Dynamic System Modelling and Adaptation Framework … · Abstract Dynamic System Modelling and Adaptation Framework for Irregular Cellular Networks Levent Kayili Doctor of Philosophy

Dynamic System Modelling and Adaptation Framework forIrregular Cellular Networks

by

Levent Kayili

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of Electrical and Computer EngineeringUniversity of Toronto

c© Copyright 2015 by Levent Kayili

Abstract

Dynamic System Modelling and Adaptation Framework for Irregular Cellular Networks

Levent Kayili

Doctor of Philosophy

Graduate Department of Electrical and Computer Engineering

University of Toronto

2015

In the decades since the development of the traditional cell concept, there has been much

research in the field of cellular networks for greater adaptability. In particular, networks

with irregular deployment of base stations (BSs) having widely different power classes,

have received much attention in industry and academia alike. In such networks, BSs of

both high and low power capabilities, are deployed in broad accordance with user traffic

demand. In this thesis, we consider irregular cellular networks from the viewpoint of the

design, adaptability and dynamism of adaptive resource allocation.

Despite extensive research in the area, to our knowledge high-level system modelling

for simulation purposes did not receive attention in the literature. We consider the devel-

opment of such a model in this thesis. In particular, we assume a network with universal

frequency allocation and power assignment. A high-level model representation and an

adaptive coordinated resource allocation strategy is developed for such a network. A

modified Monte Carlo simulation is proposed as a simulation model, which includes rep-

resentation of BS and terminal deployments in scenarios either with or without hotspots,

as well as a set of dynamics occurring at a large time scale. A representation including

such dynamics is expected to be important for the consideration of adaptive resource

allocation strategies and other adaptive functions such as Self Organization (SO) that

operate at an assumed large time scale.

A shadowing model with spatial correlation is considered as part of the system

model, and is generalized for a network with an arbitrary number of distinct BS power

ii

classes. Enhancements are then proposed to an adaptive re-configurable resource alloca-

tion framework which is based on dynamically forming scheduling cells at a large time

scale. In particular, a strategy for BS power adaptation at a large time scale is proposed

for improved interference mitigation.

The dynamic adaptation case of BS outage compensation is additionally studied as

an application of the model and adaptive resource allocation.

It is believed that the high-level model with resource allocation can serve as a skeleton

network model to be tailored to different purposes for more realistic network representa-

tion and design.

iii

Acknowledgements

First and foremost, I would like to begin by extending my deep gratitude to my supervisor,

Professor Sousa. His guidance and support throughout my studies inspired me greatly in

the pursuit of my research. I learned so much from him throughout the years, for which

I am truly grateful.

I would also like to thank the members of my defense committee Professor Adve,

Professor Leon-Garcia, and Professor Valaee, and the external examiner, Professor Ro-

drigues, for their time in reading my thesis and providing many helpful comments and

suggestions.

I cannot overstate the value of having great colleagues throughout my years as a

graduate student. For this reason, I would like to sincerely thank my fellow graduate

students for their camaraderie and for providing the opportunity for many interesting

conversations and discussions over the years.

Finally, I would like to thank my parents, who have supported me, and encouraged

me greatly throughout my Ph.D. studies. I am truly grateful for their support, and I

believe that without it, the completion of my work would not have been possible.

iv

Contents

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.5 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Background and Preliminaries 10

2.1 Cellular Network Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 User Scheduling Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.1 Total Rate Maximization . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.2 Proportional Fair Scheduling . . . . . . . . . . . . . . . . . . . . . 14

2.3 Resource Allocation Strategies . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 Adaptive Resource Allocation Framework for Irregular BS Deployment . 17

2.5 General Slow-Time-Scale Adaptive Algorithms . . . . . . . . . . . . . . . 21

2.6 The ITU System Model: High-Level Reference Model for the Regular

Cellular Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.6.1 Channel Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.7 Considerations for an Irregular Cellular Network Model . . . . . . . . . . 26

2.8 Correlated Shadowing Model for Irregular BS Deployment . . . . . . . . 28

3 Adaptation Case: BS Outage Compensation 32

3.1 Relevant Work and Contribution . . . . . . . . . . . . . . . . . . . . . . 32

3.1.1 Relevant Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.1.2 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.1.3 Key Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3 Problem Statement and Proposed Algorithmic Approach . . . . . . . . . 35

v

3.4 Proportional Fair Compensation Algorithm . . . . . . . . . . . . . . . . . 37

3.4.1 Outage User Cell Re-association . . . . . . . . . . . . . . . . . . . 38

3.4.2 Generation of Compensating Cluster . . . . . . . . . . . . . . . . 41

3.4.3 BS Power Adjustment . . . . . . . . . . . . . . . . . . . . . . . . 42

3.4.4 Overall Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4.5 Information Exchange Requirements . . . . . . . . . . . . . . . . 44

3.5 PLM Compensation Algorithm . . . . . . . . . . . . . . . . . . . . . . . 44

3.6 Simulation/Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.6.1 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4 Dynamic System Model (Single Power Class) 54

4.1 Relevant Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.1.1 Terminal and BS Deployment Models in Cellular Networks . . . . 54

4.1.2 Terminal Mobility Models in Cellular Networks . . . . . . . . . . 56

4.2 Approach and Contributions . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.4 Drop Deployment Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.5 Time Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.5.1 Terminal Arrival and Departure Models . . . . . . . . . . . . . . 62

4.5.2 BS Deployment and Outage Models . . . . . . . . . . . . . . . . . 63

4.5.3 Terminal Movement Model . . . . . . . . . . . . . . . . . . . . . . 64

4.5.4 BS Movement Model . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.6 Typical Parameter Values . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.7 The Outline of the Network Model of the Thesis . . . . . . . . . . . . . . 72

5 BS Power Adaptation (Single Power Class) 76

5.1 Relevant Work: Interference Coordination Strategies for Heterogeneous

Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.2 Contribution: Adaptive Resource Allocation Framework with Power Adap-

tation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.2.1 Key Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.4 BS Power Adaptation Algorithm . . . . . . . . . . . . . . . . . . . . . . 82

5.4.1 Total Rate Maximizing (TRM) Power Adaptation . . . . . . . . . 82

5.4.2 Proportional Fair Power Adaptation . . . . . . . . . . . . . . . . 84

5.5 Adaptive Resource Allocation Framework with Power Adaptation . . . . 85

5.6 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

vi

6 Static Model of Multiple Power Classes 97

6.1 Relevant Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.2 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.4 Deployment Model with Hotspots . . . . . . . . . . . . . . . . . . . . . . 99

6.5 Distance Dependent Path Loss Model . . . . . . . . . . . . . . . . . . . . 101

6.6 Inhomogeneous Shadowing with Correlation . . . . . . . . . . . . . . . . 102

6.6.1 Generation of Shadowing for Links between BS-Terminal Pairs . . 104

6.6.2 Shadowing Model Consistency Verification . . . . . . . . . . . . . 108

6.6.3 Implementation of Model in Practical Network Simulations . . . . 110

7 Dynamic System Model (Multiple Power Classes) 112

7.1 Relevant Work and Contribution . . . . . . . . . . . . . . . . . . . . . . 112

7.1.1 Key Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

7.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7.3 Time Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7.3.1 Terminal Arrival and Departure Models . . . . . . . . . . . . . . 115

7.3.2 BS Deployment and Outage Models . . . . . . . . . . . . . . . . . 116

7.3.3 Terminal Movement Model . . . . . . . . . . . . . . . . . . . . . . 117

7.3.4 BS Movement Model . . . . . . . . . . . . . . . . . . . . . . . . . 119

7.4 Typical Parameter Values . . . . . . . . . . . . . . . . . . . . . . . . . . 119

7.5 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

8 Conclusion 132

8.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

8.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

Bibliography 136

vii

List of Figures

2.1 Regular cellular network deployment. The black and red points represent

a BS and terminals, respectively. . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Representation of the irregular cellular network deployment. The green

circles represent terminals, and the blue squares of different sizes represent

BSs of different power classes. . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Hypothetical scenario for terminal scheduling. . . . . . . . . . . . . . . . 14

2.4 Flowchart for the adaptive resource allocation framework. . . . . . . . . . 18

2.5 Algorithm for the adaptive resource allocation framework . . . . . . . . . 19

2.6 Algorithm for fast PF resource allocation for a given subframe l ∈ F . . . 20

2.7 The flowchart for the drop-based simulation framework based on the ITU

common reference model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.8 The wireless link between a fixed base station and a moving terminal. . . 29

2.9 Wireless links with a common endpoint. . . . . . . . . . . . . . . . . . . 29

2.10 Wireless links without a common endpoint. . . . . . . . . . . . . . . . . . 29

2.11 A realization of the potential field [8]. . . . . . . . . . . . . . . . . . . . . 30

3.1 BS outage scenario. The light blue square is the outage BS (BS n0), the

light blue circles are the outage users, dark blue squares are the candidate

BSs, dark blue circles are the associated users, and the red squares and

circles are the BSs and users outside the candidate set, respectively. . . . 35

3.2 Outage user cell association update. . . . . . . . . . . . . . . . . . . . . . 41

3.3 Compensating BS power update. . . . . . . . . . . . . . . . . . . . . . . 43

3.4 Overall iteration of the PF compensation algorithm. . . . . . . . . . . . . 43

3.5 PLM Compensation Algorithm. . . . . . . . . . . . . . . . . . . . . . . . 46

3.6 User average spectral efficiency, in bps/Hz, as a function of the average

number of BSs with raised power for (a) regular and (b) irregular networks. 50

viii

3.7 Outage user QoS violation probability, in percent, as a function of the

average number of BSs with raised power for (a) regular and (b) irregular

networks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.8 Non-outage user QoS violation probability, in percent, as a function of the

average number of BSs with raised power for (a) regular and (b) irregular

networks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.1 Outline of the Simulation Framework. . . . . . . . . . . . . . . . . . . . . 60

4.2 Simulation area for deployment in networks with a single BS power class. 62

4.3 The terminal movement illustration. . . . . . . . . . . . . . . . . . . . . . 65

4.4 Algorithm for the terminal movement at the simulation subdrop. . . . . . 66

4.5 The BS movement illustration. . . . . . . . . . . . . . . . . . . . . . . . . 67

4.6 Algorithm for the BS movement at the simulation subdrop. . . . . . . . . 67

4.7 Sample realization of the evolution of the number of terminals K(b) over

subdrops b, with K(0) = 25, the arrival parameter λt = 3 terminals, and

the departure parameter µt = 5 subdrops. . . . . . . . . . . . . . . . . . 70

4.8 Sample realization of the evolution of the number of BSsN(b) over subdrops

b, with N(0) = 7, the arrival parameter λbs = 0.5 BSs, and the departure

parameter µbs = 20 subdrops. . . . . . . . . . . . . . . . . . . . . . . . . 71

4.9 A realization of the potential field [8]. . . . . . . . . . . . . . . . . . . . . 72

4.10 Flowchart of the general network model in the thesis. . . . . . . . . . . . 73

5.1 Extended adaptive resource allocation framework with power adaptation. 86

5.2 Algorithm for the adaptive resource allocation framework with TRM power

adaptation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.3 Algorithm for the adaptive resource allocation framework with PF power

adaptation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.4 Simulation area for cellular network. . . . . . . . . . . . . . . . . . . . . 89

5.5 Comparison of power adaptation schemes and the equal power level sce-

nario for varying δBS in terms of 5th percentile spectral efficiency, with

K = 28 users. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92


nario for varying δBS in terms of 50th percentile spectral efficiency, with

K = 28 users. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.7 Cumulative distribution functions for spectral efficiency at δBS = 4h/3 =

333.3 m, with (a) normal magnification and (b) increased magnification,

with K = 28 users. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

ix


nario for varying δBS in terms of the sum log utility, with K = 28 users. . 94


nario for varying δBS in terms of the user average spectral efficiency, with

K = 28 users. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94


nario for varying δBS in terms of the user average spectral efficiency, with

K = 56 users. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.1 Simulation area for deployment in networks with multiple BS power classes

and explicit hotspot modelling. . . . . . . . . . . . . . . . . . . . . . . . 99

6.2 Sample radio link in an irregular cellular network. Here, L is the total

length of the link path; k ∈ K is an index value for a terminal; n ∈ N and

p ∈ N are index values for the BSs such that n 6= p, n 6= 1, n 6= 2, p 6= 1

and p 6= 2; and lm is the length of path segment assigned to a BS m ∈ N . 105

6.3 Algorithm for assigning path segments to BSs . . . . . . . . . . . . . . . 107

6.4 Shadowing simulation scenario. . . . . . . . . . . . . . . . . . . . . . . . 109

6.5 Comparison of normalized correlation as a function of normalized distance,

d/dc,eff,2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

7.1 Outline of the Simulation Framework. . . . . . . . . . . . . . . . . . . . . 114

7.2 The terminal movement illustration. . . . . . . . . . . . . . . . . . . . . . 118

7.3 Algorithm for the terminal movement at the simulation subdrop. . . . . . 118

7.4 The BS movement illustration. . . . . . . . . . . . . . . . . . . . . . . . . 120

7.5 Algorithm for the BS movement at the simulation subdrop. . . . . . . . . 120


nario for varying terminal displacement in terms of (a) 5th percentile spec-

tral efficiency, and (b) 50th percentile spectral efficiency. . . . . . . . . . 126


nario for varying terminal displacement in terms of (a) the sum log utility,

and (b) the user average spectral efficiency. . . . . . . . . . . . . . . . . . 127


nario for varying number of users in terms of (a) 5th percentile spectral

efficiency, and (b) 50th percentile spectral efficiency. . . . . . . . . . . . . 130


nario for varying number of users in terms of (a) the sum log utility, and

(b) the user average spectral efficiency. . . . . . . . . . . . . . . . . . . . 131

x

Chapter 1

Introduction

1.1 Motivation

The cellular concept in mobile telephone networks was developed in the late 1960s. In

the original cellular paradigm, an area was divided up into cells with more or less regular

sizes, with BSs placed at cell centres. A hexagonal cell model was typically used for the

representation of the coverage areas in this network. In order to allocate the frequency

spectrum resource efficiently to mobile users, and provide acceptable user rates, it was

necessary to reduce the effects of co-channel interference. This was achieved through a

static frequency allocation strategy known as classical frequency reuse, which allowed re-

use of the spectrum resource at a fixed distance while keeping the co-channel interference

low. The network was designed for the provision of acceptable wireless service over a

given geographical area, however, it accounted for little in the form of dynamism or

adaptation. In particular, the frequency allocation strategy was static, and there was

no capability for organic growth of the network, in particular, from the viewpoint of

adaptation of the resource allocation. Such a network shall be referred to, in this work,

as a regular cellular network.

The traditional cellular network with regular deployment can be said to have a high-

level reference model for the purpose of simple Monte Carlo simulation, which provides a

high-level representation of the system. Such a model, that is well-known and commonly

used, has been described in the International Telecommunication Union (ITU) guide-

lines [1] which was developed for the IMT-Advanced and is applicable to 3GPP LTE.

According to the model, the BS deployment is regular and the coverage areas are repre-

sented by a regular (hexagonal) cell pattern. There is an allowance for a great amount

of detail for modelling the variation of multipath fading at the small time scale, but

channel modelling at the (assumed) larger time scale, in particular, the shadow fading

1

Chapter 1. Introduction 2

follows the classical lognormal model, which does not take into account more realistic

effects like the shadowing spatial correlation. In addition, the simulation methodology

for the system model does not consider the gradual evolution of the system at time scales

much larger than the multipath fading, which are due to the dynamical changes in the

network. Instead, independent snapshots known as simulation drops are typically used in

Monte Carlo type simulation. A simulation drop, in this context, is defined as a random

deployment of mobile terminals on a network area with fixed BS deployments, follow-

ing the hexagonal cell pattern. Within each drop, the set of active terminals and their

deployments as well as large-scale channel components such as path loss and shadowing

are fixed, and there are only virtual terminal movements, which result in the multipath

fading due to the Doppler effect. This high-level model can be associated with a model

for the organization of resource allocation, which is the traditional scheme of classical

frequency reuse, as already mentioned. Thus classical frequency reuse (CFR) is consid-

ered as part of the overall model. The system model based on the traditional cell concept

with regular deployment, little large-scale dynamism as well as classical frequency reuse

shall be referred to, in this thesis, as the regular cellular network model.

In the decades since the development of the traditional cell concept, there has been

much research in the field of cellular networks for greater adaptability. In order to meet

the increasing demands from mobile users while allowing the efficient use of the fre-

quency spectrum, BSs of different transmit power capabilities, known as power classes,

have started being used. The deployment of these BSs have been in broad accordance

with the user traffic demand. In particular, in areas with high density of user traffic, or

local hotspots, low-power BSs with small cell sizes have started being used. In other areas

with lower density of traffic, high-power BSs having large cell sizes were still utilized. The

heterogeneous cellular network was therefore born as a research area and received much

attention as a vast and varied field. While many definitions exist for heterogeneous net-

works, we consider the definition of a network with a single air interface and technology,

and multiple power classes such as femto, pico, micro and macro BSs. Techniques were

required for the organic growth of such networks, especially in terms of the adaptability

of the resource allocation. Strategies that were much more dynamic than the classical

frequency reuse were therefore proposed and adopted. In addition, much research work

was done to enhance the modelling of the heterogeneous cellular network with regard to

various aspects such as deployment, mobility and multipath fading channel models as

well as, in a few cases, the shadowing. However, a number of unrealistic assumptions

continued to be used in some of the work, mainly for the purpose of analytical tractability

and simplicity of the modelling. These included, e.g., the use of the Poisson point pro-


cess (PPP) as a deployment model, which was shown to be unrealistic in real networks.

They also included the use of uncorrelated lognormal shadowing and Rayleigh multipath

fading models.

Despite the extensive research on system modelling, to our knowledge high-level sys-

tem modelling for simulation purposes, like the ITU model, did not receive attention in

the literature. We consider the development of such a model in this thesis. We assume

a network with universal frequency allocation and power assignment. The main motiva-

tion of such a high-level simulation model is to provide a candidate for a new high-level

reference model, in the face of new research and practical developments such as hetero-

geneous networks as well as recent research on adaptive resource allocation strategies

as well as self organization (SO) that commonly work at an assumed large time scale.

The goal is to provide a high-level model representation, including an adaptive resource

allocation strategy, over which new and different research efforts can then make grad-

ual improvements driven by the need to represent the network with as much realism as

required or desired for each particular situation. In our view, a benefit of the model is

that researchers need not be forced to make unrealistic assumptions about deployment,

shadowing, multipath fading, large time scale evolution including movement, as well as

the interferers for the purpose of analysis and design. For instance, some works could

provide more detail on realistic deployment and shadowing, and other works more detail

on realistic multipath fading models. This contrasts with the efforts to make the models

more analytically tractable using models of varying degrees of realism. Similarly, some

works can focus on the aspect of resource allocation at the small time scale (or on multiple

antennas), and others on making small adjustments at a larger time scale to the cellular

architecture in order to adapt to the dynamism of the network. Different modifications

made to such a high-level reference model by different researchers can then be compared.

1.2 Approach

For the purpose of model development, we consider a network which includes BSs with

multiple power classes deployed according to an irregular pattern, as in a heterogeneous

cellular network. The generic type of network that is considered shall be referred to

as an irregular cellular network. In this thesis, we develop a high-level model for the

irregular cellular network as a modification of the regular cellular network model of the

ITU. This model, which shall be referred to as the irregular cellular network model,

must have characteristics that contrast with the regular cellular network model. In the

new paradigm, high- and low-powered BSs are deployed in an irregular pattern in broad


accordance with the user demand. Thus a model of arbitrary BS deployment must be

considered in place of the hexagonal deployment. Since the dynamics of the network at

time scales other than the assumed multipath fading scale are gaining a lot of importance,

the independent snapshot method of drop simulation is, inadequate, and an improved

concept is needed. The set of additional dynamics in the model occurring at a slower

scale can include terminal activations, deactivations and movements, BS deployments and

outages, gradual changes in the large-scale channel components among others. Finally, a

more realistic shadowing model is needed to reflect gradual variation of shadowing over

space and the time as well as the diverse radio propagation environments in the network.

Additionally, we consider and enhance a strategy for spectrum resource allocation [2]

for the described irregular cellular network model. The particular resource allocation

strategy is based on the idea of independent units of cells formed for coordinated resource

allocation, which are known as scheduling cells or clusters. According to the framework,

the scheduling cells, which represent a variation of the well-known cellular architecture,

are updated dynamically with the changing network conditions at a large time scale, and

the resource allocation takes place at the faster time scale of the multipath fading. Due

to the existence of BSs with multiple power classes deployed in an irregular fashion, it

is proposed that the adaptive resource allocation framework include large-time-scale BS

power adaptation for the optimization of user performance.

The parts of the work are summarized in the following:

• Initially, we discuss the BS outage as a type of dynamic change in the irregular

cellular network, formulate a BS outage compensation problem and propose an

algorithmic framework for the problem. Effectively, BS outage compensation is

used as a specific case of the dynamic system model and adaptations for irregular

cellular networks that are studied in the rest of the thesis. This work was published,

in part, in [3].

• We extend the adaptive resource allocation framework from [2], proposing BS power

adaptation as an additional form of adaptation for the irregular cellular network

with either a single or multiple power classes. This work is part of a journal paper [4]

that will be submitted for publication.

• We propose BS and terminal deployment models that are appropriate for networks

with either single or multiple power classes. This work is also part of the journal

paper [4] that will be submitted for publication.

• We propose a model for large-time-scale network evolution for either a single or


multiple power classes, which consists of such dynamic changes as terminal arrival,

departure and movements as well as BS deployments and outages. This contribu-

tion was published, in large part, in [5], and forms part of the contribution in [4].

• We propose path loss and shadow fading models for the case of multiple power

classes. This work has been accepted to be published in [6].

1.3 Contributions

The contributions of the thesis are summarized in the following.

Base Station Outage: Network Model and Compensation Algorithm

Most works in the literature consider the base station (or cell) outage compensation

(COC) for homogeneous or regular networks. Furthermore, even works that consider

irregular networks do not explicitly consider the modelling aspect of the problem. We

focus on the modelling of a BS outage for an irregular network with multiple power

classes in a general sense. Both the positions of BSs and the power classes are modelled.

In addition, we consider the adaptive resource allocation strategy as part of the overall

model. Specifically, we adopt the concept of cluster or scheduling cell for the purpose of

coordinated adaptive resource allocation. To our knowledge, the use of a scheduling cell

is new for the COC literature. Direct sum log utility maximization based on proportional

fairness is the primary formulation used in finding a compensating cluster, outage user

cell associations, and the adjusted BS power levels. The use of proportional fairness

in the COC formulation is additionally different from the literature for COC. Finally,

the shadowing model with spatial correlation for irregular deployment is considered for

greater realism. Base station outage compensation is discussed in Chapter 3.

Deployment Model

The most common model in the literature for irregular deployments is the PPP model

due to its simplicity and analytical tractability. However, the PPP model was shown

to be unrealistic in real irregular networks. Other more realistic models exist (with the

primary example being the Ginibre process [7]) however most such models have the issue

of being difficult to handle. In this work, we consider the development of a baseline

deployment model for Monte Carlo simulation with the primary consideration being the

model simplicity and a baseline realism, which is implemented through arbitrariness

or randomness of deployment with a minimum separation between the nodes and the


elements in the network. Deployment scenarios both with hotspots (multiple power

classes) and without hotspots (single power class) are considered. Since this is meant as

a generic model, guidelines for parameter values are provided for baseline realism of the

simulation scenario. Deployment models are discussed in Chapters 4 and 6 for the case

of single power class and multiple power classes, respectively.

Mobility Model

Most other models in the literature do not consider terminal movements over an arbitrar-

ily large time scale as a snapshot. We propose a generic model for individual mobility

(similar to the random walk model) that is meant as a generic model appropriate for

a time scale that is orders of magnitude larger than the multipath fading time scale.

The primary consideration for the model is its simplicity. A similar movement model is

also then considered for BSs. Deployment scenarios both with hotspots (multiple power

classes) and without hotspots (single power class) are considered for the movements.

Since this is meant as a generic model, guidelines for parameter values are provided for

the baseline realism of the simulation scenario. Mobility models are discussed in Chapters

4 and 7 for the case of single power class and multiple power classes, respectively.

Shadowing Model

Most of the work in the literature considers the uncorrelated lognormal shadowing model.

While a few authors proposed correlated shadowing models for links with a common end

and more recently, for links without a common end, to our knowledge, no work considers a

theory for variation of the shadowing parameters, i.e. standard deviation and correlation

distance, in a network with an arbitrary number of radio propagation environments.

Such a theory is proposed based on a hypothesis for the shadowing properties, and a

method of generation of shadowing for radio links is provided based on the work in [8].

Finally, model consistency is verified for a particular scenario, and the implementation

in practical simulations is discussed. The model is intended for the baseline realism of

the simulation scenario. The shadowing model for multiple power classes is discussed in

Chapter 6.

Adaptive Resource Allocation Framework with Power Adaptation

An adaptive resource allocation framework based on clustering and power control at a

large time scale and coordinated proportional fair resource allocation at a small time scale

is proposed for a generic irregular network. Note that in heterogeneous networks, the cell


association is normally determined based on maximizing long-term received power and

membership to different tiers is explicitly considered. In this work, we assume associa-

tion to a cell of any tier based on maximization of long-term channel gains. The primary

contribution is the power control algorithm for the purpose of adaptive resource alloca-

tion at a large time scale. While much of the literature considers power control at the

multipath fading time scale, the acquisition of channel information at this scale can lead

to substantial signaling requirements. In addition, much of the literature considers static

or semi-static allocation of a dedicated spectrum for the purpose of resource allocation.

Our contribution considers the use of the entire spectrum for greater spectral efficiency

and the power adaptation at a large time scale, to reduce the signaling requirement. As

the adaptive framework is to be used for a general or arbitrarily large time scale, the

availability of individual QoS requirements at such periodicity are not assumed. Instead,

a central entity does its best effort to maximize either the long-term sum rate or the

sum log utility (based on proportional fairness) over the time scale under the modest

assumption of base station power constraints. The framework is intended as a baseline

adaptive resource allocation framework for an irregular network. The adaptive resource

allocation framework with power adaptation is discussed in detail in Chapter 5.

1.4 Scope

• Multiple antenna technologies such as beamforming or MIMO are not explicitly

considered.

• Downlink communication is exclusively considered.

• Orthogonal frequency division multiple access (OFDMA) scheme such as that used

in 3GPP LTE is considered for the analysis.

• Full buffer traffic is considered for scheduling.

1.5 Outline

In Chapter 2, we provide a background for resource allocation strategies and the high-

level reference model for cellular networks. We begin by giving preliminaries for the

cellular network in Section 2.1. The basic definition is specialized to the cases of regular

and irregular cellular networks. Finally, the computation of data rates for user scheduling

in cellular networks is described. We then review the scheduling metrics and resource


allocation strategies utilized in cellular networks in Sections 2.2 and 2.3, respectively.

An adaptive resource allocation framework for irregular BS deployment is discussed in

detail in Section 2.4. General slow-time-scale adaptive algorithms in the literature are

then reviewed in Section 2.5. In the later part of the chapter, we introduce the high-

level reference model for the regular cellular network paradigm in Section 2.6, which is

represented in the ITU model. Then the considerations for the for the irregular cellular

networks are summarized in Section 2.7. Finally, the literature on a model for correlated

shadowing that is appropriate for irregular BS deployment is reviewed in Section 2.8.

Chapter 3 presents a model for a BS outage and a framework for compensation of the

outage. Section 3.1 discusses the relevant work, contribution and key results. Section 3.2

starts the chapter with an introduction. In Section 3.3, the problem statement is given

and the proposed algorithmic approach is discussed. The proportional fair algorithm for

BS outage compensation is developed in Section 3.4. The path loss minimizing com-

pensation algorithm is proposed in Section 3.5. The algorithms are evaluated through

numerical simulations in Section 3.6.

Chapter 4 presents a system model which includes time evolution for the irregular

network with a single BS power class and no power level adaptation. Sections 4.1 and 4.2

discuss the relevant work and the approach and contributions. Section 4.3 introduces the

topic. The method for drop deployment is introduced in Section 4.4. The time evolution

methodology is developed in Section 4.5. The typical parameter values for the system

model are discussed in Section 4.6. Finally, the outline of the network model of the thesis

is given in Section 4.7.

Chapter 5 presents algorithms for BS power level adaptation and an extended adaptive

resource allocation framework for networks with a single BS power class. Sections 5.1

and 5.2 discuss the relevant work, contributions and key results. Section 5.3 introduces

the topic. The power adaptation algorithms are developed in Section 5.4. The overall

adaptive resource allocation framework including the power adaptation is presented in

Section 5.5. Finally, simulations are performed in Section 5.6.

Chapter 6 presents the system model without time evolution for cellular networks

with multiple BS power classes. Sections 6.1 and 6.2 discuss the relevant work and con-

tributions, respectively. Section 6.3 introduces the topic. In Section 6.4, drop deployment

with explicit modelling of hotspots is discussed. In Section 6.5, the distance-dependent

path loss model is specialized to the network with multiple BS power classes. The model

of correlated shadowing with inhomogeneous channel parameters is detailed in Section

6.6.

Chapter 7 presents the final system model with time evolution for networks with


multiple BS power classes. Section 7.1 discusses the relevant work, contribution and key

results. Section 7.2 begins the discussion of the chapter. The modified time evolution

methodology is detailed in Section 7.3. In Section 7.4, the typical parameter values for

the model are discussed. Illustrative simulations results are presented for the network

with multiple power classes in Section 7.5.

Chapter 8 concludes the thesis with a summary of the work and possible future

research directions.

Chapter 2

Background and Preliminaries

In this chapter, we provide a background for resource allocation strategies and the high-

level reference model for cellular networks. We begin by giving preliminaries for the

cellular network in Section 2.1. We then review the scheduling metrics and resource

allocation strategies utilized in cellular networks in Sections 2.2 and 2.3, respectively.

An adaptive resource allocation framework for irregular BS deployment is discussed in

detail in Section 2.4. General slow-time-scale adaptive algorithms in the literature are

then reviewed in Sections 2.5.

In the later part of the chapter, we introduce the high-level reference model for the

regular cellular network paradigm in Section 2.6, which is represented in the ITU model.

Then the considerations for a high-level model for the irregular cellular network are

summarized in Section 2.7. Finally, the literature on a model for correlated shadowing

that is appropriate for irregular BS deployment is reviewed in Section 2.8.

2.1 Cellular Network Preliminaries

In this section, we discuss the basic cellular network model studied in the thesis. Consider

a network consisting of a set of BSs N and a set of terminals K. The positions of

each BS n ∈ N and each terminal k ∈ K are represented by the Cartesian coordinates

(xn, yn) and (xk, yk), respectively. Each BS is said to belong to a power class, s, which

is characterized by a distinct maximum transmit power capability, Ps,max. The set of all

available BS power classes is denoted by S. The power class of a given BS n is denoted

by sn. By definition of a power class, BS n’s operating power level, Pn, must satisfy

0 ≤ Pn ≤ Psn,max. Under practical network operating conditions, terminal positions,

(xk, yk), evolve due to terminal movement. Each terminal k is dynamically assigned to

a single BS, denoted by n(k), for the purpose of scheduling and resource allocation. In

10

Chapter 2. Background and Preliminaries 11Regular Network (Traditional)

Fixed cell

boundaries

Cell membership is

straightforward

Classical frequency

reuse

Scheduling is

organized in

straightforward way.

8

Figure 2.1: Regular cellular network deployment. The black and red points represent aBS and terminals, respectively.

the following, the basic model introduced in this section is specialized for the cases of

regular and irregular cellular networks.

Regular Cellular Network

Cells in the regular network are modelled with the hexagonal cell structure of Figure 2.1;

therefore, the base station coordinates, (xn, yn), coincide with the hexagon centres. The

number of power classes in the regular network (or the number of elements in set S) is

equal to one, that is, S ≡ |S| = 1, and the maximum transmit power capability of each

BS is represented by a constant, Pmax. Finally, the operating power level, Pn, of each BS

is typically held constant at Pmax.

Irregular Cellular Network

The focus of this thesis is the study and modelling of the irregular network. In the

irregular network, BSs are deployed in an inhomogeneous or arbitrary pattern, hence,

the BS coordinates, (xn, yn), are irregular as shown in Figure 2.2. In addition, BSs

belong to a number of different power classes, which are characterized by a wide range

of maximum power capabilities. Therefore, in the irregular network, S = |S| > 1. It will

also be proposed later in the thesis that the BS power levels, Pn, be dynamically adapted

in the range 0 ≤ Pn ≤ Psn,max, according to changing network conditions.

Chapter 2. Background and Preliminaries 12

Irregular Cellular Network

1. Dynamic BS assignmentsbased on

max channel gain

2. Dynamically formclusters

(scheduling cells)

3

Figure 2.2: Representation of the irregular cellular network deployment. The green circlesrepresent terminals, and the blue squares of different sizes represent BSs of different powerclasses.

Computation of Transmittable Data Rates for Scheduling

In this thesis, we adopt the organization of the time-frequency resource allocation from

3GPP LTE standards [9]. Terminal scheduling is performed at the LTE subframe (which

corresponds to the assumed time scale of multipath fading variation) and requires the

knowledge of transmittable data rates for links between all terminals and BSs. Note

that computation of transmittable data rates further requires the knowledge of channel

gains for all links. Let C be the set of available subcarriers in orthogonal frequency

division multiplexing (OFDM). The channel power gain between BS n ∈ N and terminal

k ∈ K on subcarrier c ∈ C is computed as the product of the distance-dependent path

loss component, PLn,k, the large-scale shadowing component, SFn,k, and the small-scale

multipath component, mn,k,c:

gn,k,c = PLn,kSFn,kmn,k,c. (2.1)

where the models for the channel components are elaborated later in Section 2.6. The

transmittable rate, between terminal k and BS n on subcarrier c, has been approximated

in [2] for the LTE system, through curve fitting of the Shannon rate formula with an

SINR gap, γ:


rk,n,c ∼= Bc min

(log2

(1 +

SINRk,n,c

γ

), 5.5547

), (2.2)

where

γ = 2, (2.3)

SINRk,n,c =pn,cgn,k,c

N0 +∑

j∈N, j 6=n pj,cgj,k,c. (2.4)

Here, BC is the subcarrier bandwidth, pn,c is the power transmitted on subcarrier c by

BS n and N0 is the noise power on each subcarrier at the receiver.

2.2 User Scheduling Metrics

There has been much research by various authors [10–13] on the appropriate metric to

be used for resource allocation including user scheduling. In the following, we consider

full-buffer persistent data traffic conditions. Two significant scheduling metrics known

as total rate maximization and proportional fair scheduling are described under such

conditions.

2.2.1 Total Rate Maximization

The traditional scheduling strategy has been the maximization of the total system rate in

the long term [14], which shall be referred to as the total rate maximization (TRM). Under

this strategy, the average rate performance of the system as a whole is optimized without

regard for any single individual user’s performance. The long-term TRM optimization

objective is given by

max∑k∈Kn

Rk for all n ∈ N . (2.5)

where Kn = k|n(k) = n is the set of terminals assigned to BS n for scheduling, and Rk is

the time-averaged rate for terminal k. Consider a single-frequency system. The terminal

scheduled by BS n at subframe t is determined with the knowledge of the transmittable

rate, rk(t), as

k∗n(t) = arg maxk∈Kn

rk(t) for all n ∈ N , (2.6)

which has been shown to be equivalent to the long-term optimization of (2.5). In prac-

tice, there is a significant problem with the use of the total rate maximization of (2.5)

and (2.6). Consider the simple hypothetical scenario consisting of a single BS and two


BS

Terminal 1 Terminal 2

Figure 2.3: Hypothetical scenario for terminal scheduling.

stationary terminals depicted in Figure 2.3. Terminal 1 is much closer to the BS than

Terminal 2, and therefore Terminal 1 will tend to have much higher channel gain on

average. We additionally assume that the channel gain fluctuations over time are small

in comparison to Terminal 1’s channel gain. According to the TRM criterion of (2.6),

therefore, Terminal 1 would be scheduled at all of the subframes, and Terminal 2 would

not be scheduled at any of the subframes. This situation where Terminal 2 is deprived of

service is commonly referred to as the fairness problem in the literature [10,15]. Propor-

tional fair (PF) scheduling criterion [14], which was proposed in order to overcome the

fairness problem, is discussed in the next section.

2.2.2 Proportional Fair Scheduling

According to the proportional fair (PF) criterion, scheduling should maximize the sum

log utility (SLU) function, which is computed as the sum of the logarithm of the average

user rates. Therefore, the long-term optimization objective is

max∑k∈Kn

log(Rk) for all n ∈ N . (2.7)

Since the logarithm function increases at a continually decreasing (or diminishing) rate,

PF scheduling overcomes the fairness problem encountered in TRM. In particular, con-

sider the scenario depicted in Figure 2.3. Terminal 2, i.e., the user with the weak channel,

is said to be treated fairly, because a given increase in user data rate, for example, 10

kbps, for Terminal 2 (the weak user) results in a greater increase in utility than the same

amount of data rate increase for Terminal 1 (the strong user). The concept of the de-

creasing rate of utility increase is known in economics as the law of diminishing marginal

returns or law of diminishing marginal utility [16]. Under SLU maximization, Terminal

2 is therefore expected to achieve a non-zero average data rate. In fact, as discussed


in detail in [17], both terminals achieve a long-term rate that is approximately propor-

tional to their average transmittable rate based on the respective channel gains, which

is the reason for the term proportional fairness. For more details on the properties of PF

scheduling, the reader may refer to some of the many works on the topic [10–15,17].

For a single-frequency system, the optimization of (2.7) is realized by scheduling

terminal k∗n(t) at each subframe t, with the knowledge of instantaneous achievable rate

rk(t) and time-averaged rate Rk(t), that is,

k∗n(t) = arg maxk∈Kn

rk(t)

Rk(t)for all n ∈ N (2.8)

Note that the time-averaged rate is updated at each subframe based on an exponential

moving average:

Rk(t+ 1) =

(1− 1

T0

)Rk(t) +

(1

T0

)rk(t) (2.9)

where T0 is the averaging window size selected for smooth averaging. In a system with

multiple frequencies, the scheduling is performed either independently in each subcarrier

at the subframe or through more elaborate strategies. Note that the PF scheduling will

be used in the rest of the thesis.

2.3 Resource Allocation Strategies

In the traditional cellular network, the BSs are deployed according to a regular pattern,

which is modeled with hexagons. The strategy of classical frequency reuse is utilized.

According to this scheme, the BSs are partitioned into groups of geographically adjacent

cells, known as frequency reuse clusters. The frequency reuse factor (FRF) defines the

size of the clusters, and the available spectrum is distributed among the BSs according to

pre-defined patterns. The FRF is said to be an indicator of the level of frequency reuse

in the network. In early networks, high (or conservative) reuse factors were used for the

avoidance of excessive interference. However, the scheme did not consider the variation

of traffic load in the cells. In addition, the ever growing demand for capacity necessitated

more aggressive frequency reuse, which would cause degraded performance for terminals

at the cell edges.

More recently, the fractional frequency reuse (FFR) technique was proposed as an

alternative to classical frequency reuse. In this scheme (also referred to as static FFR),

two different frequency reuse patterns are applied: a higher (or conservative) reuse factor

for the cell-edge terminals with weak channel gains, and a lower (or aggressive) reuse


factor for the stronger or cell-centre terminals. The drawback of the method was that

the terminal partitioning was based solely on distances from the BSs, and the spectrum

assignments again did not adapt to the variations of traffic load.

To overcome the limitations of static FFR, techniques have recently been proposed

in the literature that attempt to adapt to the variation in traffic, by using different

methods to organize the frequency resource allocation. This goal is commonly achieved by

coordination of resource allocation between neighboring BSs, which is also referred to in

the literature as inter-cell interference coordination (ICIC). Different types of techniques

are reviewed in the following.

Dynamic FFR

Dynamic FFR schemes attempt to adapt to variations of traffic while maintaining the

general concept of FFR in the method. Boudreau et al. [18] have proposed an adaptive

frequency reuse strategy for interference coordination suitable for 4G networks. The

scheme switches between three different frequency-power profiles according to the traffic

load in each cell.

Ali and Leung [19] present an elaborate dynamic frequency allocation technique while

maintaining the general idea of fractional frequency reuse. The frequency allocation is

determined according to the average performance of all terminals in the network on all

available frequency resources.

Two-Level Resource Allocation

A number of works in the literature deal with resource allocation at two levels (phases)

or time scales. The common idea of these methods is that a frequency reuse pattern is

decided based on slow-varying traffic distributions in the first phase, and at a faster time

scale, the second phase fine-tunes the resource allocation inside each cell. Bonald et al.

propose a two-level scheme for multi-cell coordination for the purpose of scheduling [20].

In the first phase, the activity of the interfering BSs is determined through interference

coordination with the goal of transmission rate maximization. A TDMA transmission

scheme is assumed. In the second phase, the load balancing is performed in order to

divert traffic from heavily-loaded to lightly-loaded cells.

Li and Liu extend the two-level resource allocation framework to a multi-cell OFDMA

system [21]. In the first phase, the available spectrum is assigned to the terminals in the

network by the radio network controller. In the second phase, the BSs independently

modify the channel assignment in each cell according to the buffer sizes of the active


terminals.

Scheduling and resource allocation requires the knowledge of SINRs, which depend

on the channel information for both the desired signal links and the interference links.

It has, therefore, been noted that for ideal resource allocation, BSs need to have accu-

rate network-wide channel information at every instant. However, acquiring the channel

information from out of the cell can lead to an excessively large signalling complexity in

practice. While most of the recent resource allocation techniques assume full knowledge

of the channel information in the network to simplify the problem, Chang et al. [22]

devise a scheme that takes this aspect into account. A two-level framework based on

graph theory is proposed. In the first phase, intercell interference is reduced with no

knowledge of out-of-cell interference, based solely on the geographical locations of termi-

nals. In the second phase, the resource allocation is performed according to knowledge

of instantaneous channel gains.

2.4 Adaptive Resource Allocation Framework for Ir-

regular BS Deployment

A solution for coordinated resource allocation for a network with irregular deployment,

which considered the issue of channel information signaling, was developed in [2]. An

important goal of the work was to design a resource allocation framework that could

adapt to the variations of traffic load in the cells, which is comparable to the adaptive

resource allocation strategies in the previous section. The primary design constraint

was that the resource allocation was required to be flexible and adaptive enough to be

suitable for irregular BS deployment. The case of a single BS power class, with BSs

placed uniformly at random without restriction, was primarily considered in the work.

Note that a practical application could be in a network with both small and large cells.

A workable strategy to reduce the signalling requirements is to periodically group

the BSs in the network into separate clusters, also referred to as scheduling cells (SCs).

An OFDMA system such as the 3GPP LTE was considered. Inside each SC, channel

information is exchanged and the resource allocation is coordinated. However, SCs act

independently from one another during the course of scheduling at the LTE subframe

level. The set of SCs is updated dynamically with the changing traffic distribution at

a slower time scale. A complete adaptation framework that incorporates PF scheduling

was developed. The flowchart outlining the adaptive framework is depicted in Figure 2.4.

At the large time scale, terminals are assigned to BSs according to the long-term channel


l = l + 1

Coordinated resource

allocation in each cluster

Form clusters

(scheduling cells)

Assign terminals to BSs

l mod F = 0?

No

Yes

Figure 2.4: Flowchart for the adaptive resource allocation framework.

gains, and the SCs are formed based on the interference or SINR caused by each cell on

every other cell. Then, at the subframe time scale, proportional fair resource allocation

is performed with coordination or exchange of channel information inside each SC. The

scheduling has been combined with on-off power switching and is executed in parallel

in each subcarrier. Note that the power switching is designed to allow added flexibility

due to finer variations of traffic distributions. The algorithmic framework for adaptive

resource allocation is detailed in Figure 2.5, which lists both the slow-scale adaptation and

fast-scale resource allocation steps. The details of the fast resource allocation algorithm

are given in Figure 2.6. In the following, other important details are discussed in relation

to (1) the clustering algorithm, (2) the fast resource allocation algorithm and (3) the

requirements for channel information signalling.

Clustering Algorithm

The clustering algorithm specified in Figure 2.5 operates on the principle that BSs cre-

ating a lot of interference on each others’ terminals should be assigned to the same

cluster (so that interference can then be avoided through fast scheduling). Therefore,

the algorithm needs as inputs the set of BS assignments n(k)k∈K, the long-term time-

averaged channel gains, Gn,k = PLn,kSFn,k, for all n and k, and the BS power level


INPUTS:

• Distance-dependent path loss components, PLn,k, and shadowing components,SFn,k, for all n ∈ N and k ∈ K

• Vector of operating BS powers P = [Pn] = [Psn,max]

• Clustering algorithm parameters Q and T

• Proportional fair (PF) resource allocation window size parameter T0

OUTPUTS:

• Set of assigned BSs n(k)k∈K

• Scheduling cells (SCs) N (ω) for ω = 1, . . . ,Ω, forming a partition of the set N

• Optimized instantaneous BS power vectors p(ω)c

∗(l) = [p∗n,c(l)] where n ∈ N (ω);

and sets of scheduled users K(p(ω)c

∗, l) = k∗n,c(l)n∈N (ω); corresponding to each SC

ω = 1, . . . ,Ω, each subframe l ∈ F , and each subcarrier c ∈ C

1: for all terminals k ∈ K do2: for all BSs n ∈ N do3: Compute the time-averaged channel gain according to Gn,k = PLn,kSFn,k4: end for5: Determine the assigned BS according to n(k) = arg maxn∈N Gn,k.6: end for7: Form the SCs by using Algorithm 3 of [2].8: for SCs ω = 1 to Ω do9: for subframes l = 1 to F do

10: Determine the optimized instantaneous power vectors and the sets of scheduledusers by using the algorithm of Figure 2.6.

11: end for12: end for

Figure 2.5: Algorithm for the adaptive resource allocation framework


INPUTS:

• Set of coordinated BSs A = N (ω)


• BS power level vector P = [P1, . . . , PA]

• Fast-changing channel gains, gk,n,c, ∀k ∈ K,∀n ∈ A,∀c ∈ C

OUTPUTS:

• Optimized instantaneous BS power vectors, p∗c(l) = [p∗1,c(l), . . . , p∗A,c(l)]

T , ∀c ∈ C

• Sets of scheduled users, K(p∗c , l) = k∗n,c(l)n∈A, ∀c ∈ C

1: for all subcarriers c = 1 to C do2: Initialize maxSum to 0.3: for all vectors pc(l) ∈ 0, Pn/C2A×1 do4: for all n ∈ A do5: Set k∗n,c(l) = arg maxk∈Kn

rk,n,c(l)

Rk,n(l)

6: end for7: Form the candidate user set K(pc, l) = k∗n,c(l)n∈A8: Compute newSum =

∑k∈K(pc,l)

rk,n,c(l)

Rk,n(l)

9: if newSum > maxSum then10: Set maxSum to newSum, p∗c(l) to pc(l), and K(p∗c , l) to K(pc, l)11: end if12: end for13: end for14: for all terminals k ∈ K do15: Update the time-averaged user rate based on the scheduled user rates by using

Rk,n(k)(l + 1) =(

1− 1T0

)Rk,n(k)(l) +

(1T0

)∑c|k∈K(p∗

c ,l)rk,n(k),c(l)

16: end for

Figure 2.6: Algorithm for fast PF resource allocation for a given subframe l ∈ F .


vector P = [P1, . . . , PN ], for the computation of long-term SINRs. We let F denote the

set of subframes in the analysis. According to the proposed method, the set of clusters

are determined by running the K-means clustering algorithm [23] on an SINR-based BS

similarity matrix. Note that the clustering algorithm requires two parameters, Q and T ,

as inputs: Parameter Q limits the maximum number of clusters in each iteration of the

algorithm. Parameter T adjusts the tendency of BSs to join a cluster. It is, therefore,

possible to adjust the cluster sizes by tuning parameters Q and T together. Detailed

algorithm steps are found in [2].

Fast Resource Allocation

The fast PF resource allocation specified in Figure 2.6 is performed independently at each

SC ω. Note that to simplify the exposition, the set of coordinated BSs is denoted by

A = N (ω), and the set of their assigned terminals by K in Figure 2.6. Additionally, the

SC index ω is dropped in the rest of the variables. The algorithm steps are shown for a

given subframe l ∈ F . PF scheduling discussed in Section 2.2.2 together with binary (on

or off) PF power switching is used. At the end of the power optimization and scheduling,

the time-averaged rate of each user is updated with the total scheduled user rate over

the subframe.

Channel Information

In summary, the requirements for channel information signalling is significantly reduced

due to clustering. In order to perform BS assignment and clustering, the long-term

channel gains, Gn,k, between all BSs and terminals need to be known at a central entity

at every F subframes—which is a low frequency. In contrast, the estimate of fast-varying

channel gains, gn,k,c, need to be exchanged at every subframe only for BSs and terminals

inside the given SC.

2.5 General Slow-Time-Scale Adaptive Algorithms

There has been a recent recognition in the wider community of the need for adaptation

at time scales larger than the fast multipath fading time scale (i.e. at the LTE subframe).

Such slowly-adaptive algorithms have prominently been studied in the context of 3GPP

LTE Self Organizing Networks (SON) [24]. In Sections 2.3 and 2.4, we discussed resource

allocation frameworks that commonly work at two different time scales. The adaptations

at the slow time scale such as the dynamic re-organization of the frequency reuse patterns,


and dynamic BS assignments and clustering can be considered similar to SON functions1.

More generally, the main driving force for SON has been the need to automate the

configuration, optimization, maintenance, troubleshooting and recovery of the cellular

system. Automation of tasks previously performed manually is expected to enable more

agile adaptation in face of the greater dynamism of emerging irregular cellular networks,

and simultaneously to reduce operating and other costs incurred by wireless operators. A

wide variety of SON functions have been proposed in the literature [25–29]. In the most

common taxonomy, the functions are classified based on the phases corresponding to the

life cycle of the cellular system equipment, that is, deployment, operation, maintenance,

redeployment, recovery etc. The adaptive functions and algorithms that organize these

phases are classified into (a) self configuration, (b) self optimization and (c) self healing.

Each of the categories is reviewed in the following, together with function and algorithm

contributions from the SON literature.

Self Configuration: Self configuration functions are primarily executed at the de-

ployment and re-deployment phases of the network equipment life cycle. A number of

different parameters can be configured, including radio propagation parameters such as

antenna type, antenna gain, and antenna azimuth and tilt angles. The authors in [25]

proposed an antenna tilt optimization method for the purpose of achieving higher system

capacity and better coverage. In particular, they demonstrated an approach based on

simulated annealing that uses measurements of terminal SINR information. An impor-

tant aspect of the method is that it was specifically designed to utilize readily available

terminal measurements in an online manner. The authors in [26] proposed an alterna-

tive game theoretic method for antenna tilt optimization. Furthermore, they proved the

existence of a Nash equilibrium which can be achieved in a non-cooperative game for the

optimization of system-wide utility function.

Self Optimization: Self optimization is primarily executed at the operation phase of

the network life cycle, and involves continuous optimization of system parameters after

their initial configuration, in order to ensure efficient performance of the system. A variety

of parameter optimizations for realization of distinct goals are possible. The authors in

[27] proposed a scheme for balancing the load across multiple cells, and avoiding intercell

interference, by performing intercell and intra-cell handovers in a partial frequency reuse

(PFR) scheme for OFDMA. Authors in [28] developed a method for adaptive organization

of an fractional frequency reuse (FFR) scheme for the purpose of interference control.

1There have been few attempts to formulate a precise definition of Self Organization (SO). One suchattempt [24] emphasizes the need for stability, scalability and agility of the algorithms to be consideredSO, distinguishing it from the non-self-organized type of system adaptation.


Table 2.1: Classification of time scales.

Time Scale System Dynamics Adaptation

Fastest Multipath fading User scheduling

Medium

Slow terminal movements Self optimization functions

Terminal activation Cell assignments

Terminal deactivation Clustering

Slowest

BS Deployments Self configuration

BS Outages Self healing

BS Movements Cell outage management

The basic idea of the algorithm was to dynamically create efficient FFR patterns in order

to adapt to the changing user traffic distributions and system conditions.

Self Healing: Self healing functions are primarily executed during recovery from faults

and failures that can be due to component malfunctions or natural disasters. Self heal-

ing is a systematic process which is comprised of several stages including (a) remote

monitoring, (b) detection, (c) diagnosis and (d) triggering of compensation actions for

the clearing of the fault or failure. The reference [29] considered the self healing for the

fault referred to as a cell outage. A cell outage is defined as a rare and catastrophic

failure of a BS, which results in ceasing of all signal transmission and reception at the

BS. The authors in [29] provided a full description for the management of cell outages

in LTE networks, reviewing both detection and compensation algorithms, and discussing

the role of operator policies in the design of detection and compensation schemes.

We note that self optimization functions are generally intended for execution on a

relatively frequent basis during the operation of the wireless network, e.g. after a cer-

tain number of subframes. In contrast, self configuration and self healing functions are

commonly executed infrequently as events such as BS deployments and outages tend to

be rare. A classification of dynamism and adaptation with approximate demarcation of

different time scales is provided in Table 2.1. Note that the multipath fading time scale

is included in the taxonomy. Terminal movements, activations and de-activations are

classified as medium scale dynamics due to their relative high frequency. BS dynamics

such as outages, deployments and movements are classified as slow-scale dynamics due

to the low frequency of occurrence. Note that BS movements are not common in today’s

cellular networks, however, they are included with the thought that they may become

feasible for emerging and future networks. According to classification of the adaptation

functions, user scheduling occurs at the fastest time scale, self optimization in SON oc-


curs at the medium time scale, and self configuration and self healing at the slowest time

scale. The medium- and slowest-scale dynamics can be considered as the focus of the

adaptations in this dissertation.

2.6 The ITU System Model: High-Level Reference

Model for the Regular Cellular Network

The ITU system model [1] provides a high-level reference for the modelling of a cellu-

lar system, which was designed for the regular cellular network paradigm. The model

will be used in this thesis as a starting point for developing a high-level model for the

irregular cellular network. The ITU model is based, in part, on the idea of generating a

large number of snapshots, known as simulation drops, for the purpose of Monte Carlo

simulation. A simulation drop, in this context, is defined as a random deployment of

mobile terminals on a network area with fixed BS deployments, following the hexagonal

cell pattern. Within each drop, the set of active terminals and their deployments as well

as large-scale channel components such as path loss and shadowing are fixed, and there

are only virtual terminal movements, which result in the multipath fading due to the

Doppler effect. As mentioned in Section 2.1, the time scale for multipath fading corre-

sponds to a subframe in LTE, which is the basic time unit for user scheduling. In order

to obtain statistically representative results at this time scale, a large number of LTE

subframes are simulated within each simulation drop. The flowchart of the simulation

methodology is given in Figure 2.7, where the number of drops used in the simulation is

denoted by the variable P , and the number of LTE subframes per drop by F . Notably,

the simulation drops themselves are generated independently from each other, that is, as

uncorrelated snapshots of network configurations. In addition, the model uses standard

large-scale channel models, in particular, for shadow fading, that are static in time and

statistically uncorrelated in space. In the following, we review the channel modelling

in the ITU model, considering distance-dependent path loss, shadowing and multipath

fading.

2.6.1 Channel Modelling

Distance-Dependent Path Loss

A typical expression for the distance-dependent path loss model is given as follows: Let

A be the path loss exponent parameter, B the intercept parameter, C the frequency


j = j + 1

Perform user scheduling

Generate subframe j of drop i (multipath fading)

Compute path loss and shadowing for all links

Generate simulation drop i

j mod F = 0?

No

Yes

i = i + 1

i =P? STOP

No

Yes

Figure 2.7: The flowchart for the drop-based simulation framework based on the ITUcommon reference model.

dependence parameter and X the environment-specific parameter. dn,k is the distance

between BS n and terminal k in metres, and fc is the system frequency in GHz. The

path loss from BS n to any terminal k is given by [30]

PLn,k[dB] = A log10(dn,k) +B + C log10

(fc5.0

)+X (2.10)

As an example, A = 20, B = 46.4, C = 20 and X = 0 are typical values for free space.

Shadowing

Shadowing channel gain is represented using the common log-normal model [31]. For the

link between a BS n and a terminal k, logSFn,k is, therefore, generated as a Gaussian

random variable with zero mean and standard deviation σ.


Multipath Fading

Modelling of fast-varying channel components mn,k,c at the subcarrier and subframe

granularity is discussed in this section.

Frequency variation in the wireless channel is due to the multiple path (multi-path)

components with different excess propagation delays, or different echoes of the transmit-

ted pulse. The distribution of the delay values for the echoes, weighed by signal power, is

referred to as the delay spread for the channel. The delay spread affects the rate of varia-

tion of the channel response with frequency. The frequency variation due to delay spread

is commonly modelled through statistical means with a frequency correlation function.

The channel time variation arises due to the motion of the terminal with respect

to each of the multipath channel components. In particular, the different signal multi-

path components undergo frequency shifts (Doppler shifts) that are proportional to the

relative velocity of the terminal with respect to the angle of arrival of the component.

The distribution of the Doppler shifts, weighed by signal power, is referred to as the

Doppler spread for the channel. The Doppler spread is related to the rate of variation

of the channel response in time. The time variation due to Doppler spread is commonly

modelled through statistical means with a time correlation function.

Finally, the spatial variation occurs due to the constructive and destructive interfer-

ence of the electromagnetic waves for the multipath components arriving at a receiver

with multiple antenna elements. Note that the situation with multiple antennas is not

considered in the thesis.

The literature contains many different models for time, frequency and space variation

of the wireless channel [9, 32–35]. However, the joint computation of channel variation

in all three variables can prove to be challenging. The ITU provides a channel model

appropriate for the OFDM scheme incorporating all three types of variations, which is

based on the concept known as ray clusters. Detailed explanation of the model is provided

in [1].

2.7 Considerations for an Irregular Cellular Network

Model

The irregular cellular network has characteristic that are very different from the regular

cellular network, which has the ITU model as a high-level representation. A more ap-

propriate high-level model would be able to provide a common network representation as

a reference for the consideration of different types of dynamical changes in the network


in the design of appropriate dynamic resource allocation and adaptation as well as SON

strategies.

The limitations of the existing ITU model is described in the following: First, the

hexagonal deployment pattern is inadequate due to the irregular deployment of BSs in

the new network paradigm. Second, more accurate channel models are needed due to

irregularity of BS deployment. In particular, a more realistic shadowing model than the

common lognormal model should be developed that can represent the gradual variation of

shadowing over space and the time as well as the diverse radio propagation environments

in the network. Third, the snapshot simulation method using simulation drops does not

accurately represent the dynamism in the network. Such dynamics are important for the

accurate evaluation of dynamic resource allocation frameworks, and other adaptation

and SON functions that were discussed in Sections 2.3-2.5. In particular, we note that

in implementing slowly-adaptive functions, it is important to minimize signaling com-

plexity and overhead resulting from, e.g., the exchange of channel information. This is

especially true if centralized decision making or coordination is required for the specific

implementation. The signalling complexity can potentially be reduced by considering

that the evolution of the cellular channel and environment is gradual in time, meaning

that each execution of the function can partially utilize the information from the previous

execution. It would, therefore, be possible to design algorithms with reduced signalling

complexity. To evaluate the performance of reduced-complexity adaptive algorithms, it

becomes necessary to accurately model the time evolution of the cellular system and

channel at such slow time scales.

In this thesis, we propose a new system model framework, which is appropriate as a

high-level model for the irregular network. The main considerations for the model are as

follows:

1. The deployment model should reflect both the arbitrariness of deployment and the

broad level of agreement between the positions of terminals and BSs (e.g. due to

planning).

2. Since the original simulation drop concept of the ITU model does not allow the

modelling of network time evolution at the slow time scale, an appropriate modifi-

cation of the concept should be devised.

3. The shadowing model should represent the shadowing correlation as well as the

diverse radio propagation environments present in the network.


2.8 Correlated Shadowing Model for Irregular BS

Deployment

The classical lognormal shadowing model gives no information on the correlation of shad-

owing over the network (also known as spatial correlation). Consider the special case of a

link between a fixed base station and a moving terminal as depicted in Figure 2.8: Due to

the inherent correlation in the topography of the physical environment, the channel gain

experienced by the terminal cannot change abruptly after having traveled a relatively

small distance. In 1991, Gudmundson proposed a method [36] to model the spatial cor-

relation of shadowing. According to the Gudmundson model, the correlation between the

links A-B and A-C shown in Figure 2.9, is given as a function of the distance d between

the links as

R(d) = σ2e−d/dc (2.11)

where σ is the shadowing standard deviation, and dc is an important parameter known

as the shadowing correlation distance. Note that the Gudmundson correlation model

considers the correlation between pairs of links with a common endpoint; specifically,

either between a terminal and multiple BSs, or between a BS and multiple terminals.

Other research in cellular networks [31, 37–39] expanded on Gudmundsons work, also

focusing on the correlation between pairs of links with a common endpoint. More recent

work [40] considered the mathematical feasibility of shadowing correlation models, as well

as the calculation of aggregate interference in scenarios with correlated shadowing [41–43].

However, the literature in cellular networks did not report on the correlation between

pairs of links with no common endpoint, illustrated in Figure 2.10, effectively neglecting

such correlation. Links without a common endpoint frequently have low correlation in

traditional cellular networks, due to the regular spacing and large distances between BSs.

Thus it can be acceptable to neglect such correlation in the model. However, in irregular

networks, BSs can, at times, be very close together. Therefore, the correlation between

two links with no common end cannot be neglected in such scenarios. An appropriate

correlation model incorporating non-common-endpoint links was proposed in [8], which

is discussed in the following.

We first define the potential field as a random process whereby each realization is a

function from the plane to real numbers. Thus, if we fix a point in the plane, the result is a

real random variable. According to the spatially correlated shadowing model, shadowing

is computed from a Gaussian potential field generated over the network area. Note that

the potential field is stored in a vector denoted by u. A homogeneous shadow fading


BS

Terminal at time t1

Terminal at time t2

Figure 2.8: The wireless link between a fixed base station and a moving terminal.

A

B

C

d

Figure 2.9: Wireless links with a common endpoint.

A

B

C D

Figure 2.10: Wireless links without a common endpoint.


Figure 2.11: A realization of the potential field [8].

scenario with fixed shadowing standard deviation, σ0 (in dB), and correlation distance,

dc, is considered. The value of the potential field at point A defines the potential level

of point A and is denoted by uA. For a cellular network with N BSs and K terminals,

J = N +K potential levels are generated according to the following steps:

• Step 1: Generate a J×1 vector v whose elements are independent Gaussian random

variables with zero mean and unity standard deviation.

• Step 2: Generate a J × J correlation matrix R = [rij] such that

rij =σ2

0

2e−dij/dc (2.12)

where dij is the distance between point i and point j.

• Step 3: Decompose the correlation matrix using Cholesky factorization such that

R = BBH.

• Step 4: Calculate the J × 1 potential level vector u according to

u = Bv (2.13)


A realization of the shadowing potential field is depicted in Figure 2.11. The shadowing

between point A and point B (normally, a terminal and a BS) is then computed as a

function of the two potential levels according to

SF [dB] = f (uA, uB) = sgn (uA + uB) · |uA − uB| . (2.14)

It was shown in [8] that the above potential field method of shadowing generation results

in the correct lognormal shadowing with spatial correlation, and the resulting shadowing

values satisfy a number of essential statistical properties that are detailed in the work.

We note that all the correlated shadowing models discussed in this thesis will be based

on the foundation of the potential field model discussed in this section.

Finally, it is important to mention that different radio propagation environments

would cause different radio-channel characteristics, and therefore, the values of shad-

owing channel parameters, σ0, and dc, need to be different based on the environments.

Typically, propagation environments are characterized by urban, suburban or rural, and

indoor or outdoor scenarios. Different environments also frequently have different sizes

of cells. The values of σ0, and dc (as well as other types of channel parameters for

multipath fading or distance-dependent path loss) are normally obtained by conducting

channel measurements in practical propagation scenarios [1, 30, 36, 44, 45]. Note that

the consideration of varying shadowing model parameter values will be important in the

development and implementation of the model in this chapter.

Chapter 3

Adaptation Case: BS Outage

Compensation

3.1 Relevant Work and Contribution

3.1.1 Relevant Work

A cell outage is defined as a rare and catastrophic failure of a BS, which results in ceasing

of all signal transmission and reception at the BS. Following the outage, the users assigned

to the outage BS lose service and adaptation of the network needs to be performed. The

type of adaptation has been referred to, in the context of 3GPP LTE SON, as cell outage

compensation (COC).

A small number of works in the literature have explored the COC problem. In ho-

mogeneous networks, the literature [29,46–48] focused on the development of algorithms

for cell outage compensation by tuning of various control parameters in different types

of systems, including antenna tilt, pilot power and uplink target received power, in order

to mitigate the performance effects due to the outage in accordance with operator pol-

icy. In the area of heterogeneous networks, the authors in [49] and [50] studied the cell

outage compensation problem and offered algorithm solutions based on the adjustment

of antenna gains and transmission powers.

3.1.2 Contribution

Most works in the literature consider the base station (or cell) outage compensation

(COC) for homogeneous or regular networks. Furthermore, even works that consider

irregular networks do not explicitly consider the general modelling aspect of the problem.

32

Chapter 3. Adaptation Case: BS Outage Compensation 33

We focus on the modelling of a BS outage for an irregular network with multiple power

classes in a general sense. Both the positions of BSs and the power classes are modelled.

In addition, we consider the adaptive resource allocation strategy as part of the overall

model. Specifically, we adopt the concept of cluster or scheduling cell for the purpose of

coordinated adaptive resource allocation. To our knowledge, the use of a scheduling cell

is new for the COC literature. Direct sum log utility maximization based on proportional

fairness is the primary formulation we used in finding a compensating cluster, outage user

cell associations, and the adjusted BS power levels. The use of proportional fairness in

the COC formulation is additionally different from the literature for COC.

Finally, the shadowing model with spatial correlation, discussed in Chapter 2, is

considered for greater realism especially of the irregularly deployed network.

3.1.3 Key Results

The results indicate that both COC algorithms show robustness toward cell outage in

terms of user average spectral efficiency, and outage user and non-outage user QoS vio-

lation probabilities for both a regular and an irregular network. In particular, the COC

algorithms are shown to achieve a large percentage (nearly 75%) of the user spectral

efficiency of the no outage scenario in both network types. In addition, the proportional

fairness-based COC algorithm is shown to allow the tuning, or trading off, of the average

number of BSs with raised power—which is a measure of the algorithm cost—against

different measures of performance including user spectral efficiency.

3.2 Introduction

Chapter 2 discussed adaptive resource allocation strategies that respond to network vari-

ations (such as due to traffic changes) for cellular networks. In addition, more general

adaptive functions in the 3GPP LTE SON literature were reviewed. The focus of the

discussion in this thesis is slow- and medium-time-scale adaptations. Typical adapted

parameters include transmit power, antenna gain, antenna tilt and other parameters re-

lated to the adaptive frequency re-use scheme that is employed. This thesis focuses on

dynamic adaptations with the final aim of improved spectral resource allocation, that is

appropriate for the irregular cellular network paradigm. In particular, BS assignments,

clustering and power adaptation (power control) will be mainly considered in the devel-

opment of the schemes.

In this chapter, we study BS (or cell) outage compensation as a specific case of


dynamic adaptation for the irregular cellular network. A BS outage is defined as a rare

and catastrophic failure of a BS, which results in ceasing of all signal transmission and

reception at the BS. Following the outage, the users assigned to the outage BS lose service

and adaptation of the network needs to be performed. The type of adaptation has been

referred to, in the context of 3GPP LTE SON, as cell outage compensation (COC).

Several works [29, 46–48] studied the concept of COC and proposed algorithms for its

implementation. In summary, the proposed COC algorithms aim to mitigate the effects

of the cell outage without the requirement for manual intervention. Particularly, the users

assigned to the BS that have lost service—which shall be referred to as outage users—

have their cell assignments switched to physically nearby BSs, known as compensating

BSs or cells. Adjustments are also made to a selected control parameter (e.g. BS transmit

power) in the compensating BSs.

In this thesis, a distinct approach for outage compensation is proposed, partially

based on concepts studied throughout the thesis, such as BS assignments, power adap-

tations and scheduling clusters. The authors of [46–48] showed that, in attempting to

compensate for the outage, there exists the risk of sacrificing the performance in terms of

data rate in the nearby compensating cells. For this reason, the concept of compensating

cells is refined in this work as follows: Instead of being classified, merely, as a group

of cells assisting in compensation, compensating cells explicitly form the compensating

cluster with resource allocation coordination—similar to the clustering concept discussed

in Chapter 2. In addition, the methods for outage user cell association and BS power ad-

justment in existing COC algorithms are typically based on traditional criteria, including

total rate maximization (TRM) or path loss minimization (PLM). Note that the path

loss minimization means the maximization of the long-term average (or expected value)

of the channel power gain between a BS and a user. However, for resource allocation,

proportional fairness (PF) (equivalent to maximization of the sum-log utility (SLU)) was

established to be a better criterion than TRM, as it has the property of achieving a

balance between two conflicting goals: the maximization of total rate (TRM) and the

maintenance of perfect fairness among users. In this work, we, therefore, propose a COC

algorithm based on the PF criterion, as the final aim of the work is the PF resource

allocation under the cell outage scenario.

The chapter first considers a model of the BS outage in a network with irregular

BS deployment and distinct power levels. Then methods are developed for BS outage

compensation, with an emphasis on the variation of architecture resulting from the use

of the compensating cluster. The rest of this chapter is organized as follows: The COC

problem formulation and proposed algorithmic approach is described in Section 3.3. The


BS n0d0

Figure 3.1: BS outage scenario. The light blue square is the outage BS (BS n0), thelight blue circles are the outage users, dark blue squares are the candidate BSs, dark bluecircles are the associated users, and the red squares and circles are the BSs and usersoutside the candidate set, respectively.

COC algorithm based on the PF criterion is developed in Section 3.4. The simplified COC

algorithm based on PLM is discussed in Section 3.5. Finally, the numerical evaluations

are presented in Section 3.6.

3.3 Problem Statement and Proposed Algorithmic

Approach

We consider that a BS nO has experienced an outage. Let KO be the set of users currently

associated with BS nO—otherwise known as outage users. Let NC be the set of candidate

BSs, which are specified as those BSs located within a pre-determined distance, dO, from

BS nO (based on the channel power). The BS outage is illustrated for a sample scenario

in Figure 3.1. The COC problem—consisting of joint cell association, cluster formation

and power adaptation—is stated in terms of the following three questions:

1. Which BSs should be in the compensating cluster?

Determine the set of BSs comprising the compensating cluster, denoted as NCO,

such that NCO ⊆ NC ⊂ N .


2. What are the compensating BS power levels?

For each BS n ∈ NCO, determine the operating power level, Pn.

3. Which BSs should be associated with the outage users?

For each outage user, k ∈ KO, find the associated BS n(k) ∈ NC .

In relation to point 3, we form the set of BSs newly associated with any of the outage

users:

NAS = n|Kn,O 6= ∅, (3.1)

where Kn,O = k|n(k) = n; k ∈ KO is the set of all outage users assigned to BS n. As the

goal of a utility maximizing algorithm (such as a PF or TRM procedure) is to maximize

the utility function over all outage users and the relevant non-outage users, the resource

allocation must be coordinated among some (or all) of the BSs in NAS. Equivalently,

the associations set NAS and the coordination set NCO must not be disjoint, which is

expressed mathematically as:

NAS ∩NCO 6= ∅. (3.2)

Note that the condition of (3.2) will be useful in the development of COC algorithms

later in this chapter.

We emphasize that the COC problem statement of this section considers the clus-

tering, user cell associations and BS power levels in a limited geographic area, within a

distance dO from the outage BS. According to the adaptive resource allocation framework

of Chapter 2, however, the same type of adaptations should also typically be performed

elsewhere in the network, which can have an effect on the COC solution inside the area

of question. The effect of BSs outside the candidate BS area are neglected in the present

contribution, as interference received from BSs at a sufficiently large distance tends to be

small, and the research’s primary aim is to develop an approximate, tractable algorithm

for compensation.

Our detailed study of the stated COC problem based on utility maximization (such

as PF or TRM procedures) has shown that difficulty arises in finding an optimal solution

to the COC problem, even in a limited physical area, due to the mutual dependence of

individual problem parts. In particular, the key mutual dependencies are listed in the

following:

• As per (3.2), the best choice for compensating cluster, NCO, is dependent on the

outage user cell associations, n(k).

• The best choice for NCO is, in general, dependent on the BS power levels, Pn, as

shown in the clustering algorithm development of [2].


• The actual measured other-cell interference will tend to reduce with increased clus-

ter size, which will affect the optimal choice for BS power levels Pn. Thus, the

optimal choice for Pn is itself dependent on the choice of compensating cluster,

NCO.

• The best cell association, n(k), for each outage user is, in general, dependent on

the choice of Pn as discussed in [27,51].

We conclude from the discussion that it is not straightforward to solve all three parts of

the COC problem directly. Therefore, we take the approach of decoupling the problem

parts, and providing low-complexity heuristic solutions. In particular, the main proposed

solution, which is based on the proportional fair criterion, will consist of iteration between

the following steps:

1. Fixing BS power levels, Pn, and solving for the updated outage user cell associa-

tions, n(k).

2. Fixing the BS power levels, Pn, and the user cell associations, n(k), and solving for

the compensating cluster set, NCO.

3. Fixing the user cell associations, n(k), and the compensating cluster set, NCO, and

solving for the updated BS power levels, Pn.

Additionally, the secondary COC solution method will be a further simplified algorithm

based on the TRM and PLM criteria—referred to as the PLM COC algorithm. Details

of the PF COC and PLM COC algorithms are discussed in the following sections.

3.4 Proportional Fair Compensation Algorithm

In the previous section, it was mentioned that adaptations are considered for a limited

geographic area, and the effect of BSs outside the candidate set, n /∈ NC , and their

terminals, k such that n(k) /∈ NC , are neglected. Consequently, certain assumptions are

needed for the BS power levels and cell associations for cells outside the candidate BS

set, which will be applied to both the PF and the PLM compensation algorithms.

Let E[gn,k] be the long-term average (or expected value) of the channel power gain

between BS n and user k, P0n the maximum allowed power level based on the transmission

capability of BS n and αn a power multiplier constant which depends on the BS index

n. It is assumed that

1. The BSs n /∈ NC operate at a constant power setting Pn = P0n/αn.


2. The associated BS for non-outage users (both within and outside the candidate

cells) is determined according to the traditional method of maximizing E[gn,k]:

n(k) = arg maxn

E[gn,k], (3.3)

which shall be referred to as the path loss minimization (PLM) method.

Note that it was explained in Chapter 2 that the maximum allowed BS power level, P0n,

is normally determined based on the BS power class, s. In particular, P0n is equal to the

distinct maximum power capability, Psn,max, of the power class sn. However, the present

chapter uses a simplified model of BS deployment for the irregular network, which does

not consider the complete realistic modelling of hotspot deployment. Such a model will

be developed in detail in Chapter 6. In this chapter, individual maximum BS power

capabilities, P0n, will be determined through a heuristic optimization method, to provide

a form of realism to the analysis. The power capability optimization is discussed in

Section 3.6.

In this section, the PF outage compensation algorithm is discussed, which consists of

the steps of:

1. Outage user cell re-association

2. Compensating cluster generation

3. BS power adjustment

3.4.1 Outage User Cell Re-association

The cell re-association of outage users is discussed in this section. Denote the vector of

associated BSs for outage users as nKO= [n(k1), n(k2), n(k3), . . . , n(kKO)]. The propor-

tional fair outage user cell association problem is given by

maxnKO

SCO (3.4)

subject to n(k) ∈ NC , k ∈ KO

where SCO is the sum-log utility for the compensating cluster users and is computed by

SCO =∑

k∈KCO

logRk,n(k) (3.5)


Here, KCO = k|n(k) = n;n ∈ NCO is the set of users assigned to the BSs in the

compensating cluster (which is as yet undetermined), and Rk,n(k) is the long-term average

rate for user k from BS n(k). Note that the use of the PF scheduling strategy at the

subframe time scale is assumed throughout the analysis. Under the assumption, Rk,n(k)

is approximated as [27]:

Rk,n(k) =G(Kn(k))

Kn(k)

E[rk,n(k)] (3.6)

Here, Kn(k) is the number of terminals assigned to BS n(k), G(y) is a quantity known

as the multiuser diversity scheduling gain, expressed in terms of the number of assigned

users, and E[rk,n(k)] is the long-term average of the instantaneous achievable rate of

user k from BS n, where rk,n, the instantaneous rate over the entire frequency band, is

given by rk,n =∑

c∈C rk,n,c. The derivation in [27] showed that the multiplicative factor,

G(Kn(k))/Kn(k), in (3.6) accounts for the PF scheduling in the system. In particular,

the numerator G(Kn(k))—the scheduling gain—accounts for the multiplicative increase

in the average rates due to the diversity from scheduling multiple users. The denominator

Kn(k) accounts for the relative reduction in the average rates due to division of the time-

frequency resource among the users assigned to BS n(k). It was shown in [27] that under

an easily satisfiable set of conditions, PF scheduling results in equal division of resources

among the users. Furthermore, the scheduling gain, G(y), was approximated as

G(y) =

y∑i=1

1

i(3.7)

Finally, we approximate E[rk,n(k)] as a function of total bandwidth, BT and the average

SINRs, SINRk,n(k), which in turn depend on the power levels and the channel gains:

E[rk,n(k)] = BT log2

(1 +

SINRk,n(k)

2

)

= BT log2

(1 +

Pn(k)E[gn(k),k]

2(N0C +∑

j 6=n(k) PjE[gk,j])

)(3.8)

Thus, by substitution of (3.7) and (3.8) into (3.6), the expression for Rk,n(k) is evaluated

as

Rk,n(k) =

∑Kn(k)

i=11i

Kn(k)

BT log2

(1 +

Pn(k)E[gn(k),k]

2(N0C +∑

j 6=n(k) PjE[gk,j])

)(3.9)


Finally, the substitution of (3.9) into (3.4) yields the optimization:

maxnKO

SCO (3.10)

subject to n(k) ∈ NC , k ∈ KOSCO =

∑k∈KCO

logRk,n(k)

Rk,n(k) =

∑Kn(k)

i=11i

Kn(k)

BT log2

(1 +

Pn(k)E[gn(k),k]

2(N0C +∑

j 6=n(k) PjE[gk,j])

), k ∈ KCO

The authors in [27] show that the direct solution of the optimization problem of (3.10)

is highly complex computationally. To reduce algorithmic complexity, they take the

approach of determining the initial cell associations offline by using a direct solution,

and periodically updating the associations using a simpler heuristic online solution. For

this purpose, an efficient cell association update technique is proposed, which is based

on the following property:

Proposition 3.1 [27] Assume that a user k is initially associated with BS n. Then,

under a set of modest assumptions, switching the association of user k to another BS j

will result in an increase in the overall network-wide sum-log utility if

Rk,j > Rk,n. (3.11)

In this section, we propose a heuristic iterative cell association update technique for

outage users, inspired by Proposition 3.1 and the solution approach of [27]. The detailed

algorithm is given in Figure 3.2. Note that the method requires an initial set of cell

associations, n(k), for outage users, k ∈ KO, which are typically computed according to

PLM:

n(k) = arg maxn∈NC

E[gn,k] (3.12)

The optimal BS for switching is then computed for each user based on maximizing the

increase in Rk,n. In addition, a single outage user is selected for switching at each iteration

in order to avoid oscillatory behaviour of the algorithm.


Require: The candidate set NC . Initial cell associations, n(k), for each outage user,k ∈ KO.

1: repeat2: for all outage users k ∈ KO do3: Initialize the flag: γ(k)← 04: for all BSs j ∈ NC do5: Compute Rk,j according to (3.6), with values of Kj and E[rkj] adjusted as

appropriate.6: end for7: if

Rk,jRk,n(k)

> 1 for some j ∈ NC then

8: Find j∗k , the optimal BS to switch to, from j∗k = arg maxj∈NCRk,jRk,n(k)

.

9: else10: Raise the flag: γ(k)← 111: end if12: end for13: if γ(k) = 0 for some value of k ∈ KO then14: Find user k∗ that will have its cell association switched as

k∗ = arg maxk∈KORk,j∗

k

Rk,n(k).

15: Execute the switch: n(k∗)← j∗k∗16: end if17: until no more cell association switches are possible, i.e. γ(k) = 1,∀k ∈ KO.

Figure 3.2: Outage user cell association update.

3.4.2 Generation of Compensating Cluster

Let SINRth be a pre-determined threshold for long-term average SINRs, SINRk,n(k).

We determine the set of BSs comprising the compensating cluster as:

NCO = n|Kn,O 6= ∅; NOT(SINRk,n(k) < SINRth,∀k ∈ Kn,O

) (3.13)

where the condition, NOT(SINRk,n(k) < SINRth,∀k ∈ Kn,O

), serves to exclude those

BSs that provide low average SINR for outage users. This can serve to restrict the size

of the compensating cluster.

Note that from (3.1), NAS = n|Kn,O 6= ∅. Therefore, NCO computed according to

(3.13) satisfies NCO ⊆ NAS and the required condition of (3.2):

NAS ∩NCO 6= ∅ (3.14)

In the remainder of this chapter, SINRth is set to 0 as a specialization of (3.13), that is,


the second condition in (3.13) is not utilized. Therefore,

NCO = n|Kn,O 6= ∅ ≡ NAS (3.15)

which also satisfies (3.2).

3.4.3 BS Power Adjustment

We consider the optimization of the BS power levels, Pn, based on the maximization of

a weighted sum-log utility function over all the users in the compensating cluster. Let

PNCO= [Pn1 , Pn2 , Pn3 , . . . , PnNCO ] be the compensating cluster BS power vector. The BS

power optimization problem is expressed as

maxPNCO

Sβ,CO (3.16)

subject to Sβ,CO = β∑k∈KO

logRk,n(k) + (1− β)∑

k∈KCO,k /∈KO

logRk,n(k)

0 ≤ Pn ≤ P0n, n ∈ NCO

where Sβ,CO is the weighted sum-log utility, Rk,n(k) is computed from (3.9), β is the weight

assigned to the contribution of outage users to Sβ,CO, chosen in the range 0 ≤ β ≤ 1, and

(1− β) represents the weight assigned to the contribution of non-outage users to Sβ,CO.

Due to the non-convexity of (3.16) and the anticipated solution complexity [52], a

simple heuristic greedy power increase method is proposed as an approximate solution.

It is assumed that each BS n ∈ NCO is initially operating below its maximum power level

at P0n/αn1. Details of the algorithm are given in Figure 3.3. Note that Sβ,CO(Pn = P ′n)

represents the Sβ,CO value computed after modification of a single BS’s power level, where

P ′n is a temporary variable. According to the method, a single BS is selected at each step

for power increase in greedy fashion. The step is then repeated iteratively until no more

power increases are possible.

1The reduction factor of SINR, the long-term SINR, due to operating below P0n is given by

αnN0C+

∑j 6=n,j∈NCO

P0nE[gk,j ]/αn+∑

j 6=n,j/∈NCOP0nE[gk,j ]

N0C+∑

j 6=n P0nE[gk,j ]. While this may be significant, especially for a

rare event like a cell outage, the concern of the thesis is not on the rarity of the event. The focus ofthe thesis is on providing a model for a change in the network configuration and the correspondingadaptation.


Require: Initial power levels, Pn = P0n/αn, for each BS, n ∈ NCO.1: repeat2: Form the candidate BS set for power increase according to

NCPI = n|Pn = P0n/αn,∀n ∈ NCO.3: for all BSs n ∈ NCPI do4: Assign temporary variable: P ′n ← αnPn = P0n

5: Compute Sβ,CO(Pn = P ′n).6: end for7: Find optimum power increase BS, n∗ = arg maxn∈NCPI Sβ,CO(Pn = P ′n)8: Implement power increase: Pn∗ ← αn∗Pn∗ = P0n∗

9: until NCPI = ∅.

Figure 3.3: Compensating BS power update.

Require: Maximum power level, P0n and power multiplier, αn, ∀n ∈ N . Candidate setNC .

1: for all n ∈ N do2: Pn ← P0n/αn.3: end for4: for all k ∈ K do5: n(k)← arg maxnE[gn,k]6: n(k)(0) ← n(k)7: end for8: repeat9: for all k ∈ KO do

10: n(k)← n(k)(0)

11: end for12: Use Algorithm 1 to update the cell association, n(k), for each outage user, k ∈ KO.13: Use eq. (3.15) to update the compensating cluster NCO.14: Use a single iteration of Algorithm 2 to find the optimum power increase BS n∗

and increment its power.15: until NCPI = ∅

Figure 3.4: Overall iteration of the PF compensation algorithm.

3.4.4 Overall Algorithm

The overall algorithm relies on fixing any two of cell association, clustering and power

levels, and solving for the other through the methods described in this section. The

iteration steps are shown in Figure 3.4.


3.4.5 Information Exchange Requirements

The information exchange required is similar to that of the clustering and BS assignment

procedure of [2], which was discussed in Chapter 2. In particular, the long-term average

channel gains, E[gn,k], between BSs and users need to be known at a central node.

Additionally, the central node needs to know the traffic load information for the BSs,

i.e., the number of users, Kn, assigned to each BS, for the execution of cell re-association

and BS power adjustment steps. Note that the channel and load information exchange

occurs on an infrequent basis due to the rare nature of the BS outage occurrences. In

contrast, fast-varying channel gains, gn,k,c, need to be exchanged for the purpose of fast

scheduling at the subframe time scale, for BSs inside the compensating cluster.

3.5 PLM Compensation Algorithm

A simplified PLM compensation procedure is discussed in this section. In the particular

algorithm realization, the outage user cell associations are determined according to the

PLM method:

n(k) = arg maxn∈NC

E[gn,k], ∀k ∈ KO (3.17)

The set of BSs comprising the compensating cluster is computed from (3.13), with

SINRth set to zero:

NCO = n|Kn,O 6= ∅ ≡ NAS (3.18)

Finally, we consider the power optimization problem based on maximization of a weighted

sum rate function, denoted by S ′β,CO:

maxPNCO

S ′β,CO (3.19)

subject to S ′β,CO = β∑k∈KO

Rk,n(k) + (1− β)∑

k∈KCO,k /∈KO

Rk,n(k)

0 ≤ Pn ≤ P0n, n ∈ NCO

In order to reduce algorithm signalling complexity, S ′β,CO is replaced by a simplified

sum rate function S ′′β,CO, which is defined in terms of E[rk,n(k)] in place of Rk,n(k). The


optimization becomes

maxPNCO

S ′′β,CO (3.20)

subject to S ′′β,CO = β∑k∈KO

E[rk,n(k)] + (1− β)∑

k∈KCO,k /∈KO

E[rk,n(k)]

0 ≤ Pn ≤ P0n, n ∈ NCO

We note that the computation of outage user cell associations, n(k), from (3.17) does

not require knowledge of the rates Rk and, hence is not dependent on the power levels

Pn. Similarly, the computation of compensating cluster NCO from (3.18) only requires

the knowledge of cell associations from (3.17), and not the power levels. In contrast,

the power level optimization of (3.20) requires knowledge of both the cell associations

and the compensating set. As a consequence, the outage user cell associations, n(k), the

compensating cluster, NCO, and the power levels, Pn, can be found sequentially rather

than in iterative fashion. The heuristic greedy method of the previous section is again

used for the power level adjustments. The overall PLM algorithm is detailed in Figure 3.5.

The requirements for information exchange are similar to those of the PF compensa-

tion algorithm. In particular, the long-term average channel gains, E[gn,k], need to be

known at a central node. However, one advantage is that the exchange of load informa-

tion, Kn, for each BS is not required.

3.6 Simulation/Evaluation

We simulate both a regularly-deployed and an irregularly-deployed cellular network for

the evaluation of the COC algorithms. In each case, the network is composed of 19 BSs,

and the outage BS is located at the centre of the circular simulation area. The simulation

methodology and parameters are largely based on the ITU recommendations. Note

that a single subcarrier has been assumed for the evaluation. In the irregular network

deployment, the BSs are positioned according to the uniform statistical distribution over

the network area. In addition, the spatially correlated shadowing model of [8] detailed in

Chapter 2, which considers correlation for irregular deployment, is utilized. A summary

of the model parameters is provided in Table 3.1.

As indicated in Table 3.1, the initial BS power levels are all equal to P0/α for the

regular network. In this simulation, the initial power levels for the irregular network are

computed by means of a fast heuristic optimization. In particular, we consider the total


Require: Maximum power level, P0n and power multiplier, αn, ∀n ∈ N . Candidate setNC .

1: for all n ∈ N do2: Pn ← P0n/αn.3: end for4: for all k ∈ K, k /∈ KO do5: n(k)← arg maxnE[gn,k]6: end for7: for all k ∈ KO do8: n(k)← arg maxn∈NC E[gn,k]9: end for

10: NCO ← n|Kn,O 6= ∅ ≡ NAS11: repeat12: Form the candidate BS set for power increase according to

NCPI = n|Pn = P0n/αn,∀n ∈ NCO.13: for all BSs n ∈ NCPI do14: Assign temporary variable: P ′n ← αnPn = P0n

15: Compute Sβ,CO(Pn = P ′n).16: end for17: Find optimum power increase BS, n∗ = arg maxn∈NCPI Sβ,CO(Pn = P ′n)18: Implement power increase: Pn∗ ← αn∗Pn∗ = P0n∗

19: until NCPI = ∅.

Figure 3.5: PLM Compensation Algorithm.

rate maximization problem:

maxP

∑n∈N

Rn (3.21)

subject to Rn = BT log2

(1 +

PnGnn

N0C +∑

j∈N ,j 6=n PjGnj

), n ∈ N

0 ≤ Pn ≤P0

α, n ∈ N .

where the set of users associated with each BS was replaced by a virtual representative

user indexed by n, Pn is the BS power level, Rn is a rate computed for virtual user n and

Gnj is a channel gain for the link between virtual user n and the BS j, which is generated

as the geometric mean of user channel gains:

Gnj =

(∏k∈Kn

E[gk,j]

)1/Kn

. (3.22)

Note that the powers are bounded by P0/α as the BSs are assumed to be operating


Table 3.1: Simulation parameters.

Parameter Value

Bandwidth per subcarrier 15 kHz

BS power budget per subcarrier, P0 18.22 dBm

The factor α√

10

Initial BS powers for regular network P0/α

Noise figure at terminal 7 dB

Background noise power spectral den-sity

-174 dBm/Hz

Distance-dependent path-loss (carrierfrequency = 2 GHz)

PL = 128.1 + 37.6 log10(d)[dB], d in km

Shadowing standard deviation, σ0 8 dB

Shadowing correlation distance, dc 50 m

Inter-site distance 500 m

Total number of cells 19

Number of Users in Centre Cell 66

Number of Users per Cell in the OtherCells

10

Number of subcarriers 1

Subframe size (= time slot size) 1 ms

PF scheduling window size, T0 5 subframes

below their maximum power capabilities. The optimization of (3.21) can be thought of

as a fast way to estimate realistic BS power capabilities for a given random distribution

of users and BSs. It was shown in [53] that under the assumption of high SINR (i.e. if

(PnGnn)/(N0C+∑

j∈N ,j 6=n PjGnj) 1), (3.21) reduces to a convex geometric program—

a type of problem easily solved through interior-point methods. For the current problem,

the solution is obtained by using CVX, a package for solving convex programs [54].

3.6.1 Numerical Results

In evaluating the COC algorithms, the following performance metrics are considered:

• User Average Spectral Efficiency: The total spectral efficiency (or the rate per

total bandwidth) of the users in the compensating cluster normalized by the total

number of users in the compensating cluster.


Table 3.2: NPI and NCO as a function of β for the COC algorithms in the regular network.

β NPI for PLM NPI for PF NCO for PLM NCO for PF

0.1 6.8 7.0 6.8 7.0

0.9 6.8 6.4 6.8 7.0

1.0 6.7 5.9 6.8 7.0

• Outage User QoS violation probability: The proportion of outage users in the

compensating cluster whose average spectral efficiency is lower than a threshold,

ηt.

• Non-Outage User QoS violation probability: The proportion of non-outage

users in the compensating cluster whose average spectral efficiency is lower than

ηt.

• The Average Number of BSs in the Compensating Cluster: Since NCOdenotes the set of BSs in the compensating cluster, we let NCO be a particular

realization of the number of elements in NCO. The average number of BSs in the

compensating cluster, is thus denoted as NCO.

• The Average Number of BSs with Raised Power: Similarly, let NPI be the

set of BSs with raised power. A realization of the number of elements in NPI is

denoted by NPI , and the average number of BSs with raised power, is denoted as

NPI .

The performance of the proposed PF COC and PLM COC techniques are evaluated

for various values of the parameter β and compared to a coordination scenario in the

absence of cell outage, or the no outage scenario. The value of ηt is set to 0.02 bps/Hz. For

a fair comparison, it is assumed that the cluster and BS power levels obtained through

COC are also used for the analysis of the no outage scenario2. Note that the results

will only show the no outage scenario corresponding to the COC scheme with the best

spectral efficiency and QoS violation probability, thereby serving as an upper bound on

performance.

The variation in NCO and NPI is illustrated for regular and irregular networks in

Tables 3.2 and 3.3, respectively. For both regular and irregular deployment, as the

parameter β increases, NPI decreases for the PF algorithm. Simultaneously, NPI is

reduced slightly for the PLM algorithm. NCO stays approximately constant with changes

2In the no outage scenario, the centre BS is also included in the cluster.


Table 3.3: NPI and NCO as a function of β for the COC algorithms in the irregularnetwork.

β NPI for PLM NPI for PF NCO for PLM NCO for PF

0.1 5.8 5.9 5.8 5.8

0.9 5.8 5.5 5.8 5.9

1.0 5.7 5.3 5.8 5.9

in β in all cases. In summary, the PF algorithm allows for significant adjustment of NPI—

which can be thought of as the cost of the compensation algorithm implementation—by

variation of β. The range of adjustment is generally larger for regular deployment. The

user average spectral efficiency, outage user QoS violation probability and non-outage user

QoS violation probability are plotted as functions of NPI in Figures 3.6-3.8, respectively.

Figures 3.6a and 3.6b indicate that approximately 75% of the user spectral efficiency

of the no outage scenario is achieved by both COC algorithms for regular and irregular

deployment, with NPI at a maximum (corresponding to β = 0.1). However, if a slightly

smaller user spectral efficiency is sufficient, β can be set higher, resulting in lower NPI

for the PF algorithm. Effectively, the PF algorithm allows for a trade-off between low

compensation cost (NPI) and high user spectral efficiency.

Figure 3.7a indicates that outage user QoS violation rate for the PF algorithm is

around 6-9% under regular deployment, compared to approximately 4.5% for the no

outage scenario—which represents a modest increase. In addition, the violation rate for

the PLM COC algorithm is close to 8%. Effectively, the PF algorithm is shown to achieve

a trade-off between QoS violation rate and compensation cost (NPI) under the regular

deployment scenario.

Similarly, Figure 3.7b indicates for irregular deployment that the QoS violation rate

for the PF algorithm is around 10-12%, compared to approximately 7% for the no outage

scenario. The violation rate for the PLM COC algorithm is close to 11.5%. The PF

algorithm also achieves a trade-off between QoS violation rate andNPI under the irregular

deployment scenario.

Similar trends as Figure 3.7 are shown for the non-outage user QoS violation proba-

bility in Figure 3.8. In particular, a modest increase in violation rates compared to the

no outage scenario is obtained. The PF algorithm achieves a trade-off between violation

rate and NPI . Additionally, the non-outage user QoS violation rate stays below about

2% for both COC algorithms in the two deployment scenarios.


(a)

(b)

Figure 3.6: User average spectral efficiency, in bps/Hz, as a function of the averagenumber of BSs with raised power for (a) regular and (b) irregular networks.


(a)

(b)

Figure 3.7: Outage user QoS violation probability, in percent, as a function of the averagenumber of BSs with raised power for (a) regular and (b) irregular networks.


(a)

(b)

Figure 3.8: Non-outage user QoS violation probability, in percent, as a function of theaverage number of BSs with raised power for (a) regular and (b) irregular networks.


In summary, both the PF and PLM COC algorithms show robustness toward cell

outage. Specifically, they achieve a large percentage of the user spectral efficiency of

the no outage scenario and result in only a modest increase in the QoS violation rates.

In addition, the PF COC algorithm is tunable and allows for trading off NPI , which is

related to algorithm cost, against various measures of user performance.

Chapter 4

Dynamic System Model (Single

Power Class)

4.1 Relevant Work

4.1.1 Terminal and BS Deployment Models in Cellular Net-

works

A number of spatial point processes have been used to model the deployment of BSs

and terminals in multi-tier or heterogeneous cellular networks. Spatial point processes

are the generalization of point processes indexed by time to the higher dimensions, in

particular, two-dimensional space.

Among the point processes, the simplest and most popular is the Poisson Point Pro-

cess (PPP). In the PPP model, each node is placed independently over area A with the

density of nodes given by λ. The probability that there are n nodes in area A is thus

given by the Poisson distribution and is equal to (λA)ne−λA/n! The procedure to gen-

erate the PPP is to first draw Poisson distributed number n and then to place n points

uniformly at random in A. The PPP is suitable to model a network composed of a large

number of nodes randomly and independently existing in some area. In the cellular net-

works, it has been commonly used to model the terminal deployments [55] as well as BS

deployments [56,57]. Typically, the nodes in different tiers are modelled via independent

homogeneous PPPs.

Similar to PPP as far as the independent and identically distributed node locations,

the Binomial Point Process (BPP) has the only distinguishing feature that it generates a

fixed number of nodes in a given area. It has been used to model deployment of a known

number of mobile users in a cell of known size [58].

54

Chapter 4. Dynamic System Model (Single Power Class) 55

More general models have also been considered in the literature. The Poisson Cluster

Process (PCP), in particular, models the random patterns produced by random clusters.

A PCP is constructed from a parent PPP by replacing each point with a cluster of points

where the points are independently and identically distributed in the spatial domain.

The PCP has commonly been used to model a network where nodes occur in clusters

e.g. according to certain social behaviour such as users gathered around Wi-Fi hotspots.

In cellular networks, the combination of a PPP and a PCP has been applied in the

heterogeneous networks [59] where the PPP represents the mobile users in a macrocell

and the PCP represents femtocells or hotspots.

Although the distance among BSs are random in a cellular network, we cannot find

two BSs owned by the same provider that are arbitrarily close to each other. Thus

research has considered the Matern hard core point process (MHCPP) which imposes

a minimum distance, dmin between nodes. The MHCPP is obtained by starting from a

PPP. A random score uniformly distributed in [0, 1] is assigned to each point in the PPP,

then all the points that fall within a distance less than dmin from another point with a

lower score is deleted. This results in a minimum separation between any points of dmin.

MHCPP has been used in place of the basic PPP process to model BS deployments in a

cellular network where there is a restriction on BS separation due to such constraints as

geographical constraints as well as network planning or due to the MAC protocol that

avoids nearby simultaneous transmissions [60–62]. However, dealing with MHCPPs is

relatively more complicated compared to the PPP. Therefore, the PPP model remains

appealing partially due to its simplicity and analytical tractability [63–65].

Recent research has also considered more realistic point processes that capture the

cellular network characteristics with greater accuracy. Gibbs family of processes, in

particular, model the repulsion between the BS nodes, making it less likely that BSs

are located close to each other. Gibbs processes belong to the category of Soft Core

point processes (SCPP) as they do not impose a hard minimum distance between nodes.

Research in [65, 66] has shown that the Gibbs processes model BS spatial locations in

real networks more accurately compared to PPP models.

Ginibre process is another point process that can be used to model the repulsion

between BSs. Deng et al [7] consider and study the Ginibre point process as a model for

wireless networks. They introduce the β-GPP as an intermediate process between PPP

and GPP through consideration of the accuracy, tractability and practicality tradeoffs.

They derive the coverage probability of a typical user in a cellular network and show that

it compares well with actual base station deployments.

The authors in [67] investigate the spatial modelling issue. Both the deterministic,


hexagonal model at one end and the random deployment following a Poisson point process

(PPP) at the other extreme are considered. The authors utilize a modified perturbed

hexagonal lattice model that lies in between the PPP and the hexagonal model in terms

of irregularity. A frequency reuse-1 interference network is then assumed and an upper

bound on the average total interference as a function of distance is provided. The paper

specifies the loss in coverage probability when moving from a perfect lattice to the random

BS deployment.

4.1.2 Terminal Mobility Models in Cellular Networks

There exist a number of statistical models to represent movement in cellular networks.

The most widely-used is the random walk model [68, 69], which is sometimes referred

to as Brownian motion. According to this model, a terminal moves by choosing an ar-

bitrary direction and speed from the given ranges [vmin, vmax] and [0, 2π], respectively.

Each movement in the model is performed either for a constant time or for a constant

distance travelled. Then a new speed and direction are selected. The random walk model

represents a memoryless mobility pattern as the current speed and direction are inde-

pendent of past speeds and direction. However, this model can result in unrealistically

sudden turns and stops.

A similar model as the random walk model is the random waypoint model, which is

also widely used [70,71]. According to the model, a terminal chooses a random destination

in the simulation area and a speed that is uniformly distributed between [vmin, vmax]. It

then moves to the destination at the chosen speed. After arrival at the destination, the

terminal pauses for a predetermined length of time before repeating the process.

Gauss-Markov mobility model was introduced to improve the realism of the random

walk model [72, 73]. In this model, a nodes next location is generated by its current

location and velocity. The variation of one tuning parameter allows the variation of

the degree of randomness in the mobility pattern. At one extreme of the parameter

value, the random walk model is obtained and the other extreme results in linear motion.

Intermediate values of randomness are obtained between the two extremes.

Another category of mobility models known as group mobility models represent par-

ticular types of user movement as a group [74], such as a group of young children walking

in a single file (column mobility model), a group of students touring a museum (nomadic

community mobility model) or police officers pursuing an escaped criminal (pursue mo-

bility model).


4.2 Approach and Contributions

System Model Approach of the Thesis

A system model for an irregular cellular network is considered, which is based on the

concept of Monte Carlo simulations. The model explicitly specifies the representations

of terminal and BS deployments, shadow fading as well as the large-time-scale evolution

model of the system. Scenarios with either a single or multiple BS power classes are con-

sidered. Much of the literature considers classical models particularly for the deployment

(e.g. PPP), and the shadowing (e.g. lognormal shadowing) for the purpose of simplicity

and analytical tractability despite their lack of realism.

Our primary contributions include the deployment models for BSs and terminals,

terminal movement (mobility) models, and shadowing model with correlation for multiple

power classes. The model is considered primarily as a generic reference model for the

purpose of the design of resource allocation with adaptation at a large time scale, for an

irregular cellular network. The goal of developing the model is that it can be used along

with any modifications as desired, in order to model the cellular network as realistically

as desired in terms of the interference, deployment positions, movement and channel

models (shadowing and multipath fading) among other elements.

Deployment Model

The most common model in the literature for irregular deployments is the PPP model

due to its simplicity and analytical tractability. However, the PPP model was shown

to be unrealistic in real irregular networks. Other more realistic models exist (with the

primary example being the Ginibre process [7]) however most such models have the issue

of being difficult to handle. In this work, we consider the development of a baseline

deployment model for Monte Carlo simulation with the primary consideration being the

model simplicity and a baseline realism, which is implemented through arbitrariness

or randomness of deployment with a minimum separation between the nodes and the

elements in the network. Deployment scenarios both with hotspots (multiple power

classes) and without hotspots (single power class) are considered. Since this is meant as

a generic model, guidelines for parameter values are provided for baseline realism of the

simulation scenario.

Note that this chapter deals with the deployment model without hotspots or the

single power class scenario. The deployment model including hotspots and multiple BS

power classes is elaborated in Chapter 6.


Mobility Model

Most other models in the literature do not consider terminal movements over an arbitrar-

ily large time scale as a snapshot. We propose a generic model for individual mobility

that is meant as a generic model appropriate for a time scale that is orders of magnitude

larger than the multipath fading time scale. The primary consideration for the model

is its simplicity. A similar movement model is also then considered for BSs. Deploy-

ment scenarios both with hotspots (multiple power classes) and without hotspots (single

power class) are considered for the movements. Since this is meant as a generic model,

guidelines for parameter values are provided for the baseline realism of the simulation

scenario.

Note that this chapter deals with the movement (mobility) model without hotspots

or the single power class scenario. The mobility model including hotspots and multiple

BS power classes is elaborated in Chapter 7.

4.3 Introduction

In Chapter 3, we studied the BS outage as a specific case of a dynamic change occurring in

the irregular cellular network, and BS outage compensation as an adaptation in response

to the particular type of dynamic change. In Chapters 4 through 7, we propose a full

system model that incorporates a wide array of dynamic changes occurring in the irregular

network, and study an algorithmic framework for system adaptation.

The resource allocation framework of [2] was shown to have the capability to adapt

to the slow traffic variations, and the clustering concept was designed for the irregular

deployment of BSs. The case of a single BS power class was primarily considered. The

scheduling and on-off power switching at the subcarriers further adjusts to the variations

in the traffic and channel conditions at a finer level. The adaptive resource allocation

framework has thus been adopted in this chapter for the model with network time evolu-

tion. In the following, the system model of the ITU will be modified as needed to meet

the requirements of the irregular network with a single BS power class.

Recall that the original drop-based simulation method of the ITU model does not

allow the modelling of network evolution at the slow time scale (i.e. the time scale of the

simulation drop). Therefore, we propose a modified method of drop-based simulation,

which is summarized as follows: First, we re-define the simulation drop as an independent

deployment of both terminals and BSs. Second, we define a new term subdrop as a

modified version of a given simulation drop, which is obtained by making small step-


wise changes to the initial drop configuration. Time evolution at the slow scale is thus

obtained by implementing a series of subdrops for each drop, which consist of gradual

dynamic changes. The dynamics studied include terminal arrivals and departures, BS

deployments and outages as well as terminal movements. In this thesis, BS movements

are also considered as they provide a way to represent the arbitrariness or time variation

of the BS locations with respect to the terminals in the model, and are believed to be

plausible in emerging and future networks.

The system model is developed in stages in Chapters 4 through 7. In this chapter,

the system simulation model with network time evolution is considered for the irregular

network with a single BS power class and constant BS power levels. Adaptation of BS

power levels (power control) will be discussed in Chapter 5. The static system model and

the time evolution model for a network with multiple power classes will be examined in

Chapters 6 and 7, respectively.

In the current model, the simulation proceeds as summarized in Figure 4.1. Both the

generation of drops, subdrops and subframes, together with slow-scale time evolution, as

well as the execution of slowly-adaptive functions and fast resource allocation are shown.

The framework requires details of channel models, methods of drop deployment and time

evolution as well as slow adaptation and fast resource allocation algorithms. Note that

the multipath, correlated shadowing and path loss channel models, described in Chapter

2, are used in the current model. The rest of the chapter details the proposed drop

deployment and time evolution models. The methodology of simulation drop deployment

is developed in Section 4.4. The time evolution model is discussed in Section 4.5. Finally,

typical parameter values for the models are detailed in Section 4.6.

4.4 Drop Deployment Model

Realistic modelling of simulation drop deployment is considered in this section. Recall

that a drop is defined as the independent and mathematically random deployment of

a number of terminals and BSs in a network simulation area. In practical irregular

networks, variety of restrictions are typically imposed on the positioning of BSs due to

various reasons, including broad-level deployment planning by wireless operators, and

public and private property considerations. Therefore, it is essential that the proposed

random deployment model have certain mathematical restrictions to reflect the practical

constraints of deployment. Note that the imposed restrictions are parametrized so that

they can be adapted to a large number of practical situations by the tuning of parameters.

A circular area as shown in Figure 4.2 is utilized in the deployment of the network


shadowing potential field realization as discussed in Section

3.2 is fixed. In other words, the dynamism in subdrops re-

sults solely from gradual changes relating to BSs and termi-

nals on a given and fixed topography.

Let 𝒩(𝑏)and 𝒦(𝑏) be the set of BSs and terminals active

at subdrop 𝑏, respectively. Specifically, 𝒩(0) and 𝒦(0)

denote the set of BSs and terminals active at subdrop 0. At

each subsequent subdrop 1 ≤ 𝑏 ≤ 𝐵, ΔK 𝑎(𝑏)

terminals arrive

at uniform randomly distributed locations according to the

standard Poisson distribution

𝑓(𝑖; 𝜆𝑡) = 𝑃𝑟(ΔK 𝑎(𝑏)

= 𝑖) =𝜆𝑡

𝑖𝑒−𝜆𝑡

𝑖!,

where 𝑓( ) denotes the probability mass function, and

𝜆𝑡 > 0 is the Poisson parameter. Each terminal departs the

j = j + 1

STOP

No

Generate subframe l of subdrop j

Compute path loss and shadowing for

all terminal-BS links

Execute slowly adaptive algorithms

Yes

l = l + 1 l mod F = 0?


Check arrivals,

departures and

movements

Generate subdrop j of drop i

j mod B = 0?

Yes

Yes

i = P?

No

No i = i + 1

Perform coordinated resource alloca-

tion in each cluster

Figure 4.1: Outline of the Simulation Framework.


with a single BS power class. Let R be the radius of the simulation area. N BSs are

dropped with uniform random distribution onto the circular simulation area. The two-

dimensional (2-D) uniform random distribution for the BSs on the circle is obtained by

first generating the position in polar coordinates—a, the radius, and φ, the angle—with

respective probability densities given by fA(a) and fΦ(φ):

fA(a) =

2aR2 for 0 ≤ a ≤ R,

0 otherwise,(4.1)

fΦ(φ) =1

2πfor 0 ≤ φ ≤ 2π, (4.2)

and later converting the results into rectangular coordinates x and y by using

x = a cosφ (4.3)

y = a sinφ (4.4)

In order to reflect the restrictions due to network planning and property considerations,

it must be ensured that the distances between BSs are not smaller than the minimum

allowed separation parameter, δBS, as illustrated in Figure 4.2. If the distance between

any generated pair of BSs is less than δBS, the coordinates of one of the affected BSs

must be re-computed according to (4.1)-(4.4). Finally, the coordinates of K terminals

on the circular area are also generated randomly by using (4.1)-(4.4). If any terminal is

found to have the same coordinates as any of the BSs1, the coordinates of the terminal

must be re-computed.

4.5 Time Evolution

Time evolution in irregular networks for each simulation drop is obtained through the

generation of multiple subdrops. The list of modelled network dynamics includes gradual

terminal arrivals and departures, BS deployments and outages, terminal and BS move-

ments as well as changes in the large-scale channel parameters. A detailed time evolution

methodology is discussed in this section.

Let P be the set of drops used in the simulation, Bp the set of subdrops associated

with each drop p ∈ P , and Fb the set of subframes associated with each subdrop b ∈ Bp,for p ∈ P . As the Times New Roman large letters indicate the cardinality of sets, P is

1In practice, this condition is implemented by imposing a restriction on the minimum distance betweenthe terminals and the BSs.


BS

R

Figure 4.2: Simulation area for deployment in networks with a single BS power class.

the total number of simulation drops, Bp is the number of subdrops associated with drop

p, and Fb is the number of subframes associated with subdrop b. The number of subdrops

used in each drop is equal, that is, Bp is equal to a constant B for all p ∈ P . Similarly, an

equal number of subframes is utilized in each subdrop, or Fb = F for all b ∈ Bp, p ∈ P .

For consistency of notation, the initial deployment of a drop before any time evolution is

referred to as subdrop 0. The remaining subdrops associated with the drop are numbered

from 1 to B − 1. It is important to note the relationship between the drop, subdrop and

the shadowing model random potential field [8] discussed in Chapter 2. In particular,

for all subdrops b ∈ Bp associated with a given simulation drop p ∈ P , the realization

of the random potential field is modelled to be constant. In other words, the physical

topography can be conceptualized as fixed within each simulation drop, and the changes

in shadowing over the subdrops result solely from BS and terminal movements on the

fixed topography. The detailed modelling of the dynamics is discussed in the following.

4.5.1 Terminal Arrival and Departure Models

Let K(b) be the set of terminals active at subdrop b. In particular, K(0) denotes the set

of terminals that are active at the initial subdrop, indexed by 0. At each subsequent

subdrop 1 ≤ b ≤ B− 1, [∆Ka](b) terminals arrive to the system according to the Poisson

distribution:

f(i;λt) = Pr([∆Ka](b) = i

)=λite−λt

i!, i = 0, 1, 2, . . . (4.5)


where f( ) denotes the probability mass function, and λt > 0 is the Poisson parameter,

expressed in terms of the number of terminals. The coordinates of the arriving terminals

are determined according to the random deployment procedure of Section 4.4.

Each terminal must depart the system after a survival time, τk, expressed in terms of

the number of subdrops. τk is generated according to the following steps:

• Step 1: Virtual survival time, χ, is generated as a temporary variable according to

the exponential probability mass function with mean parameter, µt > 0 (expressed

in terms of the number of subdrops):

f(χ) =

1µte− χµt if χ ≥ 0,

0 if χ < 0.(4.6)

• Step 2: The actual survival time, τk, is computed according to

τk = dχe (4.7)

The number of terminals that depart at subdrop b, [∆Kd](b), is determined with the

knowledge of terminal survival times, τk. The total number of terminals remaining at

subdrop b is thus given by

K(b) = K(b−1) + [∆Ka](b) − [∆Kd](b) (4.8)

4.5.2 BS Deployment and Outage Models

Similar to the terminal model, the simulation framework has the option for modelling

BS deployments and BS outages. Let N(b) be the set of BSs active at subdrop b. The

BS deployment is modelled as an arrival and the BS outage is modelled as a departure

process. Let Ibs be an indicator parameter which is set to 0 if BSs are not allowed to

be deployed (i.e. arrive) and have an outage (i.e. depart). This means that the set N(0)

is fixed over the duration of the drop. If the BSs are allowed to have both deployments

and outages, Ibs is then set to 1. For Ibs = 1, the number of BSs, [∆Na](b), arriving to

the system at subdrops 1 ≤ b ≤ B − 1 follow the Poisson distribution:

f(i;λbs) = Pr([∆Na](b) = i

)=λibse

−λbs

i!, i = 0, 1, 2, . . . (4.9)

where f( ) denotes the probability mass function, and λbs > 0 is the Poisson parame-

ter, expressed in terms of the number of BSs. The coordinates of the arriving BSs are


determined according to the random deployment procedure of Section 4.4.

Each BS departs the system after a survival time, τn, (expressed in number of sub-

drops), which is generated according to the following steps:

• Step 1: Virtual survival time, χ, is generated according to the exponential proba-

bility mass function with mean parameter µbs > 0:

f(χ) =

1µbse− χµbs if χ ≥ 0,

0 if χ < 0.(4.10)

• Step 2: The actual survival time, τn, is computed according to

τn = dχe (4.11)

Similar to the terminals, the number of BSs that depart at subdrop b, [∆Nd](b), is deter-

mined with the knowledge of BS holding times, τn. The total number of BSs remaining

at subdrop b is given by

N(b) = N(b−1) + [∆Na](b) − [∆Nd](b) (4.12)

4.5.3 Terminal Movement Model

Terminals are moved at every subdrop through a randomly generated displacement in

a random direction. The movement model is illustrated in Figure 4.3. The full algo-

rithm for the terminal movement at the subdrop is given in Figure 4.4. Direction of

movement over the xy-plane is represented by angle θk,b, which is drawn from a uniform

random distribution in the interval [0, 2π]. The displacement, denoted as ∆dk,b, is drawn

from a uniform random distribution in [∆dt,min,∆dt,max], where ∆dt,min and ∆dt,max

are model parameters. Terminals are moved at each subdrop through the displacement

∆dk,b in the direction represented by angle θk,b. However, it is ensured that terminals

do not go out of the bounds of the circular simulation area of radius R illustrated in

Figure 4.2. Consequently, a check condition is utilized in the realization of the termi-

nal movements2. As long as a terminal would remain within the simulation area, i.e.,

if (xk + ∆dk,b cos θk,b)2 + (yk + ∆dk,b sin θk,b)

2 ≤ R2, its coordinates (xk, yk) are updated

2While terminals going out of bounds could be treated as a departure, the check condition is used torepresent the departures and movements separately.


(xk, yk)

(xk+dk,bcosθk,b, yk

+dk,bsinθk,b)

dk,b

θk,b

y

x

Figure 4.3: The terminal movement illustration.

according to:

xk ← xk + ∆dk,b cos θk,b (4.13)

yk ← yk + ∆dk,b sin θk,b (4.14)

If the full movement would place the terminal out of the simulation area, i.e., if (xk +

∆dk,b cos θk,b)2 + (yk + ∆dk,b sin θk,b)

2 > R2; terminal is first moved through a partial

displacement α < ∆dk,b, it is then placed at a position at 180 degrees from its position

in the circular simulation area, and finally, the remainder of the movement is completed

at its new location. The updated coordinates of the terminal are therefore given by

xk ← −(xk + α cos θk,b) + (∆dk,b − α) cos θk,b (4.15)

yk ← −(yk + α sin θk,b) + (∆dk,b − α) sin θk,b (4.16)

where α is determined by solving

α2 + 2α(xk cos θk,b + yk sin θk,b) + (x2k + y2

k −R2) = 0 (4.17)

4.5.4 BS Movement Model

Similar to the terminals, the proposed simulation framework has the option for modelling

BS movements. Let πbs be a fixed probability of BS movement. At each subdrop, BSs


Require: xk, yk, for terminals k ∈ K, and parameters ∆dt,min and ∆dt,max1: for all terminals k ∈ K do2: Generate ∆dk,b ∼ U ∈ [∆dt,min,∆dt,max], and θk,b ∼ U ∈ [0, 2π].3: if (xk + ∆dk,b cos θk,b)

2 + (yk + ∆dk,b sin θk,b)2 ≤ R2 then

4: xk ← xk + ∆dk,b cos θk,b5: yk ← yk + ∆dk,b sin θk,b6: else7: Solve for α in α2 + 2α(xk cos θk,b + yk sin θk,b) + (x2

k + y2k −R2) = 0

8: xk ← −(xk + α cos θk,b) + (∆dk,b − α) cos θk,b9: yk ← −(yk + α sin θk,b) + (∆dk,b − α) sin θk,b

10: end if11: end for

Figure 4.4: Algorithm for the terminal movement at the simulation subdrop.

are moved with probability πbs through a random displacement in a random direction.

The movement model is illustrated in Figure 4.5. The detailed algorithm is described in

Figure 4.6. The direction of movement over the xy-plane, represented by angle θn,b, is

drawn from a uniform distribution in the interval [0, 2π]. The displacement, denoted as

∆dn,b, is drawn from a uniform distribution in [∆dbs,min,∆dbs,max]. As long as a BS would

remain within the simulation area, i.e., if (xn+∆dn,b cos θn,b)2+(yn+∆dn,b sin θn,b)

2 ≤ R2,

its coordinates (xn, yn) are updated according to:

xn ← xn + ∆dn,b cos θn,b (4.18)

yn ← yn + ∆dn,b sin θn,b (4.19)

If the movement would place the BS out of the simulation area, the coordinate update

is given by

xn ← −(xn + α cos θn,b) + (∆dn,b − α) cos θn,b (4.20)

yn ← −(yn + α sin θn,b) + (∆dn,b − α) sin θn,b (4.21)


α2 + 2α(xn cos θn,b + yn sin θn,b) + (x2n + y2

n −R2) = 0 (4.22)


(xn, yn)

(xn+dn,bcosθn,b, yn

+dn,bsinθn,b)

dn,b

θn,b

y

x

Figure 4.5: The BS movement illustration.

Require: xn, yn, for BSs n ∈ N , and parameters πbs, ∆dbs,min and ∆dbs,max1: for all BSs n ∈ N do2: Generate γ such that Pr(γ = 1) = πbs and Pr(γ = 0) = 1− πbs.3: if γ = 1 then4: Generate ∆dn,b ∼ U ∈ [∆dbs,min,∆dbs,max], and θn,b ∼ U ∈ [0, 2π].5: if (xn + ∆dn,b cos θn,b)

2 + (yn + ∆dn,b sin θn,b)2 ≤ R2 then

6: xn ← xn + ∆dn,b cos θn,b7: yn ← yn + ∆dn,b sin θn,b8: else9: Solve for α in α2 + 2α(xn cos θn,b + yn sin θn,b) + (x2

n + y2n −R2) = 0

10: xn ← −(xn + α cos θn,b) + (∆dn,b − α) cos θn,b11: yn ← −(yn + α sin θn,b) + (∆dn,b − α) sin θn,b12: end if13: end if14: end for

Figure 4.6: Algorithm for the BS movement at the simulation subdrop.


Table 4.1: Typical parameter values.

Parameter Typical Range and/or Value

Minimum BS separation, δBS δBS R

Numbers of subdrops, B typical value of 20

Numbers of drops, P P B, typical value of 300

Scheduling average window size, T0 typical value of 5

Numbers of subframes, F F T0, F P , typical value of 100

Terminal arrival parameter, λt typical value of 3 terminals

BS arrival parameter, λbs λbs λt, typical value of 0.5 BSs

Terminal departure parameter, µt typical value of 5 subdrops

BS departure parameter, µbs µbs µt, typical value of 20 subdrops

Maximum terminal displacement, ∆dt,max 0.1dc ≤ ∆dt,max ≤ 10dc

Minimum terminal displacement, ∆dt,min 0 ≤ ∆dt,min ≤ 14∆dt,max

Maximum BS displacement, ∆dbs,max 0.1dc ≤ ∆dbs,max ≤ 10dc

Minimum BS displacement, ∆dbs,min 0 ≤ ∆dbs,min ≤ 14∆dbs,max

BS movement probability, πbs πbs 1

4.6 Typical Parameter Values

In this section, we discuss typical values for the time evolution and drop deployment

model parameters. Although there may be many ways to set some of the parameters,

a few important suggestions will be made. Recommended values and ranges for the

parameters are summarized in Table 4.1. Justification for the recommendations are

discussed in the following.

We first consider the minimum BS separation parameter, δBS, in the drop deployment

model. Note that if δBS were on the order of the simulation area radius R, that is,

δBS ≈ R, the number of BSs fitting in the simulation area would be severely limited.

Therefore, it is recommended that δBS be small in comparison to R, or satisfy δBS R,

so that a large number of BSs can be used in the system simulation while the practical

separation constraints are still satisfied.

In the time evolution model, we begin by considering the setting of the total number

of drops, P , the number of subdrops, B, and the number of subframes, F , as well as the

scheduling averaging window size, T0. The primary consideration is that a sufficiently

large number of drops, subdrops and subframes be used in the simulation so that the

simulation results will be representative (in a statistical sense) of the different types of


possible network configurations and slow and fast network evolutions. A secondary and

competing consideration is that P , B, and F must be small enough that the simulations

can be completed on a given computer system in a desirable amount of time. (Note that

the program execution times can vary greatly depending on the system; therefore, the

issue is not considered in detail here.) In order to balance the two competing factors, we

propose the strategy of setting P as high as possible while setting B and F at a moderate

value to keep the computational complexity sufficiently low. There is an expectation that

even if the values B and F are relatively low, the values PB and PBF (i.e. the total

number of subdrops and subframes, respectively) will be high enough to be statistically

representative of evolution scenarios. A P value of 300 (which is high) and a B value of 20

(which is moderately low) are suggested and seem to work well in practice. Note that the

P value can also be set higher for greater accuracy. The setting of the F value must be

done in conjunction with the T0 value. In the exponential averaging of (2.9) in Chapter 2,

the setting of T0 affects the response of the average rates, R, to the instantaneous rates,

r. Small T0 values result in fast but not instantaneous response times; and conversely,

large T0 values result in slow response times, but smoother averaging. We recommend

a typical low T0 value of T0 = 5 from the literature [75], which results in a relatively

fast response time. In order to ensure adequate statistical averaging of results with

exponential averaging, the value of F is recommended to be higher than the B value.

Setting F = 100 is shown to work well in practice.

The setting of terminal arrival and departure, and BS deployment and outage pa-

rameters are considered next. It is important to note that in realistic networks, the BS

deployments and outages occur much less frequently compared to terminal arrival and

departure events. It is, therefore, suggested that BS and terminal model parameters, λbs,

λt, µbs and µt, satisfy the relations:

λbs λt (4.23)

µbs µt (4.24)

Setting of the actual parameter values are expected to depend on many other factors,

including the subdrop time duration and initial number of terminals, K(0), and number

of BSs, N(0) used in the model. However, nominal settings of λbs = 0.5 BSs, λt = 3

terminals, µbs = 20 subdrops and µt = 5 subdrops are suggested, and are shown to work

well in practice. Figures 4.7 and 4.8 illustrate sample realizations of the evolution of the

number of terminals and BSs, respectively, with the suggested model values.

We finally discuss the parameter settings for the terminal and BS movement models.


Figure 4.7: Sample realization of the evolution of the number of terminals K(b) oversubdrops b, with K(0) = 25, the arrival parameter λt = 3 terminals, and the departureparameter µt = 5 subdrops.

Since subdrops are designed to model small gradual changes over the network configura-

tion, we suggest setting the maximum terminal displacement parameter, ∆dt,max, large

enough to cause a small but noticeable change in shadowing over one subdrop. It is known

from correlated shadowing theory that movement on the order of correlation distance,

dc, results in a noticeable change in shadowing. Therefore, ∆dt,max needs to satisfy

0.1dc ≤ ∆dt,max ≤ 10dc. (4.25)

Furthermore, it is recommended that the minimum displacement parameter, ∆dt,min, be

set as low as possible, and possibly set to zero, to allow a wide random variation of

terminal displacement. Hence, it should satisfy

0 ≤ ∆dt,min ≤1

4∆dt,max. (4.26)

Similar to BS arrivals and departures, BS movements are expected to occur very infre-


Figure 4.8: Sample realization of the evolution of the number of BSs N(b) over subdropsb, with N(0) = 7, the arrival parameter λbs = 0.5 BSs, and the departure parameterµbs = 20 subdrops.

quently. Thus it is suggested that πbs satisfy

πbs 1. (4.27)

Finally, for the same reasons as terminal models, it is recommended that the values of

maximum and minimum BS displacements, ∆dbs,max and ∆dbs,min, satisfy similar in-

equality relations:

0.1dc ≤ ∆dbs,max ≤ 10dc (4.28)

0 ≤ ∆dbs,min ≤1

4∆dbs,max. (4.29)


4.7 The Outline of the Network Model of the Thesis

The basic structure of the general network model in the thesis is outlined in block diagram

form in Figure 4.10. Note that this is a more detailed version of the block diagram of

Figure 4.1. However, note that the model of Figure 4.10 does not include all of the

details discussed in the full model of the thesis. For instance, there is much more detail

in terms of hotspot deployment, and shadowing with multiple propagation environments

(Chapter 6); BS dynamics (Chapters 4 and 7); and the BS power adjustment algorithms

(Chapter 5) in the complete model.

Figure 4.9: A realization of the potential field [8].


j = j + 1

STOP

No

Generate subframe l of subdrop j:

- Compute the multipath fading for all

terminal-BS links.

Channel values:

- Compute the path loss values PLn,k

for all N(j)K(j) terminal-BS links.

- Compute the shadowing values SFn,k

for all N(j)K(j) terminal-BS links as a

function of the potential field (u)

values.(2)

Large-Time-Scale Adaptations

- Determine the terminal BS associa-

tions based on maximization of large-

scale channel gain, Gn,k = PLn,kSFn,k.

- Adjust BS power levels, Pn.(3)

- Partition the BSs into scheduling

cells (clusters) for scheduling coordi-

nation.(4) Yes

l = l + 1 l mod F = 0?

Generate simulation drop i:

- Randomly deploy N(0) BSs with

minimum separation, BS.

- Randomly deploy K(0) terminals.

- Generate the underlying Gaussian

shadowing potential field, u.(1)

Dynamics:

- Account for terminal

movements.

- Account for any terminal

arrivals and/or departures:

K(j)= K(j-1)+[Ka](j)- [Kd](j).

Generate subdrop j of drop i:

- Sets of terminals and BSs and their

positions.

j mod B = 0?

Yes

Yes

i = P?

No

No

i = i + 1

Fine Resource Allocation

- In each cluster, determine the user

scheduling, with coordination.(5)

Figure 4.10: Flowchart of the general network model in the thesis.


Greater explanation is given in the following for the parts marked with superscript

numbers (1) through (5) in Figure 4.10:

(1) Generation of Shadowing Potential Field

A shadowing potential field is a random process whereby each realization is obtained as

a function from the plane to the set of real numbers. Note that the potential field is

stored in a vector denoted by u. Let σ0 be the shadowing standard deviation and dc the

correlation distance. The value of the potential field at point A defines the potential level

of point A and is denoted by uA. For a cellular network with N BSs and K terminals,

J = N +K potential levels are generated according to the following steps:



• Step 2: Generate a J × J correlation matrix R = [rij] such that

rij =σ2

0

2e−dij/dc (4.30)

where dij is the distance between point i and point j.

• Step 3: Decompose the correlation matrix using Cholesky factorization such that

R = BBH.

• Step 4: Calculate the J × 1 potential level vector u according to

u = Bv (4.31)

A sample realization of the potential field over a grid of points is shown in Figure 2.11,

reproduced for convenience in Figure 4.9. It is important to note that the underlying

potential field is fixed when generating a simulation drop. Then in the generation of

individual subdrops in a drop, the terminals move, and new ones appear and old ones

disappear, on top of the given and fixed potential field.

(2) Computation of Shadowing

In particular, the shadowing between points A and B ( a terminal and a BS) is computed

as a function of the two potential levels according to

SF [dB] = f (uA, uB) = sgn (uA + uB) · |uA − uB| . (4.32)


(3) Adjustment of Power Levels

Pn is the operating power level of BS n and is determined by maximizing some utility in

the long run, in particular, either the total rate or the sum log utility. Note that detailed

algorithms are given in Chapter 5. Once the power level is fixed, the BS transmits with

a constant power in each subcarrier (or does not transmit in the subcarrier), as the LTE

is based on constant power, variable rate transmission.

(4) Clustering

BSs that are close together will tend to cause greater interference on each other and will

tend to be grouped into the same cluster. The K-means clustering algorithm is used,

where the similarity matrix is determined by a measure of the mutual interference of a

pair of BSs on each other’s users. Note that the clustering was described in Section 2.4 of

Chapter 2. Thus, it can be said that the clustering changes based on the large-time-scale

variations of traffic and will need to be adapted dynamically.

(5) Coordination in the Clusters

Coordination in each cluster means that the neighbouring BSs may not transmit at the

same time (and at the same frequency subcarrier); or if they do transmit at the same

time, they select the scheduled users with coordination. As already mentioned, the power

level is not varied at this smallest time scale. Note that the detailed algorithm is given

in Section 2.4 of Chapter 2.

Chapter 5

BS Power Adaptation (Single Power

Class)

5.1 Relevant Work: Interference Coordination Strate-

gies for Heterogeneous Networks

A variety of interference coordination methods for heterogeneous networks have been

studied in detail in the literature. Much of the work focuses on femto access points

(FAPs) and their interaction with the macro BSs (MBSs). Femtocells operate in licensed

spectrum and improve the indoor coverage where the macro signal tends to be weak. A

primary advantage of the femtocells is that they can be easily and inexpensively deployed

by users since they are connected to the network through the residential broadband wire-

less connections. A number of interference management methods exist for heterogeneous

networks involving femto and macro and other BS networks. The literature commonly

classifies the interference as either co-tier, i.e., between the network elements belonging

to the same tier in the network, or as cross-tier, i.e., between network elements belonging

to different tiers.

In orthogonal frequency division multiple access (OFDMA)-based femto networks,

there is a flexibility to assign orthogonal subcarriers to femto and macrocells. This

gives an advantage to OFDMA networks over CDMA networks. Research in OFDMA

networks has focused on effective interference management schemes in order to mitigate

and reduce the co-tier and cross-tier interferences, which would enhance the throughput

in the network.

Li et al. [76] presented a strategy for clustering of femtocells in order to mitigate the

downlink interference for an OFDMA closed access femto network. The proposed scheme

76

Chapter 5. BS Power Adaptation (Single Power Class) 77

uses a combination of dynamic frequency band allocation among femto and macro BSs

and clustering of femto BSs based on location. In particular, a portion of the band is

dedicated to the macro BS users and the rest reused by both macro and femto BSs.

The dedicated frequency band for the macro users is intended to mitigate the macro

user downlink deadzone problem, where a cell edge macro user experiences severe co-

channel interference from nearby femtocells. A clustering algorithm is used to determine

the cross-tier and co-tier frequency reuse. Femto BSs are allocated into frequency reuse

clusters based on a calculated threshold distance. If the distance between two femto BSs

is above the threshold, they are assigned to the same cluster and the femto users share

the same frequency band. If the distance is less than the threshold, they are assigned to

different clusters to avoid co-tier and cross-tier interference.

Fractional frequency reuse (FFR) has been considered as an interference management

strategy in the literature. The basic idea is that the frequency spectrum is divided into

subbands and each subband is assigned to different macrocells and sub-areas of the

macrocells and/or the femtocells. The authors in [77] propose a scheme encompassing

fractional frequency reuse with pilot sensing to reduce cross-tier interference between

femto and macrocells. In particular, a frequency reuse factor (FRF) of 3 or above is

utilized for the macrocells. The femto BSs determine the frequency reuse by sensing

the macro BS pilot signals and avoiding the band with the highest pilot signal power.

However, the frequency partitioning, in this scheme, is static which results in inefficient

spectrum use and deterioration of throughput in the system.

Juang et al. [78] propose a dynamic adaptive FFR scheme for femto networks. The

paper considers the effect of the femto BS interference on the macro users. As the

primary interference at the macrocell edges is the inter-macrocell interference, the work

focuses on the femto BSs in highly dense inner regions. The scheme uses FFR radio

resource hopping or orthogonal FFR radio resource allocation: If the femto BS is located

in a dense inner region, an orthogonal FFR resource allocation is used by the femtocell.

Otherwise, an arbitrary FFR radio resource hopping pattern is adopted.

Power control has been considered and adopted for the purpose of interference mit-

igation. The advantage of power control is that it permits interference coordination so

that the entire bandwidth can be used (universal frequency reuse) resulting in greater

efficiency. Chandrasekhar et al. [59] propose an uplink power control algorithm for a

macrocell underlaid with femtocells. A fundamental theoretical relation is derived which

provides the largest feasible cellular SINR for any set of feasible femtocell SINRs. Then

the authors propose a utility-based power adaptation algorithm where femtocells indi-

vidually maximize an objective function consisting of an SINR dependent reward, and a


penalty proportional to the interference at the macrocell. In the case that cellular users

are unable to achieve SINR targets due to femtocell interference, an algorithm reduces

SINR targets of strongest femto interferers.

The authors of [79] study power control in heterogeneous networks. In particular, the

outage probability problem was considered under Rayleigh fading. Then a total power

minimization problem with outage specification constraints was used for minimization of

the energy consumption. A minimum worst outage probability guarantee was established

for femtocell users.

Most of the works have considered power control at the multipath fading time scale,

however, this can lead to substantial signalling load due to channel information require-

ments. Nai et al. [80] use a formulation that is appropriate for the larger time scale of

the shadowing. They treat the problem of joint admission and power control in two-tier

small cell networks. The proposed algorithm maximizes the number of small cell users

admitted while guaranteeing a QoS for the macrocell users.

Jin et al [81] propose a scheme where the uplink power control for femto cells is decided

on a per cluster basis. The large-scale channel information and shadowing variation is

exploited and an interference allowance per femtocell is determined based on aggregate

interference. Both a centralized and a distributed sensing algorithm are proposed. In

centralized sensing, the macro BS estimates the number of active femtocells per cluster

and broadcasts the interference allowance to femtocells. In distributed sensing, each cell

senses the activity of others and determines a power setting.

Ha et al [82] propose a joint base station association and power control algorithm

for heterogeneous cellular networks. In particular, the power control algorithm utilizes

non-cooperative game theory. Furthermore, an adaptation mechanism is devised for the

power control algorithm so that it can support the SINR requirements of all users when-

ever possible while exploiting multiuser diversity to improve system throughput. The

proposed framework focuses on heterogeneous networks with multiple tiers and different

QoS requirements (voice and data users). An application is then presented for two-tier

macro-femto networks with a hybrid spectrum access strategy.

The combination of fractional frequency reuse and power control has also been con-

sidered as a hybrid approach. Li et al [83] propose a fractional frequency reuse and power

control scheme in order to coordinate the interference among high and low power nodes

in a heterogeneous network. The algorithm considers both the spectral efficiency and

user fairness in the design. In particular, optimization formulations based on maximizing

the sum of the long term log-scale throughput are considered for both the power control

and resource allocation schemes with fractional frequency reuse. The performance is then


studied for a particular network model with macro BSs deployed according to the hexag-

onal structure and four micro nodes uniformly deployed in each sector. In distributed

power control, the estimation of interference powers at the receiver is an important issue.

In most algorithms, perfect estimation of interference power is assumed. Karamad et

al. [84] focus on and investigate this problem. While modern algorithms assume that

perfect channel information is available locally, this is not guaranteed to be the case.

The authors consider the performance degradation due to estimation errors and quantify

the resulting transmit power and SINR errors.

Other techniques for interference mitigation including coordinated beamforming and

clustered resource allocation have also been investigated in the literature. Yu et al [85]

propose a scheme for coordinating transmit and receive beamforming and resource allo-

cation schemes across multiple base stations for interference mitigation in both macrocell

and mixed macrocell and femtocell/picocell deployment. Coordination among different

base stations is considered particularly important for a scenario with heavy overlap of

femto/pico and macro BSs. The scheme jointly optimizes the proportional fair schedul-

ing, spatial multiplexing, and power spectrum adaptation using a heuristic method.

Hosseini et al [86] propose a cluster-based coordinated beamforming and power allo-

cation scheme for MIMO heterogeneous networks. In the scheme, the small cells form

clusters based on the distance which is an indication of interference. The small cells within

a cluster share users data symbols and cancel the intra-cluster interference through zero

forcing beamforming, which is a network MIMO scheme. Furthermore, a distributed

power control scheme is proposed for inter-cluster interference mitigation, which involves

the exchange of low-rate messages among cluster representatives.

Akoum et al [87] consider interference coordination with clustering from the point of

view of adaptive limited feedback. In particular, the work studies a network modelled

by a Poisson point process to provide an analytical assessment of the performance of

coordination. An intercell interference nulling strategy is applied to coordinate the in-

terference inside the clusters. It is assumed that channel knowledge is exchanged among

coordinating base stations in a cluster and the performance of interference coordination

is evaluated. The authors propose a scheme for determining the average cluster size as

well as for adapting the amount of limited feedback to maximize the coordination gains.

Lee et al [88] study the performance of coordinated beamforming with dynamic clus-

tering using a stochastic geometry approach. They derive analytical expressions for the

signal to interference ratio (SIR) distributions in terms of the number of BSs per cluster,

the number of antennas per BS and the pathloss exponent. The benefit of coordinated

beamforming and optimal cluster cardinality are investigated.


Singh et al [89] study the effect of power control combined with load balancing, or the

offloading of users onto small cells, on the SIR or rate coverage in the uplink in a K-tier

heterogeneous network. A tractable and general model is provided to characterize the

SIR and rate distribution as a function of the association and power control parameters.

5.2 Contribution: Adaptive Resource Allocation Frame-

work with Power Adaptation

An adaptive resource allocation framework based on clustering and power control at a

large time scale and coordinated proportional fair resource allocation at a small time scale

is proposed for a generic irregular network. Note that in heterogeneous networks, the cell

association is normally determined based on maximizing long-term received power and

membership to different tiers is explicitly considered. In this work, we assume associa-

tion to a cell of any tier based on maximization of long-term channel gains. The primary

contribution is the power control algorithm for the purpose of adaptive resource alloca-

tion at a large time scale. While much of the literature considers power control at the

multipath fading time scale, the acquisition of channel information at this scale can lead

to substantial signaling requirements. In addition, much of the literature considers static

or semi-static allocation of a dedicated spectrum for the purpose of resource allocation.

Our contribution considers the use of the entire spectrum for greater spectral efficiency

and the power adaptation at a large time scale, to reduce the signaling requirement. As

the adaptive framework is to be used for a general or arbitrarily large time scale, the

availability of individual QoS requirements at such periodicity are not assumed. Instead,

a central entity does its best effort to maximize either the long-term sum rate or the

sum log utility (based on proportional fairness) over the time scale under the modest

assumption of base station power constraints. The framework is intended as a baseline

adaptive resource allocation framework for an irregular network.

5.2.1 Key Results

The performance of the system with the BS power adaptation algorithms is evaluated

for a particular system scenario and the effect of irregularity is considered by varying

the minimum BS separation, δBS. Improvements in terms of 50th percentile spectral

efficiency, sum log utility and user average spectral efficiency are obtained. In particu-

lar, both algorithms achieve nearly 10% improvement in median spectral efficiency and

between 3-9% improvement in user average spectral efficiency.


It is shown that performance degrades significantly due to the irregularity of deploy-

ment as the distance between users and BSs become larger. In addition, the improvement

from the power adaptation algorithms is observed to increase with increasing irregular-

ity, indicating that power adaptation algorithms partially compensate for the effect of

irregularity.

5.3 Introduction

In Chapter 4, we studied a dynamic system model consisting of a simulation and adap-

tation framework for the irregular network with a single power class and constant power

levels. In this chapter, we introduce the adaptation of BS power levels which will form

part of the extension of the original adaptation framework.

Recall that regular networks are characterized by the more or less regular deployment

of BSs, equal maximum BS power capabilities (due to a single BS power class) as well

as regular deployment of terminals. Therefore, in order to optimize the average received

signal and interference powers at the terminals in the long term, and hence, the SINRs

and transmittable rates, it has generally been adequate to operate the BSs at constant

power levels (i.e. usually at their maximum). Irregular networks, however, have charac-

teristics that are very different from the regular networks. BS deployment and terminal

locations are arbitrary or irregular, and BS maximum power capabilities are unequal due

to multiple power classes. Thus the BS power operating levels are not expected to be op-

timized at their respective maximum power levels. In this thesis, we propose the dynamic

adaptation of BS power levels within their allowable ranges for irregular networks. The

power adaptation shall operate at the slow time scale and form part of an extended adap-

tive resource allocation scheme based on the framework discussed in Chapter 2, which

included slow-scale BS assignments and network clustering, and fast-scale proportional

fair (PF) resource allocation.

In this chapter, we consider BS power adaptation for an irregular network with a

single BS power class, which, however, can be applicable to the case of multiple BS

power classes that is studied later in the thesis. The rest of this chapter is organized

as follows: The BS power adaptation algorithm with two variant forms is developed in

Section 5.4. The overall extended adaptive resource allocation framework with power

adaptation is described in Section 5.5. Finally, simulations are performed in Section 5.6.


5.4 BS Power Adaptation Algorithm

In this section, we propose the adaptation of BS power levels in such a way as to mitigate

the effects of intercell interference and improve the transmittable user rates under dynam-

ically changing network conditions. Similar to the discussion in Section 2.2 of Chapter 2,

problem formulations based on total rate maximization (TRM) and proportional fairness

(PF) are considered in the following. It is expected that the PF formulation will offer

better system performance due to its ability to achieve a balance between TRM and user

fairness in the long term. However, both methods are considered in terms of the mathe-

matical formulation, implementation (e.g. the signalling requirements) and the resulting

system performance in the rest of the chapter.

5.4.1 Total Rate Maximizing (TRM) Power Adaptation

Total rate maximizing (TRM) BS power adaptation is discussed in this section. Let

P(b) = [P1(b), . . . , PN(b)] be the vector of BS power levels at a given subdrop b, Rk,n(k)(b)

be the data rate for terminal k from the associated BS, n(k), averaged over subdrop b

and Rk(b) be the current long-term user rate averaged over many subdrops. In analogy

to problem (2.5) of Chapter 2, the TRM power optimization problem is first expressed

as a function of the long-term user rates, Rk(b):

maxP(b)

∑k∈K

Rk(b) (5.1)

subject to 0 ≤ Pn(b) ≤ Psn,max, ∀n ∈ N

From the TRM simplication procedure of Section 2.2, it is recognized that the problem

(5.1) should be transformed into a form that is easily solvable at each subdrop. The

equivalent optimization is given in terms of Rk,n(k)(b), the user rates averaged over the

subdrop:

maxP(b)

∑k∈K

Rk,n(k)(b) (5.2)

subject to 0 ≤ Pn(b) ≤ Psn,max ∀n ∈ N

For simplicity of notation, the subdrop index b is dropped to yield:

maxP

∑k∈K

Rk,n(k) (5.3)

subject to 0 ≤ Pn ≤ Psn,max ∀n ∈ N


Note that the use of the PF scheduling strategy at the fast subframe time scale is assumed

throughout the analysis. Under this assumption, Rk,n(k), the user rate averaged over the

single subdrop, is approximated in a similar manner as the derivation in (3.6)-(3.9) from

Chapter 3:

Rk,n(k) =G(Kn(k))

Kn(k)

BT log2(1 + SINRk,n(k))

=G(Kn(k))

Kn(k)

BT log2

(1 +

Pn(k)Gn(k),k

N0C +∑

j 6=n(k) PjGj,k

)(5.4)

where Kn(k) is the number of terminals assigned to BS n(k), SINRk,n(k) is the SINR

at terminal k averaged over the duration of the subdrop, BT is the total bandwidth,

Gn(k),k = PLn(k),kSFn(k),k is channel gain between a user and its associated BS averaged

over the subdrop, and G(y) is a quantity known as the scheduling gain, expressed as

a function of the number of users in question—which accounts for the multiplicative

increase in the average rates due to the scheduling of multiple users. Note that G(y) is

estimated as:

G(y) =

y∑i=1

1

i(5.5)

Problem Solution

Note that the optimization problem of (5.3) is known to be non-convex, hence, there is

no guarantee of a solution with standard solution methods. Therefore, we consider an

approximate solution as follows: Under high SINR conditions, i.e., for SINRk,n(k) 1 ∀k, the problem (5.3) can be expressed as a geometric program which can be solved

through convex programming techniques [53]. In this case, the optimization becomes

minP

∏k∈K

[INSRk,n(k)

]wk(5.6)

subject to INSRk,n(k) =N0C +

∑j 6=n(k) PjGj,k

Pn(k)Gn(k),k

∀k ∈ K

wk =G(Kn(k))

Kn(k)

∀k ∈ K

0 ≤ Pn ≤ Psn,max ∀n ∈ N

where INSRk,n(k) = 1/SINRk,n(k) is the reciprocal of the time-averaged SINR for user

k and wk is a specified weighting factor. We note that the high SINR condition is

not assured in general in the context of the BS power adaptation, and the geometric


programming method is used in this thesis as an approximate solution of the power

adjustment problem. Later in this chapter, the power adaptation algorithm will be

evaluated for a selected network scenario which will serve as a verification of the proposed

methodology.

5.4.2 Proportional Fair Power Adaptation

Proportional fair (PF) BS power adaptation is considered in this section. As discussed

in Section 2.2 of Chapter 2, adaptation according to the PF criterion is equivalent to the

sum log utility (SLU) maximization. The PF power optimization procedure is developed

in a similar manner as the TRM procedure in Section 5.4.1. In analogy to (2.7) of Chapter

2, the power optimization problem is first expressed as a function of the long-term user

rates:

maxP

∑k∈K

log(Rk) (5.7)


where P is the vector of BS power levels at the subdrop. Similar to the procedure for PF

scheduling described in Section 2.2, the problem (5.7) is transformed into a form that is

easily solvable at each subdrop, given by

maxP

∑k∈K

Rk,n(k)

Rk

(5.8)


In addition, the PF scheduling strategy is again used at the subframe time scale; therefore,

the user rate averaged over the subdrop, Rk,n(k), is computed according to (5.4) and (5.5).

Furthermore, the long-term user rate Rk is updated at each subdrop according to

Rk =

(1− 1

W0

)Rk +

(1

W0

)Rk,n(k), (5.9)

where W0 is the exponential averaging window size parameter. Finally, through the

high SINR approximation, the optimization of (5.8) is transformed into the geometric


program:

minP

∏k∈K

[INSRk,n(k)

]wk(5.10)

subject to INSRk,n(k) =N0C +

∑j 6=n(k) PjGj,k

Pn(k)Gn(k),k

∀k ∈ K

0 ≤ Pn ≤ Psn,max ∀n ∈ N

where weightings, wk, for the PF formulation are given by

wk =G(Kn(k))

Kn(k)

1

Rk

∀k ∈ K (5.11)

Similar to the TRM, the geometric program is solved through convex programming tech-

niques.

5.5 Adaptive Resource Allocation Framework with

Power Adaptation

In this section, we discuss the overall extended adaptive resource allocation frame-

work with power adaptation. We note that it might be possible to jointly optimize the

BS power adaptation step together with terminal-BS assignment and clustering steps

described in Chapter 2. Such absolute algorithmic optimization, however, has not been

the aim of this work, as the main goal has been the development of a workable and

practical adaptation framework. The overview of the proposed framework is shown in

Figure 5.1. As the power adaptation step requires the BS assignments as input, and

the clustering step requires both the BS assignments and the operating powers, the BS

assignment should be executed first, followed by power adaptation and clustering, re-

spectively. The detailed pseudocode of the adaptive framework is shown for the cases

of TRM and PF power adaptation in Figures 5.2 and 5.3, respectively, which follow a

similar structure as the algorithm of Figure 2.5. The extended adaptation algorithms

return P∗ = [P ∗1 , . . . , P∗N ], the optimized BS power vector, as an additional output. Fur-

thermore, the framework with PF power adaptation additionally requires W0, the power

adaptation window size, as an input parameter.

The signalling requirements of the adaptive resource allocation framework is signifi-

cantly lower than resource allocation without clustering. In particular, the BS assignment

and clustering steps require that the long-term channel gains Gn,k be known at a central


l = l + 1

Coordinated resource

allocation in each cluster

Form clusters

(scheduling cells)

Set BS powers

Assign terminals to BSs

l mod F = 0?

No

Yes

Figure 5.1: Extended adaptive resource allocation framework with power adaptation.

node at a periodicity of F subframes. The power adaptation (based on the formulations

of (5.6) and (5.10)) additionally requires the knowledge of traffic load for the BSs, i.e.,

the number of users, Kn, assigned to each BS, at the central entity. Fast-varying channel

gains, gn,k,c, need to be exchanged for the purpose of fast scheduling at the subframe

time scale, for BSs inside a given cluster.

5.6 Simulations

Simulations have been performed using the proposed model to evaluate the performance

of the BS power adaptation algorithms. The minimum BS separation parameter, δBS,

was varied in the evaluation to observe the effect of irregularity on performance. The

evaluations additionally illustrate the use of the proposed model in a particular scenario.

Note that a simplified system model is utilized in the simulation. The model param-

eters are summarized in Table 5.1. The simulation area for the scenario is illustrated in


INPUTS:




OUTPUTS:


• Optimized BS power level vector P∗ = [P ∗1 , . . . , P∗N ]



∗(l) = [p∗n,c(l)] where n ∈ N (ω);




1: for all terminals k ∈ K do2: for all BSs n ∈ N do3: Compute the time-averaged channel gain according to Gn,k = PLn,kSFn,k4: end for5: Determine the assigned BS according to n(k) = arg maxn∈N Gn,k.6: end for7: Find the optimized BS power level vector by solving the optimization problem of

(5.6).8: Form the SCs by using Algorithm 3 of [2].9: for SCs ω = 1 to Ω do

10: for subframes l = 1 to F do11: Determine the optimized instantaneous power vectors and the sets of scheduled

users by using the algorithm of Figure 2.6.12: end for13: end for

Figure 5.2: Algorithm for the adaptive resource allocation framework with TRM poweradaptation


INPUTS:


• Power adaptation window size parameter W0



OUTPUTS:


• Optimized BS power level vector P∗ = [P ∗1 , . . . , P∗N ]



∗(l) = [p∗n,c(l)] where n ∈ N (ω);




1: for all terminals k ∈ K do2: for all BSs n ∈ N do3: Compute the time-averaged channel gain according to Gn,k = PLn,kSFn,k4: end for5: Determine the assigned BS according to n(k) = arg maxn∈N Gn,k.6: end for7: Find the optimized BS power level vector by solving the optimization problem of

(5.10).8: Form the SCs by using Algorithm 3 of [2].9: for SCs ω = 1 to Ω do

10: for subframes l = 1 to F do11: Determine the optimized instantaneous power vectors and the sets of scheduled

users by using the algorithm of Figure 2.6.12: end for13: end for

Figure 5.3: Algorithm for the adaptive resource allocation framework with PF poweradaptation


2.78 h

h

Figure 5.4: Simulation area for cellular network.

Figure 5.4. The inter-BS distance in the regular deployment is taken to be 2h = 500

m based on the urban macro environment in the ITU model. The simulation area ra-

dius of R = 2.78h is used in order to equal the area of the hexagonal pattern in the

regular deployment. The time evolution in this scenario considers terminal movements

with constant displacement and uniformly random direction, and neglects other types

of dynamics. Constant terminal displacements values as listed in Table 5.1 have been

chosen according to (4.25). BSs are not allowed to power on and off, as a non-zero lower

limit, Pmin, has been imposed on the BS power levels. The difference between Pmin and

Pmax has been set to 6 dB. Note that the remainder of the parameter values were sup-

plemented from the ITU model. P = 150 simulation drops each consisting of B = 20

subdrops are used in the evaluation. Both the total rate maximization (TRM) power

adaptation of (5.6) and the proportional fair (PF) power adaptation of (5.10) are sim-

ulated for comparison against the utilization of equal BS powers, or Pn = Pmax, for all

BSs.

In order to see the effect of irregularity, the value of minimum BS separation parameter

δBS was varied in the simulation from a value of 0 to 2h in increments of 2h/3 as shown


Table 5.1: Simulation Model Parameters

Parameter Value

Simulation area radius, R 2.78h = 695m

Nominal number of terminals, K 28

Total number of BSs, N 7

Bandwidth, BT 15 kHz


Background noise power spectraldensity

-174 dBm/Hz

Path loss, PL −[ 128.1 + 37.6 log10 d ], d in km

Shadowing standard deviation, σ0 10 dB

Shadowing correlation distance, dc 50 m

Maximum BS power capability, Pmax 46 dBm

Minimum allowed BS power, Pmin 40 dBm

Terminal displacement dc(s)/3 = 16.67m

Power adjustment window size, W0 4

Table 5.2: The variation of δBS.

Deployment type δBS

Irregular

0 m

2h/3 = 166.66 m

4h/3 = 333.33 m

Regular 2h = 500 m


in Table 5.2. Note that δBS = 0 m represents the case of completely arbitrary or irregular

deployment with no restriction on separation, and δBS = 2h = 500 m represents the case

of maximum BS separation, which was implemented by using the regular hexagonal BS

placement of the regular cellular network model. The other values represent intermediate

cases of BS separation.

Overall Performance

Initially, a constant number of terminals is assumed. We look at different measures to

understand the performance in terms of different criteria. The 5th percentile or cell edge

spectral efficiency as well as sum log utility are used to as indicators of the performance in

terms of user fairness. The 50th percentile (median) and user average spectral efficiency

(defined as the total spectral efficiency, or the rate per total bandwidth, normalized per

user in the system) are used to understand typical or average user performance.

Figure 5.5 shows the 5th percentile (cell edge) spectral efficiency with varying δBS.

The equal BS power method, in this case, has equal or higher cell edge spectral effi-

ciency than the PF and TRM methods for most δBS values. In addition, the TRM has

the lowest performance. However, the complete cumulative distribution functions (for

which the plot was produced for the case of δBS = 166.6 m in Figure 5.7) show that

the equal power method has improvement only below about the 5th percentile and PF

and TRM methods have better performance at other percentiles. Figure 5.6 shows the

50th percentile (median) spectral efficiency. The TRM algorithm shows about 10% im-

provement over the equal power method, and the PF algorithm performs nearly as well

as the TRM algorithm. Figure 5.8 shows the variation of sum log utility (SLU) with

BS separation, where PF and TRM methods achieve improvement over the equal power

method. In this case, the PF method has a slightly better performance. The user average

spectral efficiency for the system is plotted in Figure 5.9, where PF and TRM methods

achieve approximately 3-7% and 5-9% improvement, respectively. It is concluded for this

scenario that while the PF algorithm does better in terms of measures of fairness and

TRM in terms of typical or average user performance, both algorithms generally obtain

improvement over the equal power assignment.

We note that it is possible for the PF and TRM curves for specific percentiles to cross

each other (i.e. the 5th and 50th percentiles in Figures Figure 5.5 and Figure 5.6), as

the amount of spread in spectral efficiency is larger for the TRM than the PF. However,

the PF and TRM curves for user average spectral efficiency and the SLU must not cross

each other; as the PF outperforms in terms of SLU, and TRM outperforms in terms of

user average spectral efficiency.


0.16

0.17

0.18

0.19

0.2

0.21

0.22

0.23

0.24

0.25

0 100 200 300 400 500

5th

%le

Sp

ectr

al E

ffic

iency

(b

ps/

Hz)

Minimum Separation between BSs

Equal Power

PF

TRM

Figure 5.5: Comparison of power adaptation schemes and the equal power level scenariofor varying δBS in terms of 5th percentile spectral efficiency, with K = 28 users.

0.7

0.8

0.9

1

1.1

1.2

0 100 200 300 400 500

50

th %

le S

pec

tral

Eff

icie

ncy

(b

ps/

Hz)

Minimum Separation between BSs

Equal Power

PF

TRM

Figure 5.6: Comparison of power adaptation schemes and the equal power level scenariofor varying δBS in terms of 50th percentile spectral efficiency, with K = 28 users.


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.5 1 1.5 2 2.5

Cu

mu

lati

ve

Dis

trib

uti

on

Fu

nct

ion

Spectral efficiency (bps/Hz)

Equal Power

PF

TRM

(a)

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Cu

mu

lati

ve

Dis

trib

uti

on

Fu

nct

ion

Spectral Efficiency (bps/Hz)

Equal Power

PF

TRM

(b)

Figure 5.7: Cumulative distribution functions for spectral efficiency at δBS = 4h/3 =333.3 m, with (a) normal magnification and (b) increased magnification, with K = 28users.


113.5

114.5

115.5

116.5

117.5

118.5

119.5

0 100 200 300 400 500

Sum

Lo

g U

tili

ty

Minimum Separation between BSs (m)

Equal Power

PF

TRM

Figure 5.8: Comparison of power adaptation schemes and the equal power level scenariofor varying δBS in terms of the sum log utility, with K = 28 users.

1.12

1.22

1.32

1.42

1.52

1.62

0 100 200 300 400 500

Use

r A

ver

age

Sp

ectr

al E

ffic

iency

(b

ps/

Hz)


Equal Power

PF

TRM

Figure 5.9: Comparison of power adaptation schemes and the equal power level scenariofor varying δBS in terms of the user average spectral efficiency, with K = 28 users.


0.76

0.81

0.86

0.91

0.96

1.01

1.06

0 100 200 300 400 500

Use

r A

ver

age

Sp

ectr

al E

ffic

iency

(b

ps/

Hz)


Equal Power

PF

TRM

Figure 5.10: Comparison of power adaptation schemes and the equal power level scenariofor varying δBS in terms of the user average spectral efficiency, with K = 56 users.

Effects of Irregularity

An important trend observed from the results is that all the metrics are lower with re-

duced δBS, and increase with higher δBS. To further investigate the effects of irregularity

of deployment, we consider the user average spectral efficiency. We additionally plot the

user average spectral efficiency for a different number of terminals (K = 56), which is

depicted in Figure 5.10. In particular, a penalty of close to 35% difference in user av-

erage spectral efficiency is obtained for completely irregular deployment (δBS = 0 m) as

compared to the regular deployment (i.e. δBS = 2h = 500 m) with the two different K

values. This result highlights the idea that performance is worse under irregular deploy-

ment since the distance between users and serving BSs become greater under increasing

irregularity.

We additionally consider the percentage improvement in user average spectral effi-

ciency over equal power assignment obtained by different power adaptation algorithms

for both K values, which are listed in Tables 5.3 and 5.4. For a user number of 56, the

improvement for PF and TRM methods are reduced compared to a user number of 28. It

is significant to note, however, that with reduced δBS, or with increasing network irregu-

larity, the improvement in spectral efficiency increases for both K values. The increased

improvement for irregular deployment indicates that the power adaptation algorithms


Table 5.3: Percentage improvement in user average spectral efficiency for different poweradaptation algorithms over equal power assignment with K = 28 users.

Deployment type δBS PF TRM

Irregular

0 m 6.9% 9.1%

2h/3 = 166.66 m 5.3% 7.9%

4h/3 = 333.33 m 3.9% 6.0%

Regular 2h = 500 m 3.6% 5.4%

Table 5.4: Percentage improvement in user average spectral efficiency for different poweradaptation algorithms over equal power assignment with K = 56 users.

Deployment type δBS PF TRM

Irregular

0 m 4.1% 5.8%

2h/3 = 166.66 m 4.0% 5.3%

4h/3 = 333.33 m 2.6% 4.3%

Regular 2h = 500 m 2.6% 3.8%

compensate, in part, for the effect of increasing irregularity in the considered scenario.

Chapter 6

Static Model of Multiple Power

Classes

6.1 Relevant Work

Deployment Model

The literature review for the deployment model in this chapter was included with Chap-

ter 4.

Correlated Shadowing Model for Irregular BS Deployment

The literature review for the correlated shadowing model in this chapter was included in

Chapter 2, Section 2.8.

6.2 Contribution

Deployment Model

The contribution for the deployment model in this chapter was included with Chapter 4.

Correlated Shadowing Model

Most of the work in the literature considers the uncorrelated lognormal shadowing model.

While a few authors proposed correlated shadowing models for links with a common end

and more recently, for links without a common end, to our knowledge, no work considers a

theory for variation of the shadowing parameters, i.e. standard deviation and correlation

distance, in a network with an arbitrary number of radio propagation environments.

97

Chapter 6. Static Model of Multiple Power Classes 98

Such a theory is proposed based on a hypothesis for the shadowing properties, and a

method of generation of shadowing for radio links is provided based on the work in [8].

Finally, model consistency is verified for a particular scenario, and the implementation

in practical simulations is discussed. The model is intended for the baseline realism of

the simulation scenario.

6.3 Introduction

In Chapter 4, we considered system modelling for an irregular network with a single BS

power class, and proposed drop deployment, channel gain and time evolution models for

such a network. Then in Chapter 5, we studied an adaptation framework that is appro-

priate for the irregular network. Starting in this chapter, we develop a system modelling

framework for the irregular network with multiple BS power classes. Note that existence

of multiple power classes imply (1) a number of distinct topographic environments, and

(2) great variation of deployment density of users and BSs, from the highly dense in

local hotspots to the highly sparse e.g. in rural areas. Appropriate drop deployment and

channel models for the particular type of network is considered in this chapter. The time

evolution methodology and system evaluation will be discussed in Chapter 7.

The developments in this chapter are summarized in the following: First, the drop

deployment model of Chapter 4 did not consider the deployment of local hotspots. There-

fore, it will be important, in the following, to develop a realistic method of drop deploy-

ment with explicit modelling of hotspots. Second, the channel models of Chapter 4 were

specified under the assumption of fixed model parameters; that is, the path loss model

assumed fixed values for A, B, C and X parameters. Similarly, the shadowing model

of [8] assumed fixed values for σ0 and dc parameters. In the network with multiple power

classes and distinct topographic environments, the parameter values are expected to be

variable rather than fixed. Note that scenarios with fixed and varying channel parameter

values will be referred to, in this thesis, as homogeneous and inhomogeneous scenarios,

respectively. In the following, it will be important to develop channel models for scenarios

of varying or inhomogeneous parameter values.

The rest of this chapter is organized as follows: The drop deployment method with

explicit modelling of hotspots is proposed in Section 6.4. The path loss and shadow-

ing channel models for inhomogeneous scenarios are developed in Sections 6.5 and 6.6,

respectively.


R

Rh

(xh,H, yh,H)

(xh,1, yh,1)

Kh,1 users

Nh,1 BSs

Kh,H users

Nh,H BSs

Rh

Kl users

Nl BSs

Figure 6.1: Simulation area for deployment in networks with multiple BS power classesand explicit hotspot modelling.

6.4 Deployment Model with Hotspots

Drop deployment with explicit modelling of hotspots is detailed in this section. As

discussed in Chapter 4, restrictions are imposed on positioning of BSs due to such reasons

as broad-level deployment planning by wireless operators, and public and private property

considerations. In addition, similar restrictions need to be enforced in the deployment

of local hotspots. We, therefore, propose a deployment method, in this section, that

is statistically random with certain mathematical conditions. The area of simulation

and the important model parameters are illustrated in Figure 6.1. A circular area with

radius R is utilized in the simulation. The area falling outside the local hotspots shall

be referred to, in this thesis, as the low-density zone. Let H be the number of hot spots,

(xh,i, yh,i) the rectangular coordinates of the centre of the ith hot spot and Rh the radius

of each hot spot. Kh,i and Nh,i are the number of terminals and the number of BSs in

the ith hot spot, respectively. Kl and Nl are the number of terminals and the number

of BSs in the low-density zone, respectively,. The number of terminals, Kl, Kh,i and K,

and the number of BSs, Nl, Nh,i and N , satisfy

Kl +H∑i=1

Kh,i = K (6.1)

Nl +H∑i=1

Nh,i = N. (6.2)


In addition, we introduce the following minimum separation parameters: δHS is the

minimum separation between hotspots, δedge is the minimum separation between the edge

of each hotspot and the edge of the simulation area, δBSH is the minimum separation

between BSs inside hotspots, and δBSL is the minimum separation between BSs in the

low-density zone. The deployment of hotspots, BSs and terminals in a given simulation

drop is generated according to the following steps

1. Set the input parameters: Set the parameters N , K, R, H, Rh, δHS, δedge, δBSH ,

δBSL, Kl, and Kh,i for each hot spot i. For the purpose of realism, it must be

assured that Rh R, δHS R, δedge R, δBSH R and δBSL R. Since the

typical distances inside hotspots are smaller than those in the low-density zone,

it is suggested that δBSH be an order of magnitude smaller than δBSL, that is,

δBSH δBSL.

2. Drop the hotspots: Generate the coordinates (xh,i, yh,i) for each of the H hot spots

randomly according to (4.1)-(4.4). If the separation between any of the hotspots

is smaller than δHS, or the separation between a hotspot and the simulation area

edge is smaller than δedge, repeat the step of coordinate generation according to

(4.1)-(4.4).

3. Drop the terminals:

(a) Create the coordinates, (xk, yk), of the Kh,i terminals inside each hot spot by

first generating random quantities a and φ with probability densities:

fA(a) =

2aR2 for 0 ≤ a ≤ Rh,

0 otherwise,(6.3)

fΦ(φ) =1

2πfor 0 ≤ φ ≤ 2π, (6.4)

and later converting the results into rectangular coordinates according to

xk = xh,i + a cosφ (6.5)

yk = yh,i + a sinφ (6.6)

(b) Generate the coordinates of the Kl low-density zone terminals according to

(4.1)-(4.4). If any of the low-density zone terminal coordinates fall inside one

of the hotspots, re-generate the coordinates of affected terminals.


4. Drop the BSs:

(a) Determine the number of base stations that will be deployed in each of the

different areas as a function of the ratio of terminals in a given area to the

total number of terminals in the simulation, that is,

Nl = Round

(NKl

K

)(6.7)

Nh,i = Round

(NKh,i

K

)(6.8)

where Round( ) returns the nearest integer for any real number.

(b) Generate the coordinates, (xn, yn), of the Nh,i BSs inside each hot spot, ac-

cording to (6.3)-(6.6). If the separation between any of the BSs is smaller than

δBSH , or if any of the BSs fall on one of the users, re-generate the coordinates

of the affected BSs.

(c) Generate the coordinates of Nl low-density zone BSs inside the simulation

area. If any of the low-density zone BSs fall inside one of the hotspots; or the

separation between any of the BSs is smaller than δBSL; or, lastly, if any of

the BSs fall on one of the users, re-generate coordinates of the affected BSs

until the situation is corrected.

6.5 Distance Dependent Path Loss Model

Modelling of distance-dependent path loss for the inhomogeneous network scenario is

considered in this section. A straightforward extension of the homogeneous model of

Chapter 2 is proposed. In inhomogeneous network scenarios, there exist a variety of

radio propagation environments as well as distinct BS power classes. We hypothesize,

in this work, that the distinct topographic environments, with unique path loss model

parameters, correspond directly to the transmitting BS power classes—as evidenced in

some of the relevant literature [1,9,30,35,90]. In particular, the path loss model parame-

ters A, B, C and X in the inhomogeneous scenario are modelled as functions of the BS’s


power class, sn,

A = A(sn) (6.9)

B = B(sn) (6.10)

C = C(sn) (6.11)

X = X(sn) (6.12)

The path loss from BS n to any terminal k is, therefore, given by [30]

PLn,k(sn)[dB] = A(sn) log10(dn,k) +B(sn) + C(sn) log10

(fc5.0

)+X(sn) (6.13)

where dn,k is the distance between BS n and terminal k in metres, and fc is the system

frequency in GHz. The detailed verification of the validity of this model under practical

conditions is recommended as a future research project. We note that the path loss

values will be required in the computation of shadowing according to the model of the

next section.

6.6 Inhomogeneous Shadowing with Correlation

In this section, we extend the correlated lognormal shadowing model of [8] (for the ho-

mogeneous scenario) to generate correlated shadowing for the inhomogeneous scenario,

where shadowing parameters, σ0, standard deviation, and dc, correlation distance, are

allowed to vary over space. Before providing the complete model, we find it necessary

to discuss the nature of spatial variation of shadowing parameters in realistic irregular

networks. Note that shadowing has traditionally been a little investigated area of wireless

channel modelling. In particular, although several authors [36,44,45] published real-world

shadowing measurement results for specific environments and situations, to our knowl-

edge, none of the literature reported on a general theory for the variation of shadowing

parameters in inhomogeneous wireless networks. We, therefore, note here some relevant

properties, as discussed in [9,40,91], to help in the development of a physically justifiable

inhomogeneous shadowing model:


Table 6.1: The variation of σ0 and dc based on the radio propagation environments [1].

Radio Environment Ps,max [dBm] Typical distance be-tween neighbouringcells [m]

dc [m]

Indoor hotspot 21 60 6

Urban micro 41 200 13

Urban macro 46 500 50

Suburban macro 46 1299 50

Rural macro 46 1732 120

1. Shadowing is a large-scale phenomenon averaged over small displacements in space,

which is dependent on the attenuation and scattering caused by objects in the

environment, and should not by nature vary quickly in space.

2. Small cells (where BSs have small Ps,max) typically show certain properties with

respect to the shadowing parameters whereas large cells (where BSs have large

Ps,max) tend to show the opposite properties with respect to the same parameters.

For instance, the correlation distance, dc, in small cells tend to be smaller due

to the large number of small electromagnetic scatterers and attenuating objects

typically present in small-cell environments. Conversely, in large cells, dc tends

to be larger due to the small number of large scatterers and attenuating objects

in such environments. Table 6.1 shows a list of the dc values for different radio

environments, (obtained from IMT-Advanced guidelines [1]), where the trend of

dc values is confirmed. The values increase with increasing Ps,max and distance

between BSs, from the indoor hotspot environment to the rural macro environment.

3. The effect of an attenuating object, e.g. a vehicle, on shadowing is different in a

short link (e.g. 5 m) and in a long link (e.g. 500 m). The attenuating effect is

stronger in the short link as the vehicle is considered long in proportion to the total

length of the link.

4. When a propagation front passes through successive areas in the wireless channel,

the gains (or losses) in shadowing are multiplicative and thus add in the logarithmic

domain.

In light of points 1 to 4, we hypothesize the following principles for the generation of

shadowing parameters in the cellular network:


Principle 1: Homogeneous Scenario

As per point 2, the topographic properties can be modeled to correspond directly to the

BS power class associated with each of the cells. Thus, in the homogeneous scenario, the

shadowing parameters σ0 and dc, should be modelled as functions of the BS power class

s:

σ0 = σ0(s) (6.14)

dc = dc(s) (6.15)

Principle 2: Inhomogeneous Scenario

In the inhomogeneous scenario, there exist cells (or BSs) of various power classes. Fur-

thermore, some radio links between BSs and terminals cross through a number of cells

of different power classes over different lengths, which is illustrated in Figure 6.2. As

per Principle 1, and point 4, effective shadowing parameters (denoted by σ0,eff and dc,eff)

for such radio links are expected to be mathematical functions of (1) the homogeneous

shadowing parameters corresponding to power classes of the cells crossed (i.e. σ0(s) and

dc(s)), and (2) the length over which the cells are crossed.

In the following, we develop the spatially correlated shadowing model for the inho-

mogeneous network scenario. The generation of shadowing for individual radio links,

including the calculation of effective shadowing parameters according to Principle 2 and

the actual computation of shadowing values, is detailed in Section 6.6.1. The model con-

sistency verification is discussed in Section 6.6.2. The practical model implementation for

a full network through the use of complexity reduction techniques is discussed in Section

6.6.3.

6.6.1 Generation of Shadowing for Links between BS-Terminal

Pairs

The inhomogeneous shadowing model with spatial correlation has a similar underlying

structure as the homogeneous shadowing model, as it is also based on the computation

of the potential level vector (potential field). The primary difference is that multiple

sets of shadowing parameters, σ0,eff and dc,eff, as well as potential fields, u, will need

to be generated. Consider the sample radio link between a BS n ∈ N and a terminal

k ∈ K, depicted in Figure 6.2, which crosses through several cells of potentially distinct

BS power classes. Shadowing is generated for each radio link according to the following


associated with each of the cells. Thus, in the homogeneous

scenario, the shadowing parameters 𝜎0 and 𝑑𝑐 , should be

modelled as functions of the BS power class 𝑠:

𝜎0 = 𝜎0(𝑠)

𝑑c = 𝑑c(𝑠)

Principle 2: In the inhomogeneous scenario, there exist cells

(or BSs) of various power classes. Furthermore, some radio

links between BSs and terminals cross through a number of

cells of different power classes over different lengths, as

illustrated in Fig. 1. As per Principle 1, and points 3 and 4,

effective shadowing parameters (denoted by 𝜎0,𝑒𝑓𝑓 and 𝑑𝑐,𝑒𝑓𝑓)

for such radio links is expected to depend on (a) the

shadowing parameters corresponding to power classes of the

cells crossed (i.e. 𝜎0(𝑠) and 𝑑𝑐(𝑠)), and (b) the length over

which the cells are crossed. Detailed expressions for 𝜎0,𝑒𝑓𝑓

and 𝑑𝑐,𝑒𝑓𝑓 will be given in the next subsection.

D. Detailed Shadowing Model

The inhomogeneous shadowing model has a similar underlying structure as the homogeneous shadowing modelreviewed in Section II in that it is also based on computation of the potential level vector (potential field). The main difference is that multiple sets of 𝜎0,eff and 𝑑c,eff are used in the model,

hence, multiple potential fields must be calculated. The steps of shadowing generation are described in the following:

1) Calculation of Effective Shadowing Standard

Deviation, 𝜎0,𝑒𝑓𝑓:The effective standard deviation will potentially have a

distinct value for each radio link. Let 𝜎0,eff(𝑛, 𝑘) be the

effective standard deviation for the link between BS 𝑛 ∈ 𝒩 and terminal 𝑘 ∈ 𝒦. Hence there are a total of 𝑁𝐾 values of

𝜎0,eff(𝑛, 𝑘) that need to be calculated. The indices (𝑛, 𝑘) are

dropped in the following discussion except where necessary.The calculation for each link consists of the following steps:

Step 1: Calculate the length of the link path, 𝐿, illustrated in

Fig. 1, according to

𝐿 = √(𝑥𝑛 − 𝑥𝑘)2 + (𝑦𝑛 − 𝑦𝑘)2

Step 2: Partition the link path into 𝑁 path segments by

assigning segments of the path to a BS 𝑚 ∈ 𝒩 . The

assignment is based on path loss minimization. The path

segment lengths are denoted by 𝑙1, 𝑙2, … , 𝑙𝑁, where not all of

𝑙1, 𝑙2, … , 𝑙𝑁 are equal to zero and ∑ 𝑙𝑚𝑁𝑚=1 = 𝐿. The detailed

procedure is given in Algorithm 1.

Step 3: Calculate 𝜎0,𝑒𝑓𝑓 for the link as the weighted average

of the homogeneous shadowing standard deviation for each

BS 𝑚’s power class, 𝜎0(𝑠𝑚):

𝜎0,eff =∑ (𝑙𝑚)𝛼𝜎0(𝑠𝑚)𝑁

𝑚=1

∑ (𝑙𝑚)𝛼𝑁𝑚=1

Fig. 1. Sample radio link in an irregular cellular network. Here, 𝐿 is the total

length of the link path, 𝑙𝑚 is the length of path segment assigned to BS 𝑚 ∈𝒩, 𝑘 ∈ 𝒦 is an index value for a terminal, and 𝑛 ∈ 𝒩 and 𝑝 ∈ 𝒩 are index

values for BSs such that 𝑛 ≠ 𝑝, 𝑛 ≠ 1, 𝑛 ≠ 2, 𝑝 ≠ 1, 𝑝 ≠ 2.

where 𝛼 > 0 is a parameter used for tuning the weighting of

longer path segments in relation to shorter path segments1.

2) Calculation of Effective Shadowing Correlation

Distance, 𝑑𝑐,𝑒𝑓𝑓:

Having calculated σ0,eff ’s, the effective correlation

distance, 𝑑𝑐,eff, is also calculated for each of the 𝑁𝐾 radio links

as the weighted average of the homogeneous correlation distance for each BS 𝑚’s power class, 𝑑c(𝑠𝑚):

𝑑c,eff =∑ (𝑙𝑚)𝛼𝑑c(𝑠𝑚)𝑁

𝑚=1

∑ (𝑙𝑚)𝛼𝑁𝑚=1

3) Computation of Potential Fields:The procedure is similar to the generation of the potential

field in the homogeneous scenario. The difference is that multiple potential fields are generated, one for each of the radiolinks as they can have distinct shadowing parameters, 𝜎0,eff and

𝑑c,eff . The potential levels are generated according to the

following steps:

Step 1: Generate a single 𝐽 × 1 vector 𝒗 whose elements are

independent Gaussian random variables with zero mean and

unity standard deviation.

Step 2: Generate a 𝐽 × 𝐽 correlation matrix 𝑹 = [𝑟𝑖𝑗] for

each radio link such that

𝑟𝑖𝑗 =𝜎0,eff

2

2e−𝑑𝑖𝑗 𝑑𝑐,eff⁄

Step 3: Decompose each correlation matrix 𝑹 using the

Cholesky factorization such that 𝑹 = 𝑩𝑩𝑯.

1 As 𝛼 increases, the relative weighting of longer path segments’ standarddeviation increases.

BS 2 → s2

BS n → sn

BS 1 → s1 BS p → sp

Terminal k

lnl1

lp L

l2 = 0

Figure 6.2: Sample radio link in an irregular cellular network. Here, L is the total lengthof the link path; k ∈ K is an index value for a terminal; n ∈ N and p ∈ N are indexvalues for the BSs such that n 6= p, n 6= 1, n 6= 2, p 6= 1 and p 6= 2; and lm is the lengthof path segment assigned to a BS m ∈ N .

steps:

• Step 1: Calculate effective shadowing standard deviation, σ0,eff, and effective shad-

owing correlation distance, dc,eff, for the link according to Principle 2.

• Step 2: Compute the Gaussian potential field vector, u, as a function of σ0,eff and

dc,eff, that is, u = u(σ0,eff, dc,eff). (A potential field was defined in [8] as a random

process whereby each realization is a function from the plane to the real numbers.

Thus, if we fix a point in the plane, the outcome is a Gaussian random variable.)

• Step 3: Compute the shadowing value, SF , for each radio link as a function of the

respective entries of the potential field vector u(σ0,eff, dc,eff).

Detailed algorithms for each of the steps are discussed in the following.

Calculation of Effective Shadowing Standard Deviation and Effective Shad-

owing Correlation Distance

Let σ0,eff(n, k) and dc,eff(n, k) be the effective shadowing standard deviation and effective

shadowing correlation distance, respectively, for the link between BS n and terminal k,

illustrated in Figure 6.2. σ0,eff(n, k) and dc,eff(n, k) will potentially have distinct values


for each radio link; therefore, there are a total of NK values of σ0,eff(n, k) and dc,eff(n, k)

that need to be calculated. The calculation for each link is detailed in the following steps:

• Step 1: Calculate the length of the link path, L(n, k) = dn,k, shown in Figure 6.2,

according to

L(n, k) =√

(xn − xk)2 + (yn − yk)2 (6.16)

• Step 2: Partition the link path into N path segments by assigning segments of the

path to a BS m ∈ N as shown in Figure 6.2 The assignment is based on path loss

minimization. The path segment lengths are denoted by lm(n, k) for m ∈ N , where

lm(n, k) are not all zero and∑

m∈N lm(n, k) = L(n, k). The detailed procedure is

given in Figure 6.3.

• Step 3: Compute the total path length corresponding to each BS power class s ∈ S

ls(n, k) =∑

sm=s,m∈N

lm(n, k) (6.17)

• Step 4: By using Principle 2, compute σ0,eff(n, k) as the weighted average of the

homogeneous shadowing standard deviation for each BS power class, σ0(s), where

the weights are proportional to the lengths, ls(n, k):

σ0,eff(n, k) =

∑s∈S [ls(n, k)]α σ0(s)∑

s∈S [ls(n, k)]α. (6.18)

Here α > 0 is a parameter used for tuning the weighting of longer path segments in

relation to shorter path segments. As α increases, the relative weighting of longer

path segments increases.

• Step 5: Similar to σ0,eff calculation, compute dc,eff(n, k) as the weighted average of

the homogeneous shadowing correlation distance for each BS power class, dc(s):

dc,eff(n, k) =

∑s∈S [ls(n, k)]α dc(s)∑

s∈S [ls(n, k)]α(6.19)

Note that the indices (n, k) in the various terms are dropped in the remainder of the

discussion to simplify the notation.


Require: Z, a large positive integer parameter1: Generate Z equally-spaced points on the link path, forming the set of points Z.2: for all points z ∈ Z do3: for all BSs m ∈ N do4: Compute the path loss, PL(sm), between point z and BS m from (6.13).5: end for6: Assign point z to the BS m∗z = arg minPL(sm).7: end for8: Use the boundaries of BS assignment between contiguous groups of points z to par-

tition the link path into path segments, indexed by the BS m (see Figure 6.2).9: for all m ∈ N do

10: return The length of each path segment, lm.11: end for

Figure 6.3: Algorithm for assigning path segments to BSs

Computation of Potential Fields

Let J = N + K be the total number of points (BSs and users) in the network. The

potential field vector u = u(σ0,eff, dc,eff), of size J × 1, is calculated according to the

following steps:



• Step 2: Compute the J × J correlation matrix R = [rij] for each pair of values of

σ0,eff and dc,eff, that is, R = R(σ0,eff, dc,eff), such that

rij =σ2

0,eff

2e−dij/dc,eff (6.20)

where dij denotes the distance between point i and point j.

• Step 3: Decompose the correlation matrix R(σ0,eff, dc,eff) using Cholesky factoriza-

tion such that

R(σ0,eff, dc,eff) = B(σ0,eff, dc,eff)[B(σ0,eff, dc,eff)]H. (6.21)

• Step 4: Compute the J × 1 potential field vector u(σ0,eff, dc,eff) according to

u(σ0,eff, dc,eff) = B(σ0,eff, dc,eff)v (6.22)


Computation of Shadowing for the Radio Link

Let uA be the potential level or the value of the potential field at some given point A. The

shadowing between a terminal and a BS, denoted as point A and point B, respectively,

is then computed as a function of the respective potential levels, uA(σ0,eff, dc,eff) and

uB(σ0,eff, dc,eff), according to

SF [dB] = sgn(uA(σ0,eff, dc,eff) + uB(σ0,eff, dc,eff)

)·∣∣uA(σ0,eff, dc,eff)− uB(σ0,eff, dc,eff)

∣∣(6.23)

6.6.2 Shadowing Model Consistency Verification

We emphasize that the inhomogeneous shadowing model with spatial correlation has a

similar underlying structure as the homogeneous shadowing model. The primary differ-

ence is that multiple sets of shadowing parameters, σ0,eff and dc,eff, as well as potential

fields, u, are generated. Thus, we consider the consistency of the proposed inhomo-

geneous model with the earlier homogeneous model. In the following, the correlation

between two links with a common end is simulated as a verification of the consistency

of the proposed shadowing model in a simple inhomogeneous scenario. The simulation

model is based on the setup used in [8]. The terminals k1 and k2 are located at a dis-

tance D from the BS as depicted in Figure 6.4. Let σ0,eff,1 and dc,eff,1 be the shadowing

parameters for link 1 between the BS and terminal k1. Similarly, let σ0,eff,2 and dc,eff,2 be

the shadowing parameters for link 2 between the BS and terminal k2. It is assumed that

the distance D is large, in particular, D max(dc,eff,1, dc,eff,2). We study the correlation

between the shadow fading channel gains g1 and g2 between the BS (the common end)

and the two terminals k1 and k2, respectively. By varying the angle θ, we observe the cor-

relation at different distances d while keeping the distance D between the terminals and

the BS fixed. Note that in varying θ (and therefore d), the relationships θ1 = θ2 = θ/2,

and d1 = d2 = d/2 are preserved at all times for consistency of analysis.

Three separate cases of shadowing parameter values are considered in the analysis,

which are described in the following: Case (a) is the homogeneous shadowing scenario,

where σ0,eff,1 = σ0,eff,2 and dc,eff,1 = dc,eff,2. In case (b), the shadowing standard deviation

of link 1 is double with respect to link 2 while the shadowing correlation distance remains

the same, that is, σ0,eff,1 = 2σ0,eff,2 and dc,eff,1 = dc,eff,2. Finally in case (c), the shadowing

correlation distance of link 1 is double with respect to link 2 while the shadowing standard

deviation remains the same, or, σ0,eff,1 = σ0,eff,2 and dc,eff,1 = 2dc,eff,2.

Figure 6.5 depicts the normalized correlation for cases (a)-(c) as a function of distance


Step 4: Calculate the 𝐽 × 1 potential level vector 𝒖 for each

radio link according to

𝐮 = 𝐁𝐯

Step 5: Construct the matrix of potential levels 𝑼 according

to

𝐔 = [𝐮(1,1), … , 𝐮(𝑛, 𝑘), … , 𝐮(𝑁, 𝐾)]

where 𝒖(𝑛, 𝑘) represents the potential level vector generated

using the parameters 𝜎0,𝑒𝑓𝑓(𝑛, 𝑘) and 𝑑𝑐,𝑒𝑓𝑓(𝑛, 𝑘).

4) Calculation of Shadowing for a Radio Link:The shadowing between point A and point B (a terminal

and a BS) is computed according to (3), where the appropriatecolumn of the 𝐔 matrix is determined by the correspondingindices, (𝑛, 𝑘), of the radio link considered.

Algorithm 1 Path Segment Assignment to BSs

1. Generate 𝑍 equally-spaced points on the link path where𝑍 is a positive integer parameter that is chosensufficiently large.

2. Use the appropriate path loss expression 𝑃𝐿(𝑠𝑚) tocalculate the path loss between each BS 𝑚 ∈ 𝒩 and the𝑍 equally-spaced points on the link path.

3. Assign each point to the BS with the smallest 𝑃𝐿(𝑠𝑚).

4. Hence, partition the link path into path segments byassigning contiguous sets of points to one of the BSs𝑚 ∈ 𝒩. The partitioning determines the path segmentlengths 𝑙1, 𝑙2, … , 𝑙𝑁, as illustrated in Fig. 1, such that notall of 𝑙1, 𝑙2, … , 𝑙𝑁 are equal to zero, and ∑ 𝑙𝑚

𝑁𝑚=1 = 𝐿.

IV. MODEL VERIFICATION

In this section, the correlation between two links with a common end is simulated as a verification of the proposedshadowing model in a simple inhomogeneous scenario. The simulation model is based on the setup used in [10]. Theterminals 𝑘1 and 𝑘2 are located at a distance 𝐷 from the BS asdepicted in Fig. 2. Let 𝜎0,eff,1 and 𝑑c,eff,1 be the shadowing

parameters for link 1 between the BS and terminal 𝑘1 .Similarly, let 𝜎0,eff,2 and 𝑑c,eff,2 be the shadowing parameters

for link 2 between the BS and terminal 𝑘2. It is assumed that

the distance 𝐷 is large, in particular, 𝐷 ≫ max(𝑑c,eff,1, 𝑑c,eff,2).

We study the correlation between the shadow fading channel gains 𝑔1 and 𝑔2 between the BS (the common end) and the twoterminals 𝑘1 and 𝑘2, respectively. By varying the angle 𝜃, weobserve the correlation at different distances 𝑑 while keepingthe distance 𝐷 between the terminals and the BS fixed. Note that in varying 𝜃 (and therefore 𝑑), the relationships 𝜃1 = 𝜃2 =𝜃 2⁄ , and 𝑑1 = 𝑑2 = 𝑑 2⁄ are preserved at all times forconsistency of analysis.

Three separate cases of shadowing parameter values are

considered in the analysis, which are described in the

following: Case (a) is the homogeneous shadowing scenario,

Fig. 2. Shadowing simulation scenario.

where 𝜎0,eff,1 = 𝜎0,eff,2 and 𝑑c,eff,1 = 𝑑c,eff,2 . In case (b), the

shadowing standard deviation of link 1 is double with respect

to link 2 while the shadowing correlation distance remains the

same, that is, 𝜎0,eff,1 = 2𝜎0,eff,2 and 𝑑c,eff,1 = 𝑑c,eff,2. Finally in

case (c), the shadowing correlation distance of link 1 is double

with respect to link 2 while the shadowing standard deviation

remains the same, or, 𝜎0,eff,1 = 𝜎0,eff,2 and 𝑑c,eff,1 = 2𝑑c,eff,2.

Fig. 3 depicts the normalized correlation for cases (a)-(c) as

a function of distance 𝑑, given by

𝜌(𝑑) =𝐸[𝑔1𝑔2]

𝜎0,eff,1𝜎0,eff,2

The correlation curve of case (a) agrees with the homogeneous

shadowing analysis performed in [10]. In addition, case (b)

differs from case (a) only in the values of 𝜎0,eff,1 and 𝜎0,eff,2.

Consequently, the normalized correlation curve of case (b) is

exactly the same as case (a) owing to the division by the factor

(𝜎0,eff,1𝜎0,eff,2) in (13). Finally, the correlation curve of case

(c) has the same maximum point as cases (a) and (b), however

it decays at a different rate as a function of distance. To study

the decay rate under inhomogeneous 𝑑c,eff conditions, we

introduce the resultant effective correlation distance—defined

as the distance at which the correlation between the

shadowing of links 1 and 2 drops to approximately 0.40 times

the maximum correlation 2 . Let 𝑑c,eff,1−2 be the resultant

effective correlation distance between links 1 and 2. Since half

of the length 𝑑 falls in the region with 𝑑c,eff,1 and the other

half in the region with 𝑑c,eff,2 , 𝑑c,eff,1−2 should intuitively

satisfy

𝑑c,eff,2 ≤ 𝑑c,eff,1−2 ≤ 𝑑c,eff,1

In case (c), 𝑑c,eff,1−2 is determined from Fig. 3 to be

approximately equal to 1.3𝑑c,eff,2 , thereby agreeing with the

inequality of (14). In conclusion, the analysis shows that the

2 The definition of resultant correlation distance is consistent with cases (a) and (b) where the effective correlation distances of the two links are equal.

k2 k1

g2

BS

g1

D

d2 = d/2 d1 = d/2

= /2 = /2

d

Figure 6.4: Shadowing simulation scenario.

d, given by

ρ(d) =E[g1g2]

σ0,eff,1σ0,eff,2

(6.24)

The Gudmundson correlations modelled by e−d/dc are additionally depicted with ap-

propriate values of dc for reference1. The correlation curve of case (a) agrees with the

homogeneous shadowing analysis performed in [8]. In addition, case (b) differs from case

(a) only in the values of σ0,eff,1 and σ0,eff,2. Consequently, the normalized correlation curve

of case (b) is exactly the same as case (a) owing to the division by the factor (σ0,eff,1σ0,eff,2)

in (6.24). Finally, the correlation curve of case (c) has the same maximum point as cases

(a) and (b), however it decays at a different rate as a function of distance. To study

the decay rate under inhomogeneous dc,eff conditions, we introduce the resultant effective

correlation distance—defined as the distance at which the correlation between the shad-

owing of links 1 and 2 drops to approximately 0.40 times the maximum correlation2. Let

dc,eff,1−2 be the resultant effective correlation distance between links 1 and 2. Since half

of the length d falls in the region with dc,eff,1 and the other half in the region with dc,eff,2,

dc,eff,1−2 should intuitively satisfy

dc,eff,2 ≤ dc,eff,1−2 ≤ dc,eff,1. (6.25)

In case (c), dc,eff,1−2 is determined from Figure 6.5 to be approximately equal to 1.3dc,eff,2,

1The main difference between Gudmundson’s and our model is that as the distance between thetwo terminals increases, our model predicts a correlation which decreases and converges to a non-zeroconstant value, which can be justified through physical arguments about obstructions in the vicinity ofa BS [8].

2The definition of resultant correlation distance is consistent with cases (a) and (b) where the effectivecorrelation distances of the two links are equal.


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4

No

rmal

ized

co

rrel

atio

n f

unct

ion

Normalized distance, d/dc,eff,2

Gudmundson

(dc,eff,2)

Gudmundson

(1.3dc,eff,2)

Case (a) and (b)

Case (c)

Figure 6.5: Comparison of normalized correlation as a function of normalized distance,d/dc,eff,2.

thereby agreeing with the inequality of (6.25). In conclusion, the analysis shows that

the proposed shadowing model produces consistent results in a simple inhomogeneous

shadowing scenario.

6.6.3 Implementation of Model in Practical Network Simula-

tions

The calculation of potential field vectors u(σ0,eff, dc,eff) in the proposed inhomogeneous

shadowing model requires a number of matrix computations as discussed earlier in the

section. Since the values of σ0,eff and dc,eff can be distinct for each BS-terminal link, it

appears as though a large number of matrix computations is required, possibly leading

to prohibitive computational complexity for the model. However, upon close examina-

tion, the number of distinct pair of values of (σ0,eff, dc,eff), hence, the number of required

matrix computations, is much reduced in a practical network simulation scenario. This

is explained below for two distinct types of radio links. The first is the type of link

whose path crosses through either a single cell or through multiple cells all belonging to

a single BS power class s. The second is the type of link whose path crosses through cells

belonging to multiple power classes.


Link Path over Cells with Single Power Class

The total number of distinct (σ0,eff, dc,eff) values is low in this case because in practical

irregular networks, the total number of power classes, S, is likely to be low. Consider the

BS power classes recommended for evaluation in the IMT-Advanced guidelines [1], which

are listed in Table 6.1. The total number of distinct power classes is five. In order to

allow for additional types of cell topographies, we assume a higher estimate of six distinct

power classes. Under this assumption, the total number of (σ0,eff, dc,eff) values as obtained

by (6.18) and (6.19), as well as the number of repetitions of matrix computations will

also be six.

Link Path over Cells with Multiple Power Classes

In this type of link, the number of distinct (σ0,eff, dc,eff) values can, strictly speaking,

be very high. However, the total number can be reduced by using approximations for

(σ0,eff, dc,eff) values while maintaining an acceptable level of precision for the shadowing

model. In particular, if σ0,eff(n, k) for the link between BS n and terminal k is sufficiently

close to σ0,eff(n1, k1) for the link between BS n1 and terminal k1, or∣∣∣∣σ0,eff(n, k)− σ0,eff(n1, k1)

σ0,eff(n1, k1)

∣∣∣∣ ≤ ε, (6.26)

where ε is a small positive real constant, then σ0,eff(n, k) is approximated as σ0,eff(n1, k1).

Similarly, if ∣∣∣∣dc,eff(n, k)− dc,eff(n1, k1)

dc,eff(n1, k1)

∣∣∣∣ ≤ ε, (6.27)

then dc,eff(n, k) is approximated as dc,eff(n1, k1). The value of ε should be tuned appro-

priately in order to have a low number of distinct (σ0,eff, dc,eff) values while at the same

time maintaining an acceptable level of shadowing model accuracy.

Chapter 7

Dynamic System Model (Multiple

Power Classes)

7.1 Relevant Work and Contribution

The literature review and the main contribution for the mobility model in this chapter

was included with Chapter 4, and for the BS power adaptation algorithm is included

with Chapter 5. A set of key results is discussed in the following:

7.1.1 Key Results

The performance of the system with BS power adaptation algorithms is studied for

different values of the terminal movement parameter and for different number of users.

The case of two different power classes is considered. Improvements in terms of 5th

percentile spectral efficiency, 50th percentile spectral efficiency, sum log utility and user

average spectral efficiency over the equal power setting have been obtained. In particular,

the algorithms achieve as much as 100% increase in 5th %le spectral efficiency, 8 %

increase in median spectral efficiency, and 5% increase in user average spectral efficiency.

A variation of the PF power adaptation algorithm is additionally studied, where the

power optimization problem is carried out separately and independently within each

power class. The independent adaptation can be important to consider as it reduces the

required exchange of channel information. The main observation is that the indepen-

dent PF method performs relatively poorly in terms of cell edge spectral efficiency and

SLU, which represent measures of fairness. However, the method still provides relative

improvement in terms of median spectral efficiency (5%) and user average spectral effi-

ciency (2%). Therefore, it remains a consideration if the reduced signalling proves to be

112

Chapter 7. Dynamic System Model (Multiple Power Classes) 113

an important criteria in the particular implementation.

7.2 Introduction

In Chapters 4 and 5, we studied the dynamic system model and the BS power adaptation,

respectively, for networks having a single BS power class. In Chapter 6, we introduced

the static model for networks having multiple BS power classes.

The use of the extended adaptive resource allocation framework with power adapta-

tion is proposed for the case of the network with multiple power classes. This can be

explained with respect to the clustering and power adaptation steps of the framework:

Firstly, the clustering concept was devised for a network with irregular deployment of

BSs, and it was designed to be self-configurable in the face of different types of dynamic

changes in the network. Secondly, the power adaptation step is applicable and expected

to be especially important in a network with a wide array of power levels.

In this chapter, we first discuss the time evolution to complete the dynamic model

for networks with multiple power classes. Later, we present evaluation results for the

complete irregular network system model with BS power adaptation.

The outline of the full dynamic system model is shown in Figure 7.1, which is the

same as the model outline of Chapter 4 given in Figure 4.1. However, the details of the

time evolution differ due to the existence of multiple BS power classes as discussed in

the following. The time evolution methodology is detailed in Section 7.3. Typical model

values are described in Section 7.4. Finally, evaluation results for the system model and

the adaptation framework are presented in Section 7.5.

7.3 Time Evolution

Time evolution in networks with multiple power classes follows the same principle as

the method of Chapter 4, Section 4.5. The list of modelled network dynamics includes

gradual terminal arrivals and departures, BS deployments and outages, and terminal and

BS movements. The main difference of the model for multiple power classes relates to

the positioning of arriving terminals and BSs due to the existence of local hotspots. The

detailed time evolution methodology is discussed in this section.

The necessary variables were previously defined in Chapter 4. In summary, P is

the total number of simulation drops, B is the number of subdrops associated with

each drop, and F is the number of subframes associated with each subdrop. We recall

that the initial deployment of a drop before any time evolution is indexed by 0, and


shadowing potential field realization as discussed in Section

3.2 is fixed. In other words, the dynamism in subdrops re-

sults solely from gradual changes relating to BSs and termi-

nals on a given and fixed topography.

Let 𝒩(𝑏)and 𝒦(𝑏) be the set of BSs and terminals active

at subdrop 𝑏, respectively. Specifically, 𝒩(0) and 𝒦(0)

denote the set of BSs and terminals active at subdrop 0. At

each subsequent subdrop 1 ≤ 𝑏 ≤ 𝐵, ΔK 𝑎(𝑏)

terminals arrive

at uniform randomly distributed locations according to the

standard Poisson distribution

𝑓(𝑖; 𝜆𝑡) = 𝑃𝑟(ΔK 𝑎(𝑏)

= 𝑖) =𝜆𝑡

𝑖𝑒−𝜆𝑡

𝑖!,

where 𝑓( ) denotes the probability mass function, and

𝜆𝑡 > 0 is the Poisson parameter. Each terminal departs the

j = j + 1

STOP

No

Generate subframe l of subdrop j

Compute path loss and shadowing for

all terminal-BS links

Execute slowly adaptive algorithms

Yes

l = l + 1 l mod F = 0?


Check arrivals,

departures and

movements

Generate subdrop j of drop i

j mod B = 0?

Yes

Yes

i = P?

No

No i = i + 1

Perform coordinated resource alloca-

tion in each cluster

Figure 7.1: Outline of the Simulation Framework.


the remaining subdrops associated with the drop are numbered from 1 to B − 1. In

addition, for all subdrops associated with a given simulation drop, the realization of the

random shadowing potential field is modelled to be constant. In other words, the physical

topography can be conceptualized as fixed within each simulation drop, and the changes

in shadowing over the subdrops result solely from BS and terminal movements on the

fixed topography. The detailed modelling of the dynamics is discussed in the following.

7.3.1 Terminal Arrival and Departure Models

The terminal arrival and departure models with local hotspots follow the same general

structure as the models without hotspots. We let K(0) denote the number of terminals

that are active at the initial subdrop, indexed by 0. The number of terminals, [∆Ka](b),

arriving to the system at each of subdrops 1 ≤ b ≤ B − 1, is generated according to

Poisson random distribution:

f(i;λt) = Pr([∆Ka](b) = i

)=λite−λt

i!, i = 0, 1, 2, . . . (7.1)

where f( ) denotes the probability mass function, and λt > 0 is the Poisson parameter,

expressed as the number of terminals.

The arriving terminals are randomly assigned to either (a) one of the hot spots i with

probability Kh,i/K or (b) the low-density zone with probability Kl/K. The terminal

coordinates are then randomly generated according to the procedure discussed in Chap-

ter 6, Section 6.4. In particular, the terminals that are assigned to one of the hotspots i

are dropped according to step 3.a, and the terminals that are assigned to the low-density

zone are dropped according to step 3.b.

Each terminal must depart the system after a survival time, τk, expressed in terms of

number of subdrops. τk is generated according to the following steps:

• Step 1: Virtual survival time, χ, is generated as a temporary variable according to

the exponential probability mass function with mean parameter µt > 0, expressed

in number of subdrops:

f(χ) =

1µte− χµt if χ ≥ 0,

0 if χ < 0.(7.2)

• Step 2: The actual survival time, τk, is computed according to

τk = dχe (7.3)


The number of terminals that depart at subdrop b, [∆Kd](b), is determined with the

knowledge of terminal survival times, τk. The total number of terminals remaining at

subdrop b is, therefore, given by

K(b) = K(b−1) + [∆Ka](b) − [∆Kd](b) (7.4)

7.3.2 BS Deployment and Outage Models

The BS deployment and outage models with local hotspots also follow the same general

structure as the models without hotspots. We let N(0) denote the number of BSs that are

active at the initial subdrop, indexed by 0. The BS deployment is modelled as an arrival

and the BS outage is modelled as a departure process. Ibs is an indicator parameter

which is set to 0 if BSs are not allowed to be deployed (i.e. arrive) and have an outage

(i.e. depart). This means that the number N(0) is fixed over the duration of the drop.

If the BSs are allowed to have both deployments and outages, Ibs is then set to 1. We

consider the modelling of BS arrivals under the setting of Ibs = 1. The number of BSs,

[∆Na](b), arriving to the system at each of subdrops 1 ≤ b ≤ B−1, is generated according

to Poisson random distribution:

f(i;λbs) = Pr([∆Na](b) = i

)=λibse

−λbs

i!, i = 0, 1, 2, . . . (7.5)

where f( ) denotes the probability mass function, and λbs > 0 is the Poisson parameter,

expressed in terms of the number of BSs.

The arriving BSs are randomly assigned to either (a) one of the hot spots i with

probability Nh,i/N or (b) the low-density zone with probability Nl/N . The BS coordi-

nates are then randomly generated according to the procedure discussed in Chapter 6,

Section 6.4. In particular, the BSs that are assigned to one of the hotspots i are dropped

according to step 4.b, and the BSs that are assigned to the low-density zone are dropped

according to step 4.c.

Each BS departs the system after a survival time, τn, (expressed in number of sub-

drops), which is generated according to the following steps:

• Step 1: Virtual survival time, χ, is generated according to the exponential proba-

bility mass function with mean parameter µbs > 0:

f(χ) =

1µbse− χµbs if χ ≥ 0,

0 if χ < 0.(7.6)


• Step 2: The actual survival time, τn, is computed according to

τn = dχe (7.7)

Similar to the terminals, the number of BSs that depart at subdrop b, [∆Nd](b), is deter-

mined with the knowledge of BS survival times, τn. The total number of BSs remaining

at subdrop b is, therefore, given by

N(b) = N(b−1) + [∆Na](b) − [∆Nd](b) (7.8)

7.3.3 Terminal Movement Model

Terminals are moved at each subdrop through a randomly generated displacement in a

random direction. The movement model is illustrated in Figure 7.2. The full algorithm

for the terminal movement at the subdrop is given in Figure 7.3. Let θk,b be the angle

(or direction) of movement over the xy-plane, and ∆dk,b the displacement over the xy-

plane of terminal k in subdrop b. In addition, ∆dt,min,h is the minimum displacement

inside a hotspot, ∆dt,max,h is the maximum displacement inside a hotspot, ∆dt,min,l is

the minimum displacement inside the low-density zone and ∆dt,max,l is the maximum

displacement inside the low-density zone. θk,b is randomly generated for each terminal

from a uniform distribution in the interval [0, 2π]. The generation of ∆dk,b is dependent

on the position of the terminal: For terminals currently inside the radius of one of the

hotspots, ∆dk,b is drawn from a uniform distribution in [∆dt,min,h,∆dt,max,h]; and for

terminals inside the low-density zone, ∆dk,b is drawn from a uniform distribution in

[∆dt,min,l,∆dt,max,l]. As long as a terminal would remain within the simulation area after

movement, the terminal coordinates are updated according to

xk ← xk + ∆dk,b cos θk,b (7.9)

yk ← yk + ∆dk,b sin θk,b (7.10)

If the movement would place the terminal out of the simulation area, the updated terminal

coordinates are given by

xk ← −(xk + α cos θk,b) + (∆dk,b − α) cos θk,b (7.11)

yk ← −(yk + α sin θk,b) + (∆dk,b − α) sin θk,b (7.12)


(xk, yk)

(xk+dk,bcosθk,b, yk

+dk,bsinθk,b)

dk,b

θk,b

y

x

Figure 7.2: The terminal movement illustration.

Require: xk, yk, for terminals k ∈ K, and parameters ∆dt,min,h, ∆dt,max,h, ∆dt,min,l, and∆dt,max,l

1: for all terminals k ∈ K do2: if terminal k is inside a hotspot then3: Generate ∆dk,b ∼ U ∈ [∆dt,min,h,∆dt,max,h], and θk,b ∼ U ∈ [0, 2π].4: else5: Generate ∆dk,b ∼ U ∈ [∆dt,min,l,∆dt,max,l], and θk,b ∼ U ∈ [0, 2π].6: end if7: if (xk + ∆dk,b cos θk,b)

2 + (yk + ∆dk,b sin θk,b)2 ≤ R2 then

8: xk ← xk + ∆dk,b cos θk,b9: yk ← yk + ∆dk,b sin θk,b

10: else11: Solve for α in α2 + 2α(xk cos θk,b + yk sin θk,b) + (x2

k + y2k −R2) = 0

12: xk ← −(xk + α cos θk,b) + (∆dk,b − α) cos θk,b13: yk ← −(yk + α sin θk,b) + (∆dk,b − α) sin θk,b14: end if15: end for

Figure 7.3: Algorithm for the terminal movement at the simulation subdrop.


α2 + 2α(xk cos θk,b + yk sin θk,b) + (x2k + y2

k −R2) = 0 (7.13)


7.3.4 BS Movement Model

Similar to the terminals, the proposed simulation framework has the option for modelling

BS movements. Let πbs be a fixed probability of BS movement. At each subdrop, BSs

are moved with probability πbs through a random displacement in a random direction.

The movement model is illustrated in Figure 7.4. The detailed algorithm for the BS

movement at the subdrop is described in Figure 7.5. Let θn,b be the angle (or direction)

of movement over the xy-plane, and ∆dn,b the displacement over the xy-plane of BS

n in subdrop b. In addition, ∆dbs,min,h is the minimum displacement inside a hotspot,

∆dbs,max,h is the maximum displacement inside a hotspot, ∆dbs,min,l is the minimum

displacement inside the low-density zone and ∆dbs,max,l is the maximum displacement

inside the low-density zone. θn,b is randomly generated for each BS from a uniform

distribution in the interval [0, 2π]. The generation of ∆dn,b is dependent on the position

of the BS: For BSs currently inside the radius of one of the hotspots, ∆dn,b is drawn

from a uniform distribution in [∆dbs,min,h,∆dbs,max,h]; and for BSs currently inside the

low-density zone, ∆dn,b is drawn from a uniform distribution in [∆dbs,min,l,∆dbs,max,l]. As

long as a terminal would remain within the simulation area after movement, the terminal

coordinates are updated according to

xn ← xn + ∆dn,b cos θn,b (7.14)

yn ← yn + ∆dn,b sin θn,b (7.15)

If the movement would place the BS out of the simulation area, the coordinate update

is given by

xn ← −(xn + α cos θn,b) + (∆dn,b − α) cos θn,b (7.16)

yn ← −(yn + α sin θn,b) + (∆dn,b − α) sin θn,b (7.17)


α2 + 2α(xn cos θn,b + yn sin θn,b) + (x2n + y2

n −R2) = 0 (7.18)

7.4 Typical Parameter Values

In this section, we discuss typical values for the time evolution model parameters. Rec-

ommended values and ranges for the parameters are summarized in Table 7.1. Note


(xn, yn)

(xn+dn,bcosθn,b, yn

+dn,bsinθn,b)

dn,b

θn,b

y

x

Figure 7.4: The BS movement illustration.

Require: xn, yn, for BSs n ∈ N , and parameters πbs, ∆dbs,min,h, ∆dbs,max,h, ∆dbs,min,land ∆dbs,max,l

1: for all BSs n ∈ N do2: Generate γ such that Pr(γ = 1) = πbs and Pr(γ = 0) = 1− πbs.3: if γ = 1 then4: if BS N is inside a hotspot then5: Generate ∆dn,b ∼ U ∈ [∆dbs,min,h,∆dbs,max,h], and θn,b ∼ U ∈ [0, 2π].6: else7: Generate ∆dn,b ∼ U ∈ [∆dbs,min,l,∆dbs,max,l], and θn,b ∼ U ∈ [0, 2π].8: end if9: if (xn + ∆dn,b cos θn,b)

2 + (yn + ∆dn,b sin θn,b)2 ≤ R2 then

10: xn ← xn + ∆dn,b cos θn,b11: yn ← yn + ∆dn,b sin θn,b12: else13: Solve for α in α2 + 2α(xn cos θn,b + yn sin θn,b) + (x2

n + y2n −R2) = 0

14: xn ← −(xn + α cos θn,b) + (∆dn,b − α) cos θn,b15: yn ← −(yn + α sin θn,b) + (∆dn,b − α) sin θn,b16: end if17: end if18: end for

Figure 7.5: Algorithm for the BS movement at the simulation subdrop.


Table 7.1: Typical parameter values.

Parameter Typical Range and/or Value

Numbers of subdrops, B typical value of 20

Numbers of drops, P P B, typical value of 300

Scheduling average window size,T0

typical value of 5

Numbers of subframes, F F T0, F P , typical value of 100

Terminal arrival parameter, λt typical value of 3 terminals

BS arrival parameter, λbs λbs λt, typical value of 0.5 BSs

Terminal departure parameter, µt typical value of 5 subdrops

BS departure parameter, µbs µbs µt, typical value of 20 subdrops

Hotspot maximum terminal dis-placement, ∆dt,max,h

0.1dc,h ≤ ∆dt,max,h ≤ 10dc,h

Hotspot minimum terminal dis-placement, ∆dt,min,h

0 ≤ ∆dt,min,h ≤ 14∆dt,max,h

Low-density zone maximum ter-minal displacement, ∆dt,max,l

0.1dc,l ≤ ∆dt,max,l ≤ 10dc,l

Low-density zone minimum ter-minal displacement, ∆dt,min,l

0 ≤ ∆dt,min,l ≤ 14∆dt,max,l

Hotspot maximum BS displace-ment, ∆dbs,max,h

0.1dc,h ≤ ∆dbs,max,h ≤ 10dc,h

Hotspot minimum BS displace-ment, ∆dbs,min,h

0 ≤ ∆dbs,min,h ≤ 14∆dbs,max,h

Low-density zone maximum BSdisplacement, ∆dbs,max,l

0.1dc,l ≤ ∆dbs,max,l ≤ 10dc,l

Low-density zone minimum BSdisplacement, ∆dbs,min,l

0 ≤ ∆dbs,min,l ≤ 14∆dbs,max,l

BS movement probability, πbs πbs 1


that many of the parameters have the same recommended values and ranges as the time

evolution model for a single power class, thus the reader is asked to refer to Chapter 4

for the justification of those recommended ranges. In the following, we discuss the ranges

of maximum and minimum displacement parameters. Similar to Chapter 4, the maxi-

mum displacement parameters should be set large enough to cause a small but noticeable

change in shadowing over the subdrop. Therefore, we apply the basic principle of (4.25)

and (4.26) for the recommended ranges of the maximum and minimum terminal displace-

ment parameters based on the correlation distance, dc, that is

0.1dc ≤ ∆dt,max ≤ 10dc, (7.19)

0 ≤ ∆dt,min ≤1

4∆dt,max. (7.20)

It was discussed in Chapter 6, however, that in inhomogeneous networks, there exist

multiple distinct dc values corresponding to the distinct power classes. We, therefore,

define the typical correlation distance inside the hotspots and outside the hotspots (or

in the low-density zone). Let dc,h be the typical correlation distance inside the hotspots

and dc,l be the typical correlation distance inside the low-density zone. dc,h and dc,l are

computed as the average of the correlation distances, dc, for the BS power classes that

are found inside the hotspots and the low-density zone, respectively. In analogy to (7.19)

and (7.20), the terminal displacement parameter ∆dt,max,h should, hence, be bounded by

multiples of dc,h, or

0.1dc,h ≤ ∆dt,max,h ≤ 10dc,h (7.21)

and the parameter ∆dt,min,h should be bounded as

0 ≤ ∆dt,min,h ≤1

4∆dt,max,h (7.22)

Similarly, the parameters ∆dt,max,l and ∆dt,min,l should satisfy the relations based on

multiples of dc,l:

0.1dc,l ≤ ∆dt,max,l ≤ 10dc,l (7.23)

0 ≤ ∆dt,min,l ≤1

4∆dt,max,l (7.24)


By similar reasoning as the terminals, the BS displacement parameters ∆dbs,max,h, ∆dbs,min,h,

∆dbs,max,l and ∆dbs,min,l should satisfy:

0.1dc,h ≤ ∆dbs,max,h ≤ 10dc,h (7.25)

0 ≤ ∆dbs,min,h ≤1

4∆dbs,max,h (7.26)

0.1dc,l ≤ ∆dbs,max,l ≤ 10dc,l (7.27)

0 ≤ ∆dbs,min,l ≤1

4∆dbs,max,l (7.28)

7.5 Simulations

In this section, system simulations are performed in order to demonstrate the use of the

proposed system modelling and the benefit of the BS power adaptation in the network

with multiple power classes. Note that a simplified system model is utilized in the simu-

lation. Hotspotted deployment with two distinct power classes is considered, consisting

of the outdoor pico power class (used exclusively inside hotspots) and the macro power

class (used exclusively inside the low-density zone). The time evolution model will con-

sider terminal movements with constant displacement and uniformly random direction

both inside and outside hotspots, and neglect other types of dynamics such as terminal

arrivals and departures.

The model parameters (largely based on the suggested values in Table 7.1 and sup-

plemented with the ITU guidelines where needed) are summarized in Tables 7.2 and 7.3.

Table 7.2 contains the deployment scenario parameters, and Table 7.3 lists other physical

model parameters.

The number of BSs in the low-density zone and in hotspots have been computed as

a function of the nominal number of terminals according to (6.7) and (6.8), respectively.

The total number of terminals will be varied in the simulation, with the proportion of

terminals inside each area kept constant. Similarly, the constant terminal displacements

listed in Table 7.3 have been chosen according to (7.21) and (7.23), and will be varied in

simulation. Note that the BSs are not allowed to power on and off, as a non-zero lower

limit, Ps,min, has been imposed on the BS power levels. The difference between Ps,min

and Ps,max is set to 6 dB for both of the power classes.

A simplified method for determining effective shadowing standard deviation, σ0,eff,

and shadowing correlation distance, dc,eff, is introduced. The procedure is described in

the following: First, the σ0,eff values are equal to 10 dB for all links, as σ0(s) in the

homogeneous scenario is equal for both BS power classes. Second, the dc,eff values are


Table 7.2: Deployment scenario parameters

Parameter Value

Simulation area radius, R 375 m

Number of hotspots, H 2

Hotspot radius, Rh 40 m

Nominal number of terminals in the low-density zone, Kl 36

Nominal number of terminals in hotspot 1, Kh,1 12

Nominal number of terminals in hotspot 2, Kh,2 36

The total nominal number of terminals, K 84

Number of BSs in the low-density zone, Nl 3

Number of BSs in hotspot 1, Nh,1 1

Number of BSs in hotspot 2, Nh,2 3

Total number of BSs, N 7

approximated by using certain properties of the specific deployment scenario considered.

In particular, from Table 7.2, the number of hotspots, H, is small and the hotspot radius,

Rh, is much smaller than the simulation radius, R, that is, Rh R. Furthermore, there

are only two available power classes, one of which (the outdoor pico) is used exclusively

for hotspots, and the other (the macro) for the low-density zone. Thus, it is concluded

that the path between certain types of links, for instance, that between a macro BS and

a terminal in the low-density zone is statistically unlikely to cross through the outdoor

pico power class. Additionally, for other types of links, such as that between a macro

BS and a terminal inside a hotspot, the path does cross through the outdoor pico power

class, however, the respective path length segment is statistically expected to be small.

Consequently, from (6.19), the dc,eff value for the two types of links is approximated to

be equal to the homogeneous dc(s) for the macro BS cell, or dc,eff∼= dc(s) = 50 m. A

similar analysis for links between pico BSs and terminals inside the same hotspot shows

that the dc,eff value is exactly equal to the homogeneous dc(s) for the outdoor pico BS

cell, or dc,eff = dc(s) = 10 m. The dc,eff values used for different link types are listed in

Table 7.4.

P = 150 simulation drops each consisting of B = 20 subdrops are used in the eval-

uation. Both the total rate maximization (TRM) power adaptation of (5.6) and the

proportional fair (PF) power adaptation of (5.10) are simulated for the purpose of com-

parison. The TRM and PF algorithms are also compared against the utilization of equal

BS powers according to the power classes, that is, where the Pn = Psn,max is used for all


Table 7.3: Simulation Model Parameters

Parameter Value

Bandwidth, BT 15 kHz


Background noise power spectraldensity

-174 dBm/Hz

Path loss for macro BS, PL(sn) −[ 128.1 + 37.6 log10 d ], d in km

Path loss for pico BS, PL(sn) −[ 140.7 + 36.7 log10 d ], d in km

Homogeneous shadowing standarddeviation, σ0(s)

10 dB

Homogeneous shadowing correlationdistance, dc(s), for macro cell

50 m

Homogeneous shadowing correlationdistance, dc(s), for pico cell

10 m

Maximum BS power capability,Ps,max, for macro BS

46 dBm

Maximum BS power capability,Ps,max, for pico BS

30 dBm

Minimum allowed BS power, Ps,min,for macro BS

40 dBm

Minimum allowed BS power, Ps,min,for pico BS

24 dBm

Terminal displacement for low-density zone (nominal value)

dc(s)/3 = 16.67m

Terminal displacement for hotspot(nominal value)

dc(s)/3 = 3.33m

Power adjustment window size, W0 4

Table 7.4: Values of Effective Shadowing Correlation Distance, dc,eff

Link Type Value

macro BS to terminal in the low-density zone 50 m

macro BS to terminal in a hotspot 50 m

pico BS to terminal in the low-density zone 50 m

pico BS to terminal in the same hotspot 10 m

pico BS to terminal in different hotspot 50 m


0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0 0.5 1 1.5 2

5th

%le

Sp

ectr

al E

ffic

iency

(b

ps/

Hz)

Terminal Displacement in Multiples of dc(s)

Equal Power

PF

TRM

(a)

0.35

0.355

0.36

0.365

0.37

0.375

0.38

0.385

0.39

0.395

0.4

0 0.5 1 1.5 2

50

th %

le S

pec

tral

Eff

icie

ncy

(b

ps/

Hz)


Equal Power

PF

TRM

(b)

Figure 7.6: Comparison of power adaptation schemes and the equal power level scenariofor varying terminal displacement in terms of (a) 5th percentile spectral efficiency, and(b) 50th percentile spectral efficiency.


300

301

302

303

304

305

306

307

308

309

310

311

0 0.5 1 1.5 2

Sum

Lo

g U

tili

ty


Equal Power

PF

TRM

(a)

0.54

0.55

0.56

0.57

0.58

0.59

0.6

0.61

0.62

0 0.5 1 1.5 2

Use

r A

ver

age

Sp

ectr

al E

ffic

iency

(b

ps/

Hz)


Equal Power

PF

TRM

(b)

Figure 7.7: Comparison of power adaptation schemes and the equal power level scenariofor varying terminal displacement in terms of (a) the sum log utility, and (b) the useraverage spectral efficiency.


BSs of each of the macro and the outdoor pico power classes.

The variation of the terminal displacement parameters was simulated for a constant

number of users, K = 84. Figure 7.6a shows the variation of 5th percentile (cell edge)

spectral efficiency. Both the TRM and PF methods show over 100% improvement in

cell edge spectral efficiency over the equal BS power method. Figure 7.6b shows the

50th percentile (median) spectral efficiency for the different power setting methods. 8%

increase in median spectral efficiency is obtained for low values of terminal displacement,

which however drops to 2% for higher values of displacement. Figure 7.7a and 7.7b show

the sum log utility (SLU) and user average spectral efficiency variation with increasing

terminal displacements. It is observed that both PF and TRM methods improve the SLU.

The methods also improve the user average spectral efficiency obtained in the network

with nearly 5% increase. We note that there is only a small difference between the PF

and TRM methods with respect to both the SLU and user average spectral efficiency

obtained.

The variation of the total number of users is considered next with the terminal dis-

placement values set to dc(s)/3. A variation of the PF power adaptation algorithm,

referred to as independent PF method, was additionally considered in this section. In

particular, the original PF and TRM power adaptation consider the optimization of (5.6)

or (5.10) jointly over the entire set of BSs and terminals. In the independent PF method,

the PF power adaptation of (5.10) was carried out separately and independently for the

pico and the macro BSs, respectively. In each case, the power adaptation considered

either the pico (or macro) BSs and the terminals assigned to the pico (or macro) BSs as

the whole set of BSs, N , and terminals, K, respectively, in the optimization of (5.10).

In other words, the interference from either the macro or pico BSs were neglected in

the solution. Terminal data rates were then computed normally with the two separately

obtained sets of BS power levels. The independent PF can be important to consider as

it limits the coordination for BS power adaptation to within one set of BSs (either the

pico or macro in the particular case), thereby limiting the required exchange of channel

information.

Figure 7.8a shows the variation of 5th percentile (cell edge) spectral efficiency with

respect to the total number of users for the different power setting methods. The TRM-

based method achieves approximately 100% improvement compared to the equal BS

power method. Furthermore, the PF-based method achieves around 110% improvement

over the equal BS power method. The independent PF method shows negligible improve-

ment over the equal power method in terms of cell edge spectral efficiency. Figure 7.8b

shows the 50th percentile (median) spectral efficiency for the different power setting


methods. Both the TRM and PF methods show close to 8% improvement in median

spectral efficiency over the equal power setting methods. The independent PF method

also achieves 5% improvement over equal power setting. Figure 7.9a and 7.9b show the

sum log utility (SLU) and user average spectral efficiency variation with increasing ter-

minal displacements. Both PF and TRM methods are shown to improve the SLU. The

independent PF method has SLU that is nearly equal to the equal power method, repre-

senting negligible improvement. The PF and TRM also show close to 5% improvement

in the user average spectral efficiency over the equal power method, and the independent

PF shows 2% improvement.

It can be concluded that the TRM and PF algorithms show improvement over the

equal power method. A smaller improvement is obtained for median spectral efficiency

and user average spectral efficiency, which represents the overall system performance.

Greater improvement is obtained for cell edge spectral efficiency and SLU, representing

a measure of fairness among users. It can be observed that the independent PF method

performs poorly in terms of cell edge spectral efficiency and SLU. However, if the reduced

signalling proves to be an important criteria in the particular implementation, the method

can be used due to the improvement it provides in median spectral efficiency and total

system rate.


0.01

0.02

0.03

0.04

0.05

0.06

20 30 40 50 60 70 80 90

5th

%le

Sp

ectr

al E

ffic

iency

(b

ps/

Hz)

Total Number of Users

Equal Power

PF

Independent PF

TRM

(a)

0.3

0.4

0.5

0.6

0.7

0.8

0.9

20 30 40 50 60 70 80 90

50

th %

le S

pec

tral

Eff

icie

ncy

(b

ps/

Hz)


Equal Power

PF

Independent PF

TRM

(b)

Figure 7.8: Comparison of power adaptation schemes and the equal power level scenariofor varying number of users in terms of (a) 5th percentile spectral efficiency, and (b) 50thpercentile spectral efficiency.


100

150

200

250

300

28 56 84

Sum

Lo

g U

tili

ty


Equal Power

PF

Independent PF

TRM

(a)

0.48

0.58

0.68

0.78

0.88

0.98

1.08

1.18

20 30 40 50 60 70 80 90

Use

r A

ver

age

Sp

ectr

al E

ffic

iency

(b

ps/

Hz)


Equal Power

PF

Independent PF

TRM

(b)

Figure 7.9: Comparison of power adaptation schemes and the equal power level scenariofor varying number of users in terms of (a) the sum log utility, and (b) the user averagespectral efficiency.

Chapter 8

Conclusion

8.1 Summary

In this thesis, we consider the development of a high-level system model for simulation

purposes for a generic irregular cellular network. We assume a network with universal

frequency allocation and power assignment. The main goal is to provide a high-level

model representation, including an adaptive resource allocation strategy. It is believed

that a high-level simulation model can allow for representation of the network with as

much realism as desired, tailored to each particular scenario.

An existing high-level reference model (represented in the ITU model) uses the hexag-

onal deployment pattern, independent simulation drop concept and the common lognor-

mal shadowing model, and is primarily suitable for regular cell deployment with little

dynamism. In contrast, the proposed model is designed for the irregular cellular net-

work having arbitrary deployment with small and large BSs and increased dynamism

and adaptation. The model considers a wide array of dynamic changes that are expected

to occur in the irregular network. As part of the model, an adaptive resource allocation

framework that works at both a slower and a faster time scale is considered.

We first discussed BS outage as a specific type of dynamic network change in Chap-

ter 3. BS outage compensation was then considered as the adaptation in response to

the BS outage. First, a model of the network in an outage was devised for an irregular

cellular network. In particular, the BS positions and power classes were modelled in a

general sense. Then, an outage compensation method that uses the idea of the cluster

or scheduling cell combined with cell re-association and power adjustment was proposed.

Simulations were performed with correlated shadowing for non-common-end links that

is appropriate for irregular deployments. Through evaluations using the proposed mod-

elling, it was demonstrated that the proposed algorithms show robustness against the

132

Chapter 8. Conclusion 133

outage in both regular and irregular networks.

The high-level model and the adaptive resource allocation framework for the irregular

network was developed and enhanced, in stages, in the remainder of the thesis:

In Chapter 4, the system model with time evolution was considered for the irregular

network with a single BS power class and constant power levels. A modification of the

ITU simulation drop concept was proposed. A drop deployment methodology considering

both the arbitrariness and the restrictions of deployment was discussed. Finally, a time

evolution methodology considering an array of dynamic changes such as terminal arrivals

and departures, BS deployments and outages as well as BS and terminal movements

was detailed. The parameters for the drop deployment and time evolution models were

considered and typical values were suggested.

Chapter 5 considered the adaptation of BS power levels which form an extension of

an adaptive resource allocation framework for irregular networks. BS power adaptation

algorithms which made use of the total rate maximization and proportional fairness

were proposed. Then, a high-level adaptive resource allocation framework based on

clustering and power control at a large time scale and coordinated proportional fair

resource allocation at a small time scale was proposed for a generic irregular network.

The simulation results illustrate the use of the system model, with the suggested ranges of

the model parameters. The results showed the improvement that was obtained by using

the power adaptation algorithms, as well as the general degradation of performance

for increased values of the BS separation parameter. Finally, the power adaptation

algorithms were shown to compensate, in part, for the effect of irregularity.

Chapter 6 considered the extension of the model to the case of multiple power classes

without time evolution. The drop deployment methodology was modified to be suitable

to model the deployment of hotspots. Then an enhanced correlated shadowing model

was proposed for the network having multiple radio propagation environments. For this

purpose, a theory was proposed for variation of the shadowing parameters, (standard

deviation and correlation distance) in a network with an arbitrary number of radio prop-

agation environments. The model consistency with an earlier model was verified for a

particular scenario, and the implementation in practical simulations was discussed.

Finally, Chapter 7 extended the time evolution methodology of Chapter 4 to the

situation with multiple power classes and hotspots. The high-level adaptive resource

allocation framework of Chapter 5 with BS power adaptation was applied for the case of

multiple power classes. Simulation results showed the use of the model and the improve-

ments obtained by the BS power adaptation in terms of different measures of performance

(5th percentile, 50th percentile and user average spectral efficiencies as well as sum log


utility) for a network with multiple power classes.

The simulation results illustrate the use of the system model, with the suggested

ranges of the model parameters. In addition, improvements obtained by the BS power

adaptation in terms of different measures of performance (5th percentile, 50th percentile

and user average spectral efficiencies as well as sum log utility) for a network with multiple

power classes were discussed, and the effect of limited coordination was evaluated.

The proposed high-level model provided an arbitrary deployment method, a large-

scale time evolution method and a correlated shadowing model appropriate for the irreg-

ular cellular network considered in the thesis. It is believed that a high-level simulation

model can serve as a skeleton network model over which different researchers can make

modifications. This can allow for representation of the network with as much realism as

needed, tailored to each particular scenario.

8.2 Future Work

• The deployment of additional BSs in an irregular cellular network would normally

improve the system performance. Consider that there are a limited number of

extra BSs that can be deployed. It would be interesting to study where is the

best location to deploy the additional BSs in order to enhance the cellular system

performance.

• Traditionally, measures such as cell capacity, coverage probability and average SINR

or rate, among others, have been used to quantify the performance in the cellular

system. Looking to the future or as a parallel area of research, a high-level simu-

lation model could facilitate the development of new measures of performance for

irregular cellular networks, that are more appropriate than the above traditional

performance measures. This is because the simulation model need not be bound

by the analytical tractability as a constraint. Instead, the constraint can be the

utility of the performance measure corresponding to the desired level of realism for

the application. For instance, new and more appropriate measures of user perfor-

mance could be developed for an irregular network where the concept of cells is

more flexible due to the dynamism of BSs (such as due to deployments, outages

and the formation of scheduling cells or clusters). Additionally, the analysis in this

thesis assumes full buffer traffic once a terminal is active. In the future, improved

performance metrics could be devised to quantify the user experience for varied

traffic patterns other than full buffer at a finer time scale.


• Throughout this work, it is assumed for the purpose of scheduling that the ter-

minals have full buffer traffic. It would be an interesting problem to consider the

proportional fair scheduling and the adaptation framework under the condition

that the traffic is not full buffer. The system model, the scheduling and the slow

adaptation algorithms could be adapted for such a traffic scenario.

• In this thesis, we were able to justify the inhomogeneous distance-dependent path

loss and shadowing channel models through physical arguments. However, it would

be a worthwhile effort to verify the channel models by gathering real-world mea-

surements under inhomogeneous network conditions.

• The dynamic system model of this thesis makes it possible to represent the grad-

ual time evolution of the irregular cellular network. This can enable the study

and evaluation of adaptive algorithms with reduced signalling in practically real-

izable scenarios. It would, therefore, be interesting to consider the reduction of

signalling for the proposed specific adaptation framework. In particular, a variety

of signalling reduction schemes could be considered and analyzed under different

practical system evolution scenarios.

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Dynamic System Modelling and Adaptation Framework … · Abstract Dynamic System Modelling and Adaptation Framework for Irregular Cellular Networks Levent Kayili Doctor of Philosophy