Optimization of Joint Cell, Channel and Power Allocation in Wireless Communication Networks

Optimization of Joint Cell, Channel and Power Allocation

in Wireless Communication Networks

Optimization of Joint Cell, Channel and Power Allocation in Wireless Communication Networks

MIKAEL FALLGREN

Doctoral Thesis

Stockholm, Sweden 2011

TRITA MAT 11/OS/05 ISSN 1401-2294

ISRN KTH/OPT/DA-11/05-SE ISBN 978-91-7501-084-7

Optimization and Systems Theory Department of Mathematics Royal Institute of Technology Se: 100 44 Stockholm, Sweden Akademisk avhandling som med tillstånd av Kungl Tekniska Högskolan framlägges till offentlig granskning för avläggande av teknologie doktorsexamen, fredagen den 7 oktober 2011 klockan 10.00 i rum D3, Lindstedtsvägen 5, Kungl Tekniska Högskolan, Stockholm.

© Mikael Fallgren, 2011

Print: Universitetsservice US AB, Stockholm, 2011

Till mina vänner

The future belongs to those who believe in the beauty of their dreams

- Eleanor Roosevelt, (1884-1962)

Abstract

In this thesis we formulate joint cell, channel and power allocation problems within wireless communication networks. The objectives are to maximize the user with mini- mum data throughput (Shannon capacity) or to maximize the total system throughput, referred to as the max-min and max-sum problem respectively. The complexity is stud- ied together with proposed optimization- and heuristic-based approaches.

In the first paper an overall joint cell, channel and power allocation max-min prob- lem is formulated. We show that the decision problem is NP-hard and that the op- timization problem is not approximable unless P is equal to NP, for instances with a sufficiently large number of channels. Further, it follows that for a feasible binary cell and channel allocation, the remaining continuous power allocation optimization problem is still not approximable unless P is equal to NP. In addition, it is shown that first-order optimality conditions give global optimum of the single channel power al- location optimization problem, although the problem is in general not convex.

In the following two papers heuristics for solving the overall problem are proposed.

In the second paper we consider the single channel problem with convex combinations of the max-min and the max-sum objective functions. This variable utility provides the ability of tuning the amount of fairness and total throughput. The third paper investi- gates the multiple channel setting. On a system with three cells, eight mobile users and three channels, we perform an exhaustive search over feasible cell and channel alloca- tions. The exhaustive search is then compared to the less computationally expensive heuristic approaches, presenting potential earnings to strive for. A conclusion is that several of the proposed heuristics perform very well.

The final paper incorporates fixed relay stations into the overall joint cell, channel and power allocation max-min problem. The complexity is inherited from the formula- tion without relay stations. Further, we propose a heuristic channel allocation approach that shows good performance, compared to an optimization based approach, in numer- ical simulations on the relay setting.

Keywords: Wireless multicell networks, Optimization, OFDMA, Shannon capacity,

Complexity, NP-hard, Relays, Allocation problems, Heuristic algorithms.

Acknowledgments

First and foremost, I would like to express my deepest appreciation to my advisor Professor Anders Forsgren for all your help, encouragement and mathematical intuition whenever needed. I am very grateful for your continuous support and inspiration during these years.

I would like to thank my co-advisor Professor Johan Håstad for sharing his expertise and many ideas. Further, I would like to thank my industrial co-advisors Dr. Gábor Fodor and Dr. Mikael Prytz at Ericsson Research for sharing their knowledge and valuable sug- gestions. An additional thanks goes to my frequent co-author Gábor for his selfless way of finding time. Many thanks goes to my reference group for all our valuable discussions throughout the work that continuously provided me with interesting and challenging prob- lems, and for helping me to approach them. I am very grateful to have had the opportunity of collaborating with all of you throughout this work.

Further, I would like to express my gratitude to Professor Andrea Goldsmith for hosting my visit at Stanford University in 2011. On a personal note, I am very grateful for the wonderful hospitality that Denise, Ethan and Antonetta showed me during my stay.

To Hildur Æsa Oddsdóttir, whom I helped to supervise in her master’s thesis project, it was very rewarding to work with you. Many thanks for contributing to this thesis.

In terms of careful proofreading of this thesis, big thanks goes to Anders Möller, Per Enqvist, Gábor Fodor and Anders Forsgren. All your valuable suggestions have very much helped to improve this text. To locate the remaining errors, solely made by me, is hereby left as an exercise for the reader.

This work has been possible thanks to the financial support by the Swedish foundation for strategic research (SSF) via the Center for Industrial and Applied Mathematics (CIAM).

I am very grateful to all parties that have made this project possible.

Special thanks goes to all my past and present colleagues at the Division of Optimiza- tion and Systems Theory for creating a marvelous workplace where every day is enjoyable.

Many thanks to the faculty with Professor Anders Lindquist, Professor Krister Svanberg, Professor Anders Forsgren, Professor Xiaoming Hu, Dr. Ozan Öktem and Dr. Per En- qvist, together with its previous members Dr. Claes Trygger, Dr. Ulf Brännlund, Dr. Amol Sasane and Professor Ulf Jönsson, for all enjoyable moments.

To present and past students - I’m very happy for each and every moment with you!

To Anders M for friendship and your positive source of enthusiasm, encouragement and inspiration. To Enrico for being my great roommate during these years with all the shared jokes and laughter. To Johan T and Johan M for our personal and professional discussions.

xi

xii A CKNOWLEDGMENTS

To Albin and Rasmus for course collaboration and snowboard assistance. To Æsa, Dr.

Corentin, Yuecheng, Henrik S and Tove for friendship and good times.

A big thanks goes to the previous students+Per for warmly welcoming me into the group. To Dr. Mats and Dr. Fredrik for incorporating me into their Ph.D. course collab- orating unit. To Dr. David and Dr. Maja for all the interesting lunch discussions. To Dr.

Johan and Dr. Stefan for being so fun to be around! To Dr. Tove for your positive spon- taneity and your genuine caring. To Dr. Per for your way of sharing experience and your sense of humor. Thanks for all cheerful moments during these years, they mean a lot to me!

I also would like to direct thanks to my other colleagues at the Department of Mathe- matics, and especially to the floorball team with Joel and André in the lead. Many thanks to the administrative staff with Leena, Marie, Ann-Britt and Maria.

In addition I would like to thank Anna L and Eva for providing me with challenges and support in my strive to become a better training instructor.

A sincere thanks to all my friends, relatives and family who continuously have been a tremendous support. Thanks for being such positive part of my life. To my "Badminton gang" with Henrik, Sebbe, Janne, Roger and Per G. I enjoy every set - especially the ones I win! To Ingemar for always being up for board games and table hockey. To Patrik and Robert for sharing almost every day in our early years. To Agge and Siv for your positive support. To Tina for your enormous kindness. To Lina for delightful café discussions about books and life in general. To Carro for being a very wonderful person.

To my parents, Kerstin and Mats, for all your continuous help, guidance and patience.

Finally, to Anna for being my lovely sister!

Stockholm, August 2011

Mikael Fallgren

Table of Contents

Abstract ix

Acknowledgments xi

Table of Contents xiii

List of Algorithms xvii

List of Figures xix

List of Tables xxiii

I Introduction 1

I.1 Background . . . . 1

I.2 Problem statement . . . . 3

I.2.1 Problem statement with relay stations . . . . 4

I.3 System model . . . . 6

I.4 Related research together with different problem formulations . . . . 7

I.4.1 Single cell setting . . . . 9

I.4.2 Multicell setting . . . . 12

I.4.3 Relay setting . . . . 14

I.5 Main contributions and limitations . . . . 15

I.5.1 Main contributions . . . . 15

I.5.2 Main limitations . . . . 15

I.6 Summary of the appended papers . . . . 16

I.7 Bibliography . . . . 18

Paper A 21 A On the Complexity of Joint Cell, Channel and Power Allocation 23 A.1 Introduction . . . . 23

A.2 System model and problem formulations . . . . 25

A.2.1 The overall problem . . . . 26

xiii

xiv T ^{ABLE OF} C ^ONTENTS

A.2.2 The power allocation problem . . . . 27

A.3 The overall decision problems are NP-hard . . . . 29

A.4 On the approximability of the overall optimization problems . . . . 35

A.5 On properties of the power allocation problem . . . . 39

A.5.1 Some power allocation observations . . . . 39

A.5.2 On convexity of the power allocation problem . . . . 41

A.6 Summary and conclusions . . . . 46

A.7 References . . . . 48

Paper B 51 B Joint Cell and Power Allocation 53 B.1 Introduction . . . . 53

B.2 System model . . . . 55

B.3 Problem formulation . . . . 56

B.3.1 The max-min problem . . . . 56

B.3.2 A generalization of the max-min problem . . . . 57

B.4 Solution approaches . . . . 58

B.4.1 Link assignment (cell assignment) . . . . 58

B.4.2 Power allocation . . . . 60

B.4.3 A simplified generalization of the max-min power allocation problem 60 B.5 Numerical results . . . . 61

B.5.1 Simulation environment . . . . 61

B.5.2 Numerical simulation results . . . . 62

B.6 Conclusions . . . . 67

B.7 References . . . . 68

Paper C 71 C Joint Cell, Channel and Power Allocation 73 C.1 Introduction . . . . 73

C.2 System model . . . . 75

C.3 Problem formulation . . . . 76

C.3.1 The max-min throughput problem . . . . 77

C.3.2 The max-sum throughput problem . . . . 78

C.4 Solution approaches . . . . 78

C.4.1 Link allocation (cell allocation) . . . . 78

C.4.2 Channel allocation . . . . 80

C.4.3 Power allocation . . . . 82

C.4.4 The overall allocation . . . . 86

C.5 Numerical results . . . . 87

C.5.1 Simulation environment . . . . 87

C.5.2 Numerical results in the larger setting, |B| = 19 . . . . 89

C.5.3 Numerical results in the smaller setting, |B| = 3 . . . . 92

xv

C.5.4 Numerical running times . . . . 96

C.6 Discussion . . . . 99

C.7 Conclusions . . . . 99

C.8 References . . . 100

Paper D 103 D Joint Cell, Channel and Power Allocation with Relays 105 D.1 Introduction . . . 105

D.2 Problem formulation . . . 107

D.3 Solution approaches . . . 111

D.3.1 Link allocation (cell allocation) . . . 111

D.3.2 Channel allocation . . . 112

D.3.3 Power allocation . . . 113

D.3.4 Allocation algorithm . . . 116

D.4 Numerical results . . . 116

D.4.1 Simulation environment . . . 117

D.4.2 Numerical results in the larger setting, with |B| = 7 . . . 118

D.4.3 Numerical results of the smaller setting, with |B| = 3 . . . 121

D.4.4 Numerical simulation statistics . . . 124

D.5 Discussion . . . 125

D.6 Conclusions . . . 126

D.7 References . . . 127

List of Algorithms

B.1 Direct Greedy Link Assignment (DGA) . . . . 58

B.2 Maximize the Total Path Gain Link Assignment (MTGA) . . . . 59

B.3 Maximize the Minimum Path Gain Link Assignment (MMG) . . . . 59

C.1 Link allocation Greedy (LaG) . . . . 79

C.2 Link allocation Optimization (LaO) . . . . 79

C.3 Link allocation All (LaA) . . . . 79

C.4 Max-min Channel allocation Greedy (CaG) . . . . 80

C.5 Max-sum Channel allocation Greedy (CaG) . . . . 81

C.6 Channel allocation Optimization (CaO) . . . . 81

C.7 Channel allocation Relaxation (CaR) . . . . 82

C.8 Power allocation Evenly (PaE) . . . . 83

C.9 Power allocation Derivatives (PaD), max-sum . . . . 84

C.10 Power allocation Simplified (PaS) . . . . 85

C.11 Power allocation Optimization (PaO) . . . . 86

C.12 The allocation Update (TaU) . . . . 86

C.13 Link Power grid Allocation (LPgA) . . . . 87

D.1 Link allocation Greedy (LaG) . . . 111

D.2 Channel allocation Heuristic (CaH) . . . 114

D.3 CaH: via RS or include t = 1 channel . . . 115

D.4 CaH: via RS or not . . . 115

D.5 Power allocation Evenly (PaE) . . . 115

D.6 Allocation . . . 116

xvii

List of Figures

I.1 Each time-frequency slot (square) denotes the assigned user (color). . . . 2 I.2 BSs and MSs are communicating on various channels (color of line) with var-

ious transmission powers (width of line). . . . 4 I.3 BSs, RSs and MSs are communicating on various channels (color of line) with

various transmission powers (width of line). There are three available channels at each time slot and cell. . . . 5 A.1 A given graph G. . . . 30 A.2 First stage of creating the problem instance. For each node i ∈ V the source

S ¯ i is represented as a square, destination ¯ D i as a circle, and the path gain ¯g between them is illustrated as a link ( ¯ S i , ¯ D i ). . . . 30 A.3 Second stage of creating the problem instance. Introduce the two blue links

( ¯ S i , ¯ D j ) and ( ¯ S j , ¯ D i ) for each edge (i, j) ∈ E of the graph G, to illustrate that the path gains between these sources and destinations are δ. . . . 31 A.4 Third stage of creating the problem instance. For each node i ∈ V of the graph

G, the source ˜ S i is represented by a square, and destination ˜ D i by a circle.

The path gain ˜g between base station ˜ B i and mobile user ˜ M i is illustrated by a black link ( ˜ S i , ˜ D i ) and the path gain ˜ G between source ˜ S i and destination ¯ D i

is illustrated by a red link ( ˜ S i , ¯ D i ). . . . 32 A.5 The Shannon capacity η = min{η ¹¹ , η 22 } is a function of the power variables

p 11 and p 21 , which both uses the single channel |C| = 1. . . . 42 A.6 A setting consisting of three base stations illustrated as squares, four mobile

users illustrated as circles and two channels illustrated as links. . . . 45 B.1 Comparing the performance of the three link assignment algorithms for the

max-min power allocation: DGA, MTGA and MMG in terms of the worst off user throughput. . . . 62 B.2 Comparing the performance of the three link assignment algorithms for max-

throughput power assignment: DGA, MTGA and MMG in terms of the total throughput. . . . 63 B.3 Comparing the performance of the max-min and the maximum throughput

power allocation in terms of the uplink SINR distribution at the base stations. 64

xix

xx L ^{IST OF} F ^IGURES

B.4 Comparing fair (α = 0) and throughput maximizing (α = 1) power allocations by plotting the ratio of the associated total throughput values. . . . 64 B.5 The total system throughput for the max-min allocation in the three user dis-

tribution cases "Uniform", "Close", "Far". . . . 65 B.6 Comparing the total system throughput when employing the maximum through-

put power allocation for the three user distribution cases. . . . 66 B.7 The total system throughput (left axis) for α along with the ratio of the mini-

mum user throughput divided by the maximum user throughput (right axis), in uplink. . . . 66 B.8 Comparing the worst off user throughput in uplink when employing the max-

min allocation simulation to the global optimal solutions. . . . 67 C.1 Comparing both the cell allocation (using CaG, CaR), and the channel alloca-

tion (using LaG, LaO), in terms of the worst off user throughput (using max- min power allocations). . . . 89 C.2 Comparing the power allocation algorithms (for LaG and LaO, together with

CaG and CaR) in terms of the worst off user throughput. . . . 90 C.3 Comparing the power allocation algorithms (using LaG, LaO and CaG), in

terms of total user throughput. . . . 91 C.4 Comparing the power allocation algorithms (using LaG, LaO and CaG), in

terms of total user throughput. Every solution has been subtracted by the solu- tion of LaG-CaG-PaE, to make the comparison easier. . . . 91 C.5 Comparing the channel allocation algorithms (using LaG, LaO and all max-

min power allocations), in terms of the worst off user throughput. . . . 92 C.6 Comparing the power allocation algorithms (using CaG), in terms of the worst

off user throughput. . . . 93 C.7 Comparing the power allocation algorithms, in terms of the worst off user

throughput. . . . 93 C.8 Comparing the power allocation algorithms in terms of total user throughput. 94 C.9 Comparing the power allocation algorithms in terms of total user throughput. 95 C.10 Comparing the power allocation algorithms in terms of total user throughput.

Every solution has been subtracted by the solution of LaG-CaG-PaE, to make the comparison easier. . . . 95 D.1 Comparing the different channel and power allocation approaches, in terms of

the worst off mobile user throughput, in the setting without RSs. . . 119 D.2 Comparing the different channel and power allocation approaches, in terms of

every mobile user throughput, in the setting with RSs (RS ^b r ). . . 120 D.3 Comparing the different channel and power allocation approaches, in terms of

the throughput of each mobile user, in the setting with RSs (RS ^r _m ). . . 120 D.4 Comparing the different channel and power allocation approaches, in terms of

the worst off mobile user throughput, in the setting with RSs (RS ^b _r ). . . 121

xxi

D.5 Comparing the different cell allocations, both in the setting without and with RSs, with the CaH-PaE approach, in terms of the worst off mobile user through- put. . . 122 D.6 Comparing the different cell allocations, both in the setting without and with

RSs, with the CaH-PaE approach, in terms of every mobile user throughput. . 123 D.7 Comparing the different cell allocations, both in the setting without and with

RSs, with the CaH-PaO approach, in terms of the worst off mobile user through- put. . . 123 D.8 Comparing the different cell allocations, both in the setting without and with

RSs, with the CaH-PaO approach, in terms of every mobile user throughput. . 124

List of Tables

B.1 Definition of the sets of the system model . . . . 55 B.2 Definition of some variables and constants . . . . 56 B.3 Sets used for power control (given y and x) . . . . 60 B.4 The main parameters in the system . . . . 61 B.5 Path gain specific parameters . . . . 62 C.1 Definition of the sets in the system model . . . . 75 C.2 Definition of the constants and variables used in our system model . . . . 76 C.3 The main parameters in the system . . . . 88 C.4 Path gain specific parameters . . . . 88 C.5 Parameter values in the simulations . . . . 89 C.6 The average solution time (in seconds) and the average number of iterations

niter over all test cases in the setting with 19 cells, when solving the max-min problem (C.5) with Algorithm TaU, for the various cell, channel and power allocation approaches. . . . 96 C.7 The average solution time (in seconds) and the average number of iterations

niter over all test cases in the setting with 19 cells, when solving the max-sum problem (C.6) with Algorithm TaU, for the various cell, channel and power allocation approaches. . . . 96 C.8 The average solution time (in seconds) and the average number of iterations

niter over all test cases in the setting with 3 cells, when solving the max-min problem (C.5) with Algorithm TaU, for the various cell, channel and power allocation approaches. . . . 97 C.9 The average solution time (in seconds) and the average number of iterations

niter over all test cases in the setting with 3 cells, when solving the max-sum problem (C.6) with Algorithm TaU, for the various cell, channel and power allocation approaches. . . . 97 C.10 The average solution time ([s] = seconds, [min] = minutes, [h] = hours, [days]

= days), over all test cases in the setting with 3 cells, when solving the max-min and the max-sum problem. Also, the average number of iterations niter when using LaA, and the percentage of solved optimization problems (opt-solve-%) within LPgA. . . . 99

xxiii

xxiv L ^{IST OF} T ^ABLES

D.1 Definition of sets . . . 108 D.2 Definition of constants . . . 108 D.3 Definition of variables . . . 109 D.4 The main parameters in the system . . . 117 D.5 Path gain specific parameters . . . 118 D.6 The average number of allocated relays, given that an RS was allocated, to-

gether with the percentage of how often not even a single RS was allocated, over all test cases in the setting with 7 cells, when solving problem (D.6). . . 124 D.7 The average solution time in seconds, and the average number of iterations

n ^iter , over all test cases in the setting with 7 cells, when solving problem (D.6). 125 D.8 The average solution time [s] and the average number of iterations n ^iter , over

all test cases in the setting with 3 cells, when solving problem (D.6) without RSs using CaH. . . 125 D.9 The average number of allocated relays, given that an RS was allocated, to-

gether with the percentage of how often not even a single RS was allocated,

and also the average solution time [s] and the average number of iterations

n ^iter , over all test cases in the setting with 3 cells, when solving problem (D.6). 125

Introduction

In less than two centuries communication networks have significantly reshaped the infras- tructure of our society. Nowadays we live in a connected world with wireless networks virtually everywhere. To meet today’s continuously growing demand for wireless access, even more advanced techniques need to be developed. The purpose of this introduction is to give a brief overview of wireless cellular communication networks and to describe how the problems in the appended papers arise and relate to known results.

I.1 Background

The first major technological communication device using electricity, the telegraph, was invented by S. Morse in 1838. As one could transmit several tones on a telegraph line, it could also transmit human speech. A. G. Bell filed his patent of the telephone in 1876. Ini- tially the telephone calls were connected manually. Already in the 1850’s, J. C. Maxwell foresaw that energy could be transported wireless, which later was demonstrated by H.

Hertz in 1888. Wireless telegraphy units followed, where the transmitter needed to gen- erate all energy that could be received. Hence, the transmitter had to use extremely high power over the wireless connection to compensate for the severe energy loss over distance, known as path loss. The electron tube amplifier by de Forest, 1915, made it possible to compensate for this path loss. This enabled radio broadcasting, later followed by TV broad- casting. The electron tube was subsequently replaced by the lightweight and low-power transistor, invented by Shockley, Bardeen and Brattain in 1948 [1, 15, 42].

A contemporary theoretical breakthrough was made on wireless channels by C. Shan- non [32], by showing that there are fundamental limitations to the maximum data rates that can be transmitted with vanishing error probability. The wireless communication systems of today are close to these limits [1, 15, 42].

In the beginning of the 1980’s early analog cellular systems appeared, often referred to as the first-generation (1G) systems, such as the Nordic Mobile Telephone (NMT) and the Advance Mobile Phone Service (AMPS). After about a decade the early digital cellular systems followed, referred to as the second-generation (2G) systems. The 2G systems were mainly designed for voice, with power control approaches that strived toward a fixed data rate target given by the sought Quality of Service (QoS). Main 2G standards are Global System for Mobile Communications (GSM), IS-95 and IS-136 [15, 42]. The succeeding generations of cellular systems, third-generation (3G) and fourth-generation (4G), support varying data rates and handle various QoS requirements.

1

2 I NTRODUCTION

The 3G is based on a wideband Code-Division Multiple Access (CDMA) standard, called International Mobile Telecommunications 2000 (IMT-2000), specified by the Inter- national Telecommunication Union (ITU). One such 3G standard is the Universal Mobile Telecommunications System (UMTS) which is a successor to GSM and standardized by the 3rd Generation Partnership Project (3GPP) [15].

Another decade followed before the ITU Radio Communications group (ITU-R) speci- fied the IMT-Advanced requirements for 4G standards, see, e.g., [5]. Technology proposals based on Long-Term Evolution Advanced (LTE Advanced) standardized by the 3GPP or 802.16m standardized by the Institute of Electrical and Electronics Engineers (IEEE) were submitted as 4G candidates [3, 15, 28]. In December 2010 the ITU announced that current versions of LTE, WiMAX and other evolved 3G technologies that despite not fulfilling IMT-Advanced requirements could be considered 4G, provided that they represent prede- cessors to IMT-Advanced with "a substantial level of improvement in performance and capabilities with respect to the initial third generation systems now deployed." [4]. In the emerging standards based on Orthogonal Frequency-Division Multiple Access (OFDMA), such as the 3GPP LTE standard and the IEEE 802.16j standard, the concept of multihop relays has been introduced, see, e.g., [28, 36].

The OFDMA technology is designed for multiple user systems and has evolved from the more basic single user Orthogonal Frequency-Division Multiplexing (OFDM) approach.

In OFDMA neither the time allocation nor the frequency allocation for any mobile user is static, unlike the situation in the 2G systems. Hence, each time-frequency slot is to be strategically scheduled among the users. An example is given in Figure I.1 where each square denotes a single frequency channel at a given time slot, i.e., one axis denotes time while the other denotes frequency. The user that has been assigned to the time-frequency slot is given by the color of the square.

Figure I.1: Each time-frequency slot (square) denotes the assigned user (color).

To maximize the efficiency of such system, the scheduling of time and frequency re-

sources needs to be coordinated with the power control that allocates the transmission

powers [15, 22, 38].

O PTIMIZATION OF WIRELESS COMMUNICATION NETWORKS 3

I.2 Problem statement

The overall goal is to optimize the performance in a communication network. This opti- mization means that we want to send with data rates in such a manner that it favors the overall system performance and its individual users. We want to study systems that are in accordance with current standards in the context of how much these systems theoretically can produce. This allows us to find the potential of the system in terms of what can be achieved. In future work this acquired knowledge can be used as a guideline when devel- oping more advanced distributed solution methods than what we see in today’s systems.

Typical for transmissions in wireless networks are variations of the radio channel. The quality of a radio channel depends heavily on the distance between the transmitter and the receiver, but there are also other important aspects to consider. For example there may be large objects, such as a house or a mountain, between the transmitter and the receiver, that are blocking a large part of the signal. Another important aspect to take into consideration is that a signal often has been reflected by several objects before reaching the receiver. The signal is affected in different ways depending on the size, orientation and material of each such object.

When the same frequency channel is used on more than one place, this means that the users of that channel will interfere with each other. To reduce the disturbance in the system one wants to ensure that nearby users communicate on different channels, since the distance has such a high impact.

The systems that we consider consist of a number of Base Stations (BS) and Mobile Stations (MS). Either the BSs are transmitting to the receiving MSs, or it is reversed. Fur- ther, we study the setting where each MS is connected to precisely one BS. Due to the increase of interference when nearby users communicate on the same channel, a BS will only use each channel once. The MSs that are connected to the same BS can thus commu- nicate simultaneously on the different channels.

To simplify, we will not consider time variations, but rather focus on the cell, channel and power allocation problems at a fixed moment in time. For a given cell and channel allocation, it remains to balance the limited transmission power resources to maximize the overall system performance. One example of the described overall problem statement is illustrated in Figure I.2.

In Figure I.2 we can see three BSs (cells), eight MSs and three different channels (colors). Each MS is connected to precisely one BS, which is illustrated by at least one line between the BS and MS. The color of the line represents the channel that is used, while the width of the line represents the amount of transmission power that is used on this specific channel.

The more transmission power that is used, the higher data rate can be transmitted and

received. However, at the same time this also increase the amount of received interference

by other users on the same channel. These contradictory properties give rise to a resource-

balancing problem. In this thesis we study various approaches on how to choose a cell and

channel allocation with good conditions together with finding a power allocation that is

able to balance the resources in a beneficial way.

4 I NTRODUCTION

Figure I.2: BSs and MSs are communicating on various channels (color of line) with vari- ous transmission powers (width of line).

In general, each BS performs the resource allocation within its cell without consider- ation of the consequences in other cells. As a result suboptimal allocations in terms of overall performance follow, which implies a decrease in achieved system rate. However, under some form of coordination among the BSs, the channel and power allocation can be studied jointly. We therefore assume that there is an Access Point (AP) in the system that has perfect information of all users. The overall multicell allocation problem is solved at the AP. Thereafter the AP informs each MS about its allocations through a feedback chan- nel, and each MS transmits its data according to that prescription. In, e.g., UMTS there are RNCs that can perform resource allocation within limited regions. Further, a RRM server can be introduced to perform resource allocations within a multicell region [2]. This moti- vates a study of centralized algorithms, that can also serve as a benchmark for distributed approaches.

In addition to the problem described above, we introduce relay stations in section I.2.1.

I.2.1 Problem statement with relay stations

A relay station (RS) serves as an intermediate station. In a setting with RSs, one is able

to relay the transmission from the source to the destination via the RSs whenever this is

preferable. As before, one can choose the direct transmission, from source to destination,

without using any of the RSs. Further, it is possible to combine the direct transmission

with RS transmissions.

O PTIMIZATION OF WIRELESS COMMUNICATION NETWORKS 5

Figure I.3: BSs, RSs and MSs are communicating on various channels (color of line) with various transmission powers (width of line). There are three available channels at each time slot and cell.

The RS makes it possible to divide a path into several shorter, as well as offering alternative routes to users with bad reception. The attainable throughput via an RS is limited by the minimum rate between the BS and RS and between the RS and its MSs.

The advantage of an RS, compared to a BS, is that it does not need a wired backbone access. This decrease the deployment cost, makes the positioning of the RSs more flexible and the installation takes less time. An alternative approach to obtain larger capacity and coverage is to shrink the cell size, by increasing the density of BSs [36].

Here, the previously introduced problem setting is extended to include fixed RSs, where each BS has a given number of RSs. To be able to model the two-step procedure of the relay transmission we choose to divide our snap-shot in time into two parts. At the first time step the source is allowed to transmit either to an RS or direct to the destination. At the second time step the transmission is directed to the destination, either from the original source or from an RS. In Figure I.3 we illustrate the relay setting in a similar manner as Figure I.2 illustrate the setting without RSs.

In Figure I.3 we can see three BSs (cells), six RSs per cell (BS), eight MSs and three

different channels (colors) for each of the two time steps. Each MS is connected to pre-

cisely one BS, where the connection can be either direct or via one or several of the relays,

or both. This connection is illustrated by at least one line from the BS and to the MS. The

color of the line represents the channel and the time step that is used, while the width of

the line represents the amount of transmission power that is used on this specific channel

and time step.

6 I NTRODUCTION

I.3 System model

In this section we introduce a mathematical model that describes the cellular system con- sidered. The cellular system is a multicell network that consists of a number of base sta- tions and mobile stations where each cell has a centralized base station.

We assume that the cellular network consists of a set B of serving BSs and a set M of MSs. We assume that there is a cell site at each BS, i.e., in the physical location of each BS there is radio equipment that provides coverage within the cell, such that each BS maintains the coverage area of its associated cell. Any MS needs to be assigned to a serving base station to maintain service coverage. Within this thesis, each MS will be connected to precisely one BS. We say that there is a link (communication link) between an MS and its serving BS. To allow for convenient handling of both downlink and uplink simultaneously, we introduce the set of transmitting sources S and the set of receiving destinations D. In downlink S = B and D = M, whereas it is vice versa in uplink.

The radio resources that are used within each cell are assumed to be orthogonal chan- nels, such that there is no intra-cell interference, i.e., no interference between mobiles in the same cell. The set of orthogonal channels is denoted C. The communication takes place in parallel on these channels. In general, a communication link may comprise multi- ple channels. The assumption on negligible intra-cell interference is valid for the modern telecommunication standard OFDMA. For ease of presentation, we consider the transmis- sion bandwidth W as the basic radio resource, and assume that it is a known constant. This bandwidth is allocated in terms of frequency channels on the communication links within each cell. We allow for a complete reuse of all orthogonal channels C in each cell, that is we consider a system with frequency reuse factor one (i.e., a Reuse-1 system). Finally, our focus is on the channels with data traffic and we omit the channels used for synchronization and channel estimation, as well as the required control channel, over which the scheduling mechanism instructs the MSs of the current time-frequency allocation.

The effective (path) gain, g ijk , is used to predict the received power at destination j, given the transmission power at source i, on channel k. The effective path gain includes the effects of path loss, fading, effects of coding etc. The path loss gives an expected loss of power related to the distance d between the transmitter and the receiver. According to analysis and measurements, the decrease in power against distance can be modeled as d ^−α , where α ∈ [2, 6], see, e.g., [15, 31]. A distant dependent path loss is a rough approxima- tion, as the received power can vary significantly between separate locations. Obstacles between the transmitter and receiver cause shadow fading (slow fading) that attenuate the signal. The shadow fading can be well modelled by a log-normal distribution. The multi- path fading (fast fading) describes that the transmitted signals propagate via different paths before reaching the receiver, where these multipath components are added together, ei- ther constructively or destructively, causing large variations in the received signal. Two common multipath fading models are the Rayleigh model and the Ricean model [15]. In the latter part of the thesis the effective gains have been generated by the RUdimentary Network Emulator (RUNE) [42]. We assume the channel gains to be constant, since we consider a fixed moment in time.

The thermal noise, that is caused by the "Brownian dance" of electrons in all materials,

O PTIMIZATION OF WIRELESS COMMUNICATION NETWORKS 7

is unavoidable [42]. We assume strictly positive additive thermal noise at each receiver j, which we denote by σ ² _j .

Further, the receiver j is also interfered by other transmissions that take place on the same channel k ∈ C. These interfering signals will be treated as additive to the noise. The amount of interference from an interfering signal on channel k is given by a combination of the effective gain between the interfering source(s) and receiver j together with the transmission power used by the interfering source(s).

In terms of achievable data rate the thermal noise, together with the interference of other users sharing the same resource, can significantly affect the signal. The performance of a transmission from source i to receiver j on channel k can be measured by the signal to interference and noise ratio (SINR), given by

γ ijk = g ijk p ik

σ _j ² + �

s∈{S\i} g sjk p sk , i ∈ S, j ∈ D, k ∈ C, (I.1) where p denotes the non-negative transmission power.

From to the groundbreaking work of C. Shannon [32], it is known, under certain condi- tions, that there exist capacity limitations that prescribe the maximum obtainable data rates over wireless channels with asymptotical small error probability. Any higher rate implies that the error probability is bounded away from zero. The Shannon capacity of channel k over the communication link i to j is given by

η ijk = W

|C| log ₂ (1 + γ ijk ), i ∈ S, j ∈ D, k ∈ C, (I.2) where |C| denotes the number of orthogonal channels. The transferable maximal bit rate with asymptotical small error probability, on channel k over the communication link i to j, is given by the Shannon capacity (I.2), under the assumption of continuous data bits.

Hence, the Shannon capacity is a mapping from the transmission powers, via the SINR, to the achievable throughput, see, e.g., [15].

This knowledge is the foundation of the problems that we choose to study. We consider the described system model to formulate relevant optimization tasks that strive towards favorable system performance.

I.4 Related research together with different problem formulations

The purpose of this section is to discuss related research and its relevance to the various resource allocation problems that are considered within this thesis.

At a fixed time instance, i.e., at a snap-shot in time, we assume that a scheduling

mechanism already has provided the MSs that are allowed to be allocated. The possibility

of improving the multicell system performance by varying the resource allocation over

time is not considered. Further, it is mainly assumed that the resource allocation is made

on a central level, i.e., that centralized optimization problems with global information are

solved.

8 I NTRODUCTION

Our task consists of allocating the available resources, i.e., the variable cell, channel and power allocations, in such a way that it maximizes the overall system performance.

As the signals interfere, see, e.g., the SINR in (I.1), the resources should be divided in a fashion that benefits the users of the system. In general, we want to take advantage of the channel diversity, or multiuser diversity, among the users’ effective gain. We strive to maximize the overall system performance for a set of base stations and mobile users. A graphical illustration of one feasible solution is given in Figure I.2.

A system requirement ensures that no intra-cell interference takes place. This is an attempt to reduce the number of nearby users that communicate over the same channel.

This system behavior is desirable due to the significant increase of interference whenever nearby users communicate on the same channel. Hence, the channel allocation is orthog- onal within each cell. The binary variable x denotes the channel allocation, where x ijk

is equal to one whenever transmitter i and receiver j is allowed to use channel k. Also, a variable y is introduced to describe the cell allocation. By constraints in the problem formulation, y will be uniquely determined by x. The transmission power is a continuous variable denoted by p, where p ik denotes the non-negative transmission power of source i at channel k.

Apart from the fact that any channel is used at most once within each cell, two other additional technological constraints ensure that each MS is assigned to precisely one BS and that the total power of each transmitter i is upper bounded by P _i ^max . Neither is the transmission power per channel limited nor are individual users’ requirements considered, though if wanted, such constraints can be included in the formulation of the optimization problem.

To be able to maximize the efficiency of the system, we need to define our objective, or utility. Many different utilities, such as QoS based on throughput, or energy efficiency, have been considered within the literature. Two well-known utilities to be maximized are

m∈M min R m , (I.3a)

�

m ∈M

R m , (I.3b)

where R m denotes the total throughput rate of mobile user m. We refer to (I.3a) as the max- min utility, as the system will be steered towards a solution that provides the user worst off with the largest possible throughput. Using (I.3b) gives that the overall throughput within the system is maximized, which we refer to as the max-sum utility. Here, the system is allowed to neglect the users worst off for the benefit of users with better channel conditions.

Two other utilities are either to maximize the proportional fairness utility [19], given

by �

m ∈M

log (R m ) , (I.4)

or to minimize the total power consumption under some QoS requirement. The propor-

tional fairness utility is a trade-off between the max-min fairness utility (I.3a) and the

max-sum throughput utility (I.3b).

O PTIMIZATION OF WIRELESS COMMUNICATION NETWORKS 9

The focus of this thesis is directed towards (I.3a) and (I.3b). In strive for fairness (I.3a) ignores total system throughput to benefit the throughput of the user worst off, while (I.3b) totally ignores fairness in its strive for the highest possible total system throughput. In this respect, these utilities correspond to two extreme cases. Further, we also consider an alternative trade-off utility approach where convex combinations of (I.3a) and (I.3b) are studied. This trade-off utility provides the ability of tuning the amount of fairness and max-sum throughput that is desirable.

In the following three parts of this section, various related works are discussed. The first part considers the single cell setting, where there is only one BS. The second part contains the multicell setting. Finally, in the third part we also include relay stations.

I.4.1 Single cell setting

Within this part, various single cell problem formulations are presented and discussed. We choose to consider the single user setting and the multiple user setting separately.

I.4.1.1 Single user setting

In a single user and multiple channel setting the transmit powers p k , k ∈ C, are the un- known variables. In this setting the downlink and uplink formulations are identical. With the objective to maximize the total user throughput, this problem can be formulated as the following convex optimization problem

maximize

p

�

k∈C

log � 1 + g k

σ ² p k

� (I.5a)

subject to �

k∈C

p k ≤ P ^max , (I.5b)

p k ≥ 0, k ∈ C, (I.5c)

where g k denotes the effective gain between the BS and MS on channel k, σ ² denotes the thermal noise and P ^max denotes the upper bound of the transmission powers. Problem (I.5) also gives the solution of the max-min and maximize proportional fairness problem.

To obtain a solution of (I.5) we may assume, without loss of generality, that (g k /σ ² ) >

0 , k ∈ C, since otherwise the corresponding transmission power is equal to zero. Further, the constraint (I.5b) has to be fulfilled with equality in the optimum. Now we let the non-negative Lagrange multipliers of (I.5b) and (I.5c) be denoted by µ and λ k , k ∈ C, respectively. The first-order necessary optimality conditions of (I.5) imply that

λ k = µ − (g k /σ ² )

1 + (g k /σ ² )p k ≥ 0, k ∈ C.

If p k = 0 , then we have µ ≥ (g k /σ ² ). With p k > 0 we have that λ k = 0, which

gives µ < (g k /σ ² ). Hence, the well known water-filling algorithm, see, e.g., [8], solves

this problem (I.5), by locating a water level µ that also fulfills the constraint (I.5b) with

equality.

10 I NTRODUCTION

The optimization problem (I.5) remains convex if the objective is modified to instead minimize the total power consumption subject to fulfilling a required throughput.

I.4.1.2 Multiple user setting

In the multiple user and multiple channel setting both the transmission powers and the binary channel allocation can be considered. First we consider only power variables, sec- ondly we present the problem with only channel variables, and finally the problem with both as unknown variables is given.

First, let us consider the setting with continuous power variables for a fixed channel allocation. The downlink problem formulation, using one of the utilities in (I.3), is given by

maximize

p either min

m∈M R m or �

m ∈M

R m (I.6a)

subject to R m = �

k ∈C

log � 1 + g k

σ ² p k

� , m ∈ M, (I.6b)

�

k ∈C

p k ≤ P ^max , (I.6c)

p k ≥ 0, k ∈ C, (I.6d)

where the set C m contains the channels that have been assigned to mobile user m. In downlink σ ² _m denotes the thermal noise of the receiving mobile user m, while the thermal noise σ _m ² = σ ² , m ∈ M, in uplink. The corresponding uplink formulation is obtained by replacing constraint (I.6c) with the constraints

�

k ∈C

p k ≤ P m ^max , m ∈ M,

where P m ^max denotes the upper bound of the transmission powers by mobile user m. Both the downlink and the uplink formulation are convex optimization problems. Once again, the water-filling algorithm solves the max-sum problem, (I.3b), in (I.6). In uplink, the problem decomposes into a separate optimization problem per mobile user m, where each of them can be solved as (I.5).

Secondly, let us consider the setting with binary channel allocation variables together with a given fixed power allocation. The problem formulation is identical in downlink and uplink, and is given by

maximize

x either (I.3a) or (I.3b) (I.7a)

subject to R m = �

k ∈C

η mk x mk , m ∈ M, (I.7b)

�

m ∈M

x mk ≤ 1, k ∈ C, (I.7c)

x mk ∈ {0, 1}, m ∈ M, k ∈ C, (I.7d)

O PTIMIZATION OF WIRELESS COMMUNICATION NETWORKS 11

where η mk denotes the Shannon capacity that is obtained if mobile user m is involved in the transmission on channel k ∈ C.

In terms of complexity, a general binary linear programming problem is known to be NP-hard, see, e.g., [14]. This tells us that there exists no known efficient, i.e., polynomial time, algorithm that solves an arbitrary instance of this problem. As all variables of (I.7) are binary, there can be at most 2 ⁿ solutions where n denotes the number of binary variables.

In theory we can always solve this intractable problem by an exhaustive search where all 2 ⁿ combinations of our binary variables are generated, then verify which candidates that are feasible, and finally let our solution be given by one of the feasible binary allocations that gives the largest objective value. In practice this approach only works on small problems as the number of combinations to investigate grows rapidly with the number of variables.

There are various approaches that solve a general binary linear programming problem, see, e.g., [27].

In [30] two heuristic approaches that strive for a good solution to the problem (I.7), with objective (I.3a), is considered. One approach relaxes the binary variables, while the other applies a greedy approach that iteratively allocates a channel to the user with currently worst throughput. The max-sum problem (I.7), with the objective function (I.3b), can be solved to global optimality by a greedy approach where the mobile m with largest η mk is allocated for each channel k ∈ C [18].

Finally, let us consider the setting where both the continuous powers and the binary channel allocation are variables. The problem formulation in downlink is given by

maximize

x,p either (I.3a) or (I.3b) (I.8a)

subject to R m = �

k∈C

x mk log

�

1 + g mk

σ _m ² p mk

�

, m ∈ M, (I.8b)

�

m∈M

x mk ≤ 1, k ∈ C, (I.8c)

x mk ∈ {0, 1}, m ∈ M, k ∈ C, (I.8d)

�

m ∈M,k∈C

p mk ≤ P ^max , (I.8e)

p mk ≥ 0, m ∈ M, k ∈ C, (I.8f)

where g mk denotes the effective gain between the BS and MS m on channel k. The cor- responding uplink formulation is obtained by replacing the constraint (I.8e) with the con-

straints �

k ∈C

p mk ≤ P m ^max , m ∈ M.

In [30] a heuristic algorithm is proposed to obtain a good solution to the problem of

maximizing the worst off user throughput, (I.3a), in (I.8). The heuristic first allocates an

equal amount of power to each channel. Then the best available channel in terms of path

gain is allocated to the mobile user with currently worst (smallest) total throughput, until

all channels have been allocated. Alternatively, a convex optimization problem is obtained

12 I NTRODUCTION

if the binary variables are relaxed [30]. Instead of merely using fixed powers, one can use some power allocation algorithm to update the powers, see, e.g, [26, 33].

To maximize the total throughput, (I.3b), of problem (I.8) in downlink is solved to global optimality by first assigning each channel k greedily to the user m with largest gain (g mk /σ ² _m ), and thereafter solving the remaining convex power allocation optimization problem by, e.g, the water-filling algorithm, see [18]. In uplink on the other hand, we have an additional combinatorial aspect to take into consideration, as we also need to decide which transmitting mobile to use on each channel. The uplink max-sum problem (I.8) is NP-hard, already with only two mobile users [16]. To obtain a good solution [20] propose a heuristic that uses a similar approach as [18] in downlink.

Examples of other related work are, e.g., [25, 40] where the binary allocation is relaxed which introduces time-sharing over the channels.

I.4.2 Multicell setting

In the multicell setting part we consider multiple BSs that introduce inter-cell interference, whenever a channel is used in different cells simultaneously. Hence, the performance of a mobile user will no longer only depend on their own power allocation, but also on the power allocation of the other users which are using the same channel. We choose to consider the single channel setting and the multiple channel setting separately.

I.4.2.1 Single channel setting

In a multicell and single channel setting the transmit powers p i , i ∈ S, are the unknown variables. In this setting the downlink and uplink formulations are identical, as a number of already connected BS-MS pairs is considered. We denote each BS-MS pair by index i, i.e., i ∈ S = D. The optimization problem is formulated as

maximize

p either min

i∈S R i , �

i∈S

R i or �

i∈S

log(R i ) (I.9a)

subject to R i = log

�

1 + g ii p i

σ ² _i + �

j ∈{S\i} g ji p j

�

, i ∈ S, (I.9b)

0 ≤ p ⁱ ≤ P ^max , i ∈ S, (I.9c)

where R i denotes the throughput over the communication link from i ∈ S to i ∈ D, i.e., BS-MS pair i. The effective gain between the source of BS-MS pair j and the destination of BS-MS pair i is given by g ji , while σ i ² denotes the thermal noise at the destination of BS-MS pair i.

The max-min, (I.3a), problem (I.9) can be solved to global optimality, see, e.g., [24, 37]. To maximize the total user throughput, (I.3b), in (I.9) is a NP-hard problem [24].

In [29] an algorithm that converges towards a global optimal solution is proposed. To maximize the proportional fairness, (I.4), in (I.9) is a convex optimization problem [24].

For systems where users do interfere with each other, power control can be applied to

adjust the transmit powers of all users with the aim of obtaining the required performance

O PTIMIZATION OF WIRELESS COMMUNICATION NETWORKS 13

of each user. In the early 1990’s, the following linear programming problem was studied in terms of power control

minimize

p

�

i∈S

p i (I.10a)

subject to γ i = g ii p i

σ _i ² + �

j ∈{S\i} g ji p j ≥ γ i ^tar , i ∈ S, (I.10b)

p i ≥ 0, i ∈ S. (I.10c)

The required threshold, over the communication link of BS-MS pair i is given by the fixed SINR target γ _i ^tar , while γ i denotes the SINR, see (I.1), with indices j and k omitted. The conditions for existence of a feasible solution together with an explicit expression of that solution was given in [41]. A decentralized approach that solves this problem is the famous iterative Distributed Power Control (DPC) algorithm, by Foschini and Miljanic [13], which updates each power i as

p i [n + 1] = (γ _i ^tar /γ i [n]) p i [n], i ∈ S,

at the nth iteration. The convergence of the DPC algorithm in a more general framework, where problem (I.10) can be solved in a distributed manner with maximum power con- straints, is given in [37].

I.4.2.2 Multiple channel setting

The complexity of a simplified class of multicell and multiple channel problems are studied in [24]. The simplification consists of focusing on the study with a number of transmitter- receiver pairs, i.e., to consider a setting with already connected BS-MS pairs. This setting gives the following power allocation optimization

maximize

p either min

i∈S R i , �

i∈S

R i or �

i∈S

log(R i ) (I.11a)

subject to R i = �

k ∈C

log

�

1 + g ii p ik

σ _i ² + �

j ∈{S\i} g ji p jk

�

, i ∈ S, (I.11b)

�

k ∈C

p ik ≤ P i ^max , i ∈ S, (I.11c)

p ik ≥ 0, i ∈ S, k ∈ C. (I.11d)

This multiple channel BS-MS pair problem (I.11) is an extension of the single channel BS-MS pair problem (I.9). However, the power allocation problem (I.11) only considers a subset of all allocations that can be obtained from the general cell and channel allocation approach.

To maximize the minimum user throughput over a number of source-destination pairs

in (I.11), with at least three channels, is known to be NP-hard. The max-sum problem (I.11)

14 I NTRODUCTION

remains NP-hard in the multiple channel setting, compare with the max-sum problem (I.9).

To maximize the proportional fairness in (I.11), with at least three channels, is NP-hard, [24].

In the general multicell and multiple channel problems we no longer require the BS-MS pair structure, but instead allow any configuration that fulfills the technological constraints.

To minimize the total power consumption subject to Shannon capacity rate constraints for each user in the multicell and multiple channel setting is proved to be NP-hard in [6].

Good solutions to the multicell problem are sought via heuristics. One approach is to relax the problem formulation, which makes it possible to decompose the problem into one subproblem per cell. The solution of the single cell problem is obtained by solving a network flow problem,

Two other multiple cell and channel settings are the following. In [21] a multicell game approach is considered, with both channel and power allocation. Each BS strives to maximize its own utility, given the interference from all the other cells. The problem turns into a power allocation problem that is approached by an iterative algorithm. In [22], a channel allocation is sought in downlink, given cell and power allocation. Iteratively the channel with highest throughput improvement is allocated to the corresponding BS.

Thereafter, each BS greedily allocates its channels to the best user in terms of throughput.

The complexity of the max-min multicell and multiple channel problem is studied in Paper A. In Paper B we perform an exploratory heuristic single channel study, where con- vex combinations of the max-min utility and the max-sum utility serve as objective func- tions. In Paper C the multiple channel problem is considered, where proposed heuristic solution approaches are solved both in the max-min and max-sum multicell setting. In paper D we extend the model to also consider fixed relay stations, see section I.4.3.

I.4.3 Relay setting

Here, in part three, we discuss the concept of fixed relay stations.

To extend our previous model into the setting with RSs some additional variables and constraints are needed. The binary variable x rk is introduced to describe the channel al- location between a BS and one of its RSs r on channel k, while the binary variable x rmk

describes between the RS r and MS m on channel k. A RS is only allowed to transmit in the second time step, with its continuous power allocation p rk . These channel and power allocation variables have the corresponding limitations introduced in constraints as the pre- vious problem formulations. With this relay model it is possible to use various paths on various channels between a source and a destination. This makes it possible to adapt the resource allocation to current diversity in terms of maximizing system performance. In [35] it has been shown, for an ad hoc wireless network, that it is not optimal to maintain a single path between a source and its destination in terms of total throughput.

Our focus is on the Decode and Forward (DF) relay approach, in which the relay first

decodes the message and then transmits the re-encoded message. DF was originally pro-

posed in [9]. The message is decoded to remove errors (error correction). If an error is

detected but can not be corrected, then the RS asks for a retransmission. An alternative

approach is to use Amplify and Forward (AF) RSs. An AF relay station amplifies the

O PTIMIZATION OF WIRELESS COMMUNICATION NETWORKS 15

received signal and then transmits the amplified signal, which means that received interfer- ence also is reinforced. "However, amplify-and-forward does not work well in a wireless setting, since each wireless link is unreliable and often introduces errors" [15].

So far in the literature, the inter-cell interference has mostly been ignored, either by considering a single cell scenario or by merely keeping it fixed. For example, the single cell setting is considered by [7, 34] where the total system throughput is maximized subject to a guaranteed throughput to each user, and by [23, 39] where the worst off user throughput is maximized. Potential relay profits have been observed.

In [17] the impact of inter-cell interference is taken into consideration in a multicell AF relay setting where the total system throughput is maximized.

In Paper D we extend the max-min multicell and multiple channel model, previously studied in Paper A to Paper C, to also consider fixed DF relay stations. The complexity is inherited, and therefore heuristic solution approaches are studied.

I.5 Main contributions and limitations I.5.1 Main contributions

In Paper A we formulate an overall joint cell, channel and power allocation optimization problem that maximizes the user worst off. We show that the decision problem is NP-hard and that the optimization problem is not approximable unless P is equal to NP. Further, we show that for a feasible binary cell and channel allocation, the remaining continuous power allocation optimization problem is not approximable unless P is equal to NP.

In Paper B and Paper C we perform numerical studies on this problem and the corre- sponding problem of maximizing the total system throughput. We perform an exhaustive search over all feasible cell allocations, and another that runs over feasible channel allo- cations. These exhaustive search methods are compared to less computationally expensive heuristic approaches, presenting potential earnings to strive for while one also can see that some of the proposed heuristics have a quite good performance.

In Paper D we derive a model by incorporating fixed relay stations into the overall joint cell, channel and power allocation problem that maximizes the user worst off. We show that this problem inherits the complexity from Paper A. Further, we propose a heuristic channel allocation approach that shows good performance in the numerical simulations on the relay setting.

I.5.2 Main limitations

The main part of the studied approaches within this thesis are not realistic to implement

and run in a real system, due to rapid channel variations over time. In terms of real system

implementations it is not desirable to assume perfect information of all users (AP), but

instead to have distributed solution approaches. However, the main focus of our study

has been to find existing potential gains within the system. In future work less complex

approaches can strive for these achievable potential gains.

16 I NTRODUCTION

I.6 Summary of the appended papers

This section contains a brief summary of each of the four appended papers. In each paper the specific problem is described and a corresponding mathematical optimization problem formulation is presented. In Paper A the main focus is to study the complexity and other mathematical properties of the considered problem. The following papers, Paper B to Paper D, are more applied and consider solution strategies that are used to obtain computational results, given realistic test problems.

Paper A: On the Complexity of Maximizing the Minimum Shannon Capacity by Joint Cell, Channel and Power Allocation in Wireless Communication Networks

In Paper A, the overall joint cell, channel and power allocation optimization problem with the objective to maximize the worst off user is formulated. Also, the power allocation problem is studied under feasible cell and channel allocation. Thereafter, the complexity of these problems are studied, together with some additional properties regarding the power allocation problem.

It is shown that the overall decision problem is NP-hard and that the overall optimiza- tion problem is not approximable unless P is equal to NP.

The power allocation optimization problem, that only has continuous variables, is not approximable unless P is equal to NP. Further, to maximize the minimum user throughput in the single channel setting is not a convex optimization problem, though, any KKT point is shown to give a global optimum. In the two channel setting, we present an example that has two different KKT points [10].

Parts of Paper A have been published in IEEE International Workshop on Quality of Service, see [11].

Paper B: An Optimization Approach to Joint Cell and Power Allocation in Wireless Communication Networks

Here, we study the multicell single channel setting, in terms of maximizing the total throughput in the system, the user with worst off throughput, or a convex combination of these two extremes. The cell allocation is based on the effective gain, and is obtained either by a greedy approach or by optimization approaches. Within this paper the channel allocation is given by the cell allocation, as the number of channels is one. The power allocation is obtained by solving the continuous nonlinear optimization problem that arise, given a feasible cell allocation.

In a narrow region of the convex combination of the two extreme objectives, it is in- deed possible to tune the trade-off between fairness and throughput. By maximizing the worst off user, all users have the same throughput, though in a rather poor SINR domain.

Maximizing the total throughput dramatically increases the the total system throughput,

but it also allocates no power to some of the users.