S TOCHASTIC G EOMETRY S MALL C ELL N ETWORKSWITH M ODELLING T WO - TIER LTE-A DVANCED

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B

UDAPEST

U

NIVERSITY OF

T

ECHNOLOGY AND

E

CONOMICS

Faculty of Electrical Engineering and Informatics

Department of Networked Systems and Services

Mobile Communications and Quantum Technologies Laboratory

M ODELLING T WO - TIER LTE-A DVANCED

S MALL C ELL N ETWORKS WITH S TOCHASTIC G EOMETRY

Ph. D. Thesis of of

Zoltán Jakó

Scientific supervisor:

Gábor Jeney, PhD.

Budapest, 2017

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c 2017, All rights reserved to the author

This document was typeset in L

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TEX 2ε applying Adobe Times font family

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i Nyilatkozat AlulírottJakó Zoltánkijelentem, hogy ezt a doktori értekezést magam készí- tettem és abban csak a megadott forrásokat használtam fel. Minden olyan részt, amelyet szó szerint, vagy azonos tartalomban, de átfogalmazva más forrásból átvettem, egyértelm˝uen, a forrás megadásával megjelöltem.

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

Budapest, Hungary, January 30, 2017

. . . . Jakó Zoltán

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Abstract

The amount of speech and data communication proceeded through mobile network is increasing rapidly and it is foreseen that this growing tendency will be continuous in the future. In LTE-Advanced (LTE-A) the conventional access network (formed by Evolved Node-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 macro cell structure should be augmented with smaller cells. The literature applies the term“Small cells”for smaller cells underlaid to an umbrella macro cell. These small cells operate on the same spectra, as the macro eNBs, however with a reduced transmission power. Due to the reduced transmission power, small cells can cover a smaller area (compared to macro eNBs). The potential deployment places of small cells are indoor (e.g. buildings) or densely populated 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 provided byStochastic geometry. The usage of small cells can increase the coverage areas and the system throughput of the mobile access network. Nevertheless, the deployment of small cells modifies the interference pattern of the given area significantly, which also modifies the probability of coverage (or the probability of service outage). The dissertation represents how to create models for these multi-layer networks and calculate mathematically (with the aid of Stochastic geometry) important and essential system parameters such as the probability of outage/coverage and the overall system throughput. Simulations are provided in the dissertation in order to validate the correctness and accuracy of the proposed mathematics forms.

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Kivonat

A mobil hálózaton keresztül lebonyolított beszéd és adatforgalom is egyre jobban növekszik és az el˝orejelzések szerint ez a növekedési tendencia folytatódik a közeljöv˝oben 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 min˝oségét, ilyen fokozott adatforgalom mellett. Ezért a makró cellák tehermentesí- tésére kisebb cellákat is üzembeállítanak, hogy a megfelel˝o min˝oség˝u szolgáltatás biztosított legyen. 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 meglév˝o makró hozzáférési hálózat nem elegend˝o: pl. épületekbe (femtocellákat), s˝ur˝un lakott városrészek- re (pikó- és mikró cellákat). A szakirodalom„kis cellák (small cells)” gy˝ujt˝onévvel illeti ezeket a kis bázisállomásokat.

Jelen Ph. D. disszertáció a LTE-Advanced hozzáférési hálózat „kis cellákkal” történ˝o ki- b˝ovítésének lehet˝oségét és annak hatásait vizsgálja. A vizsgálatokhoz felhasználom aszto- chasztikus geometria átal biztosított matematikai eszköztárat. A kis cellák alkalmazása je- lent˝osen megnöveli a mobil hálózat lefedettségét és ezáltal a mobil felhasználók adatátviteli sebességét is. Ugyanakkor módosíthatja az adott terület interferencia térképét, ami pedig befolyásolja a szolgáltatás kiesés- és a lefedettség valószín˝uséget is.

A disszertációban bemutatom, hogy lehet a sztochasztikus geometria által biztosított matematika apparátussal, modellezni kétréteg˝u LTE-A kis cellás hálózatokat és kiszámolni olyan paramétereket (pl. szolgáltatás kiesés/ lefedettség valószín˝usége, rendszerszint˝u 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 formulák helyességét és érvényességét.

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Köszönetnyilvánítás

Mindenekel˝ott szeretnék köszönetet mondani konzulensemnekDr. Jeney Gábornak. Hasz- nos tanácsai, ötletei és kritikái nélkülözhetetlen segítséget nyújtott Ph. D. hallgatóként vég- zett kutatómunkámban és disszertációm elkészítése során egyaránt. Neki tartozom azért is köszönettel, hogy színvonalas nemzetközi projektben foglalkozhattam izgalmas kutatá- si feladatokkal, és szakmai fejl˝odésemhez minden feltételt megteremtett. Különösen hálás vagyok szerz˝otársaimnak és kollégáimnak a közös munkáért és publikációkért. Végül, de nem utolsósorban szeretném megköszönni a Mobil Kommunikáció és Kvantumtechnológiák Laboratórium (MCL) tagjainak a kiváló légkört, amelyet biztosítottak számomra az eredmé- nyes munkához.

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List of Figures

1.1 Downtown LTE-A network extended with Small cells . . . 3

1.2 3GPP LTE-Advanced Two-tier Small cell architecture [1] . . . 4

2.1 An LTE-A Two-tier small cell network . . . 12

2.2 Illustration of Homogeneous Poisson point process . . . 15

2.3 PPP based Random graph model . . . 16

2.4 cluster processes as Small cell deployment modelling . . . 17

2.5 Illustration of Cross-tier and Co-tier interference and PRB assignment . . . 26

3.1 Probability density functions for different fast fadings . . . 32

3.2 Validation of the proposed forms with Monte-Carlo simulations . . . 34

3.3 The outage probabilities for Lognormal faded channel . . . 45

3.4 Probability of outage in Rayleigh, Nakagami-mfading . . . 47

3.5 Probability of Coverage with Rayleigh, Nakagami-mfading types and path loss exponents (α) . . . 48

3.6 Probability of outage in Rician fading . . . 49

3.7 Probability of Coverage with Rician fading with various path loss exponents (α) . . . 50

3.8 Probability of coverage for macro UE vs. ^P_P^cs . . . 51

3.9 Throughput enhancement with Small Cells . . . 56

4.1 Empirical c.d.f. results from Monte-Carlo simulations . . . 61

4.2 Outage probability in Poisson Point process and Poisson cluster Process cases with different nakagami-mfading . . . 68

4.3 Result for outage probabilities vs. differentγ andcvalues,kzk=100 m . . 69

4.4 Result for outage probabilities with differentT values,kzk=100 m,m=4 70 4.5 Comparing PPP and PCP node deployment . . . 70

4.6 Outage probability for different thresholds and average number of Small cells,kzk=100 m . . . 71

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xiv LIST OF FIGURES 4.7 Outage probability for differentβ and average number of small cells,kzk=

100 m,T =0dB . . . 72 4.8 Outage probability based on cluster size . . . 74 4.9 Probability of service outage in case on Matérn cluster process (T =1) . . . 76 4.10 Validating results with Monte Carlo simulations . . . 77 4.11 Results for coverage probabilities for different distances (kzk) and power

fraction

P^c P^s

. . . 78 4.12 Throughput enhancement vs. N_s (Thomas- and Matérn cluster processes) . 80 A.1 Validating Poisson point process interference and outage probability results

with simulations (Relative error) . . . 91 A.2 Validating Poisson cluster process outage probability results with simula-

tions (Relative error) . . . 92

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List of Tables

1 Common Variables and Symbols . . . xix 3.1 E{√

h}Fading moments . . . 31 3.2 Throughputs used for Average System capacity calculation in MIMO 2×2

configuration . . . 53

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xvi LIST OF TABLES

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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

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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

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LIST OF TABLES xix Table 1: Common Variables and Symbols

Name Description

P^c Emitted power of Macro cell eNB (in Watt) P^s Emitted power of small cell eNB (in Watt)

z,kzk vector of the designated UE, distance of the designated UE from origin (in meter)

x,kx−zk vector of the small cell, distance between designated user and small cell h,h_c,h_x,h_x+y fast fading, independent and identically distributed random variable g(z) path loss at the given locationz

Ψ^[dB]_log shadowing (log-normal fading) component in dB K_i Constant parameters from path loss model λ density parameter of Poisson Point process N_s Mean value of small cells

N_c Actual number of users I(z) Interference atz

U_i number of users in theith small cell

λ_p density of parent points in Poisson cluster process

Φp,Φ set of parent points in Poisson cluster process, set of small cells c mean value of the daughter points in Poisson cluster process δ² Variance of symmetrical normal distribution

R Radius of the cluster in Matérn cluster process R_s Radius of small cell coverage

K Parameter of Rician fading P^!₀(·) reduced Palm distribution P0(·) Palm distribution

GN(·) Probability Generating Functional

Gs(·) Poisson distribution’s moment-generating function f(·) probability density function

Θc PRBs assigned by macro eNB

Θ_i the set of PRBs assigned by thei^th small cell base station E{·} Operator of expected value

L{·} Operator of Laplace transform erf(·) Gauss error function

erfc(·) complementary Gauss error function IP{·} Operator of probability

IP{cov} Probability of coverage IP{out} Probability of outage

T Threshold value

Γ(·) (complete) Gamma function Γ(·,·) Incomplete lower Gamma function γ(·) Incomplete upper Gamma function

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xx LIST OF TABLES

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1 Introduction

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

Nevertheless, evolution and research process are incessantly runs in background, in order to make faster and better networks (based on principals of LTE). The updated version of LTE is the so-called LTE-Advanced (LTE-A) system. According to the literature LTE-A is 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 standardization process is not closed yet. Besides, state-of-art research tasks focus on creating and specifying a new system, that fulfils the rigorous requirements (i.e. for throughput, latency etc.), that are already given for the next generation (so-called 5G) networks. The appearance of the first 5G capable networks are expected around 2020 [2].

One the mainstream research direction focuses on throughput enhancement (based on increasing the level of coverage) [3]. The importance of coverage becomes a rather important issue, relying on the fact that the carrier frequency (and the frequency band also) used for communication shifts for higher and higher values. For example the carrier frequency in 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 increasing rapidly. Therefore, the demand for mobile network services (especially high data rate or low latency services – such as data transfer, media stream) increase extremely. Due to the carrier frequency shifting phenomenon and the rapidly increasing user population, the size of

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2 CHAPTER 1. INTRODUCTION the cells, (which is the access point for mobile devices) are reducing. In second generation GSM system the typical cell radius was around∼30 km, nowadays in a densely populated city 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 aspects, 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. The portion 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 services require sufficient throughput and low latency supported by the network. Therefore, major research efforts in next-generation wireless networks are focusing on enhancement of spectral efficiency and throughput.

An essential solution to enhance the coverage and throughput is to extend the conventional, “one-tier” macro cell structure to a multi-tier architecture. The macrocell layer can be augmented with second (underlaid) tier. This second tier is formed by the combination of several micro-, pico- and femtocells. Micro- and picocells are already widely used in densely populated areas, such as shopping malls, crowded public transport stations etc. This structure is 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 with smaller domestic and/or customer premises cells in order to satisfy user demand in densely populated areas [6]. Therefore the access network becomes heterogeneous, constituted by at least two overlapping layers, namely an over-sailing macrocellular and an underlaid small cell layer. Small cells can be created either indoors (e.g. femtocells) or outdoors (e.g. pico- or microcells) and in their most radical incarnation they operate within the same spectral band as the conventional macrocellular base station (macro eNB). However their coverage area is limited. This heterogeneous network (HetNet) structure offers benefits for both the users and for the mobile operators. The ultimate goal is to bring the eNBs/radio ports closer to the potential users for the sake of improving the attainable reception quality. This potentially facilitates supporting higher data rates. Further benefit is that – provided the majority of the high-rate users is off-loaded to the small cells – the macrocellular eNB becomes capable of supporting additional users. However, using small cells in high-velocity outdoor scenarios would 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 are installed by users, similarly to Wi-fi hotspots [7]. Femtocells are a promising cost efficient solution 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 users data. They are usually deployed in a flat or an office in order to enhance indoor coverage and provide better QoE (Quality of Experience) to local users, by bringing the base station closer to the user. Home eNB concept differs from the previously used micro- or picocell

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

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

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

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

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

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

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4 CHAPTER 1. INTRODUCTION

S1 S1

X2

X2 X2

HeNB HeNB

S1

S1 S

1

HeNB S1 S1

1S

S1

X2

X2 X2

X2 MME / S-GW

HeNB GW

MME / S-GW MME / S-GW S1

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 the covered area scales up to a few 10 meters.

It is foreseen that these outdoor cells are deployed to for example lamp- or utility posts similarly given in Figure 1.1. In this case the data can be delivered to mobile operator for example via power line communication technologies. Due to the lower distance to potential users 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 as close as possible to users, that can guarantee the good coverage (and high data rate). The augmented, two-tier LTE-A access network is illustrated in Figure 1.2. The literature uses the 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 primary access points [6, 9]. According to [10] small cell installation is driven by the following motivations:

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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 location information.

By the contrast Wi-fi hotspots are operating in ISM-bands (industrial, scientific and medical), meanwhile small cells are operating on licensed spectra. This is exactly the same spectra that the mobile service provider operates the macro e-NodeBs (in case of the frequency reuse is one). Thanks to the commonly used frequency band(s) the small cell users (users served by 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 small cells are interference sources. Due to the ad-hoc nature of femtocells and public Internet backhaul 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 by the users, thus the location and the “uptime” is also non deterministic and mobile operators do not have influence on it. One possible solution is to “upgrade” femtocells capabilities. In other words, femtocells have to have some kind of interference detection and mitigation techniques. For example the interference detector monitors the available (E-UTRA) frequency bands, that can be allocated to potential users (for communication purpose), and in case of high 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 station does not use an amount of resources for communication. These resources are reserved for the small cells, thus for a small cell user the effect of macrocells interference can be reduced. However this dissertation investigates a worst case scenario, a heavily loaded system, where all resources are allocated continuously. Therefore all resources have been allocated independently from the interference.

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

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

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6 CHAPTER 1. INTRODUCTION 2. Highly dense deployment: with high dense of small cells (e.g. hundreds of Small cells

in a macrocell),

• the interference level increases remarkably, thus interference management required.

• 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 identifier collision generates problem in handover phase i.e. during hand-in process the source eNB can not identify the target small cell correctly.

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

4. Backhaul route: hard handover generates a short interrupt in the communication, while the handover process to the target base station is completed. This interruption time mostly depends on Round Trip Time (RTT) of the message exchange between the small cell and the gateway or the EPC (Evolved Packet Core), which is generally varies 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 calculated the probability of a randomly thrown coin hits an edge of a regular mosaic paving on the floor (1733) [14]. Another phenomenon related to Stochastic geometry is the so-called shot noise. 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 cumulative noise in a vacuum tube has similar properties, then the effect of multiple gun shoots fired in different time periods. Campbell³ studied shot noise process and characterized its mean and variance. In his work, Campbell presents the moments and generating functions of the random sum of a Poisson process on the real line, but remarks that the main mathematical argument was due to G. H. Hardy⁴. Therefore the literature often refers the theorem as Campbell-Hardy theorem [16]. This theorem is the often used in probability theory and

1French mathematician, later known as Comtr de Buffon

2Walter Hermann Schottky (23 July 1886–4 March 1976): German physicist

3Norman Robert Campbell (1880–1949), English physicist

4Godfrey Harold Hardy (1877–1947), English mathematician

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1.1. BACKGROUND OF THE RESEARCH 7 Stochastic geometry. Rice in his work in [18] investigated the effect of random noise based on Campbells’ theorem. Later, Gilbert and Pollak [19] investigated the amplitude distribution of shot-noise process. Lowen and Teich [20] showed that power law base shot noise distribution 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 and Hatzinakos considered effect of fading/shadowing on interference characteristics [22].

In the last decade Stochastic geometry gain higher attention in communication network analysis. The modelling of wireless systems (relying on Stochastic geometry) has been lavishly documented [23]–[49]. Stochastic geometry is a popular investigation tool for interference characterization and outage probability analysis in mobile Ad hoc networks. Sev- eral books had been written dealing with Stochastic geometry. Among these books stands out [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 hoc and calculated several network performance parameters with the aid of Stochastic geometry.

Kim et al. in [23] investigate the outage probability in third generation femtocell networks with Poisson point process (PPP) node deployment assuming lognormal fading channels. Chenet al.in [24] gives an outage analysis for two-tier small cell networks comparing the grid structure and PPP (random) deployment also with the assumption of lognormal fading channels. Wanget al. in [25] models a PPP based two-tier femtocell connection oriented deployment.

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] Andrewset al. investigate service outage and transmission capacity for mobile cellular systems assuming spatial homogeneous PPP structure. In [30] an elegant formula is given for calculating the coverage probability in a single-tier macrocellular structure using Stochastic geometry.

This is achieved by modelling the location of the macrocellular eNBs by a 2Dhomogeneous Poisson Point Process. In [31] the results of [30] are extended to multi-tier networks subject to Rayleigh fading, where the interference power decays exponentially and the eNBs/nodes of each tier obey the PPP. Interference and service analysis for third generation mobile networks are found in [35] and [36]. Leeet al. in [42] modelled a cognitive radio network with Poisson cluster based transmitters. Nonetheless, this cluster based node deployment is suitable for modelling small cell networks in urban environment, where the base station installation depends on building and road structure. This dissertation dedicates a whole chapter for Poisson cluster modelling. Author of [43] deduces a form that allows to calculate the probability of an UE can camp on at least one base station in HetNets, modelling the base stations locations as spatial Poission point process and assuming Rayleigh fading. Another paper provides an outage analysis for relay femtocells in [44].

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8 CHAPTER 1. INTRODUCTION Hoydiset al. in [45] investigates the outage probability and the ergodic mutual information 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 of macrocellular users subjected to the interference imposed by same-frequency LTE small cell HetNets and experiencing Rician fading has not been proposed with the aid of stochastic geometry.

Motivation and Organization of the Document

The 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 mathematical tools offered by Stochastic geometry we can analyse the average behaviour of a network, meanwhile the several input parameters are handled as random variable. In this dissertation 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 random point processes.

Goal of the Thesis

The goal of this dissertation (relaying on the state-of-art literature) is to propose a mathematical model for two-tier small cell network modelling. Some known values (e.g. the mean value of the user population) are given as input parameter, meanwhile other variables are remain random. Thanks to Stochastic geometry, despite of several random parameters are given, the model is traceable. At the first view the combination two-tier LTE-Advanced Small cell networks and Stochastic geometry might seems unusual to the reader. Since mobile telecommunication is mostly a “practical oriented” and engineer based field, meanwhile in contrast Stochastic geometry is the study of random geometric structures, focusing on theoretical issues in mostly related to mathematical topics. Therefore the mixture of two independent 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 results of the proposed formulas are compared with Monte-Carlo simulation results for validation purpose and in order to ensure the precision the proposed forms. The simulations are made with MathWorks MATLAB^r. As far as I know there is no such detailed downlink interference model for LTE-A small cells in the literature, that is proposed in this thesis. The proposed 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 calculation of the probability service outage. The dissertation introduces two new two-dimensional

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1.2. MOTIVATION AND ORGANIZATION OF THE DOCUMENT 9 point processes for modelling LTE-Advanced networks. This two introduced point process are belong to the family of Poisson cluster processes, namely: Matérn- and Thomas cluster process. Since small cell installation in a chosen area might not be homogeneous, therefore these cluster based models can model these scenarios in more accurate way compared to homogeneous point processes. The properties of Matérn- and Thomas cluster process are lavishly 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-dimensional random processes that can be used for small cell modelling. Firstly, the planar Poisson Point process is presented in details, afterwards an adaptation (mapping) of PPP proposed, that can be used for modelling small cell deployment taking into consideration the road structure of the area. Secondly, the thesis introduces two Poisson cluster processes (namely Thomas cluster and Matérn cluster process). This two cluster process give a precise model for a real environment, 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 and incomplete Gamma functions, complementary Error function etc.), surveys the fundamental aspects of Stochastic geometry and highlights the connection between random variables and fading types. Finally, the path loss model and the possible interference types are explained here, 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 by the 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.), furthermore it follows alpha-stable distribution, more precisely Lévy distribution. Next, the outage probability for a macrocell user is investigated for various fading types such as: lognormal fading, Rayleigh fading, Nakagami-mfading and Rice fading. The proposed forms are 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 author has 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 cell modelling. Afterwards the thesis gives formulas to calculate the outage probability (or the complement event – the coverage probability) for a user served by macrocell. Due to the

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10 CHAPTER 1. INTRODUCTION complexity of the forms, they are approximated. However, according to the simulation results, these approximations are still accurate.

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

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2 System model

This chapter introduces the system model used for our Stochastic geometry based investigations. The research provided in the thesis is strictly restricted to the access network, thus the small cell effects related to core network of an LTE-A system is not detailed in this dissertation. In this model the access network is separated into two tiers. The first tier is the macro tier 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 a finite field denoted byR. The area of this field is represented by|R|and this parameter denotes the coverage area of a macro eNB. The macro eNB is located at the center ofR. This point is denoted as the origin point of the coordinate system. In other words, the macrocell is located at the origin of the coordinate system and covers the area|R|. The small cells are deployed by the users on this finite area|R|. An illustration of the system model is given in Figure 2.1.

Both eNB type (macro- and small eNB) operates on a fix, constant power. The emitted power of a macro eNB is denoted byP^c, meanwhile the emitted power of the small cells are given byP^s. There are no power control applied for small cells. The emitted power of the small 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 byP_c^r and P_s^r, respectively.

It is assumed that the macro eNB has an omni directional antenna. The applied scheduling algorithm is round robin (RR). The data transmission is based on ON/OFF model; however, 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 carrier frequency 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 adaptive modulation and coding (AMC) in order to achieve significant throughput and spectral ef-

11

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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 data per MHz of spectrum, resulting in a lower cost-per-bit. The OFDM-based air interface also provides much greater deployment flexibility than 3G UMTS/HSPA with support for multiple 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 the downlink and 50 Mbps in the uplink. As in the context of LTE, we also consider elementary time/frequency Physical Resource Blocks (PRB). The PRB is defined as a block of physical layer resources that spans over one slot (typically 0.5 ms) in time and over a few adjacent OFDM sub-carriers (typically 12) in the frequency domain. In this two-tier LTE system we use the maximum number of physical resource blocks with normal cyclic prefix [53, 54].

The small- and macrocell eNBs are assigning radio resources (PRBs) independently to their 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 operate independently from each other. Moreover, there is no central coordination between small cells and macrocell(s). On the other hand small cells operate on the same licensed spectra as the macrocells. It is assumed that both macro and small cell base stations have MIMO and omni-directional antennas. The location of the macrocell user (macro UE) is given by vector z. The distance between the macro UE and the macro eNB (from the origin) is the absolute value of the vectorkzk(in meter). The effect of the fast fading is also included in the system

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13 model. We include the fading effect in the received power. Parameter h denotes the fast fading, andhis an independent and identically distributed random variable. The distribution ofhdepends on the modelled fading type. The index ofhrefers to the type base station e.g.

h_cdenotes the fast fading for a macro eNB. Further information abouthis lavishly detailed in 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 to use outdoor path loss model for radio signal propagation modelling. The modelled two-tier system is interference limited, thus the effect of the thermal noise at the receiver is ignored for the sake of simplicity. The mobility of users are not considered in this thesis, it can be viewed as an actual snapshot of a loaded two-tier network. Note that, the snapshot nature does not infect the results, because random spatial processes guarantees the random location of 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 is fix (kzk).

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

g^[dB](z) =12+39·log₁₀(kzk) +Ψ^[dB]_log (2.1) whereg(z)represents the path loss (attenuation) at the given location. Path loss due to carrier frequency and other parameters are included to the model via the constant values (e.g. wall penetration 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(Ψ_log)∼N (0,10). Note thatΨ_log follows lognormal distribution, howeverΨ^[dB]_log follows normal distribution.

Note that, choosing SUI model for path loss modelling does not affect the theorems proposed by the thesis, thus other path loss models can be used also for example proposed by [57]. It is assumed that the shadowing components are i.i.d. (independent, identically distributed) 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 the particular transmitter/receiver antenna. The non-logarithmic version of (2.1) is given as a gain value:

g(z) = 1 K_i· 1

Ψ_log · kzk^−α, (2.2)

where the constant parts (e.g. propagation loss due to carrier frequency) are merged intoK_i. The outdoor path loss exponent is denoted byα.

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14 CHAPTER 2. SYSTEM MODEL An example is given here. Let us assume that the transmitter emits onP^cand the receiver is located atz. Therefore the received power at this location is:

P_c^r(z)^[dB]=P^c[dB]−g^[dB](z), (2.3) P_c^r(z) =P^c·h_c·g(z) = P^ch_ckzk^−α

K_iΨ_log . (2.4)

Location of Small Cells

The dissertation introduces three random processes, that can be used for two-tier small cell modelling. These random processes models the planar locations of the small cells on finite areaR. 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. The actual number of small cells follows Poisson distribution with densityλ. The mean value of the small cells, thereforeN_s=λ· |R|. In this model we assume that the process is homogeneous, thus the points are scattered uniformly in the planar. In this case the intensity parameterλ is constant in every part ofR. The actual location of the small cells is given by vectorx. The distance between a small cell and the macro eNB (located at the origin) is given by the absolute value of the vectorkxk. Furthermore the distance between a small cell and the user is given bykx−zk. Note that, if the process is not homogeneous, then the intensity parameter should be given asλ(x). The homogeneous version of Poisson point process widely used, accepted and popular model. Our model uses homogeneous point process, thus for the sake of simplicity we drop the word homogeneous and refer the process simply PPP hereafter.

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

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

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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 example is only the mean value.

Application of PPP model

In the average system throughput calculation (given in Section 3.4) we apply PPP model in order to calculate the system level throughput in down-town. Road structure can be represented 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. One possible graph structure represented in Figure 2.3, where the macrocell and small cells are denoted by blue and green rectangles, respectively. In this case the model should take into account 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 by N_c.

• In general case it is expected that small cells are covering a circle areas with radiusR_s (thus the area covered by one small cell isR²_sπ). The whole system area is|R|.

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16 CHAPTER 2. SYSTEM MODEL

Figure 2.3: PPP based Random graph model

• On the other hand in random graph model (city road structure) small cells are covering only road segments with distance 2R_s. The roads segments are denoted by e₁,e₂,· · ·,e_n. The total length of the road segments is given by ∑_i|e_i|. Figure 2.3 gives an illustration of the proposed random graph model. Small cells are installed along the edges to provide extended coverage for the users i.e. small cells are located on 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 distribution of the small cells thus follow a Poisson Point Process. However the area where small cells possibly occur is restricted to the road structure, following [16], PPP makes possible to analyse this wireless network. The number of users in theith small cell is denoted byU_i. U_i’s are independent, identically distributed random variables which follow Poisson distribution with parameterλ.

Poisson cluster Process

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

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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 of flats), than others (e.g. public parks). Nevertheless, the introduced cluster models remains tractable due to Stochastic geometry.

The dissertation introduces two Poisson cluster processes that can be used for small cell modelling, namely the Thomas cluster process and Matérn cluster process [50]. Both processes are belong to the family of Neyman-Scott point process [50] and composed with superposition of simple Poisson point processes. This provides the mathematical tractability.

The base for Thomas and Matérn cluster process is a Poisson point process with parameter λ_p. The literature calls this points as parent points. The set of parent points is denoted by (Φ_p={x₁,x₂, . . .}). In our model the actual location of the parent points is given by vector x. Around the parent points scattered the daughter points (or the name offspring points is also used by the literature). The daughter points are independent from each other and from the other parent points. Furthermore, they are identically distributed around the parent point.

The set of cluster is represented byNxi={Ni,x_i}, whereNi={y₁,y₂, . . .}denotes the i.i.d. set of daughter points. Note thatNi is independent from the parent process, in other words, the parent points are not members of the cluster. The whole cluster therefore can be written as follows:

Φ=^[

i

(Ni∪x_i) = ^[

x∈Φp

Nx.

The number of daughter points is random, and follows Poisson distribution. The mean value of the daughter points in a cluster is represented byc. The distance between daughter

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18 CHAPTER 2. SYSTEM MODEL points and the cluster centres (parent points) is given by vectory. In PCP the daughter points represents the small cells. Their actual position is given by vectorx_i+y. The density of the cluster process can be calculated withλ =λ_p·c, wherecrepresents the average number of small cells in a cluster.

The main difference betweenThomasandMatérn clasteris the scattering of the daughter points. In case of Thomas cluster process the daughter points scatter around the parent points according to a symmetrical normal distribution (zero mean andδ²variance) with the following density function:

f(y) = 1 2π δ²exp

−kyk² 2δ²

. (2.5)

However, inMatérn cluster processthey are scatter around the parent points uniformly in a circle with radiusR. The density function is given by [26]:

f(y) = ( 1

πR², ifkyk ≤R

0, oherwise (2.6)

An illustrationThomasandMatérn claster is given in Figure 2.4a and Figure 2.4b, respectively. To compare the processes the following parameter sets are applied. In order to introduce 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 value of the parent points (λ_p· |R|) equals 4. In case of Matérn cluster process we set the radiusR to 50 meter. Therefore, the centre of the circle represents the parent point and the small cells are scattered in a circle with diameter 100 m. In Thomas cluster the small cells are scattered according to a symmetrical normal distribution, therefore 99.7% of the observations fall into the 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 variable, only the mean value is known (c). Furthermore, the actual number of the clusters is also a random variable. However, thanks to Stochastic geometry tools the model can be handled mathematically.

Mathematical Preliminaries

This section collects the essential definitions (i.e. complete and incomplete Gamma functions, complementary Error function etc.) and connections between random variables, that

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2.2. MATHEMATICAL PRELIMINARIES 19 will be used in the following sections. The moment-generating function (MGF) and the moments of the distributions are introduced in this section. The distribution of a random variable can be characterized with the moment-generating function. The Moment-generating functions have great practical relevance not only because they can be used to easily derive moments, but also because a probability distribution is uniquely determined by its MGF. It is important to mention that, all random variables have a characteristic function, however not all random variables have a moment generating function. This section also includes the definition of the so-called Probability Generating Functional (PGFL), which will be an essential input for further calculations.

Error function and complementary Error function

The Error function is defined by the integral [60]:

erf(s) = 2

√ π

Z _s

0

e^−t²dt. (2.7)

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

erfc(s) =1−erf(s) = 2

√ π

Z _∞

s

e^−t²dt. (2.8)

Gamma function and incomplete Gamma function

The complete Gamma function is defined with the following improper integral [61]:

Γ(s) = Z∞

0

t^s−1e^−tdt. (2.9)

Ifsis a positive integer, then

Γ(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) =

∞

Z

x

t^s−1e^−tdt, (2.10)

γ(s,x) =

x

Z

0

t^s−1e^−tdt. (2.11)

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20 CHAPTER 2. SYSTEM MODEL The sum of lower- and upper incomplete gamma functions yields the complete Gamma function:

Γ(s) =γ(s,x) +Γ(s,x). (2.12) Ifsis a positive integer, then the incomplete (upper) Gamma function can be written as [61]:

Γ(s,x) = (s−1)!e^−x

s−1

∑

k=0

x^k

k!. (2.13)

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

Γ(s,0) =Γ(s) if Re{s}>0, (2.14)

Γ(s,∞) =0. (2.15)

Jensen’s inequality

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

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

E{f(X)} ≥ f(E{X}) if f is convex, (2.16) E{f(X)} ≤ f(E{X}) 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 fading with no direct line-of-sight (LOS) path [58, 59]. In this case the channel vary randomly the emitted signals amplitude according to Rayleigh distribution. When we introduced the system model (beginning ofChapter 2.) we defined that the received power and the effect of the channel is modelled with parameter h. Instead of calculating the amplitude we calculate on the power domain. It is known that the square of a random variable that follows Rayleigh(ω) distribution¹is an exponential(ω) distributed random variable. A brief proof is given inAppendix A.1.1. Let us assume that a random variableX follows Rayleigh distribution with the following probability density function, denoted by f_X(x,ω)(wherex>0 and ω>0, evidently):

f_X(x,ω) =2x

ωe(^−x²^/ω). (2.18)

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

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2.2. MATHEMATICAL PRELIMINARIES 21

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

Eh

n

e^−shAo

= 1

(1+sA), (2.19)

whereAis a non-negative parameter. For the sake of simplicity we chooseω =1 hereafter.

ωandAcould be transformed into one other. With equation (2.19) we can calculate the fractional moment²forhin 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)

whereΓ(x)is the Gamma function defined in (2.9).

Gamma Distribution

In wireless communication Nakagami distribution is used to model the effect of scattered signals that reach a receiver by multiple paths. Similarly to the previous case we investigate the 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 us assume, that a given a random variable X follows Nakagami distribution with the shape parametermand spread parameterΩ. In case ofY is a Gamma distributed random variable withY∼Gamma(m,Ω/m), thenX²∼Y is valid betweenXandY. For the sake of simplicity we calculate withΩ=1 hereafter. The probability density function for an Gamma distributed random variable is given by:

f(h,m) = h^m−1e^−hm

m^−mΓ(m), (2.21)

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

Eh

n

e^−shAo

= 1

1+s_m^Am. (2.22)

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

E{√

h}= (2m−1)!!

2^m(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.

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22 CHAPTER 2. SYSTEM MODEL where m!! denotes the double factorial of a positive integer m. The double factorial of a positive integermis a generalization of the usual factorialm! 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 parameterm. In case of Rayleigh fading (m=1) (2.23) reduces to

√ π

2 , which confirms the result of (2.20).

Non-central Chi-squared Distribution

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

f(h,K) = (K+1)e(−K−(K+1)h)·I₀

2p

K(K+1)h

. (2.24)

A brief proof given inAppendix A.1.3. ParameterKdenotes the power-ratio of the direct LOS signal and of the scattered paths of the Rician distribution. TheK=0 scenario represents the Non-Line-of-Sight (NLoS) i.e. Rayleigh fading case. I₀(x)denotes the modified zero-order Bessel function. The Moment Generating Function (MGF) of the non-central Chi-squared distribution is given by [64]:

Eh

n

e^(−shA)o

= 1+K

1+K+sA·exp

− sKA 1+K+sA

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

The square root of the received interference power√

hfollows Rician distribution, thus we invoke the definition of the expected value to evaluate the fractional moment of h. We will use this form in Section 3.1. Substituting fading distribution yields,

E{√ h}=

r 1

1+K·e⁻^K²

(1+K)I₀ K

2

+K·I₁(K/2) √

π

2 , (2.26)

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2.2. MATHEMATICAL PRELIMINARIES 23 where the Modified Bessel Function of the Zero- and First order is given byI₀(x)andI₁(x), respectively. To validate the form we substitute K =0 into (2.26). This corresponds to Rayleigh fading and yields√

π/2 (and confirms the result of (2.20)), as expected. Again, the actual value of the factional moment ofhdepends on the Rice fading parameterK.

Weibull Distribution

Due to the pleasant property of Weibull distribution if channel’s amplitude obeys Weibull distribution with shape (n) and scale parameter (γ), then thek^thpower ofhis also Weibull distribution withn/kandγ, 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, substitutingn=1 andγ =1 (Rayleigh fading case) yieldsΓ(3/2), as expected.

Lognormal Distribution

In wireless communication Lognormal distribution is used to model the effect of shadowing. It is also called slow-fading. In further investigation we should calculate the value of E{p

1/Ψ}, where Ψ follows lognormal distribution. It is known from [67], that for any real or complex numbers, thesth moment of a log-normally distributed variable X can be calculated with the following form:

E{X^s}=e^sν+¹²^s²^δ², (2.28) whereµandσare the distribution parameters for location and scale, respectively. Therefore, reshapingE{p

1/Ψ}asE{Ψ^−1/2}, wheres=−1/2 yields, E{p

1/Ψ}=e^{(−1/2·µ+σ}²^·1/8). (2.29)

Fundamentals of Stochastic geometry

In this subsection mention some essential formulas and definitions that we will refer in the thesis later.

Definition 2.2 (Palm distribution [50]). (a heuristic approach) Let us consider a Poisson point process denoted byΦ. The Palm distribution probabilities are conditional probabilities

S TOCHASTIC G EOMETRY S MALL C ELL N ETWORKSWITH M ODELLING T WO - TIER LTE-A DVANCED

B

U

T

E

Faculty of Electrical Engineering and Informatics

Department of Networked Systems and Services

Mobile Communications and Quantum Technologies Laboratory

M ODELLING T WO - TIER LTE-A DVANCED

S MALL C ELL N ETWORKS WITH S TOCHASTIC G EOMETRY

Ph. D. Thesis of of

Zoltán Jakó

Scientific supervisor:

Gábor Jeney, PhD.

Budapest, 2017

c 2017, All rights reserved to the author

This document was typeset in L

TEX 2ε applying Adobe Times font family

Abstract

Kivonat

Köszönetnyilvánítás

Contents

List of Figures

List of Tables

Abbreviations

1 Introduction

Background of the Research

Brief History of Stochastic Geometry and Related works

Motivation and Organization of the Document

Goal of the Thesis

Structure of the Thesis

2 System model

Location of Small Cells

Poisson Point Process

Application of PPP model

Poisson cluster Process

Mathematical Preliminaries

Error function and complementary Error function

Gamma function and incomplete Gamma function

∑

Jensen’s inequality

Relationship Between Random Variables and Moments

Fundamentals of Stochastic geometry