. Furthermore their investigation were limitedto a without fading and Rayleigh fading infected channels only. In this thesis the results aregiven for various fast fading type e.g. Rice fading.

If the path loss exponent is α = 4, then we can define from (3.4) the c.d.f (cumulativedistribution function) and p.d.f. (probability density function). The detailed proof of getting(3.6) from (3.4) is given in [29, Section 3.2.3]. According to our knowledge the c.d.f andp.d.f. exist only if α = 4.

Theorem 3.1. In a two-tier OFDMA based system, the cumulative small cell interferencefollows Lévy distribution, if the (interference source) small cell locations are modelled withhomogeneous Poisson Point process, and the outdoor path loss exponent (α) equals 4. Re-gardless of the channel fast fading type (i.e. Rayleigh, Nakagami-m, Rician or Weibull faded)the interference distribution remains Lévy distribution. The c.d.f and p.d.f. are given as fol-lows:

fI(x) =√

c2π

exp(− c

2x

)x3/2 , (3.5)

FI(x) = IPI ≤ x= erfc(√

c2x

). (3.6)

The distributions location parameter (µ) is 0, meanwhile the scale parameter is:

c = π3λ

2K−1i

(E√

h)2

Ps/2. (3.7)


The complementary error function is defined in (2.8). Without fast fading h= 1, thereforeE√

h = 1, evidently. Meanwhile, with Rayleigh fading E√

h = Γ(1+ 2

α

)= Γ

(32

)=√

π

2 . The moments for different fading types were calculated in Section 2.2 and collected inTable 3.1 for the sake of convenience.

The probability density functions are given in Figure 3.1 for different small cell densities

1 2 3 4 5 6 7 8 9

x 10−10

0

1

2

3

4

5

6

7

8x 10

8

x [W]

Pro

babi

lity

Den

sity

No fast fadingRayleigh fading (m=1)Nakagami fading (m=4)Rician fading (K=2)Weibull(n=10,γ=1) fading

(a) λ = 5 ·10−5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10−9

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2x 10

8

x [W]

Pro

babi

lity

Den

sity

No fast fadingRayleigh fading (m=1)Nakagami fading (m=4)Rician fading (K=2)Weibull fading (n=10,γ=1)

(b) λ = 10−4

Figure 3.1: Probability density functions for different fast fadings

3.1. INTERFERENCE DISTRIBUTION 33

values (λ = 5 ·10−5 and λ = 10−4). The results are calculated with (3.5) with substitutionof the adequate fast fadings’ moment into the Lévy distribution’s parameter c. The smallcells emit on Ps = 20mW and Ki = 1. It is important to highlight that the actual value of Ps

and Ki changes c.d.f and p.d.f shape, evidently. However, these parameters are network andpath loss specific, thus other values can be substituted also.

The cumulative distribution functions are validated with empirical results of Monte Carlosimulations. The simulation results are constructed from the evaluation of 104 experiments.In every iteration the actual location of the small cells are given as a PPP and the h is arandom parameter, which obeys to the currently applied fast fading type. The simulationconditions are defined in the beginning of Chapter 2. In order to make simulations we haveto define a finite area as the simulation area. The system area (|R|) in this case is a square|R| = 1000m× 1000m. The results from (3.6) are calculated same parameters used in thesimulation (e.g. λ , Ps). The actual position of the receiver is chosen randomly (uniformlydistributed coordinates) in |R|.

The result of (3.6) is given by markers, meanwhile the empirical results of the MonteCarlo simulations are represented as a curves in Figure 3.2. The c.d.fs are evaluated forNs = 50(λ = 5 ·10−5) and Ns = 100(λ = 10−4) with numerous fading specific parametersbelong to Nakagami-m, Rician and Weibull fading (Table 3.1). We illustrated also the resultsrelated to the case without fading in every figure as a reference point.

From the results it is visible that the cumulative distribution functions are perfectlymatches with the simulation results, for both case independently from the fading. There-fore Lévy distribution given in (3.6), with suitable parameter substitution of c, provides anaccurate form in order to characterize the interference power distribution at the receiver. Asexpected, without fast fading the received interference power is higher (thus the curve of thec.d.f is flatter). Furthermore, as it has been already mentioned, due to Slivnyak’s theorem,the actual position of the receiver (macro UE in this situation) does not modifies the c.d.fshape.


10−9

10−8

10−7

10−6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x [W]

F(x)=PrI≤

x

Without Fast Fading (analytic c.d.f)

Without Fast Fading (empirical c.d.f)

Rayleigh fading (analytic c.d.f)

Rayleigh Fading (empirical c.d.f)

Nakagami−4 fading (analytic c.d.f)

Nakagami−4 Fading (empirical c.d.f)

Ns= 100N

s=50

10−9

10−8

10−7

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

x [W]

F(x)=PrI≤

x



Rayleigh fading (analytic c.d.f)

Rayleigh Fading (empirical c.d.f)

Nakagami−4 fading (analytic c.d.f)

Nakagami−4 Fading (empirical c.d.f)

Ns= 100

Ns=50

(a) Nakagami-m faded channel

10−9

10−8

10−7

10−6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x [W]

F(x)=PrI≤

x

Without Fast Fading (analytic c.d.f)Without Fast Fading (empirical c.d.f)Rice fading K=1/2 (analytic c.d.f)Rice fading K=1/2 (empirical c.d.f)Rice fading K=2 (analytic c.d.f)Rice fading K=2 (empirical c.d.f)

Ns=50 N

s= 100

10−9

10−8

10−7

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

x [W]

F(x)=PrI≤

x



Rice fading K=1/2 (analytic c.d.f)

Rice fading K=1/2 (empirical c.d.f)

Rice fading K=2 (analytic c.d.f)

Rice fading K=2 (empirical c.d.f)

Ns= 100

Ns=50

(b) Rician faded channel

10−9

10−8

10−7

10−6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x [W]

F(x)=PrI≤

x

Without Fast Fading (analytic c.d.f)Without Fast Fading (empirical c.d.f)Weibull Fading n=2 (analytic c.d.f)Weibull Fading n=2 (empirical c.d.f)Weibull Fading n=10 (analytic c.d.f)Weibull Fading n=10 (empirical c.d.f)

Ns= 100N

s=50

10−9

10−8

10−7

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

x [W]

F(x)=PrI≤

x



Weibull Fading n=2 (analytic c.d.f)

Weibull Fading n=2 (empirical c.d.f)

Weibull Fading n=10 (analytic c.d.f)

Weibull Fading n=10 (empirical c.d.f)

Ns=50

Ns= 100

(c) Weibull faded channel

Figure 3.2: Validation of the proposed forms with Monte-Carlo simulations

3.2. OUTAGE AND COVERAGE PROBABILITY IN PPP 35

Outage and Coverage Probability in PPP

The coverage probability is defined as the probability of the Signal-to-Interference ratio(SIR) being above the threshold value (T ). The Signal-to-Interference ratio (SIR) is thefraction of the useful signal and the sum all other interfering signals. Usually it is givenwithout dimension, or in dB. SIR is one of the most important performance indices in wire-less communication.

Evidently, the outage probability and the coverage probability are complementary prob-abilities, thus the sum of outage and coverage event probabilities yields one:

IPout(z) = IPSIR(z) < T= 1− IPcov(z).

where z is the position of the macrocell user. The threshold value (T ) is a system specificparameter. In LTE the minimum required SIR value (where the spectral efficiency is notzero) equals T [dB] = −6.5 dB [72, Figure A.4. p.75]. In this case, due to the bad channelquality, the data transmission is very slow, however it is not zero. To neglect the effect ofPRB allocation, we assume a heavily loaded system, where all possible resources (PRBs) arealways allocated to greedy users. In other words, if a small cells contains at least one activeuser, then all resources are allocated to this user. If a small cell contains two or more users,then the scheduler of the small cell base station allocates the resource set between the users.Let us evaluate the SIR distribution:

IPout(z) = IP

Prc

I(z)< T

= IP

Pchc ·g(z)

I(z)< T

, (3.8)

where the sum in the denominator denotes cumulative interference power (from small cells)received at the users receiver. Pr

c represents the useful signal power (received from the macroeNB). In this thesis the interference sources are always the small cells. The interference atthis position z denoted by I(z), which is formulated in (2.33).

Lognormal fading

The method to calculate the outage probability (or the coverage probability) for lognormalfading case is given in this subsection.

Theorem 3.2. I have introduced a form, that allows the calculation of the (users’) probabilityof service outage in an OFDMA based two-tier small cell system, if the channel is infectedindependent Lognormal fading. The proposed form relies on the theory that the interference


obeys Lévy distribution:

IPout(z) =∫

∞

0

1− erfc

Nsπ3/2√

PsK−1i E

√1/Ψlog

2 · |R|

√tT

Pcg(z)

fΨlog(t)dt.

(3.9)

Proof. In this case the effect of fast fading is ignored (hc = 1and hx = 1). Firstly we shouldmodify (3.8):

IP

Pc ·g(z)I(z)

< T= IP

I(z)>

Pc ·g(z)T

=1− IP

I(z)≤ Pc ·g(z)

T

. (3.10)

After substituting (2.2) to g(z) we get:

IPout(z) = 1− IP

I(z)≤ Pc · ‖z‖−α

KiT Ψlog

. (3.11)

Of course Pc, Ki, ‖z‖ and T are input parameters to the model, meanwhile I(z) and Ψlog areremain random variables. Applying the Law of total probability for a continuous distribution:

IPout(z) =∫

∞

0

(1−Pr

I(z)≤ Pc‖z‖−α

KiTt

︸︷︷︸

c.d.f of Lévy distribution

)· fΨlog(t)dt, (3.12)

fΨlog(t) is the p.d.f of the lognormal fading (with zero mean and – as an example – 10 dBvariance):

fΨlog(t) =1

√2π

ln(10)σ10 t

exp

(− (ln(t))2

2( ln(10)σ10 )2

). (3.13)

Finally substituting the c.d.f of the Lévy distribution given in (3.6) with some algebraicmanipulation yields (3.9).

Equation (3.9) (after substituting the proper input parameters) can be numerically inte-grated. E

√1/Ψlog

is given in given in Table 3.1. The results are given in Section 3.3 in

Figure 3.3.

Rayleigh and Nakagami-m fadingIn case of Rayleigh and Nakagami-m there are two ways to calculate probability of coverage.The first one is similar to the previously introduced method by using the Lévy distribution of


the interference. On the other hand, the second way is to use the PGFL of the point process.The main drawback of the Lévy distribution based method is that in this case the outdoorpath loss exponent should be restricted to α = 4. Meanwhile with PGFL based method thevalue of α is not restricted to be four.

Rayleigh and Nakagami-m fading using Lévy distribution

Theorem 3.3. I have modified the previously proposed form (that allows the calculation ofthe users’ probability of service outage in a slow faded environment), in order to calculatethe users’ probability of service outage, when the channel is Rayleigh or Nakagami-m faded:

IPout=∫

∞

0

1− erfc

Nsπ3/2√

PsK−1i E

√hc

2 · |R|·√

hT

Pc‖z‖−α )

· hm−1e−hm

m−mΓ(m)dh.

(3.14)

Proof. To calculate the coverage (or outage) probability for a given user located at z weshould rewrite (3.8) in the following convenient form:

IPout(z) = IPSIR(z)< T= IP

Pchc ·g(z)I(z)

< T.

where hc represents the fast fading parameter on the ’macro eNB – UE path’ in power do-main. The effect of shadowing is ignored hereafter.

In order to calculate the outage probability firstly we modify the SIR distribution:

IP

Pchc ·g(z)I(z)

<T=IP

I(z)>

Pchc ·g(z)T

=1−IP

I(z)≤Pchc ·g(z)

T

. (3.15)

After substituting (2.2) to g(z) we get:

IPout(z) = 1− IP

I(z)≤ Pchc · ‖z‖−α

KiT

. (3.16)

Of course Pc, Ki and T are input parameters to the model, meanwhile I(z), z and h are remainrandom variables. Applying the Law of total probability (for continuous distributions) on(3.16) yields:

IPout =∫

∞

0

(1−Pr

I(z)≤ Pchc‖z‖−α

KiT

︸︷︷︸

c.d.f of Lévy distribution

)· f (h)dh, (3.17)

The p.d.f of h ( f (h)) is given in (2.21). Finally substituting the c.d.f of the Lévy distributiongiven in (3.6) to (3.17) yields (3.14). The values of E

√hc

for Rayleigh and Nakagami-mare given in Table 3.1.


Rayleigh and Nakagami-m fading using PGFL

To calculate the coverage (or outage) probability for a given user located at z we shouldrewrite (3.8) in the following convenient form:

IPcov(z) = IP

hc ≥T I(z)Pcg(z)

.

The interference I(z) depends from the position of the interference source (small cells) basestations (given by set Φ) and from the fast fading parameter (h). Using the Law of totalprobability again we can write:

IP

hc ≥T I(z)Pcg(z)

= EΦ,h

IP

hc ≥T I(z)Pcg(z)

∣∣∣Φ,h

.

This is the definition of the complementary cumulative distribution function (CCDF). hcrandom variable is independent from h and the process itself (Φ) and follows exponentialdistribution (with rate parameter 1). According to Section 2.2 the coverage probability is:

IPcov(z) = EΦ,h

e−(

TPcg(z)

)I(z).

This expression is equal to the Laplace transform of the interference at s = TPcg(z) .

We know from (3.2) that the Laplace transform of the interference can be described bythe PGFL. Due to the independence of h we can write IPcov as follows:

IPcov(z) = EΦ

[Eh

e−(

TPcg(z)

)I(z)]

=

= EΦ

∏x∈Φ

Eh exp(−sPshg(x− z))︸︷︷︸ν(x,z)

.The internal expression of this product (ν(x,z)) is the definition of moment generating func-tion for h and it is given in (2.19). Therefore substituting ν(x,z) to the Poisson Point ProcessPGFL (2.31) yields:

IPcov(z) = EΦ

∏x∈Φ

11+ sPsg(x− z)︸︷︷︸

ν(x,z)

= GN(ν)

∣∣∣∣ν(x,z)= 1

1+sPsg(x−z)

. (3.18)

This confirms the result in [30]1. To evaluate the coverage/outage probability for a1Note that authors of [30] investigated a one-tier system


Nakagami-m faded channel we use the same steps proposed previously, however to get thefinal formula some algebraic calculation is required.

Theorem 3.4. I have proposed a mathematical form in order to evaluate outage/coverageprobability for a user at location z, if the channel is Nakagami-m faded, and the actuallocation of small cells are modelled with homogeneous Poisson Point process:

IPcov(z) =m−1

∑k=0

(−1)k

k!

[sk dk

dsk LI(z)(s)︸︷︷︸G (ν(x,z))

]∣∣∣∣∣s= T m

Pcg(z)

. (3.19)

Proof. In case of Nakagami-m fading the received power can be characterized with a Gammadistributed random variable. From the p.d.f. of the Nakagami-m distributed random variablemake the following random variable transformation:

f (x,m,Ω) =2mm

Γ(m)Ωm x2m−1 · exp(−m

Ωx2).

Let us consider that the new random variable is z = x2

Ω. Then x =

√z ·√

Ω and dxdz =

12

√Ω√z .

Therefore the p.d.f of the transformed random variable is:

f (z,m) =mm

Γ(m)zm−1 exp(−m · z).

Afterwards, let us introduce an another random variable:

y =z ·Pcg(z)

T, thus z =

y ·TPcg(z)

anddzdy

=T

Pcg(z).

Therefore the p.d.f of the transformed random variable is:

f (y,m) =mm

Γ(m)

(T

Pcg(z)

)m

(y)m−1 exp(− mT

Pcg(z)· y).

Applying the Law of total probability:

IPcov(z) =∫

∞

0f (y,m) ·PrI(z)≤ ydy. (3.20)

This equation can be calculated via integration by parts where f ′ = f (y,m) and g =PrI(z)≤ y. After evaluating the integral on part f ′ we get:

f =1

Γ(m)

∫ y

0

mm ·T m

(Pcg(z))m tm−1 exp(− mT

Pcg(z)t)

dt


In order to show that this is the definition of incomplete Gamma function, we make anothersubstitution:

t =mT

Pcg(z)t, therefore t =

Pcg(z)mT

t anddtdt

=Pcg(z)

mT.

Then

f =1

Γ(m)

∫ mTPcg(z)y

0(t)m−1 exp(−t)dt︸︷︷︸

γ

(m, mT

Pcg(z) ·y)

=1

Γ(m)γ

(m,

mTPcg(z)

· y), (3.21)

where γ(·) is the incomplete upper Gamma function defined in (2.10). It is well-known from(2.12) that

γ

(m,

mTPcg(z)

· y)= Γ(m)−Γ

(m,

mTPcg(z)

· y).

Thus part f modifies to:

f = 1−Γ

(m, mT

Pcg(z) · y)

Γ(m).

Now, we should evaluate the integration (3.20) using f :

IPcov(z) =

1−Γ

(m, mT

Pcg(z) · y)

Γ(m)

·PrI(z)≤ y

∞

0

−

∫∞

0

1−Γ

(m, mT

Pcg(z) · y)

Γ(m)

·dPrI(z)≤ y. (3.22)

We invoke the special values of Γ(m,0) and Γ(m,∞) given in (2.14) and (2.15). Therefore

IPcov(z) =∫

∞

0

Γ

(m, mT

Pcg(z) · y)

Γ(m)d

PrI(z)≤ y. (3.23)

Substituting the serial representation of the incomplete Gamma function given in (2.13) intothe nominator of (3.23) yields,

IPcov(z) = 1Γ(m)

(m−1)!m−1

∑k=0

1k!

∫∞

0exp(− T m

Pcg(z)y)(

T mPcg(z)

y)k

d

PrI(z)≤ y.

(3.24)


The definition of the Laplace transform is

LX(s) =∫

∞

0exp(−sX) fX(x)dx =

∫∞

0exp(−sX)dFX(x) .

Furthermore:

dds

Lx(s) = −∫

∞

0 xexp(−sx)dFX(x) ,

d2

ds2 Lx(s) = (−1)2 ∫ ∞

0 x2 exp(−sx)dFX(x) ,...

dk

dsk Lx(s) = (−1)k ∫ ∞

0 xk exp(−sx)dFX(x) . (3.25)

Finally, rewriting (3.24) with (3.25) in a convenient form, yields (3.19).

According to the definition of the probability generating functional given in (2.30):

LI(z)(s) = GN(ν)

∣∣∣∣∣ν(x,z)= 1

(1+sPsg(x−z)/m)m

.

It is important to highlight the fact that substituting m = 1 (Rayleigh fading case) to (3.19)yields

IPcov(z) = LI(z)(s)∣∣∣∣s= T

Pcg(z)

= G (ν(x,z)) =

= exp

−λ

∫R2

1− 1(1+ PsT

Pc‖z‖−α ‖x− z‖−α

)dx

. (3.26)

which is related to (3.18), thus the proposed form guarantees the “backward compatibility”.

Rice fadingIn case of Rice fading the coverage probability is calculated with the PGFL. However thereis limitation, due to the complex nature of this fading type. Therefore we propose an approx-imated form (lower bound).


Theorem 3.5. In a two-tier interference limited network, where the small cell positions obeya Poisson Point Process and the channel is Rician faded, the coverage probability for amacrocell user is given by:

IPcov(z)=e−K∞

∑m=0

Km

m!

m

∑k=0

(−1)k sk

k!dk

dsk LI(z)(s)∣∣∣∣s=(K+1)T

Pcg(z)

. (3.27)

Furthermore an approximated form for coverage probability is given:

IPcov(z)≥ e−KN

∑m=0

Km

m!

m

∑k=0

(−1)k sk

k!dk

dsk GN(ν)

∣∣∣∣s= (K+1)·T

Pcg(z)

. (3.28)

Proof. Firstly, let us rewrite IPcov(z) as follows:

IPcov(z) = IP

Pchc ·g(z)I(z)

≥ T= IP

I(z)≤ hcPcg(z)T︸︷︷︸.=y

. (3.29)

Thereafter we can substitute the serial representation of the modified zero order Bessel func-tion [61] :

I0(x) =∞

∑m=0

( x2

)2m

m!Γ(m+1),

into (2.24). Then, we apply the Law of total probability. The result is

IPcov(z) =∫

∞

0 Pr I(z)≤ y · f (y)dy =

=∫

∞

0(K+1)TPcg(z) e

(−K−(K+1) y·T

Pcg(z)

)·∑∞

m=0

(K(K+1) y·T

Pcg(z)

)m

m!Γ(m+1) ·Pr I(z)≤ ydy.

(3.30)

Next, we simplify the non dependent parts from the integral and perform the integration byparts:

e−K∞

∑m=0

Km

m!

1−∫

∞

0

γ

(m+1, (K+1)T

Pcg(z) y)

Γ(m+1)dPr I(z)≤ y

, (3.31)

where γ (· , ·) denotes the lower incomplete Gamma function and Γ(·) is the completeGamma function, given in (2.10) and (2.9), respectively. Recalling the definition of the


sum of upper and the lower incomplete gamma functions (2.12) we get the following form:

e−K∞

∑m=0

Km

m!

∫∞

0

Γ

(m+1, (K+1)T

Pcg(z) y)

Γ(m+1)dPr I(z)≤ y

. (3.32)

Note, that Γ(m+ 1) = m! if m is integer. Furthermore the invoke the definition series rep-resentation of the upper incomplete Gamma function given in (2.13). Therefore, after thesubstitution and with some algebraic manipulation we get the following form for (3.32):

e−K∞

∑m=0

Km

m!

m

∑k=0

((K+1)TPcg(z)

)k

k!

∫∞

0e−

(K+1)TPcg(z) y

(y)k dPr I(z)≤ y . (3.33)

The integral in (3.33) is the definition of the Laplace transform’s kth order derivate given in(3.25), with s = (K+1)·T

Pcg(z) . Rewriting (3.33) with (3.25) yields (3.34). Therefore in a two-tierinterference limited network, where the small cell positions obey a 2D PPP and the channelis Rician faded, the coverage probability for a macrocell user is given by:

IPcov(z)=e−K∞

∑m=0

Km

m!

m

∑k=0

(−1)k sk

k!dk

dsk LI(z)(s)∣∣∣∣s=(K+1)T

Pcg(z)

. (3.34)

Evidently, for K = 0 (i.e. for the Rayleigh fading case) equation (3.27) simplifies to thefollowing form:

IPcov(z) = LI(z)(s)∣∣∣∣s= T

Pcg(z)

= GN(ν),

which confirms the results of subsection 3.2.2. and [26].

Approximated form for coverage probability: With reference to the relationship be-tween the Laplace transform of the interference and the PGFL we can substitute (2.31) into(3.27). Furthermore, since the odd-order derivatives of the PGFL are negative values2, thesum in (3.27) contains only non-negative elements. Hence we can define a lower bound for

2This statement comes from the Laplace transform of the interference, for proof see (3.25). According to(3.3) this property should be true for PGFL. Therefore the odd-order derivative of the PGFL is negative.


the coverage probability by limiting the number of terms to a finite number N, yielding:

IPcov(z) = e−K∞

∑m=0

Km

m!

m

∑k=0

(−1)k sk

k!dk

dsk GN(ν)≥

≥ e−KN

∑m=0

Km

m!

m

∑k=0

(−1)k sk

k!dk

dsk GN(ν)

∣∣∣∣s= (K+1)·T

Pcg(z)

. (3.35)

Results for Outage/Coverage ProbabilityThis section collects the results given for the probability of outage (or coverage) if the modelis PPP. All of the form results are validated with simulations. The simulation results wereaveraged over 10. 000 experiments using the parameters defined as follows. The simulationarea (and also the system area |R|) is represented by a square having dimensions of 500 m×500 m. The macro eNB is located at the center of R as it is given in Chapter 2. The thresholdvalue is T = 1. The macro eNB is positioned at the center of this square, which is assumed tobe the origin of the coordinate system and transmits continuously at a power of 20 W, whilethe small cells transmit at a constant power of 20 mW. Thus the power-ratio is Pc/Ps = 103

by default. The outdoor path loss exponent was assumed to be α=4 and we stipulated Ki = 1for the sake of simplicity. Both the macrocell and the small cell are subjected to the samefading statistics. The macro cell user is settled in z. The designated user is located 100 mand 200 m from the macro eNB. The reader might notice, that just the distance ‖z‖ to themacro eNB is restricted, the coordinates in z are random.

We assume that the system is fully loaded, hence every small cells transmits at its maxi-mum power and all the PRBs of the LTE OFDM system are actively filled with data. Due tothe heavy traffic-load the interference contaminates every PRB. No wall penetration loss isassumed between the small cell tier and the macrocell user, which would in practice mitigatethe interference inflicted by the small cells located indoors (e.g. femtocells). Therefore therather high value for probability of service outage is the consequence of this two aforemen-tioned reasons.

Results for Lognormal Fading

The form results3 for Lognormal fading is gained from (3.9). The results are compared withsimulations, and presented in Figure 3.3. The curves represent the result of (3.9), meanwhile

3By form results the author means the results obtained using the proposed forms in this thesis.

3.3. RESULTS FOR OUTAGE/COVERAGE PROBABILITY 45

1 10 100 600

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1N

s

Pr o

ut(

z)

= P

rS

IR <

T

|| z || =100 m, w/o lognormal fading

|| z || = 200 m, w/o lognormal fading

|| z || =100 m, lognormal fading (0,10dB)

|| z || = 200 m, lognormal fading (0,10dB)

|| z || = 100 m, Simulation w/o lognormal fading

|| z || = 200 m, Simulation w/o lognormal fading

|| z || = 100 m, Simulation lognormal fading (0,10dB)

|| z || = 200 m, Simulation lognormal fading (0,10dB)

Figure 3.3: The outage probabilities for Lognormal faded channel

the markers denote the values of the simulations. On the horizontal axes the mean number ofsmall cells is given and the vertical axes scales the outage probability. The scale goes from 0to 1, evidently. The scenario without lognormal fading4 is also depicted as a reference point.The dashed lines represent this scenario, meanwhile the solid lines denotes the (0,10 dB)lognormal fading case in Figure 3.3.

According to Figure 3.3 the plot from (3.9) correspond with the simulation results (bothcases, even with different ‖z‖ distances). The results of Figure 3.3 gives the service outageprobability from the view of macro users, therefore higher number of small cells yieldshigher interference and higher service outage probability. Furthermore ‖z‖ also modifiesthe results. Obviously, larger distance between the macro eNB and the macro user provideshigher service outage probability, as it is shown in Figure 3.3.

Results for Rayleigh and Nakagami-m Fading

Now we investigate the fast fading effected channels. The results for Rayleigh fading (m= 1)and Nakagami-m (m = 4) fading are gained from (3.18) and (3.19), respectively. Theseresults are marked with (PFGL) in Figure 3.4. Furthermore the results provided by (3.17)are also represented. These results are marked with (Lévy) in Figure 3.4. The evaluationof both formulas are compared with simulations given in Figure 3.4. The dashed curvesrepresent the result of (3.18) belongs to PGFL method, meanwhile the markers denotes thevalues of the method that based on Lévy distribution and the simulations. On the horizontal

4Note that, in this case (Ψlog = 1 and fΨlog = 1), therefore is no need to solve the integral, only the correctvalues should be substituted to the function erfc(·).


axes the mean number of small cells is given (in logarithmic scale) and the vertical axesscales the outage probability (also in logarithmic scale).

According to Figure 3.4, with the introduced forms we are able to calculate value ofservice loss probability. The designated user is settled in a constant ‖z‖= 100 m and ‖z‖=200 m. In this case the value of α does not need to be four, due to the fact that we havededuced form PGFL, where this restriction is not required5. Again, the simulation resultsconfirm the correctness of both proposed forms ((3.17) and (3.18)).

The probability of coverage for Rayleigh, Nakagami-m fading types and path loss expo-nents (α) are plotted in Figure 3.5. In this case the designated user is settled in a constant‖z‖ = 100 m, meanwhile the mean number of small cells is Ns = 50. The horizontal axesscales the various threshold values, meanwhile the vertical axes gives the probability of cov-erage. Like previously, simulation results (illustrated with markers) are also provided forthe sake of validation and from the plot it is visible that the proposed form is also valid fordifferent T and α values.

5In the lognormal faded channel this restriction was essential, due to the fact the the interferences’ c.d.f.exits only at α = 4


1 10 5010

−3

10−2

10−1

100

Ns

Ou

tage

Pro

ba

bili

ty [

Pr o

ut]

Analyitic || z || = 100 m (PGFL)

Analytic || z || = 200 m (PGFL)

Simulation || z || = 100 m

Simulation || z || = 200 m

Analyitic || z || = 100 m (Lévy)

Analyitic || z || = 200 m (Lévy)

(a) PPP m = 1 (Rayleigh fading)

(b) PPP m = 4 (Nakagami fading)

Figure 3.4: Probability of outage in Rayleigh, Nakagami-m fading


−10 −8 −6 −4 −2 0 2 4 6 8 100.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T [dB]

Co

ve

rag

e p

rob

ab

ility

[P

r cov]

α=3 Analytic

α=4 Analytic

α=5 Analytic

α=3 Simulation

α=4 Simulation

α=5 Simulation

(a) Rayleigh fading (m = 1)

−10 −8 −6 −4 −2 0 2 4 6 8 100.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T [dB]

Co

ve

rag

e p

rob

ab

ility

[P

r cov]

α=3 Analytic

α=4 Analytic

α=5 Analytic

α=3 Simulation

α=4 Simulation

α=5 Simulation

(b) Nakagami fading (m = 4)

Figure 3.5: Probability of Coverage with Rayleigh, Nakagami-m fading types and path lossexponents (α)


10 20 30 40 5010

−3

10−2

10−1

100

Ns

Ou

tag

e P

rob

ab

ility

[P

r out]

Analytic ||z||=100 m


Simulation ||z||= 100 m


(a) PPP K = 0.5 (Rice fading)

10 20 30 40 5010

−3

10−2

10−1

100

Ns

Ou

tag

e P

rob

ab

ility

[P

r out]





(b) PPP K = 1 (Rice fading)

Figure 3.6: Probability of outage in Rician fading

Results for Rician FadingFirst of all let us confirm the upper-bound results of (3.35) with the aid of our simulationresults in terms of the outage probability versus the average number of small cells. We optedfor N = 100 as the limit of the sum.

The corresponding results are depicted in Figure 3.6 for K = 0.5 and K = 1, respectively.The results are calculated from (3.35) and represented as dashed lines, while the markersrepresent the simulation results for ‖z‖ = 100 m and ‖z‖ = 200 m respectively. Although(3.35) is just an upper bound results the simulation results validate the (approximated) results


are close to the ones from the simulator for both K values.

Next, we investigate the coverage probability for various threshold and α parameters(Figure 3.7). Here the mobile UE is positioned at a constant ‖z‖ = 100 m distance and thepower-ratio is Pc/Ps = 103. The mean number of small cells are, Ns = 100. The Ricianparameter is K = 0.5. As expected, for lower α values the coverage probability becomeshigher, since the useful signal becomes less overwhelmed by the interference.

Previously, we applied fix transmission powers for the macro- and small cell base stations(20 W and 20 mW) and for the T = 1 threshold value in order to validate our results. Nev-ertheless, these parameters are operator and/or system specific. This form is also applicablefor modelling mobile ad-hoc networks. In this case the fraction Pc/Ps should be 1 i.e. allnodes transmit at the same power. In LTE the minimum required Signal to Interference ratioT for communication is −6.5 dB. Hence we opt for this interference-threshold parameterand investigate the achievable coverage probability as a function of the transmission powerratio of Pc/Ps. The Rician fading parameter is set to K = 0.5 and the path-loss to α = 4. Theresults are given in Figure 3.8 in terms of the coverage probability versus the power-ratio.We calculated the probability of coverage for several mobile-UE distances. The form resultsare represented by the curves, while the simulation points are given by markers. Again, thesimulation results closely match the results gained from the form. The probability of cov-erage is drastically reduced, as the power-ratio tends to 1, since in this case the interferencepower becomes comparable to the useful signal power of the BS.

−10 −8 −6 −4 −2 0 2 4 6 8 10

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T [dB]

Co

ve

rag

e P

rob

ab

ility

Pr c

ov

α = 3

α = 4

α = 5

α = 3 Simulation

α = 4 Simulation

α = 5 Simulation

Figure 3.7: Probability of Coverage with Rician fading with various path loss exponents (α)

3.4. AVERAGE SYSTEM THROUGHPUT 51

101

102

103

104

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Macro and Small cell Power ratio: Pc/Pf

Cov

erag

e pr

obab

ility

Pr co

v

||z|| = 50 m||z|| = 100 m||z|| = 150 m||z|| = 200 m

Figure 3.8: Probability of coverage for macro UE vs. Pc

Ps

Average System Throughput

The previous section showed that even in two-tier LTE systems, due to small cell deploy-ment, it is possible to have downlink interference and remarkable service outage. Thereforecareful investigation is required to ensure the QoS. On the other hand small cells can ex-tend the offered capacity significantly. In this section we propose a form that allows thecalculation of the average system capacity (and the enhancement due to small cells). Thebasic small cell location model is remained PPP, however now we will use the road structurerestricted system model, described in Subsection 2.1.2.

The average system throughput is calculated via the SIR distribution (as defined in theprevious section). Due to the high outage probability we separate the two tiers in frequencyband. Therefore the capacity offered by the macro tier does not infected by the interferencecaused by small cells. On the other hand in the small cell layer the so-called co-tier inter-ference (interference generated by small cells) still exists. Signal-to-Interference Ratio for achosen small cell user at the jth resource block (served by small cell k) can be defined by the


formula given in (3.36), where the effect of AWGN is neglected,

SIR( j,k) =Psg(z)

Ico-tier( j,k)(z)=

Prs

∑i∈Φ,i6=kPsδx, j

KiΨlog(i)· ‖x− z‖−α

. (3.36)

Probability distribution of SIR

The cumulative probability function of SIR is given as F(T ). The user has the followingCDF

F(T ) = IPSIR≤ T ,

where (3.36) can be substituted into SIR and T is the threshold value. According to Sec-tion 3.1 the interference generated by small cells (2.35) has a c.d.f. given as follows,

IPI(z)≤ x= erfc

(NsKL

√1x

),

where KL = π3/2 IE√

1/Ψlog√

Ps/Ki/|R|.As the worst case scenario, let us assume that the system has greedy users, thus all PRBs

are assigned to users if a user is settled in a cell. This means that we can leave indicator δ ,as it is always equal to 1. The shape of the interference distribution independent from the re-ceiver location, thus we write I instead of I(z) for the sake of simplicity. The mean of

√1

Ψlog

is an important parameter for c.d.f., so we calculate it here. As we mentioned previouslythe logarithm of Ψlog follows normal distribution with zero mean and 10dB variance. Using

these values6, one gets IE√

1Ψlog

= 1.0068. The interference c.d.f. is an essential part to

get SIR distribution. It has been already introduced that:

IPI > x= 1− erfc(

NsKL1x

)= erf

(NsKL

√1x

),

so reshaping of (3.36) yields:

F(T ) = IP

Prs

T≤ I= erf

(NsKL

√TPr

s

). (3.37)

6substituting µ = 0 and σ = 1 to IE√

1Ψlog

in Table 3.1

3.4. AVERAGE SYSTEM THROUGHPUT 53

SIR [dB] System Throughput [kbit/s]-12 dB 0 kbit/s-9 dB 15 kbit/s-6 dB 38 kbit/s-3 dB 70 kbit/s0 dB 110 kbit/s3 dB 170 kbit/s6 dB 250 kbit/s9 dB 340 kbit/s

12 dB 480 kbit/s15 dB 620 kbit/s18 dB 740 kbit/s21 dB 850 kbit/s24 dB 910 kbit/s27 dB 930 kbit/s

Table 3.2: Throughputs used for Average System capacity calculation in MIMO 2×2 con-figuration

Overall Throughput of a Two-tier SystemThe values given in Table 3.2 are data rates collected from a 2×2 MIMO environment [73].The authors conducted measurements which were averaged7. The numbers describe the netdata rate (after channel decoding) of one sole PRB in a 2× 2 MIMO environment in kbps.Combining (3.37) with Table 3.2, one can get an average data rate (from small cells) whichdescribes each PRB in the system.

If one wants to get the overall throughput of the system, the correct computation is thefollowing. All data rates seen by the users must be summed up. That is,

Cfull= ∑U1,...,UN f

IP

U1, . . . ,UN f

U1

∑j1=1

Csj1+ · · ·+

UNf

∑jNf =1

CsjNf+Nc−∑i Ui

∑k=1

Cmk

, (3.38)

where N f denotes the actual (random) number of small cells in the area. The first sumcollects the combination of user distribution along small cells, IP

U1, . . . ,UN f

represents

the corresponding probability (which is actually the product of Poisson probabilities) andfinally the terms in brackets show the capacities which are available for the users – they are

7The proposed form use these measured data rates as an input parameter, thus mobile service providers canuse their own measurement result or e.g. the measurement results of Ericsson [74].


summed. The last term stands for the macrocell. Note, that the sums go from one: if there isno user under a base station (the zero case), there is nothing to sum up.

The evaluation of (3.38) is difficult since we should know the actual number of smallcells, the required capacity by the users etc. Therefore to keep the model tractable we madethe following assumptions:

1. The resource allocation between users is fair, thus there are no priority order betweenusers.

2. Every user require the available offered resources. Therefore if a small cell contains atleast one user, then all resources are offered to the user(s).

3. In case of having multiple users in a small cells all resources are allocated between theusers.

4. If there is no user in the cell, then no resources had been allocated.

5. Assuming that the density of the ON/OFF model is high enough to maintain full loadin the system.

Thanks to these assumptions each term (sum) inside the bracket in (3.38) yields the fullcapacity Cs

full. Thus if a small cell is active (contains at least one user) the seen capacityby the user(s) is Cs

full. Therefore IP

U1, . . . ,UN f

in the first sum in (3.38) simplifies to

IPUi 6= 0= 1− IPUi = 0. The users are distributed on |R| according to PPP, thereforethe probability of not having user in the small cell coverage area is: IPUi = 0= 1− e−λ ,where

λ =

NcR2

s π

|R| two-dimensional PPP space width circular coverage of small cells,Nc2Rs∑i ei

graph model of PPP width line coverage of small cells.

If a general case let us assume that a small cell covers a circle with radius Rs, thus the areaof the covered zone is R2

s π . Meanwhile in the graph model the covered area is a 2Rs section.The average number of users under the small cell’s coverage equals Nc(R2

s π)/|R| andNc2Rs/(∑i ei), respectively8. The average number of active small cells equals e−λ Ns.

If k small cells are active (they have at least one active user), then the bracket termin (3.38) yields k ·Cs

full +Cmfull, for each PRB. Though inactive small cells also have free

capacities, they are not added: they do not raise the system capacity due to their inactivity.8The reader might note that this assumption is not always valid. For example, if the small cell base station

is close to a vertex or the circles overlap, the coverage might be greater or smaller depending on the numberand angle between neighbouring edges. However, the error due to this phenomenon is neglected, if the numberof small cells is low.

3.5. RESULTS OF THROUGHPUT ENHANCEMENT WITH SMALL CELLS 55

Theorem 3.6. I have proposed a form to calculate the overall two-tier system capacity in atwo-tier small cell network, if the small cell are scattered according to homogeneous Poissonpoint process. Given the average number of active small cells, the overall two-tier systemcapacity can calculated with

Cfull = NPRB ·(

e−λ ·Ns ·Csfull +Cm

full

), (3.39)

where the multiplication by NPRB represents the number of PRB’s in the system.

The values of Csfull and Cm

full are calculated from the SIR distribution given in (3.37). Theprobability that the SIR is between two threshold values (T1, T2) is given as a difference ofthe c.d.f’s:

IPT1 < SIR≤ T2= F(T2)−F(T1). (3.40)

Combining (3.37), (3.40) with Table. 3.2 provides the values of Csfull and Cm

full. The numericalresults are given in Figure 3.9a.

Results of Throughput enhancement with Small Cells

The system model contains a macrocell base station and small cell base stations varying ona scale between 0–128. The small cells are used outdoor (no wall penetration loss is as-sumed) and they are in open access mode (accessible for anybody). The transmission powerthe macro and small base stations is 20 W and 20 mW, respectively. The received powerPr

s is restricted to 1 mW. The nominal system bandwidth is 20 MHz, which means thereare 100 PRBs per slot. Every base station applies round robin scheduling algorithm. Theequipment have 2×2 MIMO antenna configuration. The system has 100 active mobile ter-minals. The model area is a 1000 m×1000 m square. In graph model the total road lengthis ∑i ei = 5600 m. The system is fully loaded i.e. all PRBs are assigned. Figure 3.9a showthe theoretical total system throughput (3.39) as a function of number of small cells. Theslow fading mean value is 0 dB and the variance is 10 dB in this scenario. The axes are log-arithmic scaled. On the horizontal axis one finds the logarithm of the number of small cells(log2 N f ), where −1 stands for the one-tier (N f = 0) case. On the vertical axis the systemthroughput is given in Mbit/s, in logarithmic scale. The results reflect the improvement dueto the introduction of small cells. According to (3.39) the total system capacity remarkablyincreases with the number of small cells, until the capacities remain the maximum possible.It is visible that the maximum system throughput is achieved around 8200 small cells. Themaximum system capacity we can have here equals 2 · 105 Mbit/s compared to the one-tiercase which is around 100 Mbit/s. This is the point where operator should stop or lower the


0 2 4 6 8 10 12 14 16 18 2010

1

102

103

104

105

106

log2(N

f): Logarithm of the number of Small cell base stations

Tota

l sy

stem

thro

ughput

[Mbps]

in l

ogar

ithm

ic s

cale

PPP model

Random graph PPP model

(a) Average System throughput vs. number of Small cells

−1 0 1 2 3 4 5

105

106

log2(N

f): Logarithm of the number of Small cell base stations

Tota

l sy

stem

thro

ughput

[kbps]

in l

ogar

ithm

ic s

cale

Simulation results (PPP)

Analytic results (PPP)

(b) Simulation throughput vs. number of Small cells

Figure 3.9: Throughput enhancement with Small Cells

3.5. RESULTS OF THROUGHPUT ENHANCEMENT WITH SMALL CELLS 57

small cell population. However for a reasonable number of small cells deployment for exam-ple 32 (25) offers around ∼ 3.2× more system level throughput compared to one-tier macronetwork. Above 213 small cells (Ns > 8200) the throughput begins to decrease: the inter-ference becomes so high at this point, that it is not possible to offer any improvement whennew small cells are installed. The advantage of the new small cell installation is exceeded bythe disadvantage it causes to other already existing small cell installations. Around Ns ≈ 219

the performance of the network returns to the starting point, and provides the same systemcapacity as a one-tier solution. After this point the system throughput does not vary with thenumber of small cells, hence deploying more small cells is superfluous. Figure 3.9b showsthe comparison of the PPP theoretic results and simulation results for the two dimensionalcase. The scaling of axes are the same as in Figure 3.9a. The simulation is limited dueto its computational burden, thus the simulated range is smaller in Figure 3.9b comparedto Figure 3.9a. The theoretic total system throughput follows an almost linear curve. Thesimulation results increases with the number of small cells, and fits well on the theoreticalresults in case of the number of small cell is low. The simulation results validate the proposedmethod.

4 Small Cell modelling with PCP

The cluster based small cell modelling is discussed in this chapter. The details of Thomas-and Matérn cluster process had been already discussed in Section 2.1.3. Due to the differentnotation of small cell locations (i.e. parent and daughter points) we should slightly modifythe interference description. Afterwards, the remaining of the chapter gives approximatedforms that allows us to calculate the coverage- or outage probability for various fast fadingtypes.

Interference CharacterizationThis section firstly gives a brief interference characterization and afterwards provides themethod to calculate the service outage probability for a macrocell user in case of high densityclosed accessed or fully loaded small cells. Our investigation focuses on downlink, small cellgenerated interference, like in the previous chapter.

The easiest way to characterize the received interference power in a given random loca-tion is as follows. Supposing that the macrocell user is located at z, then the total receivedinterference power from small cells can be calculated with the following formula:

I(z) = ∑x+y∈Φ

Pshx+yg(x+y− z). (4.1)

Vector x+ y represents the random location of interference source base stations (from setΦ). The effect of i.i.d fast fading is denoted by hx+y and the path loss between the receiverunit and the interference source is given as g(‖x+y− z‖).

The form given in (4.1) is appropriate for single carrier (e.g. Code division Multiple Ac-cess – CDMA) interference description. LTE is an Orthogonal frequency-division multiplex-ing access (OFDMA) based communication system with many subcarriers. The subcarriers

59

60 CHAPTER 4. SMALL CELL MODELLING WITH PCP

are clamped to Physical Resource Blocks (PRBs) and these PRBs are allocated to the users,therefore the correct interference description should be:

I( j)(z) = ∑x+y∈Φ

σx, j ·Ps ·hx+y ·g(x+y− z), (4.2)

where σx, j is an indicator: it equals one, if the macro and the small cell eNB (located at x+y)assigns the same jth resource block, and thus interference occurs. Otherwise (if σx, j = 0)the interference is zero:

σx, j =

1 if j ∈Θc and j ∈Θx0 otherwise.

The set of PRB indices assigned by the macro eNB to a desired user is represented by Θc. Θxdenotes the set of PRBs assigned by the small cell eNB located at position x+y to their smallcell user(s). If the macro eNB and the small cell eNB assign the same PRB(s) to their users,then interference level is increasing on a designated PRB. Nevertheless, we assume a heavilyloaded systems (same as Chapter 3), where all possible resources (PRB’s) are allocated tousers, therefore the overall interference is interpretable with (4.1). Therefore we neglect σx, j

hereafter. The further investigation requires the Laplace transform of the interference (I(z)):

LI(z)(s) = E

exp

(−s ∑

x+y∈Φ

Pshx+yg(x+y− z)

). (4.3)

Rewriting the sum in (4.3) exponent as a product we get

LI(z)(s) = E

∏

x+y∈Φ

exp(−sPshx+yg(x+y− z))

. (4.4)

The fading patterns (hx+y) are i.i.d. we can write the PGFL in the following form using(2.22) for Rayleigh and Nakagami-m case:

LI(z)(s) = Ex,h

∏

x+y∈Φ


=

= Ex

∏

x+y∈Φ

Eh


=

= Ex

∏

x+y∈Φ

1(1+ sPs

m g(x+y− z))m

. (4.5)

4.1. INTERFERENCE CHARACTERIZATION 61

10−11

10−10

10−9

10−8

10−7

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x [W]

F(x

)=P

rI<

x

Empirical CDF

Without Fast fading (c = 10)

Without Fast fading (c = 1)

Rayleigh fading (c = 10)

Rayleigh fading (c = 1)

10−11

10−10

10−9

10−8

10−7

10−6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x [W]

F(x

)=P

rI<

x

Empirical CDF

Without Fast fading (c = 1)Without Fast fading (c = 10)Nakagami-m fading (m=4) (c = 1)Nakagami-m fading (m=4) (c = 10)

(a) Thomas cluster process Interference c.d.f.

10−11

10−10

10−9

10−8

10−7

10−6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x [W]

F(x

)=P

rI<

x

Empirical CDF

Without Fast fading (c = 1)Without Fast fading (c = 10)Rayleigh fading (c = 1)Rayleigh fading (c = 10)

10−11

10−10

10−9

10−8

10−7

10−6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

F(x

)

Empirical CDF

Without Fast fading (c = 1)Without Fast fading (c = 10)Nakagami-m (m=4) fading (c = 1)Nakagami-m (m=4) fading (c = 10)

(b) Matérn cluster process Interference c.d.f.

Figure 4.1: Empirical c.d.f. results from Monte-Carlo simulations

Interference Characterization with Monte Carlo Simulations

The cumulative distribution functions are plotted from empirical results of Monte Carlo sim-ulations. The simulation results are constructed from the evaluation of 104 experiments. Inevery iteration the actual location of the small cells are given as a PCP (Thomas and Matérncluster processes) and the hx+y is a random parameter, which obeys to the currently appliedfast fading type. The simulation conditions are defined in the beginning of Chapter 2. Inorder to make simulations we have to define a finite area as the simulation area. The systemarea (|R|) in this case is a square |R|= 1000m×1000m. The actual position of the receiveris chosen randomly (uniformly distributed coordinates) in R.


The empirical results of the Monte Carlo simulations are represented as a curves in Fig-ure 4.1. Simulation c.d.fs are evaluated for Ns = 100 with fading specific parameters belongto Rayleigh and Nakagami-m. We illustrated the without fading results in every figure as areference point. Compared to PPP based small cell modelling in the PCP based model theaverage number of parent points and cluster size is also relevant input parameter. During thesimulation we have investigated two scenarios. In the first scenario the average number ofparent points Eλp|R| was 100. This implies that the average number of small cells in acluster (c) equals 1. In the second scenario the mean number of parent points was 10, there-fore the average number of small cells in a cluster 10 (c = 10). Simulations were evaluatedfor Thomas cluster process with δ = 16.67 and Matérn cluster process R = 100. The resultsare represented in Figure 4.1a and Figure 4.1b, respectively. From the results it is visible thatthe two scenarios (c = 1 and c = 10) have different cumulative distribution functions. Fur-thermore the shape of the c.d.f curves, that belong to Thomas cluster process differ from thec.d.f curves, that belong to Matérn cluster process. As expected (same as PPP case), withoutfast fading the received interference power is higher (thus the curve of the c.d.f is flatter).

Service Outage Probability in cluster based Small cellsBasically, the power of the received useful signal equals Pchcg(z), thus the service outageprobability for an UE in position z is given by the following simple form:

IPcov(z) = IP

Pchcg(z)I(z)

> T, (4.6)

where I(z) is given in (4.2). Note that, for the sake of simplicity in the path loss model theeffect of shadowing is ignored and we use Ki = 1, however the proposed form is valid withother values of Ki as well. The UE’s which is not able to connect to any small cells, has anoutage probability in case of Nakagami-m fading as given by

IPcov(z) =m−1

∑k=0

(−1)k

k!

[sk dk

dsk LI(z)(s)

]∣∣∣∣∣s= T m

Pcg(z)

. (4.7)

The detailed proof is given in Section 3.2.2. Let us utilize the relationship between the in-terference Laplace transform (4.5) and the PGFL (2.32) to calculate the coverage probabilityfor Poisson cluster process case:

IPcov(z) =m−1

∑k=0

(−1)k

k!sk dk

dsk

[exp(−λp

∫R1− exp(c(κ−1))dx

)]∣∣∣∣∣s= T m

Pcg(z)

, (4.8)


where

κ =∫R

1(1+ sPs

m g(x+y− z))m f (y)dy. (4.9)

In (4.9) x represents the locations of parent points, and y draws the daughter points aroundpoint x.

Note that in the case of Rayleigh fading where m = 1 (4.8) simplifies to:

IPcov(z) = LI(z)(s)

∣∣∣∣∣s= T

Pcg(z)

which corresponds with the results of PPP case proposed in the previous chapter.

Approximated form for coverage probability

The calculation of (4.8) is rather difficult, therefore we provide only an approximate version.Inspired by the idea given in [75], we would like to reduce the complexity of (4.9). We applythe idea of [75] for both Thomas and Matérn cluster process, although keeping attention thatthere are differences between the two cluster process.

Approximated coverage probability for Thomas cluster process:

Theorem 4.1. In case of the small cell locations are modelled with Thomas cluster process,the macro users’ probability of service outage can be calculated with the following form:

IPcov(z) =m−1

∑k=0

(−1)k

k!sk dk

dsk

[exp(−λp


)]∣∣∣∣∣s= T m

Pcg(z)

,

where

κ =1

2πδ 2

∫R

exp(−‖y‖

2

2δ 2

)(1+ sPs

m g(x+y− z))m dy.

An approximated version of the elaborated form is given as follows:

IPcov(z)≈m−1

∑k=0

(−1)k

k!

[sk dk

dsk exp(−λp

∫R

1− exp(c(κ−1))dx)]∣∣∣∣∣

s= T mPcg(z)

,


where

κ =11+Pss

m

((δ 2+‖x− z‖2)

2+2δ 4+4δ 2‖x− z‖2

)−α

4m .

According to the properties of Thomas cluster process, vector y is normally distributedaround a parent point with a two-dimensional normal probability density function f (y).Hence κ given in (4.9) is the definition of an expected value. Therefore rewriting (4.9)yields:

κ =∫R

1(1+ sPs

m g(x+y− z))m f (y)dy = Ey

[1(

1+ sPs

m g(x+y− z))m

].

Substituting the path loss model given in (2.2) (with Ki = 1 and 1Ψlog

= 1):

κ = Ey

[1(

1+ sPs

m ‖x+y− z‖−α)m

].

After a random variable transformation (U = ‖x+y− z‖2) κ yields,

κ = Ey

[1(

1+ sPs

m (U2)−α/4)m

].

Note that function 1/(

1+ sPs

m (U2)−α/4)m

is concave when U2 > 0 and α ≤ 4, hence the

Jensen’s inequality1 gives the upper bound:

κ ≤ κ =1(

1+ sPs

m (Ey U2)−α/4)m . (4.10)

Furthermore, with some algebraic manipulation and using the nice properties of raw momentdefinition [76] of a normal random variable, Ey

U2 equals

Ey

U2= [(δ 2 +‖x− z‖2)2+2δ

4 +4δ2‖x− z‖2

]. (4.11)

1Introduced in Section 2.2.3.


For detailed proof see Appendix A.3.1. Rewriting κ with the raw moment yields

κ =11+sPs

m

((δ 2+‖x− z‖2)

2+2δ 4+4δ 2‖x− z‖2

)−α

4m . (4.12)

Finally, substituting κ to (4.8) gives the approximated form for coverage:

IPcov(z)≈m−1

∑k=0

(−1)k

k!

[sk dk

dsk exp(−λp

∫R

1− exp(c(κ−1))dx)]∣∣∣∣∣

s= T mPcg(z)

. (4.13)

This equation can be easily integrated numerically, hereafter we use this form for our inves-tigation instead of (4.8).

Approximated coverage probability for Matérn cluster process: In case of Matérn clus-ter process the key point is to simplify (4.9), hence we adapt the method introduced in [75].It is visible that κ is an expected value and y follows uniform distribution with f (y) definedin (2.6):

κ = Ey

[1(

1+ sPs

m g(x′+y− z))m

].

Now, we substitute the path loss model and rewriting κ as:

κ = Ey

[1(

1+ sPs

m (U2)−α/4)m

],

where U = ‖x+ y− z‖2. According to [75] applying the Jensen’s inequality to declare anapproximate form for κ:

κ ≤ κ =1(

1+ sPs

m (Ey U2)−α/4)m . (4.14)

Furthermore, with algebraic manipulation and moving to polar coordinate system the cal-culation of Ey

U2 is feasible. The calculation of the expected value is evaluated in Ap-

pendix A.3.2. Now rewriting κ with this expected value provides an approximate form:

κ =11+sPs

m

(‖x− z‖4+2·R2‖x− z‖2 + 2

5R4

)−α

4m . (4.15)


Finally, substituting κ to the original equation provides an approximation form, to calculatecoverage probability in case of MCP:

IPcov(z)≈m−1

∑k=0

(−1)k

k!sk[

dk

dsk exp(−λp ·

∫R

1− exp(c(κ−1)

)dx)]∣∣∣∣∣

s= T mPcg(z)

.(4.16)

Due to removing the integration in the exponent, this equation can be easily integrated andhereafter we use this form for our investigation instead of (4.8).

Theorem 4.2. In case of the small cell locations are modelled with Matérn cluster process,the macro users’ probability of service outage can be calculated with the following form:

IPcov(z) =m−1

∑k=0

(−1)k

k!sk dk

dsk

[exp(−λp


)]∣∣∣∣∣s= T m

Pcg(z)

,

where

κ =1

πR2

∫R

1(1+ sPs

m g(x+y− z))m dy.

Furthermore an approximated version of the form is given as follows:

IPcov(z)≈m−1

∑k=0

(−1)k

k!

[sk dk

dsk exp(−λp

∫R2

1− exp(c(κ−1))dx)]∣∣∣∣∣

s= T mPcg(z)

,

whereκ =

11+sPs

m

(‖x− z‖4+2·R2‖x− z‖2 + 2

5R4

)−α

4m .

Results for Outage/Coverage ProbabilityIn this section we compare the form evaluated results with simulations. The simulation areais finite square with 500 m× 500 m. The macro base station is deployed to the center ofthis square (assumed that this is the origin) and emits continuously on a power of 20 W,meanwhile the small cells are emitting on a constant 20 mW. The chosen outdoor path lossexponent α equals 4. Every tier (macro and small cell) has the same fading statistics. Pa-rameter m gives the number of independent Rayleigh fading infected paths. Small cells aredeployed on this finite square plane firstly according to Thomas cluster process.


Results for Thomas cluster Process based model

First of all we would like to demonstrate that (4.13) is an accurate approximation for (macrouser) service outage probability in case of Poisson cluster small cell deployment. The densityof parent points is given by λp and the small cells are scattered with variance δ 2 = 0.2 aroundthe parent points. The macro UE is settled permanently ‖z‖ = 100 m from the macro basestation (located at the origin). The results are presented in Figure 4.2a and Figure 4.2b. Thevertical axis scales the service outage probability from 0 to 1 obviously and the horizontalaxis gives the mean number of small cells in a cluster (c). The blue and red circles denote thesimulation results and the corresponding theoretical results are represented with blue and redcurves. It is clearly visible that the form given by (4.13) provides an accurate upper boundfor service outage for both fading types.

Despite the fact that we have an approximation on the outage probability, consideringthe results from the proposed form, the simulation fits well on the curves in both cases. Asexpected, the distance between the desired user and the macro base station and the averagenumber of small cells profoundly influence the performance of the system. For example, ifthe macro user is 100 meter far from the macro base station, 300 small cells could causeservice outage with approx. 80% probability. This is a very high service outage probability,however the reader should note that the system is fully loaded, hence all resources are allo-cated and there has been no wall penetration loss applied in this scenario. Wall penetrationcan easily separate femto/small cell and macro tiers, yielding much lower levels for outageprobabilities (see e.g. [36] for 3G operation curves).

An interesting phenomenon needs further consideration here, more precisely that the out-age probability for a macrocell user is little-bit higher in the Rayleigh fading case comparedto the Nakagami-4 fading case. The reasoning behind this phenomenon is that in case ofRayleigh fading the macrocell’s signal and the (interference) signal from a small cell hasonly one (Non Line of Sight) path to the users receiver. Meanwhile, in the Nakagami case,small cell signals and the useful signal travel on multi paths to the user’s receiver, yieldinghigher interference, but better reception conditions for the macrocell signal.

In the previous cases for validation of the formulas we used constant path loss parameters(Kc = 1 and Ki = 1), with fix transmission power for macro- and small cell base stations(20 W and 20 mW) and for the T threshold value. Nevertheless, these parameters are operatorspecific and depends on the applied path loss model (e.g. in LTE the minimum requiredSignal to Interference ratio T for communication is −6.5 dB).

Therefore we introduce a new parameter γ and afterwards calculate the service outageprobabilities for different γ values. Lets assume that γ is defined as follows:

γ =PsKi

PcKcT.


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Figure 4.2: Outage probability in Poisson Point process and Poisson cluster Process caseswith different nakagami-m fading

The reader should note that, introduction of γ is just for the sake of easier notations. Accord-ing to the used, path loss constants, transmit powers (20 W and 20 mW) and threshold value(T = 1) the results given in Figure 4.2 are calculated with γ = 10−3.

We calculate the service outage probabilities for various γ values. The outage probabili-


0 1 2 3 4 5 6 7 8 9 10

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Figure 4.3: Result for outage probabilities vs. different γ and c values, ‖z‖= 100 m

ties depending on the mean number of small cells in a cluster (c). The results are calculatedfor λp = 4 · 10−4 (Figure 4.3a) and λp = 2 · 10−4 (Figure 4.3b). This implies, that if themacrocell area is 500× 500 m2, then the mean number of clusters is 100 and 50, respec-tively.

The fading parameter is m = 4. It is visible that the service outage probability highlydepends of the value of γ , e.g. in case of γ = 10−4, when we have expectedly 100 clusterswithin (expectedly) six small cells c = 6, the service outage probability is around ∼ 40 %.On the other hand, if γ = 10−3 the outage probability doubles.

If we restrict KiPs

KcPc to fix value (e.g. 10−3), then γ only depends on the value T . Let Tscales between 0.1–10. Then we get the outage probability for values of γ from 10−4 to10−2, evidently. Figure 4.4a and Figure 4.4b provide these results. Note that, the thresholdvalue is given in dB and fast fading parameter is m = 4.

Next, we compare the cluster based small cell deployment results to a homogeneous PPPdeployment scenario. We have set λ = cλp parameter of PPP to get the same mean numberlike in Thomas cluster. Hence the two deployment model are comparable. The results aregiven in Figure 4.5. As expected the PPP case gives a higher outage probability e.g. ifthe mean number of small cells is 200, the macro user is 100 meters way from the macrobase station and the channel is infected with Rayleigh fading, then the outage probability isaround 70% in cluster based deployment, meanwhile in PPP model it is almost 90%. Whydo we have difference in the results? Because clustering (pushing small cell base stationsaround points) helps to have better interference conditions in the rest of the area. Why do wehave better results inside the clusters? No, we do not. As our investigation gives expectedvalues on outage probability, without knowing the actual position of the base stations, the


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Figure 4.4: Result for outage probabilities with different T values, ‖z‖= 100 m, m = 4

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Figure 4.5: Comparing PPP and PCP node deployment

expectation over the whole space (area) yields lower outage probability on average. While– for sure – inside the clusters the Thomas cluster process yields worse performance. Thisis a limitation of Stochastic geometry. We have to start with the assumption that the actualposition of small cells is not known in advance: they are stochastically spread in the space(according to Thomas cluster). Thus, any point in the space gives the same expectation onoutage probability.

Introducing Thomas cluster process thus skews the maximum number of small cells in acurrent macrocell. Thomas cluster process is obviously better model for network operators,than PPP: in block of houses, but even in suburbs the location of small cells will not be


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Figure 4.6: Outage probability for different thresholds and average number of Small cells,‖z‖= 100 m

uniform. Simply, Thomas cluster describes the environment in a more precise way – whileit provides better (lower) upper bounds. Thus the application of cluster process (over PPP)in small cell modelling is essential.

Figure 4.6 provides the outage probability with various SIR threshold limits (T ). Thethreshold values are given in decibel (dB). In LTE the minimum required Signal to Interfer-ence ratio for communication is −6.5dB. However, due to adaptive modulation and coding,each modulation and coding scheme has an increasing threshold.

The outage probability highly depends on the number and the size of the clusters. Themean value of the small cells in a given area depends on the average number of small cellsinside a cluster (c) and the cluster density (λp). Let us assume that the average number ofsmall cells in the given R is Ns = 500. If every cluster has only (on average) c = 5 smallcell base station, then the average number of clusters equals E

λp|R|

= 100. On the other

hand, if a cluster has (on average) c = 500 small cell base stations, then the average numberof clusters in the given area equals E

λp|R|

= 1. In the previous case we have a lot of small

clusters, meanwhile here we have only one, but a huge cluster, while the average number ofsmall cells are the same. Obviously the two cases yields completely different scenarios, thusto treat this situation let us define a new parameter:

β =c

λp · |R|.


1e−31e−2

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Figure 4.7: Outage probability for different β and average number of small cells, ‖z‖ =100 m, T = 0dB

The reader should note, that any pair of the (β , c,λp,Ns) describes the other two, if |R| isknown. Thus introducing β is just for the sake of easier notations.

Small value of β denotes the case when we have many but small clusters. Meanwhilehigh β value denotes the case when the average number of cluster heads are small, butthe average number of small cells inside a cluster is high. The results with various Ns andthreshold (T ) are represented in Figure 4.8a, Figure 4.8b and Figure 4.8c. On the verticalaxis the service outage probability is given. On the horizontal axis one finds β in logarithmicscale.

If T = 0 dB is fixed, and we are interested in the maximum number of small cells thatcould live under the macrocell such that a pre-defined outage probability threshold is notexceeded. Thus Figure 4.7 is depicted. As before, the vertical axis represents the outageprobability, while horizontal axes show the average number of small cells Ns and the β

parameter. If the mobile operator measures the β in a macrocell, this figure could be usefulto answer, how many small cells could be installed. If β = 100, and the outage probabilityshould be under 10%, then Ns = 100 is the maximum number that a macrocell could livetogether with. The reader should not forget, that we also fixed the distance between theUE and the macrocell base station to be ‖z‖= 100 m. Larger distance yields higher outageprobability.

As expected, small β values yields to PPP: when c is small relatively to λp · |R|, then


daughter points around the parent points diminish, only parent point remain (as in PPP). β

could be arbitrarily low, there is always a limit (the PPP result) which cannot be hit. That isvery visible in Figure 4.8b and Figure 4.8c where additional low values of β has been added.On the other hand, the outage probability decreases as β grows. If β tends to infinity, thenthe outage probability tends to zero. The explanation of this phenomenon comes from thefact that high β means high c compared to λp · |R|. If Ns is finite, that could yield λp · |R| totend to zero. It means that there are no parent points in the area with high probability. If thereis no parent point, then there are no daughter points and thus, there is no interference. Thelower the λp value, the smaller the chance that interfering daughter points appear (althoughthere would be many of them).


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Figure 4.8: Outage probability based on cluster size


Results for Matérn cluster Process based modelThis section provides the performance analysis for a two-tier small cell structure focuses onthe coverage probability aspects. We made Monte Carlo simulations to confirm the validityof the evaluated forms (3.34) and (4.16).

The simulation area is restricted to a finite square width |R| = 500 m× 500 m. Themacro base station is located at center of this square (assumed that this point is the origin)and transmit continuously on a power of 20 W. Small cells are emitting on a constant 20 mW.The outdoor path loss exponent α is chosen for 4, following the engineering practice. Everytier (macro and small cell) has the same fading statistics. Parameter m denotes the number ofindependent Rayleigh fading infected paths. Parameter Ki is set to 1 for the sake of simplicityand the designated user location is given by vector z, with the distance ‖z‖= 100 m.

First of all (similarly to Thomas cluster case) we would like to demonstrate that (4.16)is an accurate approximation for (macro user) service outage probability in case of Matérncluster small cell deployment. The density of parent points is given by λp and the small cellsare scattered around the parent points in the circle with radius R = 40 m. The macro UE issettled permanently ‖z‖ = 100 m from the macro base station (located at the origin). Theresults are represented in Figure 4.9a and Figure 4.9b. The vertical axis scales the serviceoutage probability from 0 to 1 obviously and the horizontal axis gives the mean number ofsmall cells in R (Ns). The blue and red markers represent the simulation results and thecorresponding results are represented with blue and red curves. It is clearly visible that theform given by (4.16) provides an accurate upper bound for service outage for both fadingtypes.

Next, we compare the PPP scenario with Matérn cluster Process. Small cells are de-ployed on R firstly according to simple PPP, afterwards Matérn cluster process (MCP).Every simulation result provided here is averaged over 104 experiments. During each exper-iment run the actual location of the macro user z, the location and number of the parent anddaughter points (i.e. the small cell locations) are random variables, only the distance (‖z‖) isfixed.

Let us validate the results of the coverage probability for both spatial deployment sce-nario. The formula and simulation results are calculated with the same input parameters(Pc,Ps,T,R, etc.). The results are presented in Figure 4.10.

On the vertical axis the coverage probability is given, on the horizontal axis scales theSIR threshold values (T ) in dB. The size of a cluster is a circle with radius R = 20 m. Wecalculate the results for two mean small cell numbers. Firstly, the mean number of smallcells is 250 for PPP (in |R| 500 m× 500 m with λ = 10−3). For MCP deployment weexpect 50 parent points (λp = 2 · 10−4) and every cluster has on average c = 5 small cells.Therefore, the expected number of small cells for both cases equals 250 (Figure 4.10a) and


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Figure 4.9: Probability of service outage in case on Matérn cluster process (T = 1)

the two case is comparable. Secondly, we increase the number to have an average 500 smallcells (Figure 4.10b). To do that we double the mean number of parent points for MCP.

The curves denote the form based results and the simulation results are given by thesame colour markers. We Calculate the coverage for Rayleigh (m = 1) and Nakagami-m


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(a) Ns = 250,c = 5

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(b) Ns = 500,c = 5

Figure 4.10: Validating results with Monte Carlo simulations

faded channels (m = 4). The simulation results, in case of PPP deployment, perfectly matchwith the curves (gained from the proposed forms) for both small cell number and fadingtype, as expected. For MCP we calculated an approximation, thus only a lower bound isgiven. The simulation results validate that this approximation provides tight lower bound forboth fading types. So far the user is settled to a fix 100 m distance and the power ratios was


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(a) Ns = 250,c = 5

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P s = 102

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P s = 103

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P s = 104

(b) Ns = 500,c = 5

Figure 4.11: Results for coverage probabilities for different distances (‖z‖) and power frac-tion

(Pc

Ps

)

Pc/Ps = 103. However, the maximum transmitted power parameter depends on the smallcell device capability. Now, let us provide the results focusing on MCP case with differentpower fractions, meanwhile the threshold value is fixed to 1 (0 dB).

4.3. RESULTS OF THROUGHPUT ENHANCEMENT 79

The results are given in Figure 4.11a (for mean number 250) and Figure 4.11b (for meannumber 500). The distances between the macro user and the macro eNB is plotted on alogarithmic scale. As usual the coverage probability is presented on the vertical axes versusthe distances ‖z‖. From the results we can conclude that if the channel is Nakagami-4 faded,then the coverage probability is higher, however the transmitting power of small cell eNBssignificantly modifies the actual coverage parameters.

Results of Throughput enhancement

In this section the average system throughput enhancement is investigated via Monte Carlosimulations. Unfortunately, there is no mathematics based method is given here (like in PPPcase), thus we have to restrict the investigation of this important system performance param-eter through simulations. The system model contains a macrocell base station and averagenumber of small cell base stations varying on a scale between 0–500. The small cells areused outdoor (no wall penetration loss is assumed) and they are in open access mode (acces-sible for anybody). The transmission power the macro and small base stations is 20 W and20 mW, respectively. The nominal system bandwidth is 20 MHz, which means there are 100PRBs per slot. Every base station applies round robin scheduling algorithm. The equipmenthave 2×2 MIMO antenna configuration. The system has 100 active mobile terminals. Themodel area is a 1000 m× 1000 m square. Every user connects to the base station that hasthe highest SIR value. The throughput value is gained from Table 3.2. The results are gainedfrom an averaged value of 104 experiments. During the simulations we assumed fast fadingfree, Rayleigh and Nakagami-m faded channels. The effect of shadowing and AWGN noiseis ignored. Figure 4.12 shows the total system throughput as a function of mean numberof small cells in case of Thomas- and Matérn cluster processes. On the horizontal axis onefinds the mean number of the number of small cells (Ns). On the vertical axis the systemthroughput is given in kbit/s, in linear scale.

Simulations are evaluated for both random process the with different scattering parame-ter (δ , R) and cluster size c. In Figure 4.12a the scattering parameter (δ ) of the cluster equals8.33, meanwhile in Figure 4.12b the scattering is δ = 16.66. The same scenario is appliedfor Matérn cluster (Figure 4.12c and Figure 4.12d). In this case the scattering parameter (R)equals 50 m and 100 m, respectively. We already mentioned that in Poisson cluster processesthe average number of small cells in a cluster (c) can remarkably modify the system param-eters (e.g. outage probability), meanwhile the mean number of small cells in the system area(Ns) remains constant. Therefore we have calculated the average throughput enhancementfor two different c values: c = 5 and c = 20. In the figures the dashed curves belongs toc = 20, meanwhile the solid lines belong to c = 5. Similarly to PPP case, results reflect the


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Figure 4.12: Throughput enhancement vs. Ns (Thomas- and Matérn cluster processes)

improvement due to the introduction of small cells. According to simulation results of thecluster based deployment the total system capacity remarkably increases with the number ofsmall cells. It is visible from Figure 4.12 that the “without fast fading” scenario providesthe flattest throughput enhancement curve. When the (average) number of small cells is lowthe (average) system throughput grows with the number of small cells. Without fast fading,due to interference, the rate of throughput gain slows down and that is the reason, while thewithout fast fading case provides the lowest throughput enhancement.

5 Conclusive remarks

The aim of this dissertation is to the develop, propose and present models and methods thatallows the fast and efficient performance evaluation of two-tier (small cell augmented) LTE-Advanced networks. The goal is to provide models that can be useful for mobile service op-erators to acquire important performance parameters, without running tediously slow MonteCarlo simulations. The thesis focused on the access network part of LTE-Advanced system.One investigated performance parameter is the service outage probability or its complementevent, the coverage probability. Another parameter investigated in the thesis, namely the av-erage system throughput. Although small cells remarkably modifies the interference map ofa given area, the application of small cells increases the total system capacity. One purposeof the thesis is to quantify the throughput gain.

The models characterized in the thesis rely on the tools of Stochastic geometry. The firstmain group of results contain a Poisson Point process (PPP) based model of two-tier smallcell networks, taking into account random location of small cells, users and the specialitiesof LTE-Advanced systems. PPP model can be interpreted as a snapshot view of the accessnetwork, where the actual location of the small cells remains random. However, due toStochastic geometry the model remains traceable. In the PPP model firstly the interferencedistribution was analysed using the fundamentals of Stochastic geometry. The distribution ofthe small cell generated interference has a cumulative distribution function (c.d.f) and prob-ability density function (p.d.f), furthermore this distribution is the famous Lévy distribution.Another interesting finding of the thesis, that the distribution remains Lévy distribution, re-gardless of the fast fading type in the model. Therefore mobile operators can easily estimatethe small cell originated cumulative interference level at a given area, only the proper pa-rameters should be substitute the forms given in this thesis. The accuracy of the proposedc.d.f and p.d.f forms are tested against Monte Carlo simulations and the empirical results ofthe simulations proofed that the interference distribution is Lévy distribution, for every in-

81

82 CHAPTER 5. CONCLUSIVE REMARKS

vestigated fast fading type. The thesis analysed the service outage probability (and of coursethe coverage probability) for multiple fast fading types i.e. Rayleigh-, Nakagami-m and Ricefading. The thesis proposed two ways in order to characterize the probability service outage.The fist path was the Lévy-based method, where we applied the Interference Lévy distribu-tion and convert it to service outage probability. Another path was Probability GeneratingFunctional (PGFL) based method. The main advantage of the PGFL based method is norestriction is needed (compared to the Lévy based method), mobile operators can use theseforms only the parameters should be substituted. The results of Monte Carlo simulationsvalidated the forms of service outage in PPP model. This solution makes the performanceanalysis not only possible, but fast as well. In contrast, we get only approximate results incase of Rice fading, due to the complex nature of this fading type. Despite of the evaluatedform for Rice distribution is only an approximated form, the simulations showed that it isquite accurate. Next, the total system throughput enhancement was investigated. Accordingto the investigation of the dissertation, with the usage of few dozen small cells ∼ 3.2× moresystem level throughput can be available compared to one-tier macro network and the systemlevel capacity grows with the number of installed small cells. In the second research groupof thesis the author introduced two Poisson cluster based models, that can be useful in smallcell modelling. These are namely the Thomas- and Matérn cluster processes. With thesemodels mobile service providers can model realistic small cell area maps, where the den-sity of small cells are not homogeneous, therefore there some locations with densely smallcell deployment, meanwhile other areas are small cell free. The interference distributionand the throughput enhancement in this cluster based deployment were investigated throughsimulations.The service outage probability is investigated with models, thanks to Stochasticgeometry. Due to the complexity we get only approximate results. However, as shown bythe simulation results, that were obtained for the same system as well, the accuracy of theapproximation is reasonable. It is also shown how outage probability can be used to evaluatethe maximum number of small cells in a macrocell, assuming Poisson cluster distributedsmall cell base station locations. Compared to previous studies, where the small cells aredeployed according to Poisson Point process, this cluster based model introduces a moreaccurate model for two-tier environments. As shown in the thesis, the PCP model yieldsbetter upper bounds on the outage probability while it describes the system in a more accu-rate way. Since all parameters (e.g. variance of small cell scattering, emitted powers, pathloss parameters etc.) in the proposed model is tunable, the proposed model can be directlyapplied according to the needs of the user (the mobile operator), to determine the maximumsmall cells which could fit in an existing macrocell environment.

5.1. FURTHER WORKS 83

Further WorksAs possible future work, the author sees the following possible research directions. Onepossible research direction is to investigate the service outage for other fast fading typessuch as Rician or Weibull faded channels in the cluster models. The effect of shadowingand Additive white Gaussian noise (AWGN) can be included to the model next to fast fadingchannels. With this extension the model can be more accurate. The provided for forms forPoisson cluster models – given in (4.13) and (4.16) – are approximations. It is worth to inves-tigated mathematically, if they are upper- or lower bounds. For example a lower bound forcoverage probability should be a better performance parameter instead of an approximatedform. Although the simulations (provided in this thesis) showed that in the cluster modelsthe approximated forms are lower bound for coverage probability and upper bound for out-age probability. Other Poisson cluster models can be involved in the analysis provided bythe dissertation as a small cell location model. There are numerous, although rather difficultNeyman-Scott process based models that can be applied for small cell modelling such as e.g.Cox process.

The main findings and the curves proposed in this thesis i.e. the probability of serviceoutage and average throughput calculation can be compared with empirical, measured resultsgained from mobile service providers in order to compare the accuracy. Evidently, thisimplies, that mobile service providers have to have a test field where the small cells aredensely installed.

The proposed forms in this thesis are related to a two-tier (small cell based) LTE-Advanced system. However the model can be extended in order to model Device-to-device(D2D) communication as well. D2D communication is one of the most important feature ofa 5G system, thus this Stochastic geometry based model can be applied not only in today’smobile communication networks, but in the next-generation networks.

Another possible research direction is to extend the scope of stochastic geometry basedperformance analysis and investigate other important system performance parameters for ex-ample handover failures. Due to the specific nature of a small cell network (i.e. low coveragearea, limited simultaneous connections etc.) conventional handover algorithms might be notsufficient. Stochastic geometry allows to use the proposed Poisson based models (PPP orPCP) in order to evaluate the mean number of handovers, failure of handovers, sojourn timeetc. Of course in this case the snap shot view of the system is not enough, the mobility of themobile users should be included as well.

84 CHAPTER 5. CONCLUSIVE REMARKS

A Appendix

Transforming Random Variables

Exponential distributionLet us assume that a random variable X follows Rayleigh distribution with the followingprobability density function, denoted by fX(x,ω) (where x > 0 and ω > 0, evidently):

fX(x,ω) =2xω

e(−x2/ω).

We would like to transform random variable X ( fX(x,ω)) to H ( fH(h,ω)). The transforma-tion H = X2 is a 1-1 transformation, thus the inverse is x =

√h. Applying

dXdH

=1

2√

h,

to finish the transformation of the random variable yields:

fH(h,ω) = fX(√

h,ω) · dxdh

=2√

hω· e

(−√

h2ω

)· 1

2√

h=

1ω· e(

−hω ), (A.1)

which is the probability density function of the exponential distribution.

Gamma distribution

Firstly, we apply the definition of the mean value to evaluate E√

h invoking the p.d.f ofthe gamma random variable given in (2.21):

E√

h=∫

∞

0

√h f (h)dh =

∫∞

0

√h

hm−1e−mh

1mm Γ(m)

dh. (A.2)

85

86 APPENDIX A. APPENDIX

Multiplying out the constant parts from the integral and using the the following form [61,3.326.2, eq. 2]:

∫∞

0 xae−βxndx = Γ( a+1

n )

nβa+1

n[Reβ > 0,Rea > 0,Ren > 0] ,

the factional moment simplified to the following convenient fraction:

E√

h=Γ(1

2 +m)

Γ(m) ·m 12. (A.3)

Furthermore the Nakagami fading parameter m is a positive integer number, thus using theproperties of the Gamma function Γ(m)= (m−1)! and Γ(1/2+m)= (2m−1)!!

2m

√π we noticed

that the factional moment equals:

E√

h= (2m−1)!!2m(m−1)!

√m√

π. (A.4)

Rice distributionLet us assume that random variable X follows Rice distribution within the probability densityfunction (PDF) of:

f (x) =2(K +1)x

Ωexp(−K− (K +1)x2

Ω

)I0

(2

√K (K +1)

Ωx

).

The total power gleaned is denoted by Ω. We should calculate h = x2

Ωand with the aid of

dXdh

=

√Ω

2√

h,

the p.d.f transforms to:

f (h) = (K +1)e(−K−(K+1)h) · I0

(2√

K(K +1)h). (A.5)

Lévy distributionAccording to Theorem 3.1. the cumulative small cell interference follows Lévy distributionif the small cell locations are modelled with a homogeneous Poisson point process and the

A.2. LÉVY DISTRIBUTION 87

path loss exponent (α) equals four. This section provides the proof for Theorem 3.1. Notethat the proof given in from [29, Section 3.2.3].

First we map the two-dimensional PPP to the distances of the points (from the origin)of a two-dimensional uniform PPP of intensity λ . The mapped process is denoted withΦ ∈ ri = ‖xi‖. Therefore λ ′(r) = λπdrd−1 where d is the dimension of the process, thusd = 2. For the sake of simplicity we choose K−1

i = 1. The Laplace form of the cumulativeinterference is:

LI(s) = EΦ

∏r∈Φ

Eh

exp(−sPshr−α

). (A.6)

Relying on the connection between the Laplace form of the interference and the PGFL wecan write the following:

LI(s) = exp−λ′(r)

∫R

[1−Eh

exp(−sPshr−α

)]dr. (A.7)

Now we flip the order of integration and expectation:

LI(s) = exp−

Eh

∫R

[1− exp

(−sPshr−α

)]λ′(r)dr︸︷︷︸

A

. (A.8)

Substituting g = rα into integral "A" yields:∫R

[1− exp

(−sPshr−α

)]λ′(r)dr = λπ

∫R

[1− exp(−sPsh/g)

](2α

)r

2α−1 dr. (A.9)

To calculate this integral, we note that it is the expected value of an exponential distributedrandom variable g with mean value 1:

E((g/shPs)−1)

2α

.

The expected value can be rewritten as:

E((g/shPs)−1)

2α

= (sPsh)

2α Γ

(1−(

2α

)),

where Γ(·) is the gamma function. The Laplace form of the interference modifies to:

LI(s) = exp−λπE

h

2α

(sPs)

2α Γ

(1− 2

α

). (A.10)


which corresponds with (3.4). A Lévy distributed random variable has the following charac-teristic function:

φ(t) = exp−iµt−

√−2ict

, (A.11)

where µ is location parameter equals 0, meanwhile the scale parameter (c) is given in (3.7).Substituting ”s =−it” and α = 4 into (A.10) yields (A.11) as expected.

Calculation of Raw moments

Normal distribution

If y follows normal distribution, then E(|| x+y− z ||2

)2

equals to the second raw mo-ment of a non-central chi-squared random variable (denoted by U), if the mean value of U is2δ 2 +(|| x− z ||)2 and the variance is 4δ 4 +4δ 2 (|| x− z ||)2. According to the definition ofraw moment: E

U2= E

(|| x+y− z ||2

)2

.It is known that if γ is a Gaussian distributed random variable, then its moment can be

written with the following form:

E [γ p] =

0 if p is oddδ p (p−1)!! if p is even,

(A.12)

where δ denotes the variance and n!! is the double factorial.According to the definition of the second order raw moment of a non-central chi-squared

random variable:E

U2= (EU)2 +VarU . (A.13)

For the sake of simplicity let us use the following notations:

|| x+y− z ||=√(y1− x′1)

2 +(y2− x′2)2,

where

x′ = (x− z) =(

x′1x′2

)=

(x1− z1x2− z2

).

Thus(|| x+y− z ||2

)2 equals to:(|| x+y− z ||2

)2=[(y1− x′1)

2 +(y2− x′2)2]2 =

= (y1− x′1)4 +2 · (y1− x′1)

2(y2− x′2)2 +(y2− x′2)

4 (A.14)

A.3. CALCULATION OF RAW MOMENTS 89

From algebraic handbooks we know that

(a−b)4 = a4−4a3b+6a2b2−4ab3 +b4.

Therefore substitution to (A.14) yields(|| x+y− z ||2

)2= y4

1−4y31x′1 +6y2

1x′21 −4y1x′21 + x′41 +

+2y21y2

2−4y1y2x′2 +2y21x′22 −4y1y2

2x′1 +8y1y2x′1x′2−4y1x′1x′22 +2y22x′21 −

−4x′1x′22 y2 +2x′21 x′22 + y42−4y3

2x′2 +6y22x′22 −4yx′22 −4yx′32 + x′42 .

(A.15)

The expected value operator is a linear operator, thus we can calculate it on every partindividually. With the aid of (A.12) we can calculate the moments. The parts with odd ordergives 0, thus (A.15) modifies to:

E(|| x+y− z ||2

)2= δ

4(3!!)+6δ2(1!!)x′21 + x′41 +2δ

4(1!!)+2δ2(1!!)x′22 +

+2δ2(1!!)x′21 +2x′21 x′22 +δ

4(3!!)+6δ2(1!!)x′22 + x′42 =

= 8δ4 +8δ

2(x′21 + x′22 )+ x′41 + x′42 +2x′21 x′22 .

(A.16)

Now substituting the mean value and variance of U :

EU= 2δ2 +(|| x− z ||)2 ,

VarU= 4δ4 +4δ

2 (|| x− z ||)2

into (A.13) yields (A.16) and concludes the proof:

E

U2= (2δ2+ || x′ ||2

)2+4δ

4 +4δ2 (|| x′ ||)2

=

=(2δ

2 + x′21 + x′22)2

+4δ4 +4δ

2(x′21 + x′22 ) =

= 4δ4 +4δ

2(x′21 + x′22 )+(x′21 + x′22 )2 +4δ

4 +4δ2(x′21 + x′22 ) =

= 8δ4 +8δ

2(x′21 + x′22 )+ x′41 + x′42 +2x′21 x′22 .

(A.17)


Uniform distributionFor the sake of simplicity let us use the following notations:

(‖(x− z)+y‖2)2

=

((x1− z1)︸︷︷︸a

+y1

)2+((x2− z2)︸︷︷︸

b

+y2

)2

2

.

Let us change to Polar coordinates, then y1 = r · cos(ϕ) and y2 = r · sin(ϕ). Now we cancalculate the expected value with the p.d.f fR(r) = 2r/R2:

E(|| x+y− z ||2

)2=∫ R

0

[(a+ r · cos(ϕ)

)2+(

b+ r · sin(ϕ))2]2

· fR(r)dr. (A.18)

Now calculating the inner squaring (power of two) and after substituting the p.d.f, the integralmodifies to: ∫ R

0

(a2 +b2 + r2 +2r

(acos(ϕ)+bsin(ϕ)

))2

· 2rR2 dr.

Finally, after some algebraic manipulation and solving the definite integral yields,

E(|| x+y− z ||2

)2= ‖x− z‖4 +2R2‖x− z‖2 +

25

R2. (A.19)

A.4. RELATIVE ERRORS 91

Relative errors

This section shows the relative error between the simulation results and the results proposedin this thesis. Relative errors are calculated for the cumulative interference distribution andthe service outage probabilities for PPP and PCP scenarios. Figure A.1a shows the relativeerror between the Lévy distribution c.d.f (given in (3.6)) and the empirical c.d.f. (gainedfrom simulations). Figure A.1b illustrates the relative error for the service outage probabilityfor PPP environment Nakagami-m and Rician faded cases. Finally, Figure A.2 illustrates therelative error for the service outage probability for Thomas- and Matérn cluster environmentsRayleigh and Nakagami-m faded cases.

10-8

10-6

10-4

10-2

I [W]

-0.035

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

Rel

ativ

e er

ror

Rayleigh fading

Nakagami-m fading (m=4)

Rice fading (K=0.5)

(a) PPP model Interference distribution relative error

5 10 15 20 25 30 35 40 45 50

NS

-0.1

-0.05

0

0.05

0.1

Re

lative

err

or

PGFL based method Relative error (Nakagami-4 fading)

Lévy based method Relative error (Nakagami-4 fading)

PGFL based method Relative error (Rice fading K=0.5)

(b) PPP model Nakagami-4 and Rice fading relative error

Figure A.1: Validating Poisson point process interference and outage probability results withsimulations (Relative error)


1 10 20 30 40 50

Ns

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

Rela

tive e

rror

Rayleigh fading c = 5Rayleigh fading c = 10Nakagami-m fading (m=4) c = 5Nakagami-m fading (m=4) c = 10

(a) Thomas cluster Rayleigh and Nakagami-4 fading relative error

1 10 20 30 40 50N

s

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

Rela

tive e

rror

Rayleigh fading c = 5Rayleigh fading c = 10Nakagami-m fading (m=4) c = 5Nakagami-m fading (m=4) c = 10

(b) Matérn cluster Rayleigh and Nakagami-4 fading relative error

Figure A.2: Validating Poisson cluster process outage probability results with simulations(Relative error)

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List of Publications

Published or accepted peer-reviewed journals:

[J1] Zoltán Jakó and Gábor Jeney, “Analyzing LTE Small Cell Interference Distribution viaFading Factorial Moments”, GSTF Journal of Mathematics, Statistics and OperationsResearch (JMSOR), Vol. 2 No. 2, pp. 36–40, 2014, DOI: 10.5176/2251-3388-2.2.49,Print ISSN: 2251-3388, E-periodical: 2251-3396.

[J2] Zoltán Jakó and Gábor Jeney “Outage Probability in Poisson cluster based LTE Two-tier Femtocell Networks”, Wiley Wireless Communications and Mobile Computing,DOI: 10.1002/wcm.2485, 2014, (’Early View’ on Wiley Online library),(indexed in SCOPUS, WoS).

[J3] Zoltán Jakó and Gábor Jeney “Outage Analysis of LTE-A Femtocell Networks withNakagami-m Channels”, Springer Wireless Personal Communications, Vol. 79 No. 2,pp. 1369–1384, 2014, DOI: 10.1007/s11277-014-1934-5(indexed in SCOPUS, WoS).

[J4] Zoltán Jakó and Gábor Jeney, “Coverage Analysis for Macro Users in Two-Tier Ri-cian faded LTE/Small-Cell Networks", Springer Wireless Networks, Vol. 21, no. 7,pp. 2293–2302 , (indexed in SCOPUS, WoS).

Published or accepted Hungarian journals:

[J5] Zoltán Jakó and Gábor Jeney, “3G-s femtocellák interferencia vizsgálata”,HÍRADÁSTECHNIKA (ISSN: 0018-2028) LXVI:(2) pp. 2–9. (2011)

International Conference articles:

[C1] Zoltán Jakó and Gábor Jeney, “Downlink femtocell interference in WCDMA net-works", 17th International Workshop in Energy-Aware Communications (Eunice2011), Heidelberg: Springer, pp. 203–208. (Lecture Notes in Computer Science;

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6955.), 2011.09.05 - 2011.09.07. ISBN: 978-3-642-23540-5, DOI: 10.1007/978-3-642-23541-2-22

[C2] Zoltán Jakó and Gábor Jeney, “Downlink interference characterization in two-tierfemtocell networks", 8th IEEE, IET Int. Symposium on Communication Systems,Networks and Digital Signal Processing (CSNDSP 2012), Poznan, Poland,2012.07.18 - 2012.07.20, DOI: 10.1109/CSNDSP.2012.6292697

[C3] Ádám Knapp, Sándor Imre, Zoltán Jakó and Gábor Jeney, “Average System Capacityin a Two-tier LTE Environment with Random Waypoint Mobile Users", 18th IEEEInternational Conference on Networks (ICON 2012), Singapore,2012.12.12 - 2012.12.14. DOI: 10.1109/ICON.2012.6506530

[C4] Ádám Knapp, Sándor Imre, Zoltán Jakó and Gábor Jeney, “Application of LTE SmallCells in Urban Environments for Higher Capacity", 19th IEEE International Confer-ence on Networks (ICON 2013), Singapore,2013.12.11 – 2013.12.13. DOI: 10.1109/ICON.2013.6781965

[C5] Zoltán Jakó and Gábor Jeney, “Coverage Analysis of Matérn cluster based LTE SmallCell Networks," 8th International Conference on Next Generation Mobile Apps, Ser-vices and Technologies (NGMAST 2014), Oxford, United Kingdom,2014.09.10 - 2014.09.12, pp. 229-234. DOI: 10.1109/NGMAST.2014.16

Hungarian Conference articles:

[C6] Jakó Zoltán és Jeney Gábor, “Femtocellák alkalmazása LTE rendszerekben”,Mesterpróba Konferencia 2012. Budapest, Magyarország, pp. 37–41, 2012.05.23.

Other publications (not related to thesises)

[C7] Jakó Zoltán és Knapp Ádám, “Hálózati térkép alapú hívásátadási eljárás kétréteguLTE-ben”, Doktoranduszok Országos Szövetsége Tavaszi Szél konferencia (2014),Debrecen, Magyarország, 2014.03.21 - 2014.03.23

[C8] Fatemeh Bardestani, Zoltán Jakó, Ádám Knapp, Sándor Imre and Péter Fazekas, “Im-proving Wireless Network Throughput with Constant Dimension Subspace Codes", 9thIEEE, IET Int. Symposium on Communication Systems, Networks and Digital SignalProcessing (CSNDSP 2014), Manchester, United Kingdom, 2014.07.23 - 2014.07.25

BIBLIOGRAPHY 103

[J6] Gyozo Gódor, Zoltán Jakó, Ádám Knapp and Sándor Imre, “A Survey of HandoverManagement in LTE-based Multi-tier Femtocell Networks: Requirements, Challengesand Solutions”, Elsevier Computer Networks, Vol. 76, No. C, pp. 17–41, 2015,(indexed in WoS).

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