Dynamic Resource Allocation Using Stochastic

Dynamic Resource Allocation Using Stochastic Optimization in Wireless Communications

LI, Weiliang

A Thesis Submitted in Partial Fulfilment of the Requirements for the Degree of Doctor of Philosophy in Information Engineering

The Chinese University of Hong Kong April 2012

This work is dedicated to my parents.

Abstract The growing demand of ubiquitous wireless services has prompted the efficient utilization of scarce radio resources. Over the years, optimization techniques have been widely employed to design optimal resource allocation schemes to achieve performance improvement. Most work in this area assumes that the system parameters defining the optimization problem are precisely known. In practical systems, however, these parameters are often time varying and random. Ignoring the parameter uncertainties would easily lead to suboptimality or even infeasible solutions that violate system operation constraints. This thesis presents a stochastic optimization framework for the dynamic resource allocation in wireless communications. In particular, practice-relevant problem formulations are proposed to capture the stochastic nature of the uncertain system parameters, and efficient algorithms are developed to obtain the optimal allocation decisions.

The proposed framework has been

successfully applied in three promising wireless systems: adaptive orthogonal frequency division multiple access (OFDMA) systems, multiple-input and multiple-output (MIMO) antenna systems, and location-aware networks. Each application contains practice-relevant challenges, where the conventional designs using deterministic optimization fail to provide satisfactory quality of service (QoS). The results demonstrate that the dynamic resource allocation i

using stochastic optimization achieves more robust QoS performance and remarkably enhances the system practicality. In adaptive OFDMA systems, a slow adaptation scheme is proposed for optimal subcarrier allocation. The proposed scheme updates the resource allocation decisions on a much slower timescale than that of channel fluctuation, which drastically reduces the computational complexity and control signaling overhead. The problems are formulated into several stochastic programs based on different application scenarios. An efficient algorithm is developed for solving the chance constrained subcarrier allocation problem. In MIMO antenna systems, an antenna-and-power allocation scheme is proposed to enable the use of multiple antennas to support multiple radios co-operating on the same mobile device. The proposed scheme maximizes the long-term system throughput while satisfying the short-term data rate requirement of each radio transmission with occasional outage. The results show that both system throughput and success probability of QoS satisfaction are improved, and the optimal antenna allocation contributes to a larger portion of throughput increase comparing with the optimal power allocation. In location-aware networks, robust power allocation schemes are proposed to combat the uncertainties in network parameters including user positions and channel conditions. A novel robust optimization method is developed to obtain the optimal power allocation, which improves both localization accuracy and network energy efficiency. The results show that the robust schemes remarkably outperform both non-robust power allocation and uniform allocation. The goal of this thesis is to bridge the gap between the current designs under the deterministic optimization framework and their practical relevance. ii

Given the fact that many wireless system parameters are stochastic in nature, the proposed resource allocation methods using stochastic optimization are expected to find further applications in wireless communications.

iii

Acknowledgement I gratefully acknowledge all the people who have contributed to the success of this thesis. First and foremost, I would like to express my heartfelt gratitude to my doctoral supervisor, Professor Ying Jun (Angela) Zhang, for her constant encouragement and support throughout my Ph.D. pursuit. It is my great fortune of being a student of her. Her insightful guidance and invaluable suggestions have enlightened my research journey. In the past years, she has taught me how to conduct research with passions and patience, guided me to identify my thesis topic at the early stage, provided me with the freedom to pursue my research interests, and always offered me helps whenever I needed. I am deeply indebted to her. I would also like to sincerely thank Professor Moe Win for serving as my doctoral co-supervisor, and hosting me at Massachusetts Institute of Technology (MIT) during 2009 to 2011. It has been a wonderful time of working with Professor Win. His keen insights and invaluable suggestions have considerably broaden my horizon and tremendously helped through my research process. I was deeply impressed by his unique view and approach of performing worldclass researches. I have been benefitted a lot throughout countless enjoyable discussions with him. His rigorous attitude in research, tireless energy, and iv

great leadership will set up a role model for my future career. I would like to thank Professor Anthony Man-Cho So of The Chinese University of Hong Kong (CUHK) for introducing me to the field of stochastic optimization, and inspiring me with useful advices through our collaboration. His optimization course was one of the best courses I have ever taken. I would also like to thank Professor Marco Chiani of University of Bologna, and Professor Hyundong Shin of Kyung Hee University for insightful discussions on MIMO technology, which motivated in part the research done in this thesis. I would like to thank Professor Minghua Chen, Jianwei Huang, and WingCheong Lau of CUHK and Professor Roger Shu-Kwan Cheng of Hong Kong University of Science and Technology for serving on my thesis committee. I would also like to thank my research collaborators Dr. George Chrisikos of Qualcomm, Dr. Wesley Gifford of IBM, and in particular, Yuan Shen of MIT. They have inspired me a lot with their broad knowledge and experiences, and offered me countless time for valuable discussions. I especially want to thank my best friend Yuan, who has not only provided me with great helps in research, but also taken care of me like my brother during my visit at MIT. I would always remember the wonderful time we have spent together and the innumerable interesting discussions we had in the same office. I am grateful to all my friends at MIT, who have shared with me many valuable and enjoyable discussions, technical or nontechnical, and plenty of funs during my stay in Boston. Special thanks go to Dr. Chung Chan, Watcharapan Suwansantisuk, Dr. Santiago Mazuelas, Dr. Jowoon Chong, Dr. Jemin Lee, Tianheng Wang, Ulric Ferner, and Mengdi Wang. I would also like to thank my CUHK colleagues and friends, Dr. Liping Qian, Leiyi Yao, Suzhi Bi, Wanrong Tang, Lei Zhan, Ziyu Shao, Shuqin Li, Lingjie Duan, Zhengjia v

Fu, Kenan Zhou, and Weijie Wu, who have always accompanied with me and shared the ups and the downs in the past four years. Finally, I owe my deepest appreciation to my dear parents Minghuang Li and Huili Zhao for their steady encouragement and endless love. Any of my success in this academic journey would not be possible without their unwavering supports. My Mum and I have been through our toughest time when my Dad unfortunately passed away in 2008. I cannot express my full gratitude to my Mum who has tried her best to comfort me and persistently encouraged me to continue my journey. Your love has always been my momentum to pursue the dreams. This thesis is dedicated to you.

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Contents Abstract

i

Acknowledgement

iv

Contents

vii

List of Figures

xi

List of Tables

xv

List of Acronyms

xvi

List of Notations

xix

1 Introduction

1

1.1 Resource Allocation in Wireless Communications . . . . . . . .

2

1.2 Stochastic Optimization and Its Applications . . . . . . . . . . .

4

1.2.1

Robust Optimization . . . . . . . . . . . . . . . . . . . .

5

1.2.2

Chance Constrained Optimization . . . . . . . . . . . . .

8

1.3 Motivation and Research Focus . . . . . . . . . . . . . . . . . . 10 1.3.1

Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3.2

OFDM and OFDMA Systems . . . . . . . . . . . . . . . 14

vii

1.3.3

MIMO Antenna Systems . . . . . . . . . . . . . . . . . . 16

1.3.4

Location-Aware Networks . . . . . . . . . . . . . . . . . 18

1.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.5 Organization

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 Slow Subcarrier Allocation in Adaptive OFDMA Systems

25

2.1 System and Channel Model . . . . . . . . . . . . . . . . . . . . 29 2.1.1

Channel Model . . . . . . . . . . . . . . . . . . . . . . . 29

2.1.2

Slow Adaptive OFDMA . . . . . . . . . . . . . . . . . . 30

2.2 Slow Adaptive OFDMA with Average Rate Constraints for Elastic Traffics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2.1

Problem Formulation . . . . . . . . . . . . . . . . . . . . 33

2.2.2

Computation of Expected Average Data Rate . . . . . . 34

2.2.3

Numerical Results

. . . . . . . . . . . . . . . . . . . . . 37

2.3 Slow Adaptive OFDMA with Average Rate Constraints for Inelastic Traffics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.3.1


2.3.2

Numerical Results

. . . . . . . . . . . . . . . . . . . . . 43

2.4 Slow Adaptive OFDMA with Probabilistic Rate Constraints . . 46 2.4.1


2.4.2

Safe Tractable Constraints . . . . . . . . . . . . . . . . . 48

2.4.3

Algorithm Design . . . . . . . . . . . . . . . . . . . . . . 51

2.4.4

Problem Size Reduction . . . . . . . . . . . . . . . . . . 59

2.4.5

Numerical Results

. . . . . . . . . . . . . . . . . . . . . 61

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

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3 Dynamic Antenna-and-Power Allocation in Composite Radio MIMO Networks

72

3.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.1.1

Composite Radio System . . . . . . . . . . . . . . . . . . 76

3.1.2

Channel Model . . . . . . . . . . . . . . . . . . . . . . . 77

3.1.3

Dynamic Antenna-and-Power Allocation . . . . . . . . . 78

3.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . 80 3.2.1

MIMO Channel Capacity

. . . . . . . . . . . . . . . . . 80

3.2.2

Chance Constrained Formulation . . . . . . . . . . . . . 81

3.2.3

Safe Tractable Formulation

. . . . . . . . . . . . . . . . 82

3.3 Search for Feasible Solutions . . . . . . . . . . . . . . . . . . . . 85 3.3.1

Algorithm Design . . . . . . . . . . . . . . . . . . . . . . 87

3.4 Approach to Optimal Solution . . . . . . . . . . . . . . . . . . . 89 3.4.1

Cutting-Plane-Based Algorithm . . . . . . . . . . . . . . 91

3.4.2

Optimal Antenna-and-Power Allocation . . . . . . . . . . 95

3.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4 Robust Power Allocation for Energy-Efficient Location-Aware Networks

107

4.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.1.1

Network Settings . . . . . . . . . . . . . . . . . . . . . . 110

4.1.2

Position Error Bound . . . . . . . . . . . . . . . . . . . . 111

4.1.3

Directional Decoupling of SPEB . . . . . . . . . . . . . . 113

4.2 Optimal Power Allocation via Conic Programming . . . . . . . . 115 4.2.1

Problem Formulation Based on SPEB . . . . . . . . . . . 115

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4.2.2

Problem Formulation Based on mDPEB . . . . . . . . . 117

4.2.3

Formulations with QoS Guarantee . . . . . . . . . . . . . 120

4.3 Robust Power Allocation under Imperfect Network Topology Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.3.1

Robust Counterpart of SPEB Minimization . . . . . . . 123

4.3.2

Robust Counterpart of mDPEB Minimization . . . . . . 131

4.4 Efficient Robust Algorithm Using Distributed Computations . . 132 4.4.1

Algorithm for SPEB Minimization

. . . . . . . . . . . . 132

4.4.2

Algorithm for mDPEB Minimization . . . . . . . . . . . 136

4.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . 137 4.5.1

Power Allocation with Perfect Network Topology Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

4.5.2

Robust Power Allocation with Imperfect Network Topology Parameters . . . . . . . . . . . . . . . . . . . . . . . 140

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 5 Conclusions and Future Work

145

5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 5.1.1

Slow Adaptive OFDMA Systems . . . . . . . . . . . . . 146

5.1.2

Composite Radio MIMO Networks . . . . . . . . . . . . 147

5.1.3

Energy-Efficient Location-Aware Networks . . . . . . . . 148

5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 A Bernstein Approximation Theorem

153

B Ergodic MIMO Capacity and Moment Generating Function 155 Bibliography

157 x

List of Figures 1.1 Organization chart of the thesis: applications of stochastic optimization in wireless communications. . . . . . . . . . . . . . . 13 1.2 Multiple user diversity enables the subcarrier allocation in adaptive OFDMA system. . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3 System structure of a point-to-point MIMO system. . . . . . . . 17 1.4 Location-aware networks: the anchors (A, B, C, and D) transmit wireless signals to the two agents. . . . . . . . . . . . . . . . 19 2.1 Adaptation timescales of fast and slow adaptive OFDMA system (SCA = SubCarrier Allocation). . . . . . . . . . . . . . . . 31 2.2 Performance comparison between systems using the actual avern (t ) o age rate r¯k,n and expected average rate E r¯k,n rk,n0 in problem

P1slow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.3 Spectral efficiency comparison among fast adaptation, slow adaptation with and without stochastic programming. (Signaling overhead is considered here.) . . . . . . . . . . . . . . . . . . . . 39 2.4 Outage probability of slow adaptive OFDMA versus the deviation size ρ when Fd = 2 and m = 40. . . . . . . . . . . . . . . . 44

xi

2.5 Spectral efficiency of safe slow adaptive OFDMA with different ρ0 . (We fixed the actual maximum deviation ρ = 6.) . . . . . . . 45 fslow . . . . . . 54 2.6 Flow chart of the algorithm for solving Problem P 3

2.7 Trace of the difference of objective value ¯bi between adjacent

iterations (ǫk = 0.2). . . . . . . . . . . . . . . . . . . . . . . . . 62 2.8 Number of iterations for convergence of all the feasible windows (ǫk = 0.2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 2.9 Number of iterations for feasibility check of all the windows (ǫk = 0.2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.10 Comparison of system spectral efficiency between fast adaptive OFDMA and slow adaptive OFDMA. . . . . . . . . . . . . . . . 66 2.11 Outage probability of the 4 users over 61 independent feasible windows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.12 Spectral efficiency versus tolerance parameter ǫk . Calculated from the average overall system throughput on one window, where the long-term average channel gain σk of the 4 users are −65.11dB, −56.28dB, −68.14dB and −81.96dB, respectively. . . 68 2.13 Comparison of outage probability of 4 users with and without frequency correlations in channel model. . . . . . . . . . . . . . 69 3.1 A general model of the composite radios system. For simple illustration, we plot the receive terminals containing all types of radio. In general, terminal 1 to M contains possibly different subset of radios. . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.2 The radio resource allocation in composite radios system of which the general model is given in Fig. 3.1 (Tn = Tone). . . . 79

xii

3.3 High-level flow chart of Cutting Plane Method. . . . . . . . . . 92 3.4 System success probability resulted by optimal allocation and uniform allocation, where each receive terminal has four antennas. 99 3.5 System success probability resulted by optimal allocation and uniform allocation, where each receive terminal has eight antennas.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

3.6 Trace of β i in each iteration (δ = 10−3). . . . . . . . . . . . . . . 101 3.7 CDF of number of iterations for the convergence of Algorithm 3. 102 3.8 A time-series illustration of the optimal antenna-and-power allocation (third and fourth subplot) and the corresponding rate outage probabilities (fifth subplot) over 100 problem realizations. The link distance and long-term channel gain are given in the first two subplots. . . . . . . . . . . . . . . . . . . . . . . 103 3.9 System throughput (spectral efficiency) resulted by optimal antennaand-power allocation, optimal power allocation, and uniform allocation, where each receive terminal has four antennas. . . . . . 104 3.10 System throughput (spectral efficiency) resulted by optimal antennaand-power allocation, optimal power allocation, and uniform allocation, where each receive terminal has eight antennas. . . . . 105 4.1 Geometrical interpretation of the EFIM of agent k. . . . . . . . 114 4.2 Geometrical illustration of the proof of Proposition 9(a) where agent is inside the square region. We choose two anchors i and i′ in the shaded region. . . . . . . . . . . . . . . . . . . . . . . . 128

xiii

4.3 The topology of the location-aware network consisting ten anchors (red circle) and 1 agents (blue dot), where the anchors are uniformly distributed in the square region. . . . . . . . . . . . . 138 4.4 Comparison of the SPEB in single-agent network resulted by SPEB-minimization allocation, mDPEB-minimization allocation, and uniform allocation. . . . . . . . . . . . . . . . . . . . . . . . 139 4.5 The topology of the location-aware network consisting ten anchors (red circle) and 8 agents (blue dot), where the agents are uniformly distributed in the square region. . . . . . . . . . . . . 140 4.6 Comparison of the average SPEB in multiple-agent network (Nb = 10) resulted by SPEB-minimization allocation, mDPEBminimization allocation, and uniform allocation.

Both one-

stage and two-stage optimization are considered. . . . . . . . . . 141 4.7 The actual SPEB with respect to number of anchors, resulted by robust, non-robust schemes and uniform allocation with imperfect network topology parameters (εd = 0.2, εφ = 0.2).

. . . 142

4.8 The actual SPEB with respect to the error size on network topology parameters (εd and εφ are set to be equal), resulted by robust, non-robust schemes and uniform allocation. . . . . . 143

xiv

List of Tables 3.1 Notations of the sets of radio type and terminals. . . . . . . . . 77 3.2 Simulation parameters of channel model and system . . . . . . . 98

xv

List of Acronyms AC

analytical center

AOA

angle of arrival

AWGN

additive white Gaussian noise

BER

bit error rate

BS

base station

CCP

chance constrained program

CDF

cumulant density function

CSI

channel state information

DPEB

directional position error bound

DVB

digital video broadcasting

EFIM

equivalent Fisher information matrix

GPS

global positioning system

i.i.d.

independent and identically distributed

ISI

inter-symbol interference xvi

LP

linear program

mDPEB

maximum directional position error bound

MIMO

multiple-input and multiple-output

OFDM

orthogonal frequency division multiplexing

OFDMA

orthogonal frequency division multiple access

PDF

probability density function

QoS

quality of service

RF

radio frequency

RII

ranging information intensity

RSS

received signal strength

SCA

subcarrier allocation

SDP

semidefinite program

SDR

software defined radio

SNR

signal-to-noise ratio

SINR

signal-to-inteference-plus-noise ratio

SOCP

second-order cone program

SPEB

squared position error bound

STC

safe tractable constraint

xvii

STCP

safe tractable constrained program

TOA

time of arrival

TDOA

time difference of arrival

TOF

time-of-flight

UWB

ultra-wideband

xviii

List of Notations E{ · }

Expectation

Pr{·}

Probability

(·)T

Transpose

(·)†

Conjugate transpose

(·)−1

Matrix inverse

tr(·)

Trace of a matrix

det(·)

Determinant of a matrix

[A]i,j

The (i, j)th element of matrix A

{ai,j }i,j=1,··· ,N

N × N matrix with entry ai,j , i, j = 1, · · · , N

k·k

Euclidean norm

dim(·)

Dimension of a vector

1K

K-dimensional vector of ones

IK

K × K identity matrix

R

The set of real numbers

Z

The set of integer numbers

C

The set of complex numbers

|K|

The cardinality of set K

xix

Chapter 1 Introduction Future wireless systems will face a growing demand for broadband and multimedia services. In the past decade, we have witnessed a tremendous growth in wireless data traffic. According to the recent statistics from Cicso [1], globe mobile data traffic grew about 2.5-fold every year during 2008 to 2011, and the number of mobile-connected devices, including smartphone, laptop, etc., is estimated to be over 7.1 billion in 2015, which will be approximately equal to the world’s population at that time. To support the rapid demand growth, high-speed data connection becomes an essential requirement in future mobile networks. Over the years, numerous research efforts have been cast into this actively evolving field, and generated cutting-the-edge technologies to support the growing usage trends. The scarcity of radio resources, such as bandwidth, power, etc., however, has become a major hurdle in the development of high-speed wireless communications. The accommodation of user demands with limited system resources arises as a big challenge, especially when the number of mobile users is large and these users are with various quality of service (QoS) requirements. As such,

1

Chapter 1. Introduction

2

efficient utilization of radio resources is of critical importance for future wireless systems. Moreover, wireless environments, such as, channel conditions, user locations, etc., are usually changing rapidly. To maintain satisfactory QoS, the allocation of resources has to be frequently adapted to the current status of the network, which results in higher complexity in implementation. This thesis is built upon the motivation of balancing the user demands and the scarcity of radio resources. We focus on the design of optimal radio resource allocation schemes which are dynamically adapted to the environmental changes. Moreover, we exploit the stochastic nature in wireless networks to improve the system performance and user QoS satisfaction. In this chapter, we will give a high-level description of this thesis, and introduce the preliminary backgrounds of the systems and methodologies we studied. First, in Sec. 1.1, we review the resource allocation problems in wireless communications. In Sec. 1.2, we introduce the recent advances in stochastic optimization, which will be exploited as a fundamental math tool and play a pivotal role in solving the resource allocation problems in this thesis. In Sec. 1.3, we present the motivation of our work, and describe the three promising wireless communication systems, to which stochastic optimization methodology will be applied as the main body of this thesis. The contributions are summarized in Sec. 1.4, and the organization of this thesis is given in Sec. 1.5.

1.1

Resource Allocation in Wireless Communications

The efficient utilization of radio resources is of critical importance in future wireless systems. In the past decade, there have been a significant advance


3

in the design of wireless systems, ranging from the design of algorithms and protocols in physical-layer and medium-access control (MAC) layer, up to the system-level optimizations in network-layer. In wireless communications, diversities exist in several different dimensions, such as time, frequency, and space. Besides, multiple users also provide diversity. Particularly, different users experience different environments, such as channel conditions, locations, etc., and also have various demands and requirements. These diversities can be exploited by efficiently allocating radio resources so as to significantly enhance the system performances. On the other hand, time variation and randomness are two major properties in wireless systems. First, wireless channel is rapidly changing due to several factors, including the movement of mobile user and environmental objectives, random shadowing effect, and small-scale fading process. Moreover, mobile users are also generating randomness to the system, such as the arrival and departure, the traffic demand, and their locations, etc. If the resources are statically allocated, i.e., the decision is fixed without considering the time variation and randomness issues, it will inevitably result in failures in accommodating user QoS, as well as the waste of system resources. Therefore, it is essential to design dynamic resource allocation schemes to adapt according to the variations in system parameters. Over the years, optimization techniques have played a pivotal role in formulating and solving resource allocation problems, for which deep theories and efficient algorithms are available. In current literature, many resource allocation problem in wireless communications have been cast as optimization problems. From the design and analysis of efficient and fair protocols for network resource allocation (see, e.g., [2–4]) to the construction of efficient detectors in multiuser


4

systems (see, e.g., [5, 6]); from the design of beamformers for multiple-input and multiple-output (MIMO) systems (see, e.g., [7, 8]) to the design of finite impulse response (FIR) filters (see, e.g., [9, 10]), optimization theory has led to substantial improvement in the performance of the systems in question.

1.2

Stochastic Optimization and Its Applications

In the past decades, optimization theory has been widely applied in the design of optimal strategies for resource allocation in wireless communications. Particularly, convex optimization theory [11] has played a central role in formulating and solving problems due to its mature theory and efficient algorithms available. Many previous work focused on proposing convex formulations or convex approximations to the non-convex problems. There has been a great success in solving various challenging problems, such as bandwidth sharing, opportunistic routing and scheduling, power control and interference mitigation, TCP flow control, etc. Most work in this area consider deterministic optimizations for resource allocation. Specifically, it assumes that the data defining the optimization problems, such as channel state information, user positions, etc., are exactly known at the moment of making resource allocation decision. In practical systems, however, these data are often time-varying and random. Ignoring the stochastic nature of the data would often lead to suboptimal or even infeasible solutions that violate system operation constraints. In other words, the deterministic optimization can easily fail to reach its optimal performance or provide QoS guarantee in the realistic scenarios.


5

To bridge the gap between current designs and their realistic applications, it is essential to propose a framework to incorporate the stochastic nature of the data. To tackle such issue, we are motivated to apply the recent advance in stochastic optimization to address some realistic problems in wireless communications. Briefly speaking, in stochastic optimization, we consider the data defining the optimization are with uncertainties. More specifically, we will focus on two types of uncertainties with the knowledage that: 1) the size of uncertainty can be bounded; or, 2) the distribution or the distribution family of uncertainty is known. In the first case, we construct a uncertainty set, and optimize the resource allocation decisions over the constructed uncertainty set. By doing so, the worst-case performance is optimized or guaranteed. Such methodology is usually referred to as robust optimization [12]. In the second case, we could exploit the distributional information to enhance the system performance. A typical way to handle the resource allocation is to applying a probabilistic programming approach, which is referred to chance constrained programming [13]. In the following, we will give an introduction on these two types of stochastic optimization methodology, including the background and related work.

1.2.1

Robust Optimization

In wireless communications, we may only have a rough estimate of system parameters, e.g., channel state information, traffic pattern, and network topology, etc., before making a resource allocation decision. If the uncertainties in the data are not taken into account when carrying out the optimization, the resulting solution will most likely be sub-optimal or even infeasible for the problem at hand. Such an observation necessitates the need for an optimization frame-

6


work that can incorporate data uncertainties and at the same time admit efficient solution procedures. Recently, researchers have considered using robust optimization methodology, which is originally developed in the Operations Research community, to handle uncertainties in the data. Generally speaking, in robust optimization framework, the data defining the optimization problem are assumed to lie in a certain bounded set, called the uncertainty set. The goal is to find a solution that is feasible and preferably near-optimal for all possible realizations of the data in the uncertainty set. Such an approach has been pursued in several recent works, where the main concern is to guarantee a certain level of QoS in the network, regardless of the realizations of uncertain system parameters. The general framework of robust optimization can be formulated into the following problem P:

min f (x, ξ) x

s.t. g(x, ξ) ≤ 0, where x is the decision variable, and ξ is random parameters and lies in a uncertainty set, i.e., ξ ∈ U. Such problem can be equivalently converted into the worst-case optimization problem PR :

min x

s.t.

max f (x, ξ) ξ∈U

max g(x, ξ) ≤ 0, ξ∈U

which is referred to as the robust counterpart of its original formulation In the above formulation, we only consider the worst-case function value instead of the functions under all the realizations of ξ. It has been shown that such formulations can be converted into convex problems under some special structure


7

of the problem [12]. In particular, if P is a linear program (LP), its robust counterpart PR can be formulated as a second-order cone program (SOCP); if P is a conic program, then PR is a semidefinite program (SDP). The robust optimization methodologies in current literature have been successfully solved in numerous practical problems, e.g., [8,14,15]. However, there is no standard method to obtain a tractable formulation of the robust counterpart, when f (x, ξ) and g(x, ξ) are considered as some general functions. Indeed, the worst-case optimizations in PR are often non-convex problems, even when the functions in P are convex in x. In such case, the current methodologies cannot be directly applied, and the optimal solution of robust counterpart will be difficult to be obtained due to its non-convexity. On the other hand, the solution obtained via robust optimization can be overly conservative, since most system resources will be dedicated to provide performance guarantee in the worst case. Moreover, researchers have over the years spent considerable effort to model variations in the channel using probability distributions. Unfortunately, such information is completely ignored in the current robust optimization framework, as one is only interested in the worst-case realizations of the channel. This prevents the distributional information from being exploited to achieved better system performance. Last, the assumption of bounded uncertainty is not suitable to all the random data in practical systems. For instance, Gaussian perturbations are widely used in modeling the uncertain data, however, are excluded from the current framework. More limitations of robust optimization methodology can be referred to [16].

8


1.2.2

Chance Constrained Optimization

In practical systems, many applications do not require system operation constraints to hold all the time, i.e., it can tolerate an occasional outage probability in QoS. Especially in wireless communication, the system operation will be often subject to a small chance of violations due to the deep fading of wireless channels. Imposing a strict requirement on the user QoS, i.e., the constraints have to be satisfied with 100 percent, will easily lead the problem to be infeasible. To preserve a non-empty solution set, one have to lower down the requirement so as to find a best feasible solution, which inevitably results in conservativeness and poor system performance. Instead, a more realistic requirement is to satisfy the constraints with high probability. In general, such problem can be formulated as PC :

min f (x) x

s.t.

Pr{g(x, ξ) ≤ 0} ≥ 1 − ǫ,

(1.1)

where the probability measure Pr{·} is defined on the random variable ξ, and ǫ is a tolerance parameter characterizing the maximum outage probability. In stochastic optimization, the probabilistic constraint in (1.1) is called chance constraint, and the formulation PC is referred to as a chance constrained program (CCP). Indeed, chance constraints can be found in many other wireless system problems, such as wireless multimedia streaming, wireless network routing in the presence of time-varying link reliability, network planning, etc. Despite the practical relevance to real applications, the CCP PC is a computationally intractable problem in general. The main reason is due to the following two difficulties: 1) the convexity of the feasible set defined by (1.1)


9

is difficult to verify. Indeed, such sets are often non-convex except for very few special cases [13, 17]; 2) the computation of the probability in the chance constraint (1.1) is challenging. Even with the simple case, such as when the function g(x, ξ) is bilinear in x and ξ, and each entry in ξ is independent and identically distributed (i.i.d.) random variable with know distribution, it is still unclear how to compute the probability in (1.1) efficiently. The only generic case that avoid both difficulties is when g(x, ξ) follows a Gaussian distribution for each feasible x. Previous work on CCP mainly focused on the derivation of a closed-form expression by using approximations. Most of the work, e.g., [18–20], assumes that ξ is a Gaussian random vector, and g(x, ξ), affine in ξ, is also Gaussian for any given x. One of the few work that does not make such assumption is [21], where the authors approximated the probability function by a product of tail probabilities of chi-square distributed random variables, which are further approximated by some closed form expressions. The approximation, however, is through the use of a union bound, and therefore can be quite loose. Since the close-form expression for computing the probability in chance constraint is usually not available, an alternative to circumvent the intractability of chance constraints is to replace them with tractable approximations, i.e., the constraints are convex, so that optimization problem becomes efficient solvable. A natural way of such tractable approximation is to construct an analytic upper bound G(x) on the violation probability Pr{g(x, ξ) > 0}. Then, any x that satisfies the constraint G(x) ≤ ǫ will also satisfy the chance constraint (1.1). Moreover, such approximated constraint should be efficiently computable, i.e., the function G(x) is convex in x. The key part of such tractable approximation is how to construct the function G(x). In the recent advance of stochastic


10

optimization, Bernstein approximation is proposed as a useful technique to tackle the CCP [17], and its details will later be discussed in this thesis.

1.3

Motivation and Research Focus

This thesis aims at building a mathematically rigorous and practically relevant framework for dynamic resource allocation in wireless communications. To achieve such goal, we apply the advanced stochastic optimization methodologies in the design of wireless systems. By capturing the stochastic nature of various system parameters, the new design is expected to be more robust against the data uncertainties, and hence, achieve more desirable QoS performance. In this section, we first explain the motivation and philosophy behind our new design. Following upon that, we give a preliminary introduction of the three wireless systems that will be investigated in this thesis.

1.3.1

Motivation

The optimization theory has been widely adopted in the design and analysis of wireless communications and networking systems, and achieved significant performance improvements due to its capability of providing the most efficient utilization of system resources. In particular, deterministic optimization techniques have quickly developed during the past few decades, with numerous deep theories and efficient algorithms being created. Such technique becomes one of the most popular math tools in addressing resource allocation problems in wireless communications. However, there are several critical issues in the current framework using deterministic optimization: • System parameters with uncertainties: In most of current work apply-


11

ing deterministic optimization, it assumes that the system parameters, i.e., the data defining the optimization problems, are precisely known or static. However, such assumption is in general not true in realistic scenarios. In practical systems, these parameters can be obtained by estimation, and hence, often subject to certain errors. It could also be the case that the parameters are random in nature, i.e., they are related to some random quantities such as channel conditions. In any case, the “optimal” solution of deterministic optimization will most likely be suboptimal. Indeed, the resulting solutions could even generate very poor performance especially when the actual parameters deviates far away from its assumed value. To tackle the problem, several critical questions arise such as: what is the best resource allocation decision when uncertainties exist in system parameters? How does the data uncertainties, e.g., estimation errors, randomness, affects the system performance? How to capture the stochastic nature of the parameters to formulate the optimization problems? • QoS with practical relevance: In the optimization of resource allocation, there are often some QoS requirements to be satisfied, such as the minimum data rate, the minimum signal-to-noise ratio (SNR)/signalto-inteference-plus-noise ratio (SINR), etc. Such requirements can be imposed as system operation constraints in the problem formulations. In previous work using deterministic optimization, it only considers the strict system operation constraints, i.e., the QoS requirements have to be satisfied with one hundred percent. As we point out, the system parameters related to channel conditions are often random and timevarying. In wireless communications, channel conditions are usually


12

changing rapidly, i.e., the coherence time is in the order of milliseconds. Moreover, fluctuations of wireless channel are common in nature, and the communications quality could incur very poor channel conditions due to the deep fading. Considering such practical issues, it is very hard to require the operation constraints to be always satisfied. To make the current schemes feasible, the system has to compromise its QoS requirements which will yet lead to severely conservative solutions. To tackle the problem, two important questions need to be answered: how to design the resource allocation problem by taking account into the realistic issues, such as randomness and time-varying issues? What is the reasonable formulation of system operation constraints with practical relevance? • Utilization of distributional information: Researchers have over the years spent considerable efforts in modeling the variations in wireless channel using probability distributions, e.g., [22, 23]. Unfortunately, such information is completely ignored in the deterministic optimization framework, as only the instantaneous realizations of the random system parameters are of interests. This prevents the distributional information from being exploited to achieve better system performance. A nature application is to design a resource allocation scheme which adapts its decision according to the channel distribution. A major advantage of such scheme is its adaptation of resource allocation can be much less frequent, since the distribution of wireless channel usually varies considerably slower compared with the fluctuation of instantaneous channel conditions. By doing so, the computational complexity and signaling overhead for resource allocation can be significantly reduced, which is

13


Wireless Communications

Adaptive OFDMA Systems [Chap. 2]

Stochastic Optimization

Chance Constrained Programming

MIMO Antenna Systems [Chap. 3] Robust Optimization Location−Aware Networks [Chap. 4]

Figure 1.1: Organization chart of the thesis: applications of stochastic optimization in wireless communications.

much more attractive to industrial implementation. Therefore, the major question here is: how the channel distributional information can be incorporated into the new optimization framework for resource allocation? Given the above practical concerns, we are motivated to look for new methodologies in designing resource allocation schemes for wireless systems. After reviewing numerous literature, we realized that the stochastic optimization can be a powerful math tool to generate potential applications. This thesis is an endeavor to apply the stochastic optimization methodologies into wireless communications. Specifically, we investigated three wireless communications


14

systems: adaptive orthogonal frequency division multiple access (OFDMA) systems, MIMO antenna systems, and location-aware networks. Stochastic optimization is applied to design dynamic resource allocation schemes for these three systems. An organization chart of this thesis is given in Fig. 1.1, which indicates the methodologies we used in optimization and its corresponding applications in wireless communications. In the following, we give a preliminary introduction of the three application systems, and the details of our design will be given in the following chapters.

1.3.2

OFDM and OFDMA Systems

Orthogonal frequency division multiplexing (OFDM) is a multi-carrier transmission technology to support high-date-rate broadband communications. In high-data-rate communications, the transmission of short duration data symbols is significantly limited by the inter-symbol interference (ISI) due to the time dispersive nature of wireless channels. To address such issue, OFDM modulates the signals parallelly on orthogonal subcarriers, which allows the transmission of high-date-rate symbols while maintaining the symbol length is longer than the delay spread of wireless channels. In order to combat the effect of ISI, each OFDM symbol is extended by a repetition of the end of the modulated symbol, which is referred to as cyclic prefix or guard interval. As long as the length of cyclic prefix is longer than the maximum delay spread of wireless channels, the ISI can be completely eliminated. Besides its capability of combating ISI in high-data-rate transmission, OFDM has more substantial advantages of enabling the utilization of the diversities in the wireless systems. First, OFDM divide the channel into multiple orthogonal subcarriers. By doing so, each subcarrier experiences almost a flat fading, and

15

Channel gain


Frequency

Channel gain

Channel gain

Frequency

Frequency

Figure 1.2: Multiple user diversity enables the subcarrier allocation in adaptive OFDMA system.

hence the symbol transmitted on each subcarrier can be independently modulated according to its channel condition. Such scheme is referred to adaptive OFDM [24], which can great enhance the system spectrum efficiency by exploiting the frequency diversity. Moreover, in OFDM system, different mobile users experience mutually independent channel fading on subcarriers. It enables us to exploit the multiuser diversity to achieve significant performance improvement. In Fig. 1.2, it illustrates the idea with an example of a multiuser cellular system using OFDM transmission. The base station (BS) is communicating with two mobile users simultaneously. Since the two users experience independent channel conditions, the BS can adaptively assign the subcarriers to the user with better channel conditions. By doing so, the maximum spectrum efficiency is achieved. Such idea brings up to a new technology, which is recently referred to as adaptive orthogonal frequency division multiple access (OFDMA) [25, 26]. The


16

system can be generalized to more complicated cases, e.g., when each user is associated with certain QoS requirements, or when OFDM is combined with other techniques such as link adaptation, multiple antennas, etc. To find the optimal subcarrier allocation, optimization techniques have play a pivotal role in designing adaptive OFDMA system. A survey of recent work in adaptive OFDMA can be referred to [26, 27].

1.3.3

MIMO Antenna Systems

Multiple-input and multiple-output (MIMO) is a technique of deploying multiple antennas at both transmitter and receiver to improve communication performance. The use of multiple antennas enables us to exploit the spatial degrees of freedom and diversity to significantly enhance the spectral efficiency and link reliability. It has attracted numerous research interests in the past decade, e.g., [28–32], and has been considered as one of the major components in next-generation wireless systems. The general structure of a MIMO system is illustrated in Fig. 1.3. The input signal is demultiplexed into LTx parallel branches of signals, called streams, after some signal processing and coding, and then transmitted simultaneously by LTx antennas. Each stream goes through different path to arrive at LRx antennas of the receiver, and these paths usually experience different channel conditions. In general, the signal received at each receive antenna is a combination of the streams from all the transmit antennas multiplying with different channel gains. After receiving all LRx branches of signals simultaneously, the receiver adopts some signal processing techniques to separate the multiple stream, and give an estimation of the original data signal. In MIMO systems, multiple antennas can be utilized to transmit different

17


RF chain

RF chain

RF chain

RF chain

RF chain

Signal Processing and Decoding

Data output

...

...

...

...

Data input

Signal Processing and Coding

RF chain

Figure 1.3: System structure of a point-to-point MIMO system.

data streams simultaneously in the same frequency band. Such scheme is called spatial multiplexing, which enables the high-data-rate transmission. The spatial multiplexing gain depends on the spatial degree of freedom, i.e., the number of parallel channels, which is determined by the maximum between the number of transmit antennas and the number of receive antennas. On the other hand, multiple antennas can be used to transmit the same data stream to enhance the reliability of transmission. This is achieved by exploiting the antenna diversity. The techniques to exploit antenna diversity include diversity combing, space-time coding, beamforming, etc. MIMO system can be designed to achieve the performance gain either in spatial multiplexing or antenna diversity, or even both of them. A theoretical tradeoff between these two types of performance gains has been revealed in [33, 34]. In addition to allowing spatial multiplexing and providing diversity to each user, multiple antennas allow the simultaneous transmission/reception among multiple users. In multiuser MIMO, various techniques can be exploited, such as beamforming, spatial division multiple access (SDMA), etc., to achieve multiuser diversity. It prompts the efficient use of system resources such as bandwidth, power, and bits. The resource allocation in multiuser MIMO have been widely studied in current literature, e.g., [26, 35].


18

Besides the multiplexing and diversity gains, there are some other practical issues in MIMO systems regarding the complexity and cost. One important issue is that the radio frequency (RF) chains associated with multiple antennas (as shown in Fig. 1.3) are costly in terms of size, power, and hardware. In practical systems, it is quite possible that the number of available RF chains is less than the number of transmit/receive antennas. In such case, antenna selection, i.e., select a subset of antennas to connect to the RF chains, becomes a necessary and affordable alternative to exploit the advantages of MIMO systems. The selection of antennas can be optimized according to the MIMO channel. The scheme design and more conclusion on the performances of antenna selection can be found in [36, 37].

1.3.4

Location-Aware Networks

Location-awareness has the potential to revolutionize the future technologies in many aspects related to commercial, public safety, military, and social activities. Many applications and services will require reliable and accurate positional information of mobile nodes to interact with their surroundings. Examples can be found in numerous scenarios including next-generation cellular services, sensor networks, military target tracking, search-and-rescue, healthcare monitoring, logistics, etc. In location-aware networks, some mobile/static nodes in the network have prior knowledge about their own locations, are known as anchors. The remaining mobile nodes, known as agents, must determine their locations through the process of localization. An example of location-aware network is illustrated in 1.4. The purpose of localization algorithms is to find the agents’ unknown positions given a set of measurements, e.g., the distances from their neighboring

19


C A

B D

Figure 1.4: Location-aware networks: the anchors (A, B, C, and D) transmit wireless signals to the two agents. nodes. It involves two major steps including the measurements between nodes, and processing these measurements to determine the agents’ positions. Localization techniques can be classified based on the type of measurement, such as range-based, angle-based, and proximity-based localization. Among them, range-based systems (i.e., based on distance estimates) are more suitable for high-definition localization with sub-meter accuracy. The distance between nodes is usually calculated based on the metrics, e.g., time of arrival (TOA), time difference of arrival (TDOA), and received signal strength (RSS), extracted from the received signals. Then, agents’ positions can be calculated using triangulation [38]. In angle-based localization system, antenna array needs to be deployed to estimate the angle of arrival (AOA) of received signals, which characterize the directional information of agents. In current location-aware networks, global positioning system (GPS) is a widely adopted technique, which utilizes a constellation of medium-earth-orbit satellites to determine agents’ positions on the earth. The satellites transmit signals that enable a GPS receiver to determine its 3D location after locking


20

onto four satellites. However, such systems do not work in indoor environments, since GPS signals cannot penetrate most obstacles. This occurs in many industrial applications, urban combat scenarios, collapsed buildings, and civil emergency situations, though various techniques have been proposed to improve the performance of ordinary GPS localization [39,40]. Ultra-wideband (UWB) transmission technology is inherently well suited for GPS-challenged environments, since the use of extremely large bandwidths results in desirable capabilities such as fine delay resolution and robustness in harsh environments [41, 42]. Such unique advantages enable UWB to provide accurate and reliable range (distance) measurements. The related work of location-aware networks using UWB can be found in [43–47]. More advanced techniques in recent years for location-aware networks include the cooperative localization [47, 48], indoor tracking and navigation [49–51], etc.

1.4

Contributions

This thesis studies the dynamic resource allocation problems in wireless communications. The ultimate goal is to bridge the gap between the current designs under the deterministic optimization framework and their practical relevance. Motivated by the recent advances in stochastic optimization, we introduce such useful methodology to address several challenging problems in wireless system designs. We develop a framework that contains: • practice-relevant problem formulations to capture the stochastic nature of the uncertain system parameters; • efficient-and-robust algorithm designs to achieve remarkable improvement on system performance.


21

We practice our idea of designs through the investigation of three promising wireless application systems. Each of them contains some practice-relevant challenges, which lead the conventional designs using deterministic optimization infeasible in practice. We address these hurdles by applying the proposed framework using stochastic optimization. The key contributions of this thesis are summarized into the following two aspects. On the perspective of methodology innovations: • We introduce the chance constrained programming methodology into wireless communications. Chance constraints arise naturally in many wireless applications, as most practical wireless systems can tolerate occasional outage in the QoS. We propose chance constrained formulations for the resource allocation problems in OFDMA and MIMO systems, where the short-term data rate requirements of individuals are satisfied with high probability. Moreover, the chance constrained formulation is known as computationally intractable problem despite its practical relevance. To tackle the problem, we construct safe tractable constraints based on recent advances in chance constrained programming. We then develop a polynomial-time algorithm for computing the optimal solution to the reformulated problem. To the best of our knowledge, this is the first work that uses chance constrained programming in the context of resource allocation in wireless systems. • We develop a novel robust optimization methodology to solve the nonconvex optimization problem with parameter uncertainties in wireless system. Generally, conventional robust optimization methodologies can only cope with few typical convex formulations, such as LP, conic program, etc. However, the example that we demonstrate in location-aware


22

networks is a non-convex optimization problem. We develop robust formulations by exploiting the geometric properties of the original problem, and optimize the worst-case performance. The proposed robust formulations retain the same structure of its non-robust counterpart, and can be efficiently solved via conic programming techniques. On the perspective of technology innovations: • We propose a slow adaptation scheme for dynamic resource allocation in wireless communications. In contrast to the common belief that radio resource allocation should be readapted once the instantaneous channel conditions change, the proposed scheme update the resource allocation decisions on a much slower timescale than that of channel fluctuation. The allocation decisions are fixed and remains optimal for a certain duration which spans the length of many coherence times. By doing so, computational cost and control signaling overhead, both considered as the major hurdles in the implementation of current fast adaptation schemes, can be drastically reduced. • We propose an antenna allocation scheme for composite radio MIMO networks. The coexistence of multiple radios is a desirable trend in future wireless networks. The proposed scheme addresses a critical issue in exploiting multiple antennas to support concurrent communications among multiple radios. Specifically, we allocate transmit antennas, together with transmit power, for each radio using the distributional information of wireless channels. Efficient algorithms are designed to obtain the optimal allocation decisions which maximize the long-term system throughput while satisfying the short-term data rate requirement for each radio transmission. To the best of our knowledge, this is the first


23

work considering antenna allocation schemes in MIMO systems. • We design optimal and robust power allocation schemes for locationaware networks. The energy efficiency is of great importance since it not only affects the network lifetime, but also determines the localization accuracy. In general, how to optimally allocate the transmit power in location-aware networks remains as an open problem. We adopt conic programming techniques to formulate power allocation problems, and show that the optimal solutions can be efficiently obtained by offthe-shelf optimization tools. Furthermore, we extend our optimization framework into a more realistic scenario with imperfect network topology parameters. We develop robust power allocation schemes to combat the parameter uncertainties. Efficient algorithms are designed to obtain the robust power allocation decisions which are shown to remarkably outperform their non-robust counterparts.

1.5

Organization

This thesis is organized as follows. Chap. 2 investigates the subcarrier allocation problems in adaptive OFDMA systems. We proposed a slow adaptation scheme to address the hurdles in the previous work and enhance the practicality of such promising technique. We propose three problem formulations for different application scenarios. Specifically, we consider the cases when mobile users are subject to average data rate constraints for elastic and inelastic traffic, and probabilistic data rate constraints, respectively. All the problems are formulated into stochastic programs, and efficient algorithm is developed to solved the optimal solutions.


24

Chap. 3 investigates the composite radio MIMO networks where an essential task is to support the concurrent communications among multiple radios. We propose a novel antenna-and-power allocation scheme to support the operation of such system. We apply the advance stochastic optimization techniques to design algorithms which can not only efficiently determine the feasibility of the problem, but also obtain the optimal solutions to achieve the best system performance. Chap. 4 investigates the power allocation problems in location-aware networks. First, we solve the optimal power allocation problem via conic programming techniques. Then, we generalize our work into realistic scenarios by considering the imperfect network parameters. We develop a robust optimization method to combat the parameter uncertainties, and design efficient algorithms to exploit distributed computations among the nodes. The results show the proposed schemes yield satisfactory performance in realistic networks. Finally, Chap. 5 summarizes the key conclusions in this thesis, and discuss several potential directions in future work.

Chapter 2 Slow Subcarrier Allocation in Adaptive OFDMA Systems Orthogonal frequency division multiplexing (OFDM) has been identified as one of the leading technologies to support broadband and multimedia services in future wireless systems. The inherent multicarrier nature of OFDM facilitates flexible use of subcarriers to significantly enhance system capacity. Adaptive subcarrier allocation, recently referred to as adaptive orthogonal frequency division multiple access (OFDMA) [25, 52], has been considered as a primary contender in next-generation wireless standards, such as IEEE802.16 WiMAX [53] and 3GPP-LTE [54]. In the existing literature, adaptive OFDMA exploits time, frequency, and multiuser diversity by quickly adapting subcarrier allocation (SCA) to the instantaneous channel state information (CSI) of all users. Such “fast” adaptation suffers from high computational complexity, since an optimization problem required for adaptation has to be solved by the BS every time the channel changes. Considering the fact that wireless channel fading can vary quickly

25

Chapter 2. Slow Subcarrier Allocation in Adaptive OFDMA Systems

26

(e.g., at the order of milli-seconds in wireless cellular system), the implementation of fast adaptive OFDMA becomes infeasible for practical systems, even when the number of users is small. Recent work on reducing complexity of fast adaptive OFDMA includes [55–59], etc. Moreover, fast adaptive OFDMA requires frequent signaling between the BS and mobile users in order to inform the users of their latest allocation decisions. The overhead thus incurred is likely to negate the performance gain obtained by the fast adaptation schemes. To date, high computational cost and high control signaling overhead are the major hurdles that prevent adaptive OFDMA from being deployed in practical systems. We propose a slow adaptive OFDMA scheme, which is motivated by [60], to address the aforementioned problem. In contrast to the common belief that radio resource allocation should be readapted once the instantaneous channel conditions change, the proposed scheme updates the SCA on a much slower timescale than that of channel fluctuation. Specifically, the allocation decisions are fixed for the duration of an adaptation window, which spans the length of many coherence times. By doing so, computational cost and control signaling overhead can be dramatically reduced. However, this implies that channel conditions over the adaptation window are uncertain at the decision time, thus presenting a new challenge in the design of slow adaptive OFDMA schemes. An important question is how to find a valid allocation decision that remains optimal and feasible for the entire adaptation window. Such a problem can be formulated as a stochastic programming problem, where the channel coefficients are random rather than deterministic. This chapter develops a framework of slow adaptive OFDMA by considering two major types of formulations according to the QoS requirements of users:


27

• Average-Rate Constrained Formulations: We consider each user has a minimum requirement on its time-average data rate. This is a natural formulation which adapts the SCA based on the time average channel conditions. Indeed, slow adaptation schemes that have recently been studied in other contexts such as slow rate adaptation [60–62] and slow power allocation [14], considers adaptation decisions made solely based on the long-term average channel conditions. Therein, random channel parameters are replaced by their ergodic mean values, resulting in a deterministic rather than stochastic optimization problem. However, it can lead to a severe throughput loss especially for fast fading channels. To address such problem, we propose a more flexible design of slow adaptation OFDMA. Rather than simply adapting to the long-time average channel conditions, we allow the period for each adaptation, referred to as an adaptation window, to be adjustable. By tuning the length of adaptation window, we could engineer a desirable tradeoff between spectral efficiency and computational complexity. The SCA is updated according to the channel statistics within each window instead of instantaneous channel conditions. Furthermore, we show that the proposed scheme can be modified to accommodate inelastic traffic, which ensures worst-case average data rates to all users. • Probabilistic-Rate Constrained Formulation: With the increasing popularity of wireless multimedia applications, there will be more and more inelastic traffic that require a guarantee on the minimum short-term data rate. As such, slow adaptation schemes based on average channel conditions cannot provide a satisfactory QoS. Fortunately, most inelastic traffic can tolerate an occasional dip in the instantaneous data rate


28

without compromising QoS. This presents an opportunity to enhance the system performance. In particular, we employ chance constrained programming techniques by imposing probabilistic constraints on user data rate. Although this formulation captures the essence of the problem, chance constrained programs are known to be computationally intractable except for a few special cases [13]. To circumvent the difficulty, we formulate safe tractable constraints for the problem based on recent advances in the chance constrained programming. A polynomial-time algorithm is developed for efficiently computing the optimal solution to the reformulated problem. The proposed formulation guarantees the short-term data rate requirements of individual users except in rare occasions. To the best of our knowledge, this is the first work that uses chance constrained programming in the context of resource allocation in wireless systems. This chapter is organized as follows. In Sec. 2.1, we describe the channel model and introduce the slow adaptive OFDMA scheme. In Sec. 2.2, we discuss the formulation with average rate constraints for elastic traffic. In Sec. 2.3, we propose a “safe” slow adaptive OFDMA scheme by adopting robust optimization to accommodate inelastic traffic. In Sec. 2.4, we formulate the slow adaptive OFDMA into a chance constrained program, which maximizes the long-term system throughput meanwhile satisfying the short-term data rate requirement of individual users with high probability. Finally, conclusions are given in Sec. 2.5.


2.1

29

System and Channel Model

In this section, we first describe the channel model, and then, introduce the slow adaptive OFDMA scheme with a comparison of the conventional fast adaptive OFDMA scheme.

2.1.1

Channel Model

We consider a single-cell multiuser OFDM system with K users and N subcarriers. We assume that the instantaneous channel coefficients of user k and (t)

subcarrier n are described by complex Gaussian1 random variables hk,n ∼ CN (0, σk2 ).

The parameter σk can be used to model the long-term aver −γ age channel gain as σk = ddk0 · sk , where dk is the distance between the

BS and subscriber k, d0 is the reference distance, γ is the amplitude pathloss exponent and sk characterizes the shadowing effect. Hence, the channel (t) 2 (t) gain gk,n = hk,n is an exponential random variable with probability density function (PDF) given by

1 ξ fgk,n (ξ) = . exp − σk σk

(2.1)

The transmission rate of user k on subcarrier n at time t is given by ! (t) p g t k,n (t) , rk,n = W log2 1 + ΓW N0 (t)

where pt is the transmit power of a subcarrier, gk,n is the channel gain at time t, W is the bandwidth of a subcarrier, N0 is the power spectral density of Gaussian noise, and Γ is the capacity gap that is related to the target bit error rate (BER) and coding-modulation schemes. 1

Although the techniques used in this chapter are applicable to any fading distribution,

we shall prescribe to a particular distribution of fading channels for illustrative purposes.


2.1.2

30

Slow Adaptive OFDMA

In traditional fast adaptive OFDMA systems, SCA decisions are made based on instantaneous channel conditions in order to maximize the system throughput. As depicted in Fig. 2.1a, SCA is performed at the beginning of each time slot, where the duration of the slot is no larger than the coherence time of (t)

the channel. Denoting by xk,n the fraction of airtime assigned to user k on subcarrier n, fast adaptive OFDMA solves at each time slot t the following linear programming problem: P

fast

:

max (t) xk,n

s.t.

K X N X

(t)

(t)

xk,n rk,n

(2.2)

k=1 n=1

N X

n=1 K X k=1 (t)

(t)

(t)

xk,n rk,n ≥ qk , (t)

xk,n ≤ 1,

xk,n ≥ 0,

∀k

(2.3)

∀n

∀k, n,

where the objective function in (2.2) represents the total system throughput at time t, and (2.3) represents the data rate constraint of user k at time t with qk denoting the minimum required data rate. We assume that qk is known (t)

(t)

by the BS and can be different for each user k. Since gk,n (and hence rk,n ) varies on the order of coherence time, one has to solve the Problem P fast at the beginning of every time slot t to obtain SCA decisions. Thus, the above fast adaptive OFDMA scheme is extremely costly in practice. In contrast to fast adaptation schemes, we propose a slow adaptation scheme in which SCA is updated only every adaptation window of length T . More precisely, SCA decision is made at the beginning of each adaptation window as depicted in Fig. 2.1b, and the allocation remains unchanged till the


SCA SCA

SCA SCA SCA

slot slot

31

SCA SCA

slot

time (a) fast adaptive OFDMA

SCA

SCA

SCA

window

window

window

...

...

...

slot slot

slot

time (b) slow adaptive OFDMA

Figure 2.1: Adaptation timescales of fast and slow adaptive OFDMA system (SCA = SubCarrier Allocation).

next window. We consider the duration T of a window to be large compared with that of fast fading fluctuation so that the channel fading process over the window is ergodic; but small compared with the large-scale channel variation so that path-loss and shadowing are considered to be fixed in each window. Unlike fast adaptive systems that require the exact CSI to perform SCA, slow adaptive OFDMA systems rely only on the distributional information of channel fading and make an SCA decision for each window. Let xk,n ∈ [0, 1] denote the SCA for a given adaptation window2 . Then, the time-average throughput of user k during the window becomes ¯bk =

N X

xk,n r¯k,n ,

n=1

2

It is practical to assume xk,n as a real number in slow adaptive OFDMA. Since the

data transmitted during each window consists of a large mount of OFDM symbols, the timesharing factor xk,n can be mapped into the ratio of OFDM symbols assigned to user k for transmission on subcarrier n.


where r¯k,n

1 = T

Z

32

(t)

T

rk,n dt

is the time-average data rate of user k on subcarrier n during the adaptation window. The time-average system throughput is given by ¯b =

K X N X

xk,n r¯k,n .

(2.4)

k=1 n=1

When T approaches infinity (or much larger than the channel coherence time in practice), the time-average data rate r¯k,n converges to the ensemble (t)

average of the random process rk,n , if the underlying channel is an ergodic o n (t) process. That is, r¯k,n = E rk,n , where the expectation is taken over all

possible channel states. If T is not sufficiently long, then r¯k,n differs from its ensemble mean. Before leaving this section, note that we have assumed that the data trans(t)

(t)

mission rate rk,n is a function of the instantaneous channel gain gk,n . In other words, fast rate adaptation is adopted, although subcarrier allocation is performed on a slower time scale. The reason is that the computational complexity and control overhead needed for fast rate adaptation is very low compared with adaptive subcarrier allocation. As a matter of fact, fast rate adaptation has already been implemented in many practical systems.

2.2

Slow Adaptive OFDMA with Average Rate Constraints for Elastic Traffics

In this section, a slow adaptive OFDMA is formulated to maximize the expected average overall system throughput and, at the same time, to satisfy a minimum expected average data rate of each mobile user. We first formulate


33

such problem into a stochastic program, where the expected mean is considered over an adaptation window with flexible length. Then, we show how to calculate the statistic mean of data rate within an adaptation window. The performance of proposed schemes is investigated through simulations.

2.2.1

Problem Formulation

To accommodate the elastic traffic which is not very sensitive to the delay in transmitted data, we realize that the average data rate can actually be exploited instead of the instantaneous data rate as system QoS. Specifically, we consider the average system throughput ¯b and average user data rate ¯bk during an adaptation window [t0 , t0 +T ]. Since the BS cannot estimate all the channel gains for the entire window, these average values appear randomly to the BS at the beginning of each window. However, it is reasonable to assume that the BS (t )

(t )

knows gk,n0 (or equivalently rk,n0 ) when it makes subcarrier-allocation decision for adaptation window [t0 , t0 + T ]. Therefore, the problem can be designed to maximize the expected average system throughout meanwhile satisfying the minimum requirement of expected average user data rate, where the expecta(t )

tions are conditioning on the initial CSI gk,n0 . The slow adaptive subcarrier allocation is mathematically formulated as a stochastic programming problem: P1slow :

max

{xk,n }

o n (t ) E ¯b rk,n0

o n (t ) s.t. E ¯bk rk,n0 ≥ qk , K X k=1

xk,n ≤ 1,

xk,n ≥ 0,

∀n

∀k, n

∀k


34 (t)

where the expectation is taken over the random channel process g = {gk,n} for t ∈ [t0 , t0 + T ]. The expectations in the objective and the rate constraints can be calculated as follows K X N o X n n (t0 ) o (t0 ) ¯ xk,n E r¯k,n rk,n , E b rk,n =

o n (t ) E ¯bk rk,n0 =

k=1 n=1 N X n=1

n (t ) o xk,n E r¯k,n rk,n0 ,

∀k.

n (t ) o The problem P1slow can be easily solved using linear programming, if E r¯k,n rk,n0

is known.

2.2.2

Computation of Expected Average Data Rate

To solve P1slow , the major difficulty is nothing but to acquire the expected n (t ) o values E r¯k,n rk,n0 . In the following, we consider Rayleigh fading channel as one of the typical channel models. The calculation method we adopt here can

also be applied to other distribution but with more complicated expressions. When T is very large, we have r¯k,n = E{rk,n } due to ergodicity, and hence n (t0 ) o E r¯k,n rk,n = E{rk,n }, which can be easily calculated from the distribution of gk,n which is given in (2.1). Thus, we have E{rk,n } =

Z

0

∞

W log2 1 +

pt ξ 1 ξ dξ. exp − ΓW N0 σk σk

(2.5)

In a general case where T is not necessarily long enough, the calculation n (t0 ) o of E r¯k,n rk,n is much more involved. As will be proved in Theorem 1, n (t ) o E r¯k,n rk,n0 can be calculated from Eqn. (2.6). (t )

Theorem 1. Given a window [t0 , t0 + T ] and the initial data rate rk,n0 = (t0 ) pt gk,n (t ) W log2 1 + ΓW N0 where gk,n0 follows the exponential distribution in (2.1),


35

the expected average data rate of user k on subcarrier n can be computed as Z ∞ Z T n (t0 ) o E r¯k,n rk,n = α · r · 2r/W · exp(−β · 2r/W ) 0 0 p r/W − 1 dτ dr (2.6) · I0 γ · R(τ ) 2 where R(τ ) = J0 (2πfd τ ) denotes the time correlation of the channel, where J0 (·) is the zero-order Bessel function and fd is the Doppler frequency of the channel. Likewise, I0 (·) is zero-order modified Bessel function, i.e. I0 (x) = J0 (jx). α, β and γ are all functions of τ = t − t0 given as (t0 ) σk ΓN0 ln 2 ΓW N0 [(δ − σk2 /4)2rk,n /W − δ + σk2 /2] α= exp 4T pt δ σk pt δ q (t0 ) ΓW N0 2rk,n /W − 1 σk2 − R2 (τ ) σk ΓW N0 , γ= , δ= . β= 4pt δ 2pt δ 4

Proof. The following proof applies to all n and k. Hence, we omit the subscript k, n for simplicity of notation. From (2.1), the joint exponential distribution of g (t1 ) and g (t2 ) can be derived as p (t ) (t ) 1 g 1g 2 σ (t1 ) (t2 ) f (g , g ) = · I0 g +g exp − R(τ ) . 4δ 4δ 2δ pt g (t) , we can obtain the following distributions: Knowing r (t) = W log2 1 + ΓW N0 (t1 )

(t2 )

f (r (t) ) = (g (t) )′ f (g (t) ) and f (r (t1 ) , r (t2 ) ) = J · f (g (t1 ) , g (t2 ) )

∂(g(t1 ) ,g(t2 ) ) where J = ∂(r(t1 ) ,r(t2 ) ) is the Jacobian determinant. Hence, the conditional


36

rate distribution is f (r (t2 ) |r (t1 ) ) = f (r (t2 ) , r (t1 ) )/f (r (t1 ) ) r (t2 ) r (t1 ) r (t2 ) 2 W σΓN0 ln 2 ΓW N0 (2 W − 1) σΓW N0 r(t1) = − exp (2 W + 2 W − 2) 4pt δ σpt 4pt δ q r (t1 ) r (t2 ) ΓW N0 (2 W − 1)(2 W − 1) R(τ ) · I0 2pt δ Finally, we can calculate the conditional expectation of mean rate as follow, E r¯|r (t0 ) Z 1 E r (t) |r (t0 ) dt = T T Z Z ∞ 1 (t) (t) (t0 ) (t) dt r f (r |r )dr = T T 0 r (t0 ) Z TZ ∞ 2 2 ΓW N0 [(δ − σ4 )2 W − δ + σ2 ] r (t) σΓN0 ln 2 = · r (t) · 2 W exp 4T pt δ σpt δ 0 0 {z } | α q q r (t0 ) σΓW N0 r(t) ΓW N0 2 W − 1 r (t) · exp − 2 W · I0 · R(τ ) 2 W − 1 dr (t) dτ. 4pt δ 2pt δ {z } | {z } | β

γ

Problem P1slow is formulated into a LP which can be efficiently solved by standard methods, e.g. the path-following algorithm. The complexity involved depends on the number of variables (NK) and constraints (NK + N + K). We note that there is already plenty of research on fast adaptive OFDMA. The work focusing on LP formulation (e.g. [52, 63]) can also be applied to the slow adaptation here, including their efficient heuristic algorithms, whereas our scheme only affords 1/T computational complexity of fast schemes within each


37

window. Moreover, computing the expectation in (2.6) requires O(M 2 ) · NK computations where M is the number of function evaluations required for onedimensional integral. In practice, however, we may pre-compute the data and store them in a lookup table, and then only a table lookup is needed at the beginning of each window. Hence, the computation of the expectation will not be the major burden of the computational complexity in slow adaptation.

2.2.3

Numerical Results

In this section, we investigate the performance of the slow adaptive OFDMA formulated in P1slow . The average received SNR pt σk /N0 is set to be 22dB for all users and the channel fading follows Rayleigh distribution. The target bit error rate (BER) is set to 10−4 . In Fig. 2.2, we investigates the spectral efficiency of the slow adaptive OFDMA, given by ¯b/N where ¯b is defined in (2.4), using the closed-form expression of expected average rate derived in Theorem 1. Assume there are three users and 16 subcarriers. We fix the Doppler shift fd to 50Hz and vary the adaptation window size T , so that the normalized Doppler shift, defined as Fd = fd · T , varies from 0.05 to 10. The line with triangles corresponds to the n (t0 ) o ideal case where the actual realization of r¯k,n is used instead of E r¯k,n rk,n in Problem P1slow . The line with circles corresponds to the case where the expected average data rate calculated by (2.6) is used in Problem P1slow . The figure shows that the gap between the two curves is negligible. This implies that the simple mean-rate estimation given in Theorem 1 performs almost as well as the ideal case even when the adaptation window is large. In Fig. 2.3, we compare the system spectral efficiency of three adaptive OFDMA systems: fast adaptive OFDMA in P fast , the proposed slow adaptive


38

Spectrum Efficiency (bps/Hz/subcarrier)

5.6 By Perfect Mean Rate By Predicted Mean Rate

5.4 5.2 5 4.8 4.6 4.4 4.2

−1

0

10

10 Normalized Doppler Shift F

1

10 d

Figure 2.2: Performance comparison between systems using the actual average o n (t ) rate r¯k,n and expected average rate E r¯k,n rk,n0 in problem P1slow . OFDMA with stochastic programming in P1slow , and an intuitive slow adaptation scheme without stochastic programming, i.e., to solve Problem P fast at (t )

time t0 and then apply xk,n0 to the whole window. We consider an OFDMA system with four users and 64 subcarriers, and each user has a minimum rate constraint of 64 bps. The plot is averaged over independent 4000 simulation runs. We fix fd to 50Hz and vary T , so that the normalized Doppler shift Fd varies from 0.05 to 200. Presumably, the larger T , the less frequently the subcarrier allocation is updated. To quantify this, we assume that time unit per fast adaptation T0 = 1000µs is the maximum time unit within which the wireless channel does not change. Then, m = T /T0 quantifies how much times slower the slow adaptation is compared with the fast one. Here, we assume that the control overhead for subcarrier allocation consumes a bandwidth equiva-


5.6

m=4

Slow SCA with SP Slow SCA without SP Fast SCA

Spectrum Efficiency (bps/Hz/subcarrier)

m=2 5.4

39

m=10

5.2 m=1 m=20 5 m=40 4.8 m=100 m=200

4.6

m=500 m=800 m=4000 m=1000 m=2000

4.4

4.2

−1

10

0

1

10 10 Normalized Doppler Shift Fd

2

10

Figure 2.3: Spectral efficiency comparison among fast adaptation, slow adaptation with and without stochastic programming. (Signaling overhead is considered here.)

lent to 10% of T0 every time the subcarrier allocation is updated [64]. From the figure, we can see that by tuning T , we can engineer a tradeoff between spectral efficiency and computational complexity as well as control overhead. Due to high control overheads, the traditional fast adaptive OFDMA suffers from throughput degradation. Hence, when T is relatively small (Fd smaller than 1), slow adaptation achieves an even higher spectral efficiency than the fast one, while enjoying a much lower computational cost. It is worth noting that when m = 100, the proposed slow adaptation scheme achieves 92% throughput of the fast scheme with only 1/100 computational complexity. Comparing with the intuitive slow adaptation scheme without stochastic programming,


40

the proposed scheme in P1slow achieves a much higher throughput (about 10% when m = 40) with a similar computational cost.

2.3

Slow Adaptive OFDMA with Average Rate Constraints for Inelastic Traffics

In this section, we propose a safe slow adaptive OFDMA scheme for inelastic applications which requires the actual user throughput to satisfy its minimum requirement in each adaptation window. We adopt robust optimization to provide such worst-case guarantee. The proposed scheme is formulated into a conic linear program, which can be solved efficiently via interior point methods [11]. The performance is investigated through simulations.

2.3.1

Problem Formulation

The formulation of slow adaptive OFDMA in (P1slow ) guarantees the expected average throughput of users to be higher than qk per unit time during each adaptation window. The actual realized throughput ¯bk could be higher or lower than qk in different application windows. While this is good enough for elastic traffic where users value their long-term average data rates, it may not work for inelastic traffic that requires the actual throughput to be higher than a certain threshold during each period of interest. To address the need of inelastic traffic, we modify the formulation P1slow to propose a safe slow adaptive OFDMA scheme that guarantees the actual throughput ¯bk to be higher than qk for each and every application window.


41

Let vector n n (t0 ) o (t ) o , ··· , r0 = E r¯1,1 r1,10 , · · · , E r¯N,1 rN,1

n n (t0 ) o (t0 ) o T E r¯1,K r1,K , · · · , E r¯N,K rN,K ∈ RN K

denote the statistical mean of time-average data rate estimated at the beginning of an application window. Likewise, let T r = r¯1,1 , · · · , r¯N,1 , · · · , r¯1,K , · · · , r¯N,K ∈ RN K denote the actual realization of the time-average data rate. Typically, r¯k,n

varies around its statistical mean. Therefore, at time t0 , it is reasonable to assume that r will lie in an ellipsoid centered around r0 during the application window that follows [65] n o U = r ∈ RN K : r = r0 + Rv, kvk ≤ ρ0 .

(2.7)

In the above, R is a NK × NK symmetric positive semidefinite matrix, and ρ0 corresponds to the maximum deviation of r from r0 . To ensure that the minimum data rate constraints are satisfied for all possible realizations r, the slow adaptation problem P1slow is reformulated as follows P2slow :

max x

min r∈U

K X N X

xk,n r¯k,n

k=1 n=1

s.t.

N X

n=1 K X k=1

xk,n r¯k,n ≥ qk , xk,n ≤ 1,

xk,n ≥ 0,

∀n

∀k, n.

∀k, ∀r ∈ U


42

Let x = [x1,1 , · · · , xN,1 , · · · , x1,K , · · · , xN,K ]T ∈ RN K . We write P2slow into a more condensed form as min t x

s.t. rT x ≥ −t,

∀r ∈ U

(Bk r)T x ≥ qk ,

∀k, ∀r ∈ U

D·x≥d where 

  Bk =  



D=

0

B1 ..

.

0

BK

−IN

···



   ∈ RN K×N K  − IN

IN K

with Bi =



   IN ,

  0 ∈ RN ×N ,

if i = k otherwise,

 ∈ R(N +N K)×N K ,

d = [−1, · · · , −1, 0, · · · , 0]T ∈ RN +N K . {z } | {z } | N

NK

In the sequel, we show that P2slow is equivalent to a SOCP that can be

efficiently solved. Theorem 2. P2slow is equivalent to the following SOCP: min x

s.t.

t

rT0 x + t T , R x Q 0 ρ0 T T r0 Bk x − qk T T , R Bk x Q 0, ρ0

∀k = 1, · · · , K

D·x≥d where Q is the partial order defined on the second order cone Q = {(t, x) ∈ R × Rn : t ≥ kxk}, i.e. we have (t, x) Q 0 iff (t, x) ∈ Q.


43

Proof. The first constraint in P2slow rT x + t = (r0 + Rv)T x + t ≥ 0,

∀r ∈ U

can be expressed as min

v∈RNK , kvk≤ρ0

T (r0 x + t) + vT (RT x) = (rT0 x + t) − ρ0 kRT xk ≥ 0.

Hence, we obtain a SOC constraint as T r0 x + t T , R x Q 0. ρ0 Similarly, the second constraint in P2slow yields (Bk r)T x = (r0 + Rv)T (BTk x) ≥ qk ,

∀k, ∀r ∈ U.

It is equivalent to min

v∈RNK ,kvk≤ρ0

T T r0 (Bk x) + vT (RT BTk x) = rT0 BTk x − ρ0 kRT BTk xk ≥ qk ,

Hence, we have the SOC constraints T T r0 Bk x − qi T T , R Bk x Q 0, ρ0

∀k.

∀k.

Remark 1. SOCP is a convex problem, and hence the optimal solution of P2slow can be efficiently solved via interior point method [11], which is a polynomialtime algorithm and can be applied into real-time adaptation.

2.3.2

Numerical Results

The simulation settings are similar to that in Fig. 2.3. We consider an OFDMA system with four users and 64 subcarriers, and each user has a minimum rate


44

0.7 ρ0=0 ρ0=1

0.6

ρ0=2 ρ =3 0

Outage Probabiltiy

0.5

ρ0=6

0.4

0.3

0.2

0.1

increasing ρ

0

0

0

5

10

Size of Uncertainty Set ρ

15

Figure 2.4: Outage probability of slow adaptive OFDMA versus the deviation size ρ when Fd = 2 and m = 40.

constraint of 64 bps. The average received SNR is 22 dB and the channel fading follows Rayleigh distribution. In Fig. 2.4 and 2.5, we investigate the performance of the safe slow adaptive OFDMA scheme proposed in Sec. 2.3. We are interested in the occurrence of outage events when the actual data rate enjoyed by user k falls below qk during some application window. In Fig. 2.4, we plot the probability a user perceives an outage event3 versus the deviation of r from its statistical mean r0 , denoted by ρ. In particular, we vary ρ0 in (2.7) and plot four curves. For comparison, the outage probability of the proposed scheme in P1slow is also 3

In the simulation, user k is said to be in outage when ¯bk ≤ qk (1 − ε), where ε = 0.001

is for error tolerance.


45

5.6 ρ0=0 ρ0=1

Spectral Efficiency (bps/Hz/subcarrier)

5.4

ρ0=2 ρ =3 0

5.2

5

4.8

4.6

4.4

4.2

increasing ρ0

−1

10

0

1

10 10 Normalized Doppler Shift Fd

2

10

Figure 2.5: Spectral efficiency of safe slow adaptive OFDMA with different ρ0 . (We fixed the actual maximum deviation ρ = 6.)

plotted, which corresponds to ρ0 = 0. From the figure, it can be seen that by safe adaptive OFDMA in P2slow , outage probability is significantly reduced compared with the scheme in P1slow . Moreover, it is not surprising that the outage probability is always zero as long as ρ0 is larger than ρ. One concern of adopting the safe slow adaptive OFDMA scheme is that the system spectral efficiency may decrease with the increase of safety margin ρ0 . Fortunately, Fig. 2.5 shows that the throughput degradation with the increase of ρ0 is negligible. It can be seen that the scheme in P1slow (which corresponds to ρ0 = 0) yields an upper bound in the throughput, as there is no safety margin provided. When ρ0 = 3, the throughput gap is as small as 2% compared with the upper bound. Together with Fig. 4, the results


46

imply a tradeoff between the spectral efficiency and outage probability when underlying traffic is inelastic.

2.4

Slow Adaptive OFDMA with Probabilistic Rate Constraints

The previous sections have considered the minimum average data rate requirement of individual users. Such constraints can be viewed as strict QoS requirement, i.e., the constraints have to be satisfied with 100%. However, it is too strict to be deployed in real applications. Indeed, most inelastic traffic such as that from multimedia applications can tolerate an occasional dip in the instantaneous data rate without compromising QoS. In this section, we take a different approach by imposing probabilistic QoS constraints which is more meaningful in practical systems. A novel slow adaptive OFDMA scheme is proposed aiming at maximizing the long-term system throughput meanwhile satisfying short-term data rate requirements of individual users with high probability. Such an objective is naturally formulated into a chance constrained program, which is yet known to be computationally intractable. To tackle the problem, we exploit the special structure of the probabilistic constraints in our problem to construct safe tractable constraint (STC) based on recent advances in the chance constrained programming literature. Moreover, we design an interior-point algorithm that is tailored for the slow adaptive OFDMA problem, since the formulation with STC, although convex, cannot be trivially solved using off-the-shelf optimization software. Our algorithm can efficiently compute an optimal solution to the problem with STC in polynomial time. Last, the performance is investigated through extensive


47

simulations.

2.4.1

Problem Formulation

Suppose that each user has a short-term data rate requirement qk defined on P (t) each time slot. If N n=1 xk,n rk,n < qk , then we say that a rate outage occurs

for user k at time slot t, and the probability of rate outage for user k during the window [t0 , t0 + T ] is defined as ( N ) X (t) Pout xk,n rk,n < qk , k , Pr n=1

∀t ∈ [t0 , t0 + T ],

where t0 is the beginning time of the window. Inelastic applications, such as voice and multimedia, that are concerned with short-term QoS can often tolerate an occasional dip in the instantaneous data rate. In fact, most applications can run smoothly as long as the shortterm data rate requirement is satisfied with sufficiently high probability. With the above considerations, we formulate the slow adaptive OFDMA problem as follows: P3slow

:

max xk,n

s.t.

K X N X

o n (t) xk,n E rk,n

k=1 n=1 ( N X

Pr

n=1

K X k=1

(t)

xk,n rk,n ≥ qk

xk,n ≤ 1,

xk,n ≥ 0,

(2.8) )

≥ 1 − ǫk ,

∀k

(2.9)

∀n

∀k, n,

where the expectation4 in (2.8) is taken over the random channel process g = (t)

{gk,n } for t ∈ [t0 , t0 + T ], and ǫk ∈ [0, 1] in (2.9) is the maximum outage 4

n o (t) In (2.8), we replace the time-average data rate r¯k,n by its ensemble average E rk,n

due to the ergodicity of channel fading over the window.


48

probability user k can tolerate. In the above formulation, we seek the optimal SCA that maximizes the expected system throughout meanwhile satisfying each user’s short-term QoS requirement, i.e., the instantaneous data rate of user k is higher than qk with probability at least 1 − ǫk . The above formulation is a chance constrained program since a probabilistic constraint (2.9) has been imposed.

2.4.2

Safe Tractable Constraints

Despite its utility and relevance to real applications, the chance constraint (2.9) imposed in P3slow makes the optimization highly intractable. The main reason is that the convexity of the feasible set defined by (2.9) is difficult to verify. Indeed, given a generic chance constraint Pr{F (x, r) > 0} ≤ ǫ where r is a random vector, x is the vector of decision variable, and F is a realvalued function, its feasible set is often non-convex except for very few special cases [13, 17]. Moreover, even with the nice function in (2.9), i.e., F (x, r) = P (t) (t) qk − N n=1 xk,n rk,n is bilinear in x and r, with independent entries rk,n in r whose distribution is known, it is still unclear how to compute the probability in (2.9) efficiently. To circumvent the above hurdles, we propose the following formulation fslow by replacing the chance constraints (2.9) with a system of constraints P 3

H such that (i) x is feasible for (2.9) whenever it is feasible for H, and (ii) the constraints in H are convex and efficiently computable5 . The new formulation 5

Condition (i) is referred to as “safe” condition, and condition (ii) is referred to as

“tractable” condition.


49

is given as follows: fslow : P 3

K X N X

max xk,n

k=1 n=1

s.t.

o n (t) xk,n E rk,n

inf qk + ̺

̺>0

K X k=1

N X n=1

xk,n ≤ 1,

xk,n ≥ 0,

(2.10) −1

Λk (−̺ xk,n ) − ̺ log ǫk

≤ 0,

∀n

∀k

(2.11) (2.12)

∀k, n,

(2.13) (t)

where Λk (·) is the cumulant generating function of rk,n , −1

Λk (−̺ xˆk,n ) = log

Z

∞

0

1+

pt ξ ΓW N0

− W̺xˆlnk,n 2

ξ 1 dξ . (2.14) exp − · σk σk

In the following, we first prove that any solution x that is feasible for the STC fslow is also feasible for the chance constraints (2.9). Then, we prove (2.11) in P 3

fslow is convex. that P 3

(t)

(t)

Proposition 1. Suppose that gk,n (and hence rk,n ) are independent6 random (t)

variables for different n and k, where the PDF of gk,n follows (2.1). Furthermore, given ǫk > 0, suppose that there exists an x ˆ = [ˆ x1,1 , · · · , xˆN,1 , · · · , x ˆ1,K , · · · , xˆN,K ]T ∈ RN K such that

(

Gk (ˆ x) , inf qk +̺ ̺>0

N X n=1

Λk (−̺−1 xˆk,n )−̺ log ǫk ≤ 0,

Then, the allocation decision x ˆ satisfies (N ) X (t) Pr xˆk,n rk,n ≥ qk ≥ 1 − ǫk , n=1

6

)

∀k.

∀k.

(2.15)

(2.16)

The case when frequency correlations exist among subcarriers will be discussed in Sec.

2.4.5.


50

Proof. Our argument will use the Bernstein approximation theorem proposed in [17].7 Suppose there exists an x ˆ ∈ RN K such that Gk (ˆ x) ≤ 0, i.e., ( ) N X inf qk + ̺ Λk (−̺−1 xˆk,n ) − ̺ log ǫk ≤ 0. ̺>0

(2.17)

n=1

The function inside the inf ̺>0 {·} is equal to N X (t) −1 − ̺ log ǫk qk + ̺ log E exp − ̺ xˆk,n rk,n n=1

(

= qk + ̺ log E exp ̺−1 − (

= ̺ log E exp ̺−1 qk −

N X

(t) xˆk,n rk,n

n=1

N X n=1

(t) xˆk,n rk,n

)

)

(2.18)

− ̺ log ǫk

− ̺ log ǫk ,

(2.19) (2.20)

where the expectation E{·} can be computed using the distributional infor(t)

mation of gk,n in (2.1), and (2.19) follows from the independence of random (t)

variable rk,n over n. Let Fk (x, r) = qk −

PN

(t) n=1 xk,n rk,n .

Then, (2.17) is equivalent to

inf ̺E exp ̺−1 Fk (ˆ x, r) − ̺ǫk ≤ 0.

(2.21)

̺>0

According to Theorem 4 in Appendix A, the chance constraints (2.16) hold if there exists a ̺ > 0 satisfying (2.21). Thus, the validity of (2.16) is guaranteed by the validity of (2.15). Now, we prove the convexity of (2.11) in the following proposition. Proposition 2. The constraints imposed in (2.11) are convex in x ∈ RN K . Proof. The convexity of (2.11) can be verified as follows. We define the function inside the inf ̺>0 {·} in (2.15) as Hk (x, ̺) , qk + ̺

N X n=1

7

Λk (−̺−1 xk,n ) − ̺ log ǫk ,

∀k.

(2.22)

For the reader’s convenience, both the theorem and a rough proof are provided in

Appendix A.


51

It is easy to verify the convexity of Hk (x, ̺) in (x, ̺), since the cumulant generating function is convex. Hence, Gk (x) in (2.15) is convex in x due to the preservation of convexity by minimization over ̺ > 0.

2.4.3

Algorithm Design

fslow . In P fslow , In this section, we propose an algorithm for solving Problem P 3 3

the STC (2.11) arises as a subproblem, which by itself requires a minimization

fslow cannot be trivover ̺. Hence, despite its convexity, the entire problem P 3

ially solved using standard solvers of convex optimization. This is due to the fact that the subproblem introduces difficulties, for example, in defining the barrier function in path-following algorithms or providing the (sub-)gradient in primal-dual methods (see [66] for details of these algorithms). Fortunately, fslow we can employ interior point cutting plane methods to solve Problem P 3

(see [67] for a survey). Before we delve into the details, let us briefly sketch the principles of the algorithm as follows. Suppose that we would like to find a point x that is feasible for (2.11)fslow , (2.13) and is within a distance of δ > 0 to an optimal solution x∗ of P 3

where δ > 0 is an error tolerance parameter (i.e., x satisfies ||x − x∗ || < δ). We maintain the invariant that at the beginning of each iteration, the feasible set is contained in some polytope (i.e., a bounded polyhedron). Then, we generate a query point inside the polytope and ask a “separation oracle” whether the query point belongs to the feasible set. If not, then the separation oracle will generate a so-called separating hyperplane through the query point to cut out the polytope, so that the remaining polytope contains the feasible set.8 Otherwise, the separation oracle will return a hyperplane through the query 8

Note that such a separating hyperplane exists due to the convexity of the feasible set [68].


52

Algorithm 1 Structure of the Proposed Algorithm fslow is a compact set X deRequire: The feasible solution set of Problem P 3 fined by (2.11)-(2.13).

1: 2:

Construct a polytope X 0 ⊃ X by (2.12)-(2.13). Set i ← 0.

Choose a query point (Subsection 2.4.3.1-1 ) at the ith iteration as xi by

computing the analytic center of X i . Initially, set x0 = e/K ∈ X 0 where e is an N-vector of ones. 3:

Query the separation oracle (Subsection 2.4.3.1-2 ) with xi :

4:

if xi ∈ X then

5:

generate a hyperplane (optimality cut) through xi to remove the part

of X i that has lower objective values 6: 7:

else generate a hyperplane (feasibility cut) through xi to remove the part of X i that contains infeasible solutions.

8:

end if

9:

Set i ← i + 1, and update X i+1 by the separation hyperplane.

10:

if termination criterion (Subsection 2.4.3.2) is satisfied then

11: 12: 13: 14:

stop else return to step 2. end if

point to cut out the polytope towards the opposite direction of improving objective values. We can then proceed to the next iteration with the new polytope. To keep track of the progress, we can use the so-called potential value of the polytope. Roughly speaking, when the potential value becomes large, the


53

polytope containing the feasible set has become small. Thus, if the potential value exceeds a certain threshold, so that the polytope is negligibly small, then we can terminate the algorithm. As will be shown later, such an algorithm will in fact terminate in a polynomial number of steps. We now give the structure of the algorithm. A detailed flow chart is shown in Fig. 2.6 for readers’ interest. 2.4.3.1

The Cutting-Plane-Based Algorithm

1) Query Point Generator: (Step 2 in Algorithm 1) In each iteration, we need to generate a query point inside the polytope X i . For algorithmic efficiency, we adopt the analytical center (AC) of the containing polytope as the query point [69]. The AC of the polytope X i = {x ∈ RN K : Ai x ≤ bi } at the ith iteration is the unique solution xi to the following convex problem: i

max

{xi ,si }

M X

log sim

(2.23)

m=1

s.t. si = bi − Ai xi . We define the optimal value of the above problem as the potential value of the polytope X i . Note that the uniqueness of the analytic center is guaranteed by P i i the strong convexity of the potential function si 7→ − M m=1 log sm , assuming

that X i is bounded and has a non-empty interior. The AC of a polytope can be viewed as an approximation to the geometric center of the polytope, and thus any hyperplane through the AC will separate the polytope into two parts with roughly the same volume. Although it is computationally involved to directly solve (2.23) in each iteration, it is shown in [70] that an approximate AC is sufficient for our purposes,


54

Initialize: The set X 0 : (A0 , b0 ) and x0 = e/K.

Oracle

̺∗ = arg inf [H(xi , ̺)] ̺>0

N

H(xi , ̺∗ ) ≤ 0

Adding feasibility cut (23)

Y

Adding optimality cut (24)

X

Update : (Ai+1,bi+1)

i+1

Compute the analytical center of X i+1 .

xi+1 , Ai+1, bi+1

Query Point Generator

xi+1 (Optional) Atkinson and Vaidya Modification [19] on (Ai+1 , bi+1 ).

N

Termination? Y Has any optimality cut been generated?

N

Y The problem is infeasible.

The problem is feasible. The optimal solution x∗ = xi .

End

fslow . Figure 2.6: Flow chart of the algorithm for solving Problem P 3 and that an approximate AC for the (i + 1)st iteration can be obtained from an approximate AC for the ith iteration by applying O(1) Newton steps.


55

2) The Separation Oracle: (Steps 3-8 in Algorithm 1) The oracle is a major component of the algorithm that plays two roles: checking the feasibility of the query point, and generating cutting planes to cut the current set. • Feasibility Check fslow in a condensed form as follows: We write the constraints of P 3 Gk (x) = inf {Hk (x, ̺)} ≤ 0, ̺>0

∀k

A0 x ≤ b0

(2.24) (2.25)

where   IN IN · · · IN  ∈ R(N +N K)×N K , A0 =  −IN K b0 = [eTN , 0TN K ]T ∈ RN +N K

with IN and eN denoting the N × N identity matrix and N-vector of ones respectively, and (2.25) is the combination9 of (2.12) and (2.13). Now, we first use (2.25) to construct a relaxed feasible set via X 0 = {x ∈ RN K : A0 x ≤ b0 }.

(2.26)

fslow by checking Given a query point x ∈ X 0 , we can verify its feasibility to P 3

if it satisfies (2.24), i.e., if inf ̺>0 {Hk (x, ̺)} is no larger than 0. This requires

solving a minimization problem over ̺ > 0. Due to the unimodality of Hk (x, ̺) in ̺, we can simply take a line search procedure, e.g., using Golden-section search or Fibonacci search, to find the minimizer ̺∗ . The line search is more 9

To reduce numerical errors in computation, we suggest normalizing each constraint in

(2.25).


56

efficient when compared with derivative-based algorithms, since only function evaluations10 are needed during the search. • Cutting Plane Generation In each iteration, we generate a cutting plane, i.e., a hyperplane through the query point, and add it as an additional constraint to the current polytope X i . By adding cutting plane(s) in each iteration, the size of the polytope keeps shrinking. There are two types of cutting planes in the algorithm depending on the feasibility of the query point. If the query point xi ∈ X i is infeasible, then a hyperplane called feasibility cut is generated at xi as follows: i,¯κ T u (x − xi ) ≤ 0, ||ui,¯κ ||

¯ ∀¯ κ ∈ K,

(2.27)

¯ = {k : Hk (xi , t∗ ) > 0, k = 1, 2, · · · , K} is the set of users whose where K chance constraints are violated, and i,¯ κ i,¯ κ i,¯ κ i,¯ κ T ui,¯κ = [u1,k , · · · , uN,1 , . . . , u1,K , · · · , uN,K ] ∈ RN K

is the gradient of Gκ¯ (x) with respect to x, i.e., ∂Hκ¯ (x, ̺∗ ) i,¯ κ uk,n = ∂xk,n xk,n =xi k,n

i

=

− W̺∗xlnk,n2 R ∞ pt ξ pt ξ W − ln 2 0 1+ ΓW N0 ln 1+ ΓW N0 σ1κ¯ exp − σξκ¯ dξ R∞ 0

i

1+

pt ξ ΓW N0

− W̺∗xlnk,n2

1 σk

.

exp − σξκ¯ dξ

The reason we call (2.27) a feasibility cut(s) is that any x which does not satisfy (2.27) must be infeasible and can hence be dropped. 10

The cumulant generating function Λk (·) in (2.14) can be evaluated numerically, e.g.,

using rectangular rule, trapezoid rule, or Simpson’s rule, etc.


57

If the point xi is feasible, then an optimality cut is generated as follows: T v (x − xi ) ≤ 0, (2.28) ||v|| T (t) (t) (t) (t) where v = − E{r1,1 }, · · · , −E{rN,1 }, . . . , −E{r1,K }, · · · , −E{rN,K } ∈ RN K fslow in (2.10) with respect to x. The is the derivative of the objective of P 3 reason we call (2.28) an optimality cut is that any optimal solution x∗ must

satisfy (2.28) and hence any x which does not satisfy (2.28) can be dropped. Once a cutting plane is generated according to (2.27) or (2.28), we use it to update the polytope X i at the ith iteration as follows X i = {x ∈ RN K : Ai x ≤ bi }. Here, Ai and bi are obtained by adding the cutting plane to the previous polytope X i−1 . Specifically, if the oracle provides a feasibility cut as in (2.27), then 

Ai−1



i−1

¯

 ∈ R(M +|K|)×N K , Ai =  (uik /||uik ||)T   i−1 b ¯  ∈ RM i−1 +|K| bi =  (uik /||uik ||)T xi

where Mi−1 is the number of rows in Ai−1 , and | · | is the number of elements contained in the given set; if the oracle provides an optimality cut as in (2.28), then 

Ai−1



i−1

 ∈ R(M +1)×N K , Ai =  (v/||v||)T   i−1 b  ∈ RM i−1 +1 . bi =  (v/||v||)T xi


2.4.3.2

58

Global Convergence & Complexity (Step 10 in Algorithm 1)

In the following, we investigate the convergence properties of the proposed algorithm. As mentioned earlier, when the polytope is too small to contain a full-dimensional closed ball of radius δ > 0, the potential value will exceeds a certain threshold. Then, the algorithm can terminate since the query point fslow . Such an is within a distance of δ > 0 to some optimal solution of P 3

idea is formalized in [70], where it was shown that the analytic center-based

cutting plane method can be used to solve convex programming problems in polynomial time. Upon following the proof in [70], we obtain the following result: Theorem 3. (cf. [70]) Let δ > 0 be the error tolerance parameter, and let m be the number of variables. Then, Algorithm 1 terminates with a solution x fslow and satisfies kx − x∗ k < δ for some optimal solution that is feasible for P 3 fslow after at most O((m/δ)2 ) iterations. x∗ to P 3

fslow within O((NK/δ)2 ) Thus, the proposed algorithm can solve Problem P 3

iterations. It turns out that the algorithm can be made considerably more efficient by dropping constraints that are deemed “unimportant” in [71]. By incorporating such a strategy in Algorithm 1, the total number of iterations needed by the algorithm can be reduced to O(NK log2 (1/δ)). We refer the readers to [67, 71] for details. 2.4.3.3

Complexity Comparison between Slow and Fast Adaptive OFDMA

It is interesting to compare the complexity of slow and fast adaptive OFDMA fslow and P fast , respectively. To obtain an optimal schemes formulated in P 3


59

√ solution to P fast , we need to solve a LP. This requires O( NKL0 ) iterations, where L0 is number of bits to store the data defining the LP [72]. At first glance, the iteration complexity of solving a fast adaptation P fast can be lower fslow when the number of users or subcarriers are large. than that of solving P 3

fslow needs to be solved for each However, it should be noted that only one P 3

adaptation window, while P fast has to be solved for each time slot. Since the length of adaptation window is equal to T time slots, the overall complexity of the slow adaptive OFDMA can be much lower than that of conventional fast adaptation schemes, especially when T is large. Before leaving this section, we emphasize that the advantage of slow adaptive OFDMA lies not only in computational cost reduction, but also in reducing control signaling overhead. We will investigate this in more detail in Sec. 2.4.5.

2.4.4

Problem Size Reduction

fslow can be reduced from In this section, we show that the problem size of P 3

NK variables to K variables under some mild assumptions. Consequently, the computational complexity of slow adaptive OFDMA can be markedly lower than that of fast adaptive OFDMA. In practical multicarrier systems, the frequency intervals between any two subcarriers are much smaller than the carrier frequency. The reflection, refraction and diffusion of electromagnetic waves behave the same across the (t)

subcarriers. This implies that the channel gain gk,n is identically distributed over n (subcarriers), although it is not needed in our algorithm derivations in the previous sections. (t)

When gk,n for different n are identically distributed, different subcarriers become indistinguishable to a user k. In this case, the optimal solution, if


60

fslow , we obtain the exists, does not depend on n. Replacing xk,n by xk in P 3 following formulation: f′ slow : P 3

max xk

s.t.

K X N X k=1 n=1

n o (t) xk E rk,n

inf qk + ̺NΛk (−̺−1 xk ) − ̺ log ǫk ≤ 0,

̺>0

K X k=1

∀k

xk ≤ 1,

xk ≥ 0,

∀k.

f′ slow is exactly the same as that of Note that the problem structure of P 3

fslow , except that the problem size is reduced from NK variables to K variP 3 ables. Hence, the algorithm developed in Sec. 2.4.3 can also be applied to solve

f′ slow , with the following vector/matrix size reductions: A0 = [eN , −IK ]T ∈ P 3

i,¯ κ T R(1+K)×K , b0 = [1, 0, · · · , 0]T ∈ R1+K in (2.25), ui,¯κ = [u1i,¯κ , · · · , uK ] ∈ RK (t) (t) T in (2.27), and v = − E{r1 }, · · · , −E{rK } ∈ RK in (2.28). Compared with

fslow , the iteration complexity of P f′ slow is now reduced to O(K log2 (1/δ)). P 3 3

Indeed, this can even be lower than the complexity of solving one P fast — √ O( NKL0 ), since K is typically much smaller than N in real systems. Thus, the overall complexity of slow adaptive OFDMA is significantly lower than that of fast adaptation over T time slots. Before leaving this section, we emphasize that the problem size reduction f′ slow does not compromise the optimality of the solution. On the other in P 3

fslow is more general in the sense that it can be applied to systems in hand, P 3

which the frequency bands of parallel subchannels are far apart, so that the channel distributions are not identical across different subchannels.


2.4.5

61

Numerical Results

In this section, we demonstrate the performance of our proposed slow adaptive OFDMA scheme through numerical simulations. We simulate an OFDMA system with 4 users and 64 subcarriers. Each user k has a requirement on its short-term data rate qk = 20bps. The 4 users are assumed to be uniformly distributed in a cell of radius R = 100m. That is, the distance dk between user k and the BS follows the distribution11 f (d) =

2d . R2

The path-loss exponent

γ is equal to 4, and the shadowing effect sk follows a log-normal distribution, i.e., 10 log10 (sk ) ∼ N (0, 8dB). The small-scale channel fading is assumed to be Rayleigh distributed. Suppose that the transmit power of the BS on each subcarrier is 90dB measured at a reference point 1 meter away from the BS, which leads to an average received power of 10dB at the boundary of the cell12 . In addition, we set W = 1Hz and N0 = 1, and the capacity gap is Γ = − log(5BER)/1.5 = 5.0673, where the target BER is set to be 10−4 . Moreover, the length of one slot, within which the channel gain remains unchanged, is T0 = 1ms.13 The length of the adaptation window is chosen to be T = 1s, implying that each window contains 1000 slots. Suppose that the path loss and shadowing do not change within a window, but varies independently from one ′ window to another. For each window, we solve the size-reduced problem P˜slow , 11

The distribution of user’s distance from the BS f (d) =

distribution of user’s position f (x, y) =

1 πR2 ,

2d R2

is derived from the uniform

where (x, y) is the Cartesian coordinate of the

position. 12 The average received power at the boundary is calculated by 90dB + −4 dB = 10dB due to the path-loss effect. 10 log10 100 1 9c 13 , where c is the speed of light, fc is the carrier The coherence time is given by T0 = 16πf cv frequency, and v is the velocity of mobile user. As an example, we choose fc = 2.5GHz, and if the user is moving at 45 miles per hour, the coherence time is around 1ms.


62

120 100

∆¯b = ¯b i − ¯b i − 1

80

60 40

20 0 −20

−40

0

5

10

15 Iteration

20

25

30

Figure 2.7: Trace of the difference of objective value ¯bi between adjacent iterations (ǫk = 0.2).

and later Monte-Carlo simulation is conducted over 61 independent windows ′ that yield non-empty feasible sets of P˜slow when ǫk = 0.1.

In Fig. 2.7 and Fig. 2.8, we investigate the fast convergence of the proposed algorithm. The error tolerance parameter is chosen as δ = 10−2 . In Fig. 2.7, we record the trace of one adaptation window14 and plot the improvement in the objective function value (i.e., system throughput) in each iteration, i.e., ∆¯b = ¯bi − ¯bi−1 . When ∆¯b is positive, the objective value increases with each

iteration. It can be seen that ∆¯b quickly converges to close to zero within only 27 iterations. We also notice that fluctuation exists in ∆¯b within the 14

The simulation results show that all the feasible windows appear with similar conver-

gence behavior.


63

first 11 iterations. This is mainly because during the search for an optimal solution, it is possible for query points to become infeasible. However, the feasibility cuts (2.27) then adopted will make sure that the query points in subsequent iterations will eventually become feasible. The curve in Fig. 2.7 verifies the tendency. As P˜slow is convex, this observation implies that the proposed algorithm can converge to an optimal solution of P˜slow within a small

number of iterations. In Fig. 2.8, we plot the number of iterations needed for convergence for different application windows. The result shows that the proposed algorithm can in general converge to an optimal solution of P˜slow within 35 iterations. On average, the algorithm converges after 22 iterations, where each iteration takes 1.467 seconds.15 Moreover, we plot the number of iterations needed for checking the feasibility of P˜slow . In Fig. 2.9, we conduct a simulation over 100 windows, which consists of 61 feasible windows (dots with cross) and 39 infeasible windows (dots with circle). On average, the algorithm can determine if P˜slow is feasible or not after 7 iterations. The quick feasibility check can help to deal with the admission of mobile users in the cell. Particularly, if there is a new user moving into the cell, the BS can adopt the feasibility check to quickly determine if the radio resources can accommodate the new user without sacrificing the current users’ QoS requirements. In Fig. 2.10, we compare the spectral efficiency of slow adaptive OFDMA with that of fast adaptive OFDMA16 , where zero outage of short-term data 15

We conduct a simulation on Matlab 7.0.1, where the system configurations are given as:

Processor: Intel(R) Core(TM)2 CPU [email protected] 2.27GHz, Memory: 2.00GB, System Type: 32-bit Operating System. 16 For illustrative purpose, we have only considered Pfast as one of the typical formulations of fast adaptive OFDMA in our comparisons. However, we should point out that there


64

40 35

Number of Iteration

30

25 20

15 10 5

0

10

20

30 40 Window Number

50

60

Figure 2.8: Number of iterations for convergence of all the feasible windows (ǫk = 0.2).

rate requirement is ensured for each user. In addition, we take into account the control overheads for subcarrier allocation, which will considerably affect the system throughput as well. Here, we assume that the control signaling overhead consumes a bandwidth equivalent to 10% of a slot length T0 every time SCA is updated [64]. Note that within each window that contains 1000 slots, the control signaling has to be transmitted 1000 times in the fast adaptation scheme, but once in the slow adaptation scheme. In Fig. 2.10, the line with circles represents the performance of the fast adaptive OFDMA scheme, while are some work on fast adaptive OFDMA which impose less restrictive constraints on user data rate requirement. For example, in [58], it considered average user data rate constraints which exploits time diversity to achieve higher spectral efficiency.


65

30 feasible infeasible

Number of Iteration

25

20

15

10

5

0

10

20

30

40 50 60 Window Number

70

80

90

100

Figure 2.9: Number of iterations for feasibility check of all the windows (ǫk = 0.2).

that with dots corresponds to the slow adaptive OFDMA. The figure shows that although slow adaptive OFDMA updates subcarrier allocation 1000 times less frequently than fast adaptive OFDMA, it can achieve on average 71.88% of the spectral efficiency. Considering the substantially lower computational complexity and signaling overhead, slow adaptive OFDMA holds significant promise for deployment in real-world systems. As mentioned earlier, P˜slow is more conservative than the original problem Pslow , implying that the outage probability is guaranteed to be satisfied if

subcarriers are allocated according to the optimal solution of P˜slow . This is illustrated in Fig. 2.11, which shows that the outage probability is always lower than the desired threshold ǫk = 0.1.


66

18


16

fast adaptation slow adaptation (ǫ k = 0. 1)

14 12 10 8 6 4 2 0

10

20

30 40 Window Number

50

60

Figure 2.10: Comparison of system spectral efficiency between fast adaptive OFDMA and slow adaptive OFDMA.

Fig. 2.11 shows that the subcarrier allocation via P˜slow could still be quite conservative, as the actual outage probability is much lower than ǫk . One way to tackle the problem is to set ǫk to be larger than the actual desired value. For example, we could tune ǫk from 0.1 to 0.3. By doing so, one can potentially increase the system spectral efficiency, as the feasible set of P˜slow is enlarged. A question that immediately arises is how to choose the right ǫk , so that the actual outage probability stays right below the desired value. Towards that end, we can perform a binary search on ǫk to find the best parameter that satisfies the requirement. Such a search, however, inevitably involves high computational costs. On the other hand, Fig. 2.12 shows that the gain in spectral efficiency by increasing ǫk is marginal. The gain is as little as 0.5


67

outage probability of user 1 0.1 0.05 0

10

20 30 40 outage probability of user 2

50

60

10


50

60

10


50

60

10

20

50

60

0.1 0.05 0 0.1 0.05 0 0.1 0.05 0

30 ǫ k = 0. 1

40 ǫ k = 0. 3

Figure 2.11: Outage probability of the 4 users over 61 independent feasible windows.

bps/Hz/subcarrier when ǫk is increased drastically from 0.05 to 0.7. Hence, in practice, we can simply set ǫk to the desired outage probability value to guarantee the QoS requirement of users. In the development of the STC (2.11), we considered that the channel gain gk,n are independent for different n’s and k’s. While it is true that channel fading is independent across different users, it is typically correlated in the frequency domain. We investigate the effect of channel correlation in frequency domain through simulations. A wireless channel with an exponential decaying power profile is adopted, where the root-mean-square delay is equal to 37.79ns. For comparison, the curves of outage probability with and without frequency correlation are both plotted in Fig. 2.13. We choose the tolerance parameter


68

5.5


5.4 5.3 5.2 5.1 5 4.9 4.8 4.7 4.6 4.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ǫk

Figure 2.12: Spectral efficiency versus tolerance parameter ǫk . Calculated from the average overall system throughput on one window, where the long-term average channel gain σk of the 4 users are −65.11dB, −56.28dB, −68.14dB and −81.96dB, respectively.

to be ǫk = 0.3. The figure shows that with frequency-domain correlation, the outage probability requirement of 0.3 is violated occasionally. Intuitively, such a problem becomes negligible when the channel is highly frequency selective, and is more severe when the channel is more frequency flat. To address the problem, we can set ǫk to be lower than the desired outage probability value17 . For example, when we choose ǫk = 0.1 in Fig. 2.13, the outage probabilities all decreased to lower than the desired value 0.3, and hence the QoS requirement 17

Alternatively, we can divide N subcarriers into

N Nc

subchannels (each subchannel consists

Nc subcarriers), and represent each subchannel via an average gain. By doing so, we can treat the subchannel gains as being independent of each other.


69

outage probability of user 1 0.4 0.3 0.2 0.1 0

10


50

60

10


50

60

10


50

60

10

20

50

60

0.4 0.3 0.2 0.1 0

0.4 0.3 0.2 0.1 0

0.4 0.3 0.2 0.1 0

independent(ǫk = 0.3)

30

40

correlated(ǫk = 0.3)

correlated(ǫk = 0.1)

Figure 2.13: Comparison of outage probability of 4 users with and without frequency correlations in channel model.

is satisfied (see the line with dots).


2.5

70

Summary

This chapter proposed a framework of slow adaptive OFDMA that allocates the subcarriers to multiple mobile users on a much slower timescale than that of channel fluctuation. The proposed schemes can achieve a throughput close to that of fast adaptive OFDMA schemes, while significantly reducing the computational complexity and control signaling overhead. Three types of stochastic optimization problems are formulated according to the applications and QoS requirements of users, including: • Average system throughput maximization with average user date rate constraints for elastic traffic. The results show that the computational costs and control overhead are greatly reduced compared with the conventional fast adaptive OFDMA. By tuning the timescale of SCA, the proposed scheme can provide us a flexible tradeoff between spectral efficiency and computational complexity as well as overhead. • Average system throughput maximization with average user date rate constraints for inelastic traffic. A tradeoff between the spectral efficiency and outage probability of data rate requirement is offered by tuning the parameter which is related to the size of assumed uncertainty set. • Long-term average system throughput maximization with probabilistic constraints on short-term user data rate. We constructed safe tractable constraints for the chance constrained formulation, and developed a polynomial-time algorithm to compute the optimal solution. We show that the proposed scheme satisfies user data rate requirement with high probability. A tradeoff is present between spectral efficiency and the rate outage by tuning the tolerance parameter (i.e., the maximum out-


71

age probability allowed by each user). The proposed slow adaptive OFDMA schemes can be viewed as an initial attempt to apply stochastic optimization methodologies, e.g., the chance constrained programming, to wireless system designs. Indeed, stochastic constraints arise quite naturally in many wireless systems due to the randomness in channel conditions. In the following chapters, we will demonstrate the potential of such promising optimization technique with further applications in wireless communications.

Chapter 3 Dynamic Antenna-and-Power Allocation in Composite Radio MIMO Networks Multiple-input multiple-output (MIMO) system is a key technology component for future wireless networks. The use of multiple transmit and receive antennas significantly enhances the system network capacity and transmission reliability [28,30,73]. MIMO also affords an increase in the cell coverage area and cell-edge throughput which translates to fewer base stations (BS) for a given coverage area, or conversely, more coverage per BS. Besides its substantial gain in pointto-point communications, MIMO technology has a greater potential in multiuser networks by exploiting spatial and multi-user diversity [32, 74]. These benefits have spurred significant research and development efforts in smart antennas leading to MIMO technology being adopted as a key component in upcoming wireless standards, e.g., IEEE 802.11n WLAN [75] and 3GPP LTE [76, 77].

72

Chapter 3. Dynamic Antenna-and-Power Allocation in Composite Radio MIMO Networks 73

Future wireless networks will employ the use of multiple wireless standards, or radio access technologies (RATs) that are each designed and intended for specific purposes. For example, current commercial mobile communication devices contain GSM, GPRS, EDGE, UMTS, WCDMA, HSPA, LTE, LTE-A, CDMA, EV-DO, WiMAX, TD-SCDMA, BT, WLAN, GPS, FM, DTV radios and emerging technologies such as RFID, NFC, etc., for voice, video, data, localization, and a host of other applications [22, 78]. The multiple cellular standards are supported for instance due to legacy reasons of deployed networks, while other wireless standards are supported for dedicated use cases and application. Concurrent operation and interoperability of the various RATs is also a requirement. Integration strategies and single-chip solutions widely available today enable the terminal complexity to fit into smaller and smaller form factors [79]. The inclusion of the multitude of heterogeneous radios on a single terminal creates significant operational issues, particularly under conditions of concurrency where the radios operate simultaneously. Foremost, the number of antennas on a terminal is limited due to size and space limitations, as well as coupling issues between antennas. The addition of multi-order MIMO requirements as specified in standards such as 802.11, 802.16, and LTE gives rise to, e.g., 4 × 4 and 8 × 8 antenna configurations. Dedicated antennas for each radio quickly become prohibitive and exhaust the space limitations of the mobile terminal as the number of radios and MIMO order increases. As such, antenna sharing and allocation amongst radios is desirable. To support the concurrent operation of multiple radios, we consider a fixed set of antennas on each terminal and dynamically allocate them based on operating conditions. The antenna array is assumed to consist of reconfigurable


elements that are controlled by matching networks such that their operational bandwidth and frequency characteristics are tunable to the particular mode required by each radio [80]. Further, the antenna array is reconfigurable so that the mapping from antenna elements to radios is fully connected [81]. Moreover, we consider a network of mobile terminals, in the context of a peerto-peer network deployed for communication content such as voice, video, data, messaging, or localization information. Each mobile terminal supports multiple orthogonal frequency division multiplexing (OFDM) RATs, agnostic to a particular air interface. We refer such system supporting multiple radios on each single terminal to as composite radio networks [82–86]. The current literature addresses antenna selection as a technique to select a subset of antennas for communication due to the lack of dedicated RF chains to associate with each antenna [36,37,87–91]. Despite its relevance to our problem, such technique is difficult to be applied to support multiple radios. First, its solution set quickly expands as the number of radios or antennas increases, which renders the exhaustive search infeasible. Second, the antenna selection has to be frequently updated once the channel changes, which requires a huge amount of signaling overhead due to the rapid wireless channel fluctuation. To address these issues, we consider allocating the antennas based on the channel statistics. Since the channel distribution varies much slower than the instantaneous channel condition does, the update of antenna allocation decision can be much less frequent, which drastically reduces the computational complexity and signaling overhead. Given that many wireless applications, e.g., inelastic traffic, can tolerate occasional dip in their quality of service (QoS), we impose the constraint that the short-term data rate of each radio transmission satisfies its minimum require-


ment with high probability. Such problem has a natural formulation of chance constrained program [13], which is yet known as computationally intractable. The main reason is that the convexity of the probabilistic constraint is often hard to verify. To tackle the problem, we construct a safe tractable constraint (STC), of which the solution guarantees the satisfaction of the probabilistic constraint. Moreover, we consider the antenna allocation together with power adaptation for each radio transmission. Such joint optimization is more challenging due to the existence of mixed-integer variables, i.e., number of antennas is integer. In this chapter, we propose a dynamic resource allocation scheme for MIMO networks, where the transmit terminal allocates its antennas to different radios and meanwhile adapts the power for each radio transmission. The key contributions are summarized as follows: • We design an adaptive antenna-and-power allocation scheme to support concurrent operation of multiple radios on a single terminal. The allocation decision is adapted to channel statistics instead of instantaneous channel information, which drastically reduces the computational complexity and signaling overhead. • We formulate the dynamic resource allocation problem into a stochastic program, where the long-term system throughput is maximized while the short-term data rate of each radio satisfies a minimum requirement with high probability. • We propose an approach to determine the feasibility of the antenna-andpower allocation problem by exploiting the monotonicity in the formulation. The proposed method can efficiently find all the feasible solutions of antenna allocation.


• We develop an optimal algorithm to find the best antenna-and-power allocation decision. The results show that the proposed scheme outperforms the uniform allocation which equally assigns antennas and power to each radio, and a larger portion of throughput gain is attributed to antenna allocation. The rest of the chapter is organized as follows. In Sec. 3.1, we describe the system model and introduce the resource allocation scheme in MIMO networks. In Sec. 3.2, we formulate the resource allocation problem into a chance constraint program. In Sec. 3.3, we propose an algorithm to determine the problem feasibility and provide all the feasible antenna allocation decisions. In Sec. 3.4, we develop an algorithm to obtain the optimal allocation decision. In Sec. 3.5, the performance of proposed scheme is investigated through extensive simulations. Finally, the chapter is concluded in Sec. 3.6.

3.1

System Model

This chapter considers a wireless MIMO network, in which terminals employ composite radio containing different types of radio modules.

3.1.1

Composite Radio System

We let m and k denote the terminal and radio type, respectively. We consider the scenario of M + 1 terminals composed of a transmitter indexed by m = 0 and M receivers indexed by m ∈ M = {1, 2, · · · , M}. There are K radio types, indexed by k ∈ K = {1, 2, · · · , K}, where the transmitter contains all types of radios and receivers contain possibly different subset of radios. We let (k, m) denote the radio module of radio type k in the terminal m. For the


convenience, the sets of radio types and terminals are listed in Table 3.1. Set

Definition

K = {1, 2, · · · , K}

the set of all the radio types

Km ⊂ K

the set of radio types contained in terminal m

M = {1, 2, · · · , M}

the set of all the receiver terminals

Mk ⊂ M

the set of receiver terminals containing radio type k

Table 3.1: Notations of the sets of radio type and terminals. The terminals in composite radio systems are equipped with multiple antennas. There are Ltot antennas at the transmitter side, and Lkm antennas at the radio module (k, m). Due to the physical limitations1 , each antenna can only connect to a single radio type at a given time. Therefore, the terminal 0 has to allocate transmit antennas among different radio types. Fig. 3.1 gives a general illustration of the composite radio system.

3.1.2

Channel Model

We consider that radio module (k, m) utilizes Nkm tones, where each tone is assigned to a unique radio module such that there is no interference among the radio modules. Given that the terminal 0 allocates Lk antennas to its radio type k, it forms a MIMO channel between radio module (k, 0) and (k, m) for each tone n, characterized by an Lkm ×Lk matrix αm Hkmn . The parameter αm 1

We consider that the frequency isolation between two radio modules can be larger than

the working bandwidth of an antenna. In such case, the antenna can only be tuned to operate at one frequency to achieve its best performance.

Chapter 3. Dynamic Antenna-and-Power Allocation in Composite Radio MIMO Networks 78 Terminal 1 ...

Radio Module (1, 1) Type 1

...


...

...

...

Radio Module (K, 1) Type K

...


...


Terminal 2 Terminal 0 Type 1

...

Radio Module (1, 0)

Radio Module (2, M ) Type 2

...

... Radio Module (1, M ) Type 1

...

...

Radio Module (K, 2) Type K

...

...

...

...

... Radio Module (K, 0) Type K

...


Terminal M

...

...

...

Radio Module (K, M ) Type K

Figure 3.1: A general model of the composite radios system. For simple illustration, we plot the receive terminals containing all types of radio. In general, terminal 1 to M contains possibly different subset of radios. characterizes the large-scale channel variation including path loss and shadowing effect, and the matrix Hkmn characterizes the fast fading fluctuation. We model αm to be independent across M terminals, Hkmn to be i.i.d. across Nkm tones, and the entries of Hkmn to be i.i.d. complex Gaussian random variable, i.e., [Hkmn ]i,j ∈ CN (0, 1).

3.1.3

Dynamic Antenna-and-Power Allocation

Besides allocating the transmit antennas to different radios types, the transmitting terminal can also adapt the transmit power of each radio. Fig. 3.2


Type 1 Radio Module

(1, 1)

(1, 2)

N11 Tn’s N12 Tn’s

P11 L2

P12

...

Radio Module (1, M ) N1M Tn’s

... P 1M

Type 2 Radio Module

Radio Module

(2, 1)

(2, 2)

N21 Tn’s N22 Tn’s

P21

P22

...

Radio Module (2, M ) N2M Tn’s

... P 2M

.

..

...

Transmit Antenna

L1

Radio Module

Type K

LK

Radio Module

Radio Module

(K, 1)

(K, 2)

NK1 Tn’sNK2 Tn’s

Transmit Power

PK1

...

Radio Module (K, M ) NKM Tn’s

PK2 ... PKM

Figure 3.2: The radio resource allocation in composite radios system of which the general model is given in Fig. 3.1 (Tn = Tone). provides an illustration of the resource allocation in composite radio system corresponding to the general model in Fig. 3.1. The horizontal axis represents the transmit power, which is adjustable for each radio module; the vertical axis represents the number of transmit antennas, which is adjustable for each radio type. In the figure, different radios modules with the same radio type have been placed in the same row, and they will receive the signals from the same set of transmit antennas allocated to that same radio type. In this chapter, we propose a dynamic antenna-and-power allocation scheme for the composite radio MIMO networks. The allocation of both antenna and transmit power is adapted to the long-term channel variations. In particular, the duration of each adaptation is considered to be large compared with that of fast fading fluctuation so that the channel fading process over the window


is ergodic; but smaller than the coherence time of large-scale channel variation so that the path loss and shadowing effect are considered to be fixed in each window. Such scheme relies only on the statistical characterization of channel fading, and does not require the exact CSI to perform resource allocation. The adaptation based on large-scale fading has recently drawn more attention due to its substantial practicality, e.g., [60,62,92,93]. Compared with conventional designs which require adaptation of resource allocation to the instantaneous channel condition, it can be carried out with much lower computational complexity and signaling overhead.

3.2 3.2.1

Problem Formulation MIMO Channel Capacity

The data rate of a link from radio modules (k, 0) and (k, m) employing Lk transmit and Lkm receive antennas, respectively, can be written as [31] 2 Pkm αm † Hkmn Hkmn , Rkm (Lk , Pkm) = W log2 det I + SN W NkmLk n=1 N km X

(3.1)

∀k ∈ Km , ∀m ∈ M where W is the bandwidth of each tone, SN is the noise spectrum density (W/Hz). Note that the quantity

Pkm α2m SN W Nkm Lk

in (3.1) is the average received

SNR per tone and per antenna, and Rkm (Lk , Pkm) is the summation of all the data rates over Nkm i.i.d. tones.


3.2.2

Chance Constrained Formulation

Consider that each radio module (k, m) has a minimum date rate requirement qkm , i.e., if Rkm (Lk , Pkm ) < qkm , radio module (k, m) is in a rate outage, and the probability of rate outage over an adaptation window of radio module (k, m) is defined as Pout km

, Pr Rkm (Lk , Pkm) < qkm .

Due to occasional deep fades in wireless channel, such outage is inevitable in slow adaptation systems, unless the minimum rate requirement is considerably low. However, inelastic applications, such as voice and multimedia that are concerned with short-term QoS, can often tolerate occasional dips in the instantaneous date rate, and it is adequate in many applications as long as the minimum rate requirement is satisfied with sufficiently high probability. With the above consideration, we formulate the slow antenna-and-power allocation problem as follows: max

{Lk ∈Z,Pkm ∈R}

s.t.

X X

m∈M k∈Km

EH Rkm (Lk , Pkm )

Pr Rkm (Lk , Pkm ) < qkm X

Lk = Ltot ;

k∈K

X X

m∈M k∈Km

Lk ≥ Lk ,

Pkm ≤ P tot ;

≤ ǫkm ,

(3.2) ∀k ∈ Km , ∀m ∈ M (3.3)

∀k ∈ K

Pkm ≥ P km ,

(3.4) ∀k ∈ Km , ∀m ∈ M (3.5)

where qkm and ǫkm ∈ [0, 1], respectively, are the minimum required rate and the maximum allowable outage probability of radio module (k, m), Ltot is the total number of transmit antennas at terminal 0, Lk is the minimum required


number of transmit antennas for radio type k, P tot is the total transmit power available at terminal 0, and P km is the minimum required transmit power for radio module (k, m). In the above formulation, we aim at seeking the solution for both antenna and power allocation that maximizes the expected system throughput while satisfying minimum rate requirement of each radio module with high probability. Specifically, the chance constraint in (3.3) implies that the instantaneous date rate of radio module (k, m) must be higher than qkm with probability larger than 1 − ǫkm . To highlight the probabilistic nature of (3.3), such an optimization problem (3.2)–(3.5) is referred to as chance constrained program.

3.2.3

Safe Tractable Formulation

Despite its relevance to real applications, the chance constrained program is known to be an intractable optimization problem. In general, there are two major hurdles: first, the probability in the chance constraints is difficult to compute efficiently even when the distributions of underlying random variables are given, e.g., Rayleigh fading in our case of (3.3); second, the verification of convexity of the feasible set defined by the chance constraints is not attainable, making the global optimization intractable. To circumvent the above hurdles, we replace the chance constraints in (3.3) with safe tractable constraint (STC) satisfying two conditions: STC is efficiently computable; the feasible set of STC is contained in that of the original chance constraint. The proposed STC is given in the following. ˘ k and P˘km such Proposition 3. Given ǫkm > 0, suppose that there exists L


that n o −1 ˘ k , P˘km ) − ̺ log ǫkm ≤ 0, inf qkm + ̺ log EH exp − ̺ Rkm (L

̺>0

(3.6)

∀k ∈ Km , ∀m ∈ M.

˘ m , P˘km ) must satisfy Then, the allocation decision (L ˘ ˘ Pr Rkm (Lk , Pkm ) < qkm ≤ ǫkm , ∀k ∈ Km , ∀m ∈ M.

(3.7)

Proof. The function inside inf ̺>0 {·} in (3.6) is equal to

n o ˘ k , P˘km ) − ̺ log ǫkm . ̺ log EH exp ̺−1 qkm − Rkm (L

If follows that (3.6) is equivalent to n o −1 ˘ ˘ inf ̺EH exp ̺ qkm − Rkm (Lk , Pkm ) −̺ǫkm ≤ 0, ̺>0

∀k ∈ Km , ∀m ∈ M. (3.8)

According the the Bernstein Approximation Theorem [17] in Appendix A, the chance constraints (3.7) holds if there exists a ̺ > 0 satisfying (3.8). Thus, the validity of (3.7) is guaranteed by the validity of (3.6). Now, we let l = [L1 , L2 , · · · , LK ]T ∈ RK , p = [· · · , P1m , · · ·, · · · , · · · , P2m , · · ·, · · · , · · · , PKm, · · ·]T ∈ RK | {z } {z } {z } | | m∈M1

m∈M2

tot

m∈MK

l = [L1 , L2 , · · · , LK ]T ∈ RK ,

p = [· · · , P 1m , · · ·, · · · , · · · , P 2m , · · ·, · · · , · · · , P Km , · · ·]T ∈ RK | | | {z } {z } {z } m∈M1

with K tot = 2

P

m

|Km | =

m∈M2

P

k

tot

m∈MK

|Mk |. We first derive the closed-form expressions2

The derivation is based on the result in [31] given in Appendix B.


for the functions in (3.2) and (3.6) as X X

EH Rkm (Lk , Pkm )

m∈M k∈Km

=

X X

m∈M k∈Km



min {Lk ,Lkm }

Nkm W Lf (Lk ) · −1

X l=1

qkm + ̺ log EH exp −̺ Rkm (Lk , Pkm )



det(Al ) ,

(3.9)

− ̺ log ǫkm

= qkm + ̺Nkm log Lf (Lk ) det(B) − ̺ log ǫkm ,

(3.10)

where the entries of the matrices Al ∈ Rmin {Lk ,Lkm }×min {Lk ,Lkm } and B ∈ Rmin {Lk ,Lkm }×min {Lk ,Lkm } are  R ∞ max {L ,L }−min {L ,L }+j+i−2 −ξ  Pkm α2m  k km k km ξ · e · log2 1 + Lk SN W Nkm ξ dξ,  0    [Al ]i,j = if l = j      R ∞ ξ max {Lk ,Lkm }−min {Lk ,Lkm }+j+i−2 · e−ξ dξ, if l 6= j 0 − ̺Wln 2 Z ∞ 2 Pkmαm max {Lk ,Lkm }−min {Lk ,Lkm }+j+i−2 −ξ dξ, ξ [B]i,j = ξ ·e · 1+ Lk SN W Nkm 0 and 

min {Lk ,Lkm }

Lf (Lk ) = 

Y i=1

−1 min {Lk , Lkm } − i ! max {Lk , Lkm } − i ! .

(3.11)

For notational simplicity, we define the right side of (3.9) and (3.10) as   min {Lk ,Lkm } X X X Nkm W Lf (Lk ) · det(Al ) , (3.12) F (l, p) , m∈M k∈Km

l=1

Gkm (l, p, ̺) , qkm + ̺Nkm log Lf (Lk ) det(B) − ̺ log ǫkm .

(3.13)


Replace (3.3) with the proposed STC, we obtain the following formulation max F (l, p)

(3.14)

{l,p}

s.t.

inf {Gkm (l, p, ̺)} ≤ 0,

̺>0

tot 1T Kl = L ,

∀k ∈ Km , ∀m ∈ M

l ≥ l,

tot 1T , K tot p ≤ P

p ≥ p.

(3.15) (3.16) (3.17)

The above optimization problem is referred to as safe tractable constrained program (STCP). In the remanence of the chapter, we will focus on solving the optimal solution to this STCP due to the computational tractability. The STCP in (3.14)–(3.17) is more tractable since objective and constraints are in explicit form, meanwhile its solutions is guaranteed to be feasible to the optimization problem (3.2)–(3.5). Despite the tractability of STCP, the optimal solution of (3.14)–(3.17) is still not trivial to obtain due to the existence of mixed-integer variables, i.e., Lk is integer while Pkm is continuous. Instead of directly tackling the STCP, we first consider a feasibility problem (3.15)–(3.17) in the following section. We will then find the optimal solution in Sec. 3.4.

3.3

Search for Feasible Solutions

The STCP depends on specific radio resources and channel realizations in the network, and finding a feasible solution implies that these radio resources are sufficient to accommodate the data rate requirements of all the radios in the network. In the following we exploit the monotonicity of objective and constraints as function of Pkm , and develop an algorithm to 1) check the feasibility of (3.15)–(3.17) and 2) find its feasible solutions. We describe the details of Algorithm 2.


Algorithm 2 Algorithm for Finding Feasible Solutions Require: matrix Smin ∈ RK×L

tot

Lk = 1, · · · , Ltot )

1: 2: 3:

with each entry smin k,Lk = ∞ (k = 1, · · · , K,

for k ∈ K do for m ∈ Mk do

for Lk ∈ {Lk , · · · , Ltot } do

Take a bisection search on P ∈ [P km , P tot ] to find the root P ∗

4:

of the equation inf qkm + ̺Nkm log Lf (Lk ) det(B) − ̺ log ǫkm = 0

̺>0

if P ∗ exists then

5:

min set Pkm (Lk ) ← P ∗

6:

else

7:

min set Pkm (Lk ) ← ∞

8:

end if

9: 10:

end for

11:

end for

12:

Set smin k,Lk ←

13: 14:

end for

P

m∈Mk

min Pkm (Lk ), ∀Lk

min

Find a sequence of K entries in S such that

n oK min ˘ k ∈ {Lk , · · · , Ltot }, with L : sk,L˘ k

k=1

 PK min  tot    ˘k ≤ P k=1 sk,L   PK tot ˘ k=1 Lk = L      ˘k ≥ L L k


Algorithm 2 Algorithm for Finding Feasible Solutions (Cont’d) oK n min 15: if the sequence s ˘ exists then k,L k

17:

 Pkm = P min(L ˘ k ), ∀k ∈ Km , ∀m ∈ M km

else

the problem (3.14)–(3.17) is infeasible.

18: 19:

k=1

the problem (3.14)–(3.17) is feasible, and feasible solution is   ˘ k , ∀k ∈ K Lk = L

16:

end if

3.3.1

Algorithm Design

3.3.1.1

min Determine the Minimum Power Pkm (L) (Step 4–8 in Algo-

rithm 2) Consider the problem of finding minimum transmitting power that satisfies (3.15) as follows: min Pkm (Lk ) = arg min Pkm (3.18) s.t inf qkm + ̺Nkm log Lf (Lk ) det(B) − ̺ log ǫkm ≤ 0 ̺>0

(3.19)

P km ≤ Pkm ≤ P tot . Note that the function in (3.19) is monotonically decreasing in Pkm . This is more easily seen from the left side of (3.10) since the date rate Rkm (Lk , Pkm) is monotonically increasing in Pkm . For a fixed number of transmit antennas Lk ∈ {Lk , Lk + 1, · · · , Ltot }, monotonicity property enables the use of bisection search [94] to find the minimum transmit power satisfying the data rate requirement of radio module (k, m).

Chapter 3. Dynamic Antenna-and-Power Allocation in Composite Radio MIMO Networks 88 tot

min We remark that the sequence of minimum power {Pkm (Lk )}LLk =Lk is in a

descending order, since more transmit power is needed to satisfy the data rate requirement when less number of transmit antennas is used. 3.3.1.2

Generate the Minimum Total Power Matrix Smin (Step 12 in Algorithm 2)

Since the radio modules with the same radio type will receive signals from the same set of transmit antennas, we can sum up the minimum transmit min power Pkm (Lk ) over m ∈ Mk to obtain the minimum total transmit power to

guarantee the data rate requirements of all the radio modules of radio type k, i.e., smin k,Lk ,

X

min Pkm (Lk ).

m∈Mk tot

L By doing so, we obtain descending sequence {smin k,Lk }Lk =Lk for each k ∈ K.

Note that there are K sequence corresponding to each radio type, and these sequences have different lengths when Lk for k ∈ K are different. We alleviate L −1

k this by adding {smin k,Lk = ∞}Lk =1 to the beginning of each sequence, so that

all the sequences have the same length. By stacking all the K sequences, we obtain the following the minimum total power matrix as follows:   min min min s1,1 s1,2 · · · s1,Ltot    smin smin · · · smintot  tot 2,2  2,1 2,L  Smin =   ∈ RK×L .   ···     min min smin s · · · s K,1 K,2 K,Ltot

(3.20)


3.3.1.3

Obtain Feasible Solution via Matrix Smin (Step 14–19 in Algorithm 2)

Finding a solution of the feasibility problem (3.15)–(3.17) now leads to the ˘ k }K , problem of picking a column index from each row to form a sequence {L k=1 such that corresponding sequence satisfies the following constraints:   P  tot ˘  K k=1 Lk = L   ˘k ≥ L . L k

Note that the number of candidate sequence

˘ k }K {L k=1



is 

Ltot −

The feasible solution is the candidate sequence that satisfies

3.4

(3.21)

 Lk + K − 1 3 . K −1

PK

k=1

PK

min ˘k k=1 sk,L

≤ P tot .

Approach to Optimal Solution

While we have formulated our problem with explicit expressions for objective and constraints, finding the optimal solution is still difficult since it requires the search in both discrete and continuous spaces due to the mixed-integer variables, i.e., Lk is integer variable and Pkm is real variable. However, Algorithm 2 enables us to find all the feasible solutions of antenna allocation, i.e., ˇlj = [L ˇ 1,j , L ˇ 2,j , · · · , L ˇ K,j ]T ,

j = 1, 2, · · · , J,

(3.22)

˘ k ≥ L , i.e., each radio types should be allocated at least L Due to the constraint L k k PK tot among K radio antennas, there are totally L − k=1 Lk antennas left to be allocated  3

types. Thus, the number of combinations is



(Ltot −

PK

Lk ) + K − 1 . K −1

k=1


where J is the number of feasible combinations. Given an antenna allocation, the optimization over power is a continuous problem, i.e., ˇ j = arg max F (ˇlj , p) p

(3.23)

p

s.t.

inf Gkm (ˇlj , p, ̺) ≤ 0,

̺>0

∀k ∈ Km , ∀m ∈ M

p ∈ P,

where P is a linear set given by      T tot   1 P tot K tot  . p ≤  P , p ∈ RK :   −IK tot −p 

(3.24) (3.25)

(3.26)

Now, we investigate the convexity of the above problem in the following proposition. Proposition 4. The problem (3.23)–(3.25) is a convex optimization problem. Proof. We first verify the concavity of the objective (3.23). Note that the function Rkm (Lk , Pkm) in (3.1) is concave in Pkm , due to the concavity of a general function log det(I + tX) in t [11]. Since the expectation and linear summation preserve convexity, F (l, p) is convex in p. The convexity of the STC (3.24) can be verified as follows. First, the function log EH {exp(−Rkm (Lk , Pkm ))} is convex in Pkm , since it is convex and nonincreasing in Rkm (Lk , Pkm ), and Rkm (Lk , Pkm) is concave in Pkm . Note that ̺ log EH {exp(−̺−1 Rkm (Lk , Pkm ))} is its perspective function [11], and hence, is convex in (Pkm , ̺). Since convexity is preserved by the minimization over ̺ > 0, the STC (3.24) is a convex constraint. Since the constraint (3.25) is linear, we conclude that the problem (3.23)– (3.25) is convex.


3.4.1

Cutting-Plane-Based Algorithm

In this section, we propose an algorithm to solve the continuous convex optimization problem (3.23)-(3.25). Note that the constraint (3.24) arises as a subproblem of minimization over ̺, hence, the entire problem cannot be trivially solved using standard convex optimization solvers. In the following, we employ the interior cutting plane methods [67, 72] to solve the problem. A high-level flow chart of such algorithm is provided in Fig. 3.3. We give the details in Algorithm 3, and its major steps are elaborated in the following. 3.4.1.1

Query Point Generator (Step 3 in Algorithm 3)

We adopt the analytical center (AC) of the containing polytope as the query point. The AC of the polytope P i = {p ∈ RK

tot

: Ai p ≤ bi } at the ith iteration

is the unique the optimal solution pi to the following convex program: max

{pi ,si }

i 1T dim(si ) log(s )

(3.29)

s.t. si = bi − Ai pi . The AC can be viewed as an approximation to the geometric center of the polytope. The objective value is called as potential value of the polytope i P i , i.e., Ωi = 1T dim(si ) log(s ). Generally, the potential value decreases as the

volume of the polytope shrinks. 3.4.1.2

Separation Oracle (Step 4–10 in Algorithm 3)

The separation oracle plays two roles: checking the feasibility of the query point, and generating cutting planes to cut the current set. • Feasibility Check (Step 4-5 in Algorithm 3)


Initialize: Construct a ploytope P X00 containing the feasible set; Set iget0 i←0 i

p Generate a query point xi inside the polytope Xi Pi

Separation Oracle N

If the query point pxii is feasible

Generate an optimality cut through p xii to remove part of P Xii with lower objective value

Y

Generate a feasibility cut through p xii to remove part of P Xii containing infeasible solutions

←i + 1 Set iigeti+1 Update the polytope Xi Pi by adding the feasibility cut or the optimality cut N

Termination Y End

Figure 3.3: High-level flow chart of Cutting Plane Method. Given a query point pi , we can verify its feasibility by only checking if it satisfies (3.24).4 Due to the unimodality of Gkm (l, p, ̺) in ̺, we can simply use a line search procedure, e.g., Golden-section search or Fibonacci search [95], to find the minimizer ̺∗ .5 If Gkm (ˇl, p, ̺∗ ) ≤ 0, the query point pi is feasible; otherwise, it is infeasible. 4

Note that p already satisfies (3.25), since we construct the initial polytope P 0 to be the

set of p satisfying (3.25). 5 The unimodality in ̺ can be seen from the convexity of Gkm (l, p, ̺) in (p, ̺), which has been shown in the proof of Proposition 4.


Algorithm 3 Cutting-Plane-Based Algorithm Require: The feasible solution set is a compact set defined by (3.24) and (3.25), and a feasible solution ˇl. 1:

Set i ← 0. Construct a polytope P i = P by (3.25).

2:

repeat

3: 4:

Compute the analytical center of P i as pi . Find the minimizer ̺∗ = arg inf ̺>0 Gkm (ˇl, pi , ̺) by taking a line

search over ̺ > 0. 5:

if Gkm (ˇl, pi , ̺∗ ) ≤ 0 then

Generate a hyperplane (optimality cut) through pi to remove the

6:

part of P i that has lower objective values, i.e.,

7:

else

!T ∂F (ˇl, p) − (p − pi ) ≤ 0. ∂p p=pi

(3.27)

Generate a hyperplane (feasibility cut) through pi to remove the

8:

part of P i that contains infeasible solutions, i.e.,

9: 10:

end if

!T ∂Gkm (ˇl, p, ̺) (p − pi ) ≤ 0. i ∂p p=p

(3.28)

Set i ← i + 1, and update P i by adding the separation hyperplane (3.27) or (3.28).

11:

until the termination criterion is satisfied, i.e., the polytope P i is negligibly small. • Cutting Plane Generation (Step 6-10 in Algorithm 3) In each iteration, we generate a cutting plane, which is a hyperplane through the query point, depending on the feasibility of the query point.


– If the query point pi ∈ P i is feasible, we generate a cutting plane to cut out the polytope towards the opposite direction of improving objective values. Specifically, we add a hyperplane called optimality cut, which is given in (3.27). The derivatives of F (l, p) in (3.27) with respect to p is given by min {Lk ,Lkm } X ∂Al ∂F (l, p) tr adj(Al ) · , = Nkm W Lf (Lk ) · ∂Pkm ∂Pkm l=1

(3.30)

∀k ∈ Km , ∀m ∈ M where adj(·) denotes the adjugate of a matrix, and

∂Al ∂Pkm

i,j

 R ∞ max {L ,L }−min {L ,L }+j+i−2  k km k km  ξ  0    α2m ξ −ξ −1 = ·e (ln 2) 2 dξ, S W N  N km Lk +Pkm αm ξ     0,

if

l=j

if

l 6= j. (3.31)

– If the query point pi ∈ P i is infeasible, we generate a cutting plane to cut out the polytope so that the remaining polytope contains the feasible set. Specifically, we add a hyperplane called feasibility cut, which is given in (3.28). The derivatives of Gkm (l, p, ̺) in (3.28) with respect to p are given by ∂Gkm (l, p, ̺) ̺Nkm ∂B = , tr adj(B) · ∂Pkm det(B) ∂Pkm

∀k ∈ Km , ∀m ∈ M (3.32)


where

∂B ∂Pkm

= i,j

Z

∞

ξ max {Lk ,Lkm }−min {Lk ,Lkm }+j+i−2e−ξ

0

̺−W 2 ln 2 W Pkm αm ξ − · 1+ SN W NkmLk ̺ ln 2 2 αm ξ · dξ. (3.33) 2 ξ SN W Nkm Lk + Pkm αm

Then, we add the hyperplane (3.27) or (3.28) as an additional constraint to the current polytope P i . By doing so, we maintain that the feasible set is always contained in a polytope at the beginning of each iteration. 3.4.1.3

Termination Criterion (Step 11 in Algorithm 3)

In Sec. 3.4.1.1, the potential value of the polytope P i is defined to be the value of the objective function in (3.29) evaluated at the AC. Potential value is monotonic with the number of iterations, and decreases with the size of the polytope that contains the feasible set. Here, we define a metric based on the potential value as β i , exp (Ωi / dim(si )). We use β i < δ as the termination criterion where δ is an arbitrarily small number. Note that β i is the geometric mean of si , and hence, it decreases as a positive number towards zero. Thus, if β i reach a certain threshold, so that the polytope is negligibly small, then we terminate the algorithm. It has been shown in [70] that the AC-based cutting plane algorithm will be terminated in polynomial time.

3.4.2

Optimal Antenna-and-Power Allocation

ˇ j for each feasible Using Algorithm 3, we find the optimal power allocation p ˇ j ), we antenna allocation ˇlj (j = 1, 2, · · · , J). Among these candidates (ˇlj , p


select the one with the largest objective value, i.e., ˇj ) , j ∗ = arg max F (ˇlj , p j

(3.34)

ˇ j ∗ ). and the optimal solution for our STCP (3.14)–(3.17) is given as (ˇlj ∗ , p

3.5

Simulation Results

In this section, we investigate the performance of the proposed antenna-andpower allocation scheme for MIMO networks through numerical simulations. We consider a MIMO network with one transmit terminal (m = 0) and three receive terminals (m = 1, 2, 3). The terminal 0 has three radios with different type, and each receive terminal contains one type of the radio. The receive terminals are uniformly distributed in a circle area centering at the terminal 0. Hence, the link distance dm between terminal 0 and m follows the distribution f (d) = 2d/d2max, where the maximum link distance dmax is set to be 1 km. The channel model and simulation parameters are chosen according to the 3GPP LTE standard [76] so as to generate the results closed to realistic systems. Specifically, we choose the urban microcell (UMi) line-of-sight (LOS) 2 model6 as the channel propagation model. The long-term channel gain αm in

(3.1) is modeled as 2 αm = 10−0.1(P Lm (dB)+Xg )

where P Lm (dB) and Xg characterize the path loss and shadowing effect, re6

The UMi LOS model is suitable for our proposed composite radio MIMO networks, due

to its: 1) similar heights between transmitter and receivers; 2) applicable link distance (10m to 5km) and frequency range (450MHz to 6GHz).


spectively. Here, the path loss gain is given by P L(dB)    22 log10 (d) + 20 log10 (fc ) + 28, if 10m < d < dBP =   40 log10 (d) − 18 log10 ((hT − 1)(hR − 1)) + 2 log10 (fc ) + 7.8, if d > dBP

where d is the distance between transmitter and receiver in meters, fc is the carrier center frequency in GHz, hT is the height of receiver, hR is the height of transmitter, and dBP =

40 (hT 3

− 1)(hR − 1)fc is the break point distance.

The log-normal shadowing effect is considered, i.e., Xg is a Gaussian random variable with zero mean and standard deviation σXg . We perform Monte Carlo simulation over 1000 problem realizations. The long-term channel gain 2 αm ’s, including link distance dm ’s and shadowing effect Xg ’s, are indepen-

dently generated for all the realizations. The detailed parameter settings for our system-level simulation are listed in Table 3.2. In Fig. 3.4 and 3.5, we investigate the feasibility of antenna-and-power allocation as the data rate requirement qkm changes. The feasibility is defined as the existence of an allocation solution that satisfies (3.15)–(3.17) in the STCP. Given a problem realization, its feasibility can be determined by using Algorithm 2. Then, we perform Monte Carlo simulation to calculate the probability of feasible problems, referred to as system success probability. We consider two scenario settings: one with four antennas on each receive terminal, and the other with eight antennas on each receive terminal. In Fig. 3.4 and 3.5, the system success probabilities are plotted for Lkm = 4 and Lkm = 8, respectively. As shown in the figures, the system success probability decreases as qkm becomes large. It is reasonable since the increase of qkm will tighten the chance constraint which results in smaller feasible solution set. In addition, we


Table 3.2: Simulation parameters of channel model and system Channel Parameters fc

2.4 GHz

hT

5m

hR

1.5 m

dBP

64 m

dmax

1 km

Xg

N (0, 3dB)

[Hkmn ]i,j

CN (0, 1) System Parameters

K

3

M

3

Radio (k, m)

Tx: (1, 0), (2, 0), (3, 0) Rx: (1, 1), (2, 2), (3, 3)

SN

Thermal noise: −174 dBm/Hz

W

15 kHz (per tone)

Nkm

1024

ǫkm

0.2

Lk

1

Ltot

6

P km

0

P tot

23 dBm

Noise Figure: 8 dB

compared the proposed scheme with uniform allocation, where both antenna and power are equally allocated to each radio modules. In our case, each radio


1 0.9 0.8 0.7

System Success Probability

0.6 0.5 0.4 0.3

0.2

Al l oc ati on v i a STCP Uni form Al l oc ati on 0.1

0

5

10

15

20

25

qkm (Mbps)

Figure 3.4: System success probability resulted by optimal allocation and uniform allocation, where each receive terminal has four antennas. module at terminal 0 is assigned with two antennas and 18.23 dBm transmit power. The feasibility of uniform allocation can be determined by directly checking the satisfaction of each constraint. It can be seen that the success probability of uniform allocation is much lower that of proposed scheme, which implies more violations on chance constraints resulted by uniform allocation. For instance, when qkm = 10 Mbps, the system success probability of optimal allocation is higher than that of uniform allocation about 31% for Lkm = 4 and 15% for Lkm = 8, respectively. Moreover, the results indicate that the proposed scheme is able to accommodate each radio module with higher data rate requirement qkm , given the system success probability as a QoS requirement. For instance, if we set the maximum system success probability to be 80%, with the aid of the proposed scheme, each radio module is allowed to increase


1 0.9 0.8 0.7

System Success Probability

0.6 0.5 0.4 0.3

0.2

Al l oc ati on v i a STCP Uni form Al l oc ati on 0.1

0

5

10

15

20

25

qkm (Mbps)

Figure 3.5: System success probability resulted by optimal allocation and uniform allocation, where each receive terminal has eight antennas. its qkm by around 5 Mbps for Lkm = 4 and 7 Mbps for Lkm = 8, respectively. In the following, we demonstrate the performance of the optimal antennaand-power allocation proposed in Sec. 3.4. First, we investigate the convergence speed of Algorithm 3. In Fig. 3.6, we plot the trace of β i for one problem realization7 . The metric β i is used in the termination criterion of Algorithm 3, i.e., β i ≤ δ where δ is set to be 10−3 . The figure shows that β i quickly decreases towards zero within 25 iterations. Since β i characterizes the volume of the polytope, and it will keep decreasing as the polytope is being cut. In Fig. 3.7, we perform the Monte Carlo simulation to plot the cumulant density function (CDF) of the number of iterations for convergence. It shows that Algorithm 3 converges within 25 iterations with 85% probability, and on average 7

The simulation results show the similar behavior for all the feasible problem realizations.


0.25

0.2

βi

0.15

0.1

0.05

0

0

5

10

15

20

25

ith iteration

Figure 3.6: Trace of β i in each iteration (δ = 10−3 ).

the number of iterations is 22. Such observation validates the fast convergence of the algorithm. In Fig. 3.8, we give an illustration of the optimal antenna-and-power allocations and its rate outage probabilities, given the link distances and long-term channel gains over 100 problem realizations. Given that there are six antennas at transmit terminal 0, there are ten feasible solutions for antennal allocation. The choice among these solutions depends on the long-term channel gain. It can be seen that the radio module with better channel condition is more likely to be assigned with more resources, i.e., antennas or power. However, assigning the “best” radio module with the most resources is not necessarily optimal, especially for the power allocation as shown in the fourth subfigure. This is due to the nonlinearity of our objective (the ergodic MIMO capacity) in the


Empirical CDF 1 0.9 0.8

Probability

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 10

12

14

16

18

20

22

24

26

28

30

Number of iterations

Figure 3.7: CDF of number of iterations for the convergence of Algorithm 3.

number of antennas and power. This confirms the necessity of adopting the proposed algorithm to search for the optimal solution. In addition, we plot the rate outage probability in the last subfigure, which shows that the chance constraints are always guaranteed, i.e., Pout km ≤ ǫkm . In Fig. 3.9 and 3.10, we compared the expected system throughput (spectral efficiency) of optimal antenna-and-power allocation and uniform allocation. For fair comparison, we choose qkm = 1 Mbps so that all the adaptation windows are feasible for both antenna-and-power allocation and uniform allocation.8 We plot the CDF’s of the expected system throughput for two 8

Since the uniform allocation has no guarantee on data rate, the chance constraint can

be easily violated when qkm is high. In such case, it is not fair to compare the throughput of these two systems.


dm (km)

1 0.5 0

0

10

20

30

40

50

60

70

80

90

100

0

10

20

30

40

50

60

70

80

90

100

0

10

20

30

40

50

60

70

80

90

100

0

10

20

30

40

50

60

70

80

90

100

0

10

20

30

40

50

60

70

80

90

100

α2m (dB)

−70 −100

Pkm (dBm)

Lk

−130 4 3 2 1 23 15 8

Pout km

0.1 0.05 0

Radio Module (1, 1)

Radio Module (2, 2)

Radio Module (3, 3)

Figure 3.8: A time-series illustration of the optimal antenna-and-power allocation (third and fourth subplot) and the corresponding rate outage probabilities (fifth subplot) over 100 problem realizations. The link distance and long-term channel gain are given in the first two subplots. scenarios: Lkm = 4 in Fig. 3.9 and Lkm = 8 in Fig. 3.10. A noticeable gap in expected system throughput is observed between the optimal antenna-andpower allocation and the uniform allocation. In particular, when Lkm = 4, the throughput of optimal allocation is about 50Mbps (1bps/Hz) higher than that of uniform allocation with 50% probability, and the gap is up to about 100Mbps (2bps/Hz) with 90% probability; when Lkm = 8, the gap increases


Empirical CDF 1 0.9

Uni form Al l oc ati on

0.8 Opti mal Powe r Al l oc ati on

Probability

0.7 0.6 0.5

Opti mal Ante nna-and-Powe r Al l oc ati on

0.4 0.3 0.2 0.1

0 (Mbps) 0 (bps/Hz ) 0

Figure 3.9:

100 2

200 4

300 6

400 8

500 10

600 12

System throughput (spectral efficiency) resulted by optimal

antenna-and-power allocation, optimal power allocation, and uniform allocation, where each receive terminal has four antennas. to 66Mbps (1.3bps/Hz) with 50% probability, and up to 200Mbps (4bps/Hz) with 90% probability. It is reasonable to observe higher throughput in Fig. 3.10, since more receive antennas bring larger diversity gain in MIMO system. Furthermore, we investigate the performance of optimal power allocation, where only power are optimized whereas antennas are equally allocated. The CDF curves are also plotted in Fig. 3.9 and 3.10. It shows that, with 50% probability, optimal power allocation achieves about one-third (Lkm = 4) and one-fifth (Lkm = 8) of the throughput gain obtained by optimal antennasand-power allocation, respectively. On average, the throughput gain of power allocation is 17% (Lkm = 4) and 30% (Lkm = 8) of that of joint antenna-andpower allocation. It implies that a much larger portion of the throughput gain


Empirical CDF 1 0.9

Uni form Al l oc ati on

0.8 0.7

Probability

Opti mal Powe r Al l oc ati on 0.6 0.5 0.4

Opti mal Ante nna-and-Powe r Al l oc ati on

0.3 0.2 0.1 0 (Mbps) 0 (bps/Hz ) 0

100 2

200 4

300 6

400 8

500 10

600 12

Figure 3.10: System throughput (spectral efficiency) resulted by optimal antenna-and-power allocation, optimal power allocation, and uniform allocation, where each receive terminal has eight antennas. is contributed by antenna allocation, and hence, the observation confirms the importance of antennas allocation.


3.6

Summary

In this chapter, we proposed a dynamic resource allocation scheme to support multiple radios co-operating in MIMO networks. Specifically, we optimized the allocation of both antennas and power based on recent advances in stochastic programming techniques. The proposed allocation scheme maximizes the system throughput and meanwhile satisfies user data rate requirements with high probability. An efficient approach is first proposed to determine the feasibility of the resource allocation problem, and is further developed into an algorithm to obtain the optimal allocation decision. The simulation results demonstrated that the proposed scheme achieves both higher system success probability and spectral efficiency than uniform allocation. Moreover, we showed that antenna allocation contributes a larger portion of throughput increase than power allocation.

Chapter 4 Robust Power Allocation for Energy-Efficient Location-Aware Networks Positional information is of critical importance for future wireless networks, which will support an increasing number of location-based applications and services [48, 96–101]. Example applications include cellular positioning, search and rescue work, blue-force tracking in battlefield, etc., covering from civilian life to military operations. Typically, wireless localization is referred to as a process to determine the positions of mobile nodes (agents) based on the measurements with respect to mobile/static nodes with known positions (anchors), as illustrated in Fig. 1.4. With the rapid development of advanced wireless techniques, wireless localization has attracted numerous research interests in the past decades [102–114]. Localization accuracy is a critical performance measure of wireless locationaware networks. In recent work [46, 47], fundamental limits of wideband local-

107

Chapter 4. Robust Power Allocation for Energy-Efficient Location-Aware Networks

108

ization have been derived in terms of the squared position error bound (SPEB) and directional position error bound (DPEB). It shows that localization accuracy is related to several aspects of design, including anchors’ positions, signal waveforms, and transmit power. The control of transmit power is of great importance, since it affects not only localization accuracy but also network lifetime, throughput, and interference, especially for the scenarios where mobile nodes are subject to limited power resources [115–117]. However, few work has addressed the problem of power resource allocation for localization, especially for the case of multiple-agent networks. The authors in [118] formulated several optimization problems of anchor power allocation for wideband localization systems, and derived the optimal solution only for single-agent network. In [119], it exploited the geometrical interpretation of localization information to minimize the maximum DPEB (mDPEB).1 In [120], it investigated the localization using MIMO radar, and adopted the constraint relaxation and domain decomposition methods to obtain sub-optimal solutions for power allocation. In general, how to optimally allocate the transmit power in location-aware networks still remains as an open problem. On the other hand, power allocation should be adapted to the instantaneous network conditions, such as network topology and channel qualities, for optimizing the localization performance. Previous work on power allocation in location-aware networks assumes that the network parameters such as nodes’ positions and channel conditions are perfectly known [118–120]. However, these parameters are obtained through estimation and hence subject to errors. The power allocation based on imperfect network parameters often leads to 1

Geometrically, mDPEB is the projection of SPEB on one dimension that has the max-

imum value, and characterizes the maximum position error of an agent over all directions.


109

sub-optimal or even infeasible solutions in realistic networks [14, 121]. Therefore, it is essential to design a robust scheme to combat the uncertainties in network parameters. In this chapter, we present an optimization framework for robust power allocation in network localization to tackle imperfect network topology parameters. Specifically, we treat the fundamental limits of localization accuracy, i.e., SPEB and maximum directional position error bound (mDPEB), as the performance metrics. The key contributions are summarized as follows: • We formulate optimization problems for power allocation by minimizing SPEB/mDPEB. We prove that these formulations are conic programs, which can be efficiently solved by off-the-shelf optimization tools [122, 123]; • We propose a robust optimization method for the worst-case SPEB/mDPEB minimization in the presence of parameter uncertainties. The proposed robust formulations retain the same structures of conic programs, and hence, can be solved with same complexities as their non-robust counterparts; • We develop a distributed algorithm for robust power allocation, which decomposes the original problem into several subproblems enabling parallel computations among all the agents. The proposed scheme improves the computational efficiency without loss of optimality. The rest of this chapter is organized as follows. In Sec. 4.1, we describe the system model and introduce the performance metrics. In Sec. 4.2, we formulate the power allocation problems into conic programs. In Sec. 4.3, robust power allocation schemes are proposed to combat the uncertainties in network topology parameters. In Sec. 4.4, we further decompose our robust


110

formulation into several subproblems that can be independently solved by each agent. In Sec. 4.5, the performance of the proposed schemes is investigated through simulations. Finally, the chapter is concluded in Sec. 4.6.

4.1

System Model

In this chapter, we describe the system model, and introduce two performance metrics of location-aware networks.

4.1.1

Network Settings

Consider a 2-D location-aware network consisting of Na agents and Nb anchors, where the sets of agents and anchor are denoted by Na = {1, 2, . . . , Na } and Nb = {Na +1, Na +2, . . . , Na +Nb }, respectively. The 2-D position of node k is denoted by pk = [xk yk ]T . The distance and angle between nodes k and j are given by dkj = kpk − pj k and φkj = arctan [(yk − yj )/(xk − xj )], respectively. The anchors are the mobile/static nodes with known positions, and subject to limited power resources. The agents try to determine their positions based on the radio signals transmitted from the anchors. For instance, agents can obtain the signal metrics such as TOA and RSS from the received signals, and then calculate their positions via triangulation [38]. The multipath received waveform at agent k from anchor j is modeled as rkj (t) =

Lkj X l=1

(l) (l) αkj · s t − τkj + zkj (t), (l)

t ∈ [0, Tob ) (l)

where s(t) is a known transmit waveform, αkj and τkj are the amplitude and delay, respectively, of the lth path, Lkj is the number of multipath components,


111

zkj (t) represents additive white Gaussian noise (AWGN) with two-side power spectral density N0 /2, and [0, Tob ) is the observation interval. We consider that the measurements between anchors and agents do not interfere each other by using medium access control, and the network is synchronized such that the inter-node distance is estimated using one-way timeof-flight (TOF).2 Our work can be extended to asynchronous networks where round-trip TOF is employed for distance estimation, and it will be discussed in Section 4.2.

4.1.2

Position Error Bound

The SPEB introduced in [46] is a performance metric that characterizes the localization accuracy, defined as P(pk ) , tr J−1 (p ) k e

(4.1)

where Je (pk ) is the equivalent Fisher information matrix (EFIM) for agent k’s position. Using the information inequality [124], we can show that the position error is bounded by the SPEB, i.e., E kˆ pk − pk k2 ≥ P(pk ) ˆ k is an unbiased estimate of the position pk . The EFIM in (4.1) can where p be derived as a 2 × 2 matrix [46] Je (pk ) =

X

j∈Nb 2

λkj · Jr (φkj )

(4.2)

There are two common ways for inter-node distance estimation: one-way TOF (only

anchor transmits) or round-trip TOF (both anchor and agent transmit). The former requires anchors and agents to be synchronized for distance estimation.


112

where Jr (φkj ) = q(φkj )q(φkj )T is a 2×2 matrix with q(φkj ) = [cos φkj sin φkj ]T , and λkj is defined as ranging information intensity (RII) of agent k with respect to anchor j, given by λkj = Pkj ·

ξkj 2β dkj

(4.3)

in which ξkj is a positive coefficient determined by the properties of the channel and transmit signal, β is a positive coefficient denoting the amplitude loss exponent,3 and Pkj is the power of the transmit waveform.4 Proposition 5. The SPEB is a monotonically non-increasing function of the RII λkj . Proof. Given λ1kj ≥ λ2kj , we have J1e − J2e =

X

j∈Nb

λ1kj − λ2kj · q(φkj )q(φkj )T 0.

−1 −1 It follows that tr (J1e ) ≤ tr (J2e ) .

Since the SPEB characterizes the fundamental limit of localization accuracy

and is achievable in high SNR regimes, we will use it as a performance metric for location-aware networks, and allocate the transmit power to optimize the system performance by minimizing the SPEB. 3

Note that the amplitude loss exponent is β, while the corresponding power loss exponent

is 2β. 4 Although derived based on the received waveforms for wideband systems in [46], the structure of SPEB is also observed in other TOA- or RSS-based localization systems, e.g., [113, 125–127].


4.1.3

113

Directional Decoupling of SPEB

We then introduce the definition of DPEB and mDPEB. The EFIM (4.2) can be written, by eigen analysis, as [46] 

Je (pk ) = Uθk 

µ1,k

0

0

µ2,k



 UT θ

k

where µ1,k and µ2,k are the two eigenvalues of EFIM (µ1,k ≥ µ2,k ), given by

X 1 X

µ1,k , µ2,k = λkj q(2φkj ) , λkj ± 2 j∈N j∈N b

b

and Uθk is a rotation matrix with angle θk , given by   cos θk − sin θk . Uθk =  sin θk cos θk

Geometrically, the EFIM of agent k can be viewed as an information ellipse √ √ given by {x ∈ R2 : xT J−1 e (pk )x = 1} (see Fig. 4.1), where 2 µ1,k and 2 µ2,k represent the major axis and minor axis, respectively. Such an ellipse provides the insight that the localization information can be decoupled into two orthogonal directions. Consequently, we only need to consider the calculation of the SPEB in two decoupled directions. Definition 1 (DPEB). The DPEB of agent k along the direction ϕ is defined as P(pk ; u) , uT [J−1 e ]u where u = [cos ϕ sin ϕ]T . Proposition 6. The mDPEB of agent k is max

{u:uT u=1}

{P(pk ; u)} =

1 . µ2,k

(4.4)


114

y

θk √

√

x

µ1,k

µ2,k

Figure 4.1: Geometrical interpretation of the EFIM of agent k. Proof. The maximization on DPEB in (4.4) follows that: max

{u:uT u=1}

{P(pk ; u)} = = =

max

{u:uT u=1}

max

{u:uT u=1}

max

{v:vT v=1}

uT [J−1 e ]u 

µ−1 T  1,k uT (U−1 ) θk

vT [J−1 e ]v

0

0 µ−1 2,k



 U−1 θ u. k

(4.5)

where v = U−1 θk u is also a unit vector since Uθk is a rotation matrix and u is a unit vector. Now, let v∗ = [cos θk sin θk ]T and substitute it into (4.5), then we have max

{u:uT u=1}

2 −1 2 {P(pk ; u)} = max µ−1 1,k cos θk + µ2,k sin θk θk

−1 −1 2 = max µ−1 + (µ − µ ) sin θ k 2,k 1,k 2,k θk

= µ−1 2,k

where the last equation is due to µ1,k ≥ µ2,k . Proposition 6 can also be understood via the information ellipse of EFIM. The localization information achieves maximum along the major axis and minimum along the minor axis. Due to the reciprocal, the SPEB is dominated by the mDPEB, which is the inverse of the smaller eigenvalue of the EFIM.


115

Therefore, in order to improve the localization performance, it is more helpful to maximize the smaller eigenvalue of EFIM, equivalently to minimize the mDPEB that characterizes the maximum position error of an agent over all directions. We will use mDPEB as another important metric of localization accuracy.

4.2

Optimal Power Allocation via Conic Programming

In this section, we formulate the power allocation problem using SPEB and mDPEB as the objective functions, respectively. We show that the SPEB minimization is a semidefinite program (SDP) and the mDPEB minimization is a second-order cone program (SOCP).

4.2.1

Problem Formulation Based on SPEB

We first consider the problem of optimal power allocation that minimizes the total SPEB while the network is subject to a budget of power consumption. The problem can be formulated as5 P1 :

min

{Pkj }

s.t.

X

k∈Na

tr

X ξkj

X X

k∈Na j∈Nb

Pkj ≥ 0, 5

d 2β j∈Nb kj

Pkj Jr (φkj )

−1

Pkj ≤ P tot ∀k ∈ Na , ∀j ∈ Nb

(4.6) (4.7) (4.8)

The structure of the problem retains with additional linear constraints, such as the

maximum transmit power from anchor j to agent k, or the maximum total transmit power from anchor j, etc. See Remark 3 for detailed discussion.


116

where the constraint (4.7) gives the upper bound of the total transmit power of all the anchors. We first show the convexity of the above problem in the following proposition. Proposition 7. The problem P1 is convex in Pkj . Proof. Since (4.7)–(4.8) are all linear constraints, we only need to show the objective in (4.6), i.e., the SPEB, is a convex function in Pkj . We write the RII of agent k as a vector λk = [λk1 λk2 · · · λkNb ]T , and the SPEB can be expressed as a function of λ, given by X −1 [λk ]j Jr (φkj ) f (λk ) , tr j∈Nb

where [λk ]j denotes the jth element of vector λk . We choose two arbitrary b e k ∈ RN λk , λ + . Given any α ∈ [0, 1], it follows that X −1 e k ]j Jr (φkj ) e k ) = tr α[λk ]j + (1 − α)[λ f (αλk + (1 − α)λ

j∈N

b X −1 X e [λk ]j Jr (φkj ) [λk ]j Jr (φkj ) + (1 − α) = tr α

j∈Nb

e k ). ≤ αf (λk ) + (1 − α)f (λ

j∈Nb

(4.9)

The inequality (4.9) holds since the function tr {X−1 } is convex in X ≻ 0 [11]. If the matrix X is singular, the inequality (4.9) still holds. Hence, f (λk ) is 2β 2β convex in λk . Since λkj = ξkj /dkj · Pkj and ξkj /dkj is a positive scaler, the

SPEB is a convex function of Pkj . Since P1 is a convex problem, the optimal solution can be achieved by using the standard convex optimization algorithms, e.g., interior point method. Furthermore, we show that such problem can be converted to a SDP problem, which is a more favorable formulation since many fast real-time optimization solvers are available for SDP [128, 129].


117

To obtain an equivalent formulation to P1 , we replace the EFIM’s in (4.6) with matrices Mk , and add another constraint Mk J−1 e (pk ). Since Je (pk ) is a positive semidefinite matrix, due to the property of Schur complement, the above inequality is equivalent to   Mk I  0.  I Je (pk )

Then, we obtain a SDP formulations P1SDP equivalent to P1 , i.e., P1SDP :

min

{Pkj }, Mk

X

k∈Na

tr {Mk }

 Mk  P I j∈Nb

s.t.

I ξkj

P J (φ ) d 2β kj r kj kj

(4.7) – (4.8).



  0,

∀k ∈ Na

Hence, the optimal solution of P1 can be efficiently obtained by solving the SDP formulation P1SDP .

4.2.2

Problem Formulation Based on mDPEB

We now consider the minimization of total mDPEB as our objective. The problem can be formulated as P2 :

min

{Pkj }

X 1 µ2,k k∈N a

s.t. (4.7) – (4.8),


118

which can be equivalently converted to P2SOCP :

min

{Pkj ,rk }

X

k∈Na

P

1 ξkj

j∈Nb d 2β Pkj kj

− rk

X ξ

kj s.t. rk ≥ P q(2φ ) kj , 2β kj d j∈Nb kj

∀k ∈ Na

(4.10)

(4.7) – (4.8).

The constraints (4.10) define Na second-order cones given by Qk = {(rk , xk ) ∈ R × R2 : rk ≥ kxk k}, where xk =

P

j∈Nb

∀k ∈ Na

2β ξkj /dkj · Pkj q(2φkj ). Moreover, the objective is convex in

{Pkj , rk }, since the reciprocal of a positive function preserves convexity [11]. Thus, we obtain a nonlinear SOCP problem which is convex in Pkj . Remark 2. Both SDP and SOCP are well-known convex optimization problems, and hence, the optimal solutions of P1SDP and P2SOCP can be obtained efficiently by many off-the-shelf optimization tools. Remark 3. Additional linear constraints on transmit power can be imposed depending on the realistic requirements of location-aware networks. For example, we can consider Pkj ≤ P¯kj where P¯kj is the upper bound of the transmit P power from anchor j to agent k, or k∈Na Pkj ≤ Pjtot where Pjtot is the upper bound of the total transmit power from anchor j, etc. Due to the linearity

of these constraints, the convexity of the problem is retained, and the optimal solution can be obtained via conic programming. Remark 4. For the asynchronous networks where round-trip TOF is employed for distance estimation, we need to allocate the transmit power of both anchors ′ denote the power of transmit waveform from agent k to and agents. Let Pkj


119

anchor j. Besides the total anchor power constraint in (4.7), we also impose a total power constraint on agents, i.e., X X

k∈Na j∈Nb

′ Pkj ≤P

′ tot

,

(4.11)

where ′ Pkj ≥ 0,

∀k ∈ Na , ∀j ∈ Nb .

(4.12)

It can be shown that the RII of agent k with respect to anchor j is given by ′ λkj = g(Pkj , Pkj )·

ξkj 2β dkj

−1 ′ −1 ′ where the equivalent power g(Pkj , Pkj ) = 4 Pkj +Pkj

(4.13) −1

′ . Note that g(Pkj , Pkj )

′ is a concave function in (Pkj , Pkj ), and the SPEB in (4.1) is convex and non-

increasing in λkj , and hence, the convexity of SPEB in Pkj is preserved. Similarly, we can show the mDPEB with the λkj in (4.13) is also convex in Pkj . Therefore, the SPEB/mDPEB minimization for asynchronous networks is still a convex problem in Pkj , of which the optimal solution can be efficiently solved via interior point methods [72]. Furthermore, we show that the power allocations for asynchronous and synchronous networks are equivalent. First, we can derive the maximum total equivalent power by considering the following problem X X

max′

{Pkj ,Pkj }

s.t.

′ g(Pkj , Pkj )

k∈Na j∈Nb

(4.7) – (4.8) (4.11) – (4.12).

Using the Karush-Kuhn-Tucker conditions [11], it can be proved that the optimal value is reached as a constant g(P tot , P ′

′ Pkj

′ tot

) if and only if

P tot = tot · Pkj . P


120

Hence, in order to achieve the maximum total equivalent power, power allocated on anchors and agents should be proportional and consequently, the RII for asynchronous network is ′

λkj

ξkj 4P tot · 2β = Pkj · ′ tot tot P +P dkj

which is with the same structure as the RII of synchronous network in (4.3). Therefore, the power allocation on both anchors and agents in asynchronous networks can be equivalently converted into anchor power allocation in synchronous networks.

4.2.3

Formulations with QoS Guarantee

Besides P1 and P2 , we show the proposed framework also applies to the other two types of problem formulations based on different QoS requirements. 4.2.3.1

Energy-efficient Formulation

The objective is to minimize the total transmit power, while each agent has a minimum requirement on its SPEB, i.e., min

{Pkj }

X X

Pkj

k∈Na j∈Nb

s.t. tr J−1 e (pk ) ≤ γk ,

∀k ∈ Na

(4.14)

(4.7) – (4.8).

Similarly, a formulation based on mDPEB can be proposed by replacing (4.14) with 1 ≤ γk , µ2,k

∀k ∈ Na .

(4.15)


4.2.3.2

121

Min-max SPEB Formulation

The objective is to minimize the maximum SPEB among all the agents, i.e., min

{Pkj }

max{tr J−1 e (pk ) } k

s.t. (4.7) – (4.8). It can be equivalently converted into min

{Pkj }, γ

γ

s.t. tr J−1 e (pk ) ≤ γ,

∀k ∈ Na

(4.7) – (4.8),

which turns out to be with the same structure as the energy-efficient formulation. Similarly, we can also propose the min-max mDPEB formulation by replacing SPEB with mDPEB. Note that since the above formulations with QoS guarantee have the same structure as P1 or P2 , which can be solved efficiently by conic programing, we will only focus on P1 and P2 in the following. 4.2.3.3

Chance Constrained Formulation

Consider a probabilistic constraint on position estimation error, i.e, Pr {||ˆ pk − pk ||2 ≤ γk } ≥ 1 − ǫk ,

∀k ∈ Na .

(4.16)

which requires agent k’s position estimation error is below γk with probability at least 1 − ǫk . Such chance constraint has practical applications and is more robust in the real system. It has been shown in [19] that the above chance


122

constraint can be equivalently converted into linear constraints on the mDPEB µ2,k : µ2,k ≥ ζ(γk , ǫk ), where ζ(γk , ǫ) =

    22 ln 1 , ǫk η k

   γ 22·ǫ , k k

∀k ∈ Na .

(4.17)

ˆ k follows Gaussian distribution if p ˆ k follows an arbitrary distribution. if p

In [19], it adopted an approximation on the mDPEB µ2,k . Since we have derived the close-form expression for µ2,k , we are able to cope with the original chance constraints (4.16). In particular, a SOCP problem can be formulated containing Na second-order conic constraints corresponding to Na agents, as imposed in (4.10). To obtain the optimal solution of P1 and P2 , it requires the network topology parameters, i.e., the distance dkj and the angle φkj . However, dkj ’s and φkj ’s are usually not perfectly known in realistic networks, and only estimated values are available. When estimation errors exist, the formulation P1 or P2 may fail to provide reliable solutions, since the actual SPEB/mDPEB is not necessarily minimized. Therefore, it is essential to design a power allocation scheme which is robust to the uncertainties in network topology parameters.

4.3

Robust Power Allocation under Imperfect Network Topology Parameters

In this section, we consider the location-aware networks with imperfect network topology parameters, and propose robust optimization methods to minimize the worst-case SPEB/mDPEB.


4.3.1

123

Robust Counterpart of SPEB Minimization

Let dˆkj and φˆkj denote distance and angle estimates, respectively. We consider the actual distance and angle lie in linear sets, i.e.,6 o n d dkj ∈ Skj , dkj dˆkj − εdkj ≤ dkj ≤ dˆkj + εdkj o n φ φkj ∈ Skj , φkj φˆkj − εφkj ≤ φkj ≤ φˆkj + εφkj

where εdkj and εφkj are both small positive numbers denoting the maximum estimation errors on the distance and angle estimates, respectively.7 To deal with the estimation errors, we adopt robust optimization techniques to consider the worst-case performance. Instead of using the estimated values, we consider minimizing the largest SPEB over the possible set of actual network topology parameters, i.e., PR-0 :

min

{Pkj }

max

d , φ ∈S φ } {dkj ∈Skj kj kj

X

k∈Na

tr J−1 e (pk )

s.t. (4.7) – (4.8). Note that the maximization over dkj and φkj can be separated, and the maximization over dkj simply follows that dekj , arg max

d } {dkj ∈Skj

d ˆ tr J−1 e (pk ) = dkj + εkj

since tr {J−1 e (pk )} is a monotonically non-decreasing function of dkj . On the We consider the distance dkj to be always positive, i.e., dˆkj − ǫdkj > 0. 7 The estimation error in ξkj can be equivalently accounted for in the distance. Moreover,

6

if uncertainties exist in anchor positions, it can be equivalently converted into the estimation errors in channel qualities [47].


124

other hand, however, the maximization over φkj is not trivial, because arg max

φ } {φkj ∈Skj

tr J−1 e (pk )

= arg max

φ } {φkj ∈Skj

= arg max

φ {φkj ∈Skj }

4

P

ξkj j∈Nb d 2β Pkj kj

P

P − kj 2β j∈Nb d

j∈Nb kj

2

X ξ

kj P q(2φkj )

2β kj d j∈Nb kj P

ξkj

2

2

2β Pkj q(2φkj ) dkj ξkj

(4.18)

but the right-hand side of (4.18) is not a convex problem. Hence, it is difficult to obtain a close-form solution of {φekj } since it depends on {Pkj }.

We next consider a relaxation for the robust optimization with respect to

{φkj } and introduce a new matrix Q(φˆkj , δkj ) = Jr (φˆkj ) − δkj · I to replace Jr (φkj ) in the SPEB. We will show that the worst-case SPEB over φkj can be upper bounded by the new function for sufficiently large δkj . The details are given in the following proposition. Proposition 8. If

P

j∈Nb

2β ξkj /dkj · Pkj Q(φˆkj , δkj ) 0 and δkj ≥ sin(εφkj ), the

maximum SPEB over the actual angle φkj is always upper bounded as follow X X −1 −1 ξkj ξkj max tr P J (φkj ) ≤ tr P Q(φˆkj , δkj ) . 2β kj r 2β kj φ d d } {φkj ∈Skj j∈Nb kj j∈Nb kj

(4.19)

Moreover, the tightest upper bound in (4.19) is attained by X −1 ξkj φ P Q(φˆkj , δkj ) . sin(εkj ) = arg min tr 2β kj δkj d j∈Nb kj


125

Proof. Since Jr (φkj ) − Q(φˆkj , δkj )   ˆ ˆ ˆ ˆ δkj − sin(φkj + φkj ) sin(φkj − φkj ) cos(φkj + φkj ) sin(φkj − φkj ) , = ˆ ˆ ˆ ˆ cos(φkj + φkj ) sin(φkj − φkj ) δkj + sin(φkj + φkj ) sin(φkj − φkj )

we can show that Jr (φkj ) − Q(φˆkj , δkj ) is positive semidefinite if   δkj ≥ sin(φkj + φˆkj ) sin(φkj − φˆkj ),  δ 2 ≥ sin2 (φkj − φˆkj ). kj

Since |φkj − φˆkj | ≤ εφkj , the above two inequality conditions are guaranteed by δkj ≥ sin(εφkj ). Given that tr

P

j∈Nb

2β ξkj /dkj Pkj Q(φˆkj , δkj ) 0, we have

X ξkj j∈Nb

2β dkj

Pkj Jr (φkj )

−1

≤ tr

X ξkj

P Q(φˆkj , δkj ) 2β kj

j∈Nb

dkj

−1

φ for all φkj ∈ Skj . Furthermore, we can show that Q(φˆkj , δ1 ) Q(φˆkj , δ2 ) for P 2β ˆkj , δkj ) −1 0 ≤ δ2 ≤ δ1 , which implies that the function tr ξ /d ·P Q( φ kj kj kj j∈Nb

is a non-decreasing function of δkj . Hence, the minimum value of the righthand side of (4.19) is obtained when δkj = sin(εφkj ). In the rest of the chapter, we take the minimizer δkj = sin(εφkj ) and denote

the matrix Q(φˆkj ) = Jr (φˆkj ) − sin(εφkj ) · I by omitting the variable δkj in the matrix Q(φˆkj , δkj ) for simplicity. Then, we replace the matrix Jr (φkj ) with Q(φˆkj ) in the previous formulation, and


propose a robust counterpart of P1 as follow: −1 X X ξkj P Q(φˆkj ) tr PR-1 : min e 2β kj {Pkj } d j∈Nb kj k∈Na X ξkj s.t. Pkj Q(φˆkj ) 0, ∀k ∈ Na de 2β j∈Nb

126

(4.20)

kj

(4.7) – (4.8).

Again by the property of Schur complement, the problem PR-1 is equivalent to a SDP formulation, given by X SDP tr {Mk } PR-1 : min {Pkj }, Mk

s.t.

k∈Na



Mk  P I j∈Nb

I ξkj Pkj Q(φˆkj ) de 2β

(4.7) – (4.8).

kj



  0,

∀k ∈ Na

(4.21)

Thus, its optimal solutions can be efficiently solved by standard solvers. Remark 5. The formulation with QoS guarantee proposed in Section 4.2.3 can also be extended to its robust formulation using the above method. By such, the SPEB of each agent is always guaranteed to satisfy its requirement of maximum positional error. However, if using the deterministic formulation, the agent’s requirement, e.g., (4.14) or (4.15), can easily be violated due to imperfect network topology parameters. Note that from Proposition 8, the new formulation PR-1 is a valid relaxation for PR-0 when the condition (4.20) holds. Since Q(φˆkj ) is not positive definite due to det Q(φˆkj ) = sin(εφkj )(sin(εφkj ) − 1) ≤ 0, such a condition does not necessarily hold for all power allocation {Pkj }. However, we will show that it holds for the optimal power allocation of PR-0 with high probability (w.h.p.) when the number of anchors is large or the estimation error is small.


127

Proposition 9. Consider a random network where all the nodes are uniformly located in a R × R square region and the minimum distance between two nodes is r0 . The coefficient ξkj has a support on [ξmin ξmax ] where 0 < ξmin ≤ ξmax ≤ ∗ 1. Let {Pkj } be the optimal solution of PR-0 , and δ = sin(εφ ) where εφ =

max{εφkj }, then

(a) when Nb → ∞ and δ ≤ δmax , where δmax is the smallest positive root of equation 4δ 4 − 4δ 2 − 2ξmax /ξminδ + 1 = 0, we have ) ( X ξkj P ∗ Q(φˆkj ) 0 = 1 − O (exp(−α·Nb )) , Pr 2β kj e j∈Nb dkj

∀k ∈ Na

where α is a fixed positive number;

(b) when εφ → 0, we have ( ) X ξkj ∗ ˆkj ) 0 = 1 − O (εφ )Nb /2 , Pr P Q( φ kj e 2β j∈Nb dkj

∀k ∈ Na .

Proof. We first consider the network with a single agent, and then extend the

proof to the multiple-agent case. For a given k ∈ Na , we need to show that

∗ the condition (4.20) holds for {Pkj } w.h.p. for both cases (a) and (b). Note

that since X ξkj X ξkj ∗ ∗ ˆkj ) ˆkj ) − ξmax P tot δkj I, P Q( φ P J ( φ r kj kj e 2β e 2β r02β j∈Nb dkj j∈Nb dkj

it is sufficient to show that w.h.p. X −1 r02β 2 ξkj ∗ ˆ P J (φ ) ≤ tr e 2β kj r kj ξmax P tot δ j∈N dkj

(4.22)

b

where δ = sin(εφ ) with εφ = max{εφkj }.

For (a): we pick two anchors i and i′ in the region (see Fig. 4.2) such that

1. r0 ≤ deki , deki′ ≤ ζr0 with ζ > 1;


128

∆φ i′

φ i ∆

r0 ζr0

k

R×R

Figure 4.2: Geometrical illustration of the proof of Proposition 9(a) where agent is inside the square region. We choose two anchors i and i′ in the shaded region. 2. 0 ≤ φki ≤ ∆φ and π/2 − ∆φ ≤ φki′ ≤ π/2 for a small positive ∆φ . Note that if the agent is at the corner or on the boundary of the square area, we can rotate the angles accordingly to find such a region. It can be shown that there exists at least one such pair of anchors with probability 1 + (1 − 2p0 )Nb − 2(1 − p0 )Nb , where p0 = (ζ 2 − 1)r02 ∆φ /2R2 . Since the probability goes to 1 exponentially with Nb , such a pair of anchors can be found w.h.p. Consider a power allocation scheme {Peki = Peki′ = P tot /2}, and we show

this scheme satisfies the condition (4.22) for a sufficiently small δ. Based on


129

the definition of the optimal power allocation, we have X X −1 −1 ξ ξkj ∗ kj P J (φˆ ) ≤ max tr Pe J (φ ) tr e 2β kj r kj e 2β kj r kj {φkj } d d j∈Nb kj j∈Nb kj −1 ξmin P tot (Jr (φki ) + Jr (φki′ )) ≤ max tr {φkj } ζ 2β r02β 2 =

ζ 2β r02β 2 2 . 2 tot ξmin P sin (π/2 − 2∆φ − 2εφ )

Therefore, a sufficient condition of (4.22) is given as r02β 2 ζ 2β r02β 2 2 ≤ ξmin P tot sin2 (π/2 − 2∆φ − 2εφ ) ξmax P tot δ which is equivalent to ξmin 2ζ 2β sin(εφ ) ≤ 2 φ φ cos (2∆ + 2ε ) ξmax

(4.23)

given δ = sin(εφ ). Note the left-hand side of (4.23) is an increasing function in ζ, ∆φ and εφ , when ∆φ and εφ are both small positive numbers. Thus, the maximum εφ (or equivalently, maximum δ) to satisfy (4.23) can be obtained by taking the limit ζ → 1 and ∆φ → 0. It follows that 2 sin(εφ ) ξmin ≤ 2 φ cos (2ε ) ξmax and the inequality holds when 0 < δ = sin(εφ ) ≤ δmax , where δmax is the smallest positive root of the equation 4δ 4 − 4δ 2 − 2

ξmax δ + 1 = 0. ξmin

We give some numerical examples as follow: δmax = 0.318 when ξmax /ξmin = 1; δmax = 0.096 when ξmax /ξmin = 5. For (b): Consider a small angle

√

2aεφ (as εφ → 0), where a = (2β+1 R2β ξmax )/(r02β ξmin).

The probability that all Nb anchors locate in such a small angle of the R × R √ region is at most ( 2aεφ )Nb , which goes to 0 at the rate of polynomial power


130

Nb /2 as εφ → 0. Hence, we can find two anchors, i and i′ , whose angle sepa√ √ ration is larger than 2aεφ and smaller than π − 2aεφ w.h.p. We allocate the power equally on these two anchors, and it follows X X −1 −1 ξkj ∗ ξ kj tr P J (φˆ ) ≤ max tr Pe J (φ ) e 2β kj r kj e 2β kj r kj {φkj } d d j∈Nb kj j∈Nb kj −1 ξmin P tot √ (Jr (φki ) + Jr (φki′ )) ≤ max tr {φkj } ( 2R)2β 2 2 2β R2β 2 √ . = tot 2 ξmin P sin ( 2aεφ − 2εφ )

Finally, we need to show that 2β R2β 2 r02β 2 2 √ ≤ , tot tot 2 ξmin P sin ( 2aεφ − 2εφ ) ξmax P sin εφ or equivalently, √ sin2 ( 2aεφ − 2εφ ) a≤ . sin εφ The above inequality holds as εφ → 0, since the limit of its right-hand side is 2a. Now, we extend the above proof to the multiple-agent case. In Section 4.4, SDP we decomposed the one-stage problem PR-1 into two-stage optimizations. Let I II ρ∗kj and Pk∗ denote the optimal solution of PR-1 and PR-1 , respectively. Since I the first stage problem PR-1 is formulated for each single agent, we can show

by the above proof that X ξkj ρ∗ Q(φˆkj ) 0 2β kj e d

j∈Nb

kj

holds w.h.p. for agent k. Moreover, the optimal power allocation is given ∗ in (4.34) as Pkj = ρ∗kj Pk∗ , where Pk∗ obtained in the second stage does not

affect ρ∗kj . Hence, we can show that the condition (4.20) holds w.h.p. for multiple-agent networks.


131

Remark 6. Proposition 9 implies that the condition (4.20) holds w.h.p. at the rate indicated by the O notation, where O(f (n)) means that the function value is on the order of f (n) [130]. ∗ Remark 7. Note that Proposition 9 holds for {Pkj }, which implies that the

optimal solution of the original robust formulation PR-0 is included in the SDP feasible set of the proposed formulation PR-1 (or PR-1 ) w.h.p.

4.3.2

Robust Counterpart of mDPEB Minimization

We investigate the robust power allocation based on mDPEB formulation P2 . To circumvent the intractable maximization in (4.18), we consider the robust SPEB formulation PR-1. Specifically, the objective of PR-1 can be written as X −1 1 1 ξkj ˆ + Pkj Q(φkj ) = tr 2β e µ e1,k µ e2,k j∈N dkj

(4.24)

b

where µ e1,k and µ e2,k are the two eigenvalues of the matrix Pkj Q(φˆkj ), given by µ e1,k , µ e2,k

1 = 2

P

j∈Nb

ξkj /dekj2β ·

X

X ξ

ξkj

kj φ ˆ Pkj (1 − 2 sin(εkj )) ± Pkj q(2φkj ) . 2β 2β e e j∈Nb dkj j∈Nb dkj

(4.25)

Geometrically, µ e1,k and µ e2,k are similar to the DPEB’s in two orthogonal di-

rections. Using Proposition 9, we can show that µ e2,k ≥ 0 w.h.p. when Nb is large or εφ is small. Since µ e1,k ≥ µ e2,k , the smaller eigenvalue µ e2,k dominates

the function in (4.24). Hence, we formulate the problem based on µ e2,k , and


132

obtain a formulation similar to P2SOCP given by SOCP PR-2 :

X

{Pkj ,rk }

P

ξkj P de 2β kj

j∈Nb

1

φ 1 − 2 sin(ε ) − rk k∈Na kj j∈Nb kj

X ξ

kj ˆ Pkj q(2φkj ) , ∀k ∈ Na s.t. rk ≥ 2β de

min

(4.26)

(4.27)

kj

(4.7) – (4.8).

Note that the estimation error εφkj only exists in the objective, and does not affect the second-order conic constraint (4.27). According to Proposition 9, the function in the denominator of (4.26) is positive w.h.p. when εφ is a small SOCP number close to zero. In such a case, the problem PR-2 is convex, and its

optimal solution can be efficiently solved.

4.4

Efficient Robust Algorithm Using Distributed Computations

In this section, we designed a distributed robust algorithm for both SPEB and mDPEB minimization, which decomposes the original formulation into two-stage optimization problems and enables parallel computations among all the agents. The proposed algorithms achieve global optimal solution with improved computational efficiency.

4.4.1

Algorithm for SPEB Minimization

SDP Despite the convexity of the robust SDP formulation PR-1 , there are multiple

positive semidefinite constraints imposed for multiple agents, and the computational complexity depends on the number of SDP constraints. To efficiently


133

obtain the power allocation decision for multi-agent networks, we design a SDP distributed implementation for PR-1 , which can be solved using parallel com-

putations among the agents. Specifically, we let Pkj = ρkj Pk where Pk is the total power assigned for locating agent k, and ρkj ∈ [0, 1] is a fractional number denoting the percentage of Pk allocated to anchor j. By introducing the two variables ρkj and Pk , the robust formulation for power allocation can be written as −1 X 1 X ξkj ˆ tr min ρ Q(φkj ) e 2β kj {ρkj ,Pk } P k d j∈Nb kj k∈Na X ρkj ≤ 1 s.t.

(4.28)

j∈Nb

ρkj ≥ 0, ∀k ∈ Na , ∀j ∈ Nb X Pk ≤ P tot

(4.29)

Pk ≥ 0,

(4.31)

(4.30)

k∈Na

∀k ∈ Na .

Since the constraints on ρkj and Pk are separable, and Pk and ρkj are only related to the SPEB of agent k, we can decompose the above problem into two stages. In the first stage, given the total power budget Pk for agent k, we consider the optimal allocation of Pk among all the anchors, i.e., I PR-1 :

min

{ρkj }, Mk

s.t.

tr {Mk } /Pk  Mk  P I j∈Nb

I



 0

ξkj ρ Q(φˆkj ) de 2β kj

(4.28) – (4.29).

kj

I The optimal solution of PR-1 is denoted by ρ∗kj , and it is independent of the

total power for agent k since Pk only appears as a scaler in the objective and


134

I can be removed. Since the problem PR-1 is formulated for agent k, there are

totally Na problems to be solved in the first stage. In the second stage, we allocate the total Pk for localizing agent k. The objective is the total SPEB of the agents, where the parameter ρ∗kj ’s are from the n P −1 o 2β ∗ I e ˆ first stage PR-1 . In particular, we let Tk = tr ξ / d ρ Q ( φ ) kj kj kj kj j∈Nb kj and formulate the problem as: II PR-1 :

min {Pk }

X Tk Pk k∈N a

s.t. (4.30) – (4.31). II The problem PR-1 is convex in Pk , and the optimal solution is given in a closed

form as follows. I Proposition 10. Given that ρ∗kj is the optimal solution of PR-1 , the optimal II solution of PR-1 is given by

Pk∗

√ P tot Tk √ . =P Tk k∈Na

(4.32)

Proof. The Lagrangian function is given by X X Tk X tot L(Pk , uk , v) = Pk − P − u k Pk + v Pk k∈N k k∈N a

a

where uk , v ≥ 0. The KKT conditions [11] can be derived as ∂L Tk = − 2 − uk + v = 0 ∂Pk Pk

v

X

k∈Na

u k Pk = 0 tot = 0. Pk − P

Since Pk is always positive, we have uk = 0, which leads to Pk =

(4.33)

p

Tk /v in

(4.33). Moreover, the objective is monotonically decreasing in Pk , which imP plies the optimal allocation must use all the power resources, i.e., k∈Na Pk =

P tot . Hence, the optimal solution is given by (4.32).


135

Algorithm 4 Robust power allocation algorithm for multiple-agent network Require: the distance dˆkj and the angle φˆkj between anchor j (j ∈ Nb ) and agent k (k ∈ Na ) 1:

Set Pk ← 1, ∀k ∈ Na

2:

I Solve the first-stage problem PR-1 which gives the optimal solution ρ∗kj

3:

Set ρkj ← ρ∗kj , ∀k ∈ Na , ∀j ∈ Nb

4:

II Solve the second-stage problem PR-1 by using (4.32) to compute the op-

timal solution Pk∗ 5:

∗ Set Pkj ← ρ∗kj Pk∗ , ∀k ∈ Na , ∀j ∈ Nb

The optimal power allocation for the location-aware network is ∗ Pkj = ρ∗kj Pk∗

(4.34)

where Pk∗ is given in (4.32). The detailed algorithm is described in the Algorithm 4. I Remark 8. Since each first stage problem PR-1 in Algorithm 4 is with a sin-

gle SDP constraint, its complexity is much lower than the original problem SDP PR-1 which contains Na SDP constraints. Moreover, the Na first-stage probI lem PR-1 ’s can be independently solved by the Na agents, since each agent

itself does not require any information from other agents. Thus, the computation efficiency can be improved by Na times using the parallel computations among the agents. Remark 9. The proposed distributed algorithm can also be applied to the robust P power allocation with individual power constraint, e.g., k∈Na Pkj ≤ Pjtot . In P particular, we replace such constraint with k∈Na ρkj Pk ≤ Pjtot in the second-

II I stage formulation PR-1 , while the first-stage formulation PR-1 remains the

same. In such case, the close-form solution in (4.34) is not available, however,


136

the optimal solution of the second-stage problem can still be efficiently obtained since the problem is convex. Consequently, we can obtain a sub-optimal solution for the overall problem.

4.4.2

Algorithm for mDPEB Minimization

A similar decomposition method can be applied to the mDPEB minimization PR-2 , i.e., by introducing two variables ρkj and Pk . Instead of solving SDP’s in SPEB minimization, each agent will independently solve a SOCP problem with linear objective for the mDPEB minimization. Specifically, we rewrite (4.25) as µ e2,k

Pk = 2

X

X ξ ξkj

kj φ ρ 1 − 2 sin(εkj ) − ρ q(2φˆkj ) . 2β kj 2β kj e e j∈Nb dkj j∈Nb dkj

Then, the two-stage formulations are given by I PR-2 :

max µ e2,k /Pk {ρkj }

s.t. (4.28) – (4.29)

and II PR-2 :

min {Pk }

X 1 µ e2,k k∈N a

s.t. (4.30) – (4.31). respectively. The optimal power allocation is the product of the optimal solutions of the two-stage problems, given by (4.34). The algorithm for mDPEB minimization is similar to that of Algorithm 4, and hence, we omit the details here.


4.5

137

Simulation Results

In this section, we investigate the localization performance by the proposed power allocation schemes. The total power for localization is normalized to P tot = 1. The proposed optimization of power allocation, i.e., SDP and SOCP, are solved by the standard optimization solver CVX [131].

4.5.1

Power Allocation with Perfect Network Topology Parameters

First, we investigate the SPEB with power allocation as the number of anchors or agents changes. Three schemes of power allocation are compared: the allocation via SPEB minimization formulated in P1SDP , the allocation via mDPEB minimization formulated in P2SOCP , and the uniform allocation which equally assigns P tot over all the anchors. Given the number of anchors and agents, we run Monte Carlo simulation to generate 103 deployments of agents or anchors that are uniformly distributed in a squared region, i.e., U([−10, 10]×[−10, 10]), and then compute the average SPEB obtained by each scheme. In Fig. 4.3 and 4.4, we consider the network with a single agent at the center and anchors uniformly distributed. An example of the network topology is illustrated in Fig. 4.3. We plot the SPEBs resulted by the above-mentioned three schemes in Fig. 4.4. A decreasing tendency in SPEB is observed as the number of anchors increases. This is reasonable since the agent has more freedom to choose “good” anchors when there are more anchors. Moreover, the results show that the mDPEB minimization outperforms the uniform allocation by about 40%, and achieves the SPEB close to the one obtained by SPEB


138

10 8 6 4

[ m]

2 0 −2 −4 −6

Anchor Agent

−8 −10 −10

−8

−6

−4

−2

0

2

4

6

8

10

[ m]

Figure 4.3: The topology of the location-aware network consisting ten anchors (red circle) and 1 agents (blue dot), where the anchors are uniformly distributed in the square region. minimization. Next, we consider a network with multiple agents. Ten anchors are placed with fixed locations, and the agents are uniformly distributed in the region (see Fig. 4.5). Similarly, we compare the SPEB resulted by the three schemes with respect to the number of agents in Fig. 4.6. It shows that, even in multipleagent case, the mDPEB minimization still achieves the similar SPEB as the SPEB minimization, and remarkably outperforms the uniform allocation. It implies that mDPEB is a meaningful performance metric for the optimization of power allocation. In addition, we observed that the average SPEB per agent increases linearly with the number of agents. As indicated by the slope, the speed of SPEB increase of optimized allocation is about 60% slower than that


139

0.45 SP E B mi ni mi z ati on 0.4

mD P E B mi ni mi z ati on Uni form al l oc ati on

0.35

SP E B [ m 2 ]

0.3 0.25 0.2 0.15 0.1 0.05 0

2

4

6

8

10

12

14

16

Numb e r of anchors

Figure 4.4: Comparison of the SPEB in single-agent network resulted by SPEB-minimization allocation, mDPEB-minimization allocation, and uniform allocation. of uniform allocation. Furthermore, we investigate the performance of the two-stage optimization proposed in Section 4.4 which exploits the distributed computations among multiple agents. In Fig. 4.6, we plot the SPEB obtained by the two-stage optimization for both SPEB and mDPEB minimization. The results show that the SPEB solved by two-stage optimization perfectly matches that of one-stage optimization, which validates that the two-stage scheme is able to reach the optimal solution while requiring much less computational time.


140

10 8 6 4

[ m]

2 0 −2 −4 −6

Anchor Agent

−8 −10 −10

−8

−6

−4

−2

0

2

4

6

8

10

[ m]

Figure 4.5: The topology of the location-aware network consisting ten anchors (red circle) and 8 agents (blue dot), where the agents are uniformly distributed in the square region.

4.5.2

Robust Power Allocation with Imperfect Network Topology Parameters

We then investigate the performance of the power allocation with imperfect network topology parameters. We compared the following schemes: allocation SDP SOCP by the robust formulation PR-1 and PR-2 , allocation by the non-robust

formulation P1SDP and P2SOCP , and uniform allocation. We evaluate the actual SPEB of each scheme by Monte Carlo simulation. Specifically, we first solve the power allocation decision of each scheme based on (dˆkj , φˆkj ). For each deployment of (dˆkj , φˆkj ), we generate 105 pairs of uniform errors edkj ’s and eφkj ’s


141

1.4 One -stage SP E B mi ni mi z ati on Two-stage SP E B mi ni mi z ati on

1.2

One -stage mD P E B mi ni mi z ati on Two-stage mD P E B mi ni mi z ati on

1 SP E B [ m 2 ]

Uni form al l oc ati on 0.8

0.6

0.4

0.2

0

2

4

6

8

10

12

14

16

18

20

Numb e r of age nts

Figure 4.6: Comparison of the average SPEB in multiple-agent network (Nb = 10) resulted by SPEB-minimization allocation, mDPEB-minimization allocation, and uniform allocation. Both one-stage and two-stage optimization are considered. which are bounded by εd and εφ , respectively.8 There are totally 103 pairs of (dˆkj , φˆkj ) randomly generated in simulation. Then, we compute the actual SPEB under each realization of (dkj , φkj ) and determine its average value for comparison. In Fig. 4.7, we investigate the actual SPEB with respect to the number of anchors. We consider the single-agent network, and set εd = εφ = 0.2. SDP The results show that the robust SPEB minimization (PR-1 ) outperforms the

non-robust SPEB minimization (P1SDP ) by 34%, and outperforms uniform alSOCP location by 60%; the robust mDPEB minimization (PR-2 ) outperforms the 8

Without loss of generality, we set εdkj = εd and εφkj = εφ for all k, j.


142

0.9 SP E B mi ni mi z ai on (non-robust) 0.8

SP E B mi ni mi z ati on (robust) mD P E B mi ni mi z ati on (non-robust)

0.7

SP E B [ m 2 ]

mD P E B mi ni mi z ati on (robust) 0.6

Uni form al l oc ati on

0.5 0.4 0.3 0.2 0.1

3

4

5

6

7

8

9

10

Numb e r of anchors

Figure 4.7:

The actual SPEB with respect to number of anchors, resulted

by robust, non-robust schemes and uniform allocation with imperfect network topology parameters (εd = 0.2, εφ = 0.2). non-robust mDPEB minimization (P2SOCP ) by 22%, and outperforms uniform allocation by 65%. Moreover, we observed that the actual SPEB of robust mDPEB minimization is smaller than that of robust SPEB minimization, and the same observation is on the non-robust schemes. It implies that, when the network topology parameter error exists, the mDPEB minimization is more robust compared with the SPEB minimization. This is because the mDPEBbased formulation minimizes the maximum positional error over all the directions, which can be viewed as the robust optimization in another dimension. In Fig. 4.8, we investigate the actual SPEB with respect to the error size on network topology parameters, i.e., εd and εφ . We consider a single-agent network with ten anchors deployed on a circle (similar to Fig. 4.5). The distance


143

0.7 SP E B mi ni mi z ati on (non-robust) 0.6

SP E B mi ni mi z ati on (robust) mD P E B mi ni mi z ati on (non-robust)

SP E B [ m 2 ]

0.5

mD P E B mi ni mi z ati on (robust) Uni form al l oc ati on

0.4

0.3

0.2

0.1

0

0

0.05

0.1

0.15

0.2

0.25

0.3

Ne twork top ol ogy parame te r e rror si z e

Figure 4.8: The actual SPEB with respect to the error size on network topology parameters (εd and εφ are set to be equal), resulted by robust, non-robust schemes and uniform allocation. error size εd and angle error size εφ are set to be equal and increase simultaneously. As we observed, for the SPEB minimization, the actual SPEB of nonrobust scheme quickly increases as the error size goes large, while the actual SPEB of the robust scheme increases slowly. In other words, the improvement on SPEB by robust SPEB optimization becomes more remarkable than that by non-robust optimization as the network topology parameters error goes larger. For the mDPEB minimization, the actual SPEB of both robust and non-robust schemes increase slowly. The robust mDPEB minimization outperforms the non-robust mDPEB minimization and robust SPEB minimization by 17% and 6%, respectively. Both Fig. 4.7 and 4.8 have demonstrated the advantage of the proposed robust power allocation schemes, especially the mDPEB minimiza-


144

tion, in the practical location-aware networks with imperfect network topology parameters.

4.6

Summary

In this chapter, we presented an optimization framework for robust power allocation in network localization based on the performance metrics SPEB and mDPEB. We first showed that the optimal power allocation with perfect network parameters can be efficiently obtained via conic programming, and then proposed robust power allocation schemes to combat uncertainties in network topology parameters for practical systems. Moreover, we designed an efficient algorithm for robust power allocation that allows distributed computations among agents. The simulation results demonstrated that the power allocation with robust optimization remarkably outperforms the non-robust power allocation and uniform allocation. Furthermore, we showed that, compared with the SPEB minimization, the mDPEB minimization is more robust to imperfect network topology parameters for power allocation.

Chapter 5 Conclusions and Future Work This thesis focused on the design of dynamic resource allocation schemes for practical wireless communications. We presented a stochastic optimization framework for problem formulation and algorithm designs of radio resource allocation. We applied the proposed framework into three promising wireless systems and achieved remarkable performance improvements compared with the conventional designs. In this chapter, we first review the results and contributions of this thesis, and then discuss several potential directions in future works.

5.1

Conclusions

We summarize the conclusions of this thesis by reviewing the three proposed applications of stochastic optimization in wireless communications.

145

Chapter 5. Conclusions and Future Work

5.1.1

146

Slow Adaptive OFDMA Systems

The first application (Chap. 2) we investigated is the adaptive OFDMA system which has been identified as one of the leading candidates for providing broadband and multimedia services in future wireless systems. Despite years of efforts to improve the practicality of adaptive OFDMA, such promising technique is still far from real implementation due to the prohibitively high computational complexity and excessive control overhead. We proposed a slow adaptation framework for adaptive OFDMA systems, which adapts the subcarrier allocation on a much slower timescale than that of the fluctuation of wireless channel fading. The results showed that the proposed schemes can achieve a throughput close to that of fast adaptive OFDMA schemes, while significantly reduce the computational complexity and control signaling overhead (e.g., achieving 92% throughput with 1% computational cost and signaling overhead of the fast adaptation scheme). In the proposed framework, we considered the problem formulations with QoS guarantee in three different scenarios. The first problem is to accommodate average user data rate requirements for elastic traffic. We imposed constraints to meet the minimum expected average data rate of users. The second problem extends the previous problem to the inelastic traffic case. We applied robust optimization methodology to guarantee the worst-case throughput of each user. The third problem considers the instantaneous data rate requirement of individual users to be accommodated on the fast timescale with high probability. We formulated the problem using the chance constrained programming techniques. To solve the problem, we proposed a STC formulation, and a polynomial-time algorithm is developed for computing the optimal solution


147

of STCP. Simulation results showed that the proposed algorithm converges with 22 iterations on average. Moreover, the proposed framework demonstrated three types of tradeoffs: (i) tradeoff between spectral efficiency and computational cost by tuning the length of adaptation window; (ii) tradeoff between spectral efficiency and outage probability for inelastic traffic by tuning the deviation parameter ρ0 in the safe slow adaptation scheme; (iii) tradeoff between spectral efficiency and rate outage probability by tuning the tolerance parameter ǫk .

5.1.2

Composite Radio MIMO Networks

The second application (Chap. 3) focused on the resource allocation problem in MIMO networks. Future wireless networks will face a growing demand of supporting multiple radios collocated on mobile devices. Multiple antenna technique is considered as a key component in future wireless standards to enhance the system capacity and reliability. A major challenge arises in the use of multiple antennas to support concurrent operation of multiple radios. To tackle issue, we proposed a dynamic resource allocation scheme for MIMO networks, which adaptively allocates antennas and transmit power among different radios. We formulated the problem into a chance constrained program. Despite the STC technique we developed in the slow adaptive OFDMA for solving chance constrained program, the problem here is more challenging due to two major difficulties. First, the STC technique, including both formulation and algorithm, requires both the cumulant generating function of MIMO capacity and its derivatives with respective to the decision variables. This is more complicated in MIMO systems since the capacity function is associated with


148

multiple random channel coefficients. We derived the close-form expressions for such quantities based on previous work on MIMO capacity. Second, the antenna-and-power allocation is a mixed-integer non-convex problem, since the number of antennas assign to radios must be an integer. Hence, the convex optimization algorithms, such as interior point method, cannot be directly applied. We exploited the special structure of our problem, to separate the allocation of antennas and power into two stage problems. In first stage, we proposed an algorithm to efficiently find all the feasible antenna allocations to meet all the data rate constraints. In the second stage, we optimize its power allocation and find the optimal antenna allocation decision which yields the highest system throughput. The simulation results validated that the data rate requirement of each radio is satisfied with high probability. Moreover, it showed that the proposed allocation scheme outperforms uniform allocation in the satisfaction of probabilistic rate constraints. For instance, the system success probability can be increased by 31% using the proposed allocation when Lkm = 4. On the other hand, we demonstrated that the dynamic resource allocation achieves higher spectral efficiency compared with uniform allocation, and a larger portion of the throughput gain is attributed to the antenna allocation. The Monte Carlo simulation showed that the the throughput gap between the proposed scheme and uniform allocation is up to 100 Mbps when Lkm = 4 and 200 Mbps when Lkm = 8 with 90% probability.

5.1.3

Energy-Efficient Location-Aware Networks

The third application (Chap. 4) targeted at achieving energy efficiency in location-aware networks. Future wireless networks will support an increas-


149

ing number of location-based applications and services. In wireless locationaware networks, the transmit power allocation not only affects network lifetime, throughput, and interference, but also determines the localization accuracy. However, few work has investigated the power resource allocation for localization. In general, how to optimally allocate the transmit power in location-aware networks remains as an open problem. Moreover, the previous work on power allocation in location-aware networks ignores the uncertainties in network parameters such as nodes’ positions and channel conditions. As a result, it often leads to sub-optimal or even infeasible solutions in practice. We presented an optimization framework for robust power allocation in network localization to tackle imperfect network topology parameters. We formulated the problem to minimize the SPEB and the mDPEB, respectively, both of which characterize the fundamental limits of localization accuracy. We showed that the optimal solutions of such formulations can be efficiently solved via conic programming. To tackle imperfect network parameters, we proposed a novel robust optimization method to minimize the worst case SPEB/mDPEB. Furthermore, a distributed algorithm for robust power allocation scheme is developed to improve the computational efficiency by allowing parallel computations among agents. The simulation results showed that the proposed schemes significantly outperform uniform power allocation, and the robust schemes outperform their non-robust counterparts when the network topology parameters are subject to uncertainties. In addition, we showed that, compared with the SPEB minimization, the mDPEB minimization is more robust to imperfect network topology parameters for power allocation. Another conclusion in this work is that the synchronous and asynchronous


150

location-aware networks are equivalent. In asynchronous networks, round-trip TOF is employed, and hence, both anchors and agents needs power allocation. We proved that optimal power allocation on agents and anchors should be proportional, which enables us to simply focus on the anchor power allocation.

5.2

Future Work

The work presented in this thesis offers many possibilities for future extensions. In particular, the following topics are of interest: • In slow adaptive OFDMA systems, it would be worthwhile to investigate the chance constrained subcarrier allocation problem when frequency correlation exists, or when the channel distribution information is not perfectly known at the BS. Another interesting direction is to consider discrete data rate and exclusive subcarrier allocation, which is more practical and easier for implementation. In fact, the proposed algorithm based on cutting plane methods can be extended to incorporate integer constraints on the variables (see e.g., [67]). • In composite radio MIMO networks, we have investigated the resource allocation problem in a downlink network model. The extension of the proposed framework in uplink model is straightforward. However, since there are more radio modules in transmit nodes, the computational complexity can be much higher when the number of decision variables is large. Therefore, it would be essential to design more efficient algorithms for solving the allocation problem, especially when there are a large number of mobile users. Moreover, we have focused on the antenna allocation on transmit node. It would be interesting to consider the joint allocation


151

of the antennas on both transmit and receive nodes. Besides, we can further consider a more advanced system where the node/radio is allowed to transmit and receive simultaneously. It motivates the allocation of both transmit and receive antennas on the same node. There could be more scenarios for consideration. The further study of the antenna-and-power allocation in more general composite radio MIMO networks would facilitate the design of the emerging cognitive radio techniques. With the application of supporting co-operating of multiple radios, it can prompt more efficient utilization of the scarce spectrum resources. • In energy-efficient location-aware networks, we have proposed a twostage distributed algorithm, which enables parallel computations among the agents in the first stage. Although the computational complexity mainly lies in the first stage, the second-stage problem is still solved in a centralized way. A fully distributed algorithm would be more appealing, especially in a large scale network where the number of agents and anchors is large. Moreover, it would be interesting to extend the power allocation to cooperative localization. In cooperative networks, agents can help each other to gain more positional information besides that from anchors. By doing so, the localization accuracy can be significantly improved. Since the agents are usually mobile nodes with limited power resources, the efficient power allocation would be more essential. On the other hand, the tracking and navigation are of great interest in recent years, and have wide applications in civilian life and military operations. Hence, it would also be a worthwhile topic to investigate. • In the development of stochastic optimization methodology, there are several open problems to be addressed. First, in the use of the Bernstein


152

approximation to construct STC for chance constraint, the tightness of such approximation has not been quantified. Analysis on such issue would enable us to evaluate the performance of STC. Second, the convexity of chance constraint is worthwhile to be further investigated. If the convexity can be identified, then it would be necessary to design efficient algorithms for solving convex chance constraint. How to accurately evaluate the probabilistic function in chance constraint would be an critical question. Third, a more general framework for robust optimization will benefit the design of many practical wireless systems. The applications of robust optimization are ubiquitous in realistic systems. However, conventional robust methodologies are only able to handle few typical problem formulations, such as LP, SOCP, and SDP. In many cases, the robust counterpart of convex optimization problem can be non-convex. Moreover, the conventional robust methods could be overly conservative and lead to poor performance. A potential direction to address such issue to combine it with chance constrained programming, and incorporate risk measures in the design of resource allocation schemes. Our work in this thesis can be viewed as an initial attempt to apply the stochastic optimization methodology to design dynamic resource allocation schemes for wireless systems. As randomness is ubiquitous in numerous wireless system parameters, due to the stochastic nature of wireless channels, user movements, etc., we hope that the proposed methodologies will find further applications in wireless communications.

Appendix A Bernstein Approximation Theorem Theorem 4. Suppose that F (x, r) : Rn × Rnr → R is a function of x ∈ Rn

and r ∈ Rnr , and r is a random vector whose components are nonnegative. For every ǫ > 0, if there exists an x ∈ Rn such that inf {Ψ(x, ̺) − ̺ǫ} ≤ 0,

̺>0

(A.1)

where Ψ(x, ̺) , ̺E exp(̺−1 F (x, r)) ,

then Pr {F (x, r) > 0} ≤ ǫ.

Proof. (Sketch) The proof of the above theorem is given in [17] in details. To help the readers to better understand the idea, we give an overview of the proof here. It is shown in [17] (see section 2.2 therein) that the probability Pr{F (x, r) ≥ 0} can be bounded as follows: Pr{F (x, r) > 0} ≤ E ψ(̺−1 F (x, r)) . 153

Appendix A. Bernstein Approximation Theorem

154

Here, ̺ > 0 is arbitrary, and ψ(·) : R → R is a nonnegative, nondecreasing, convex function satisfying ψ(0) = 1 and ψ(z) > ψ(0) for any z > 0. One such ψ is the exponential function ψ(z) = exp(z). If there exists a ̺ˆ > 0 such that E exp(ˆ ̺−1 F (x, r)) ≤ ǫ,

then Pr{F (x, r) > 0} ≤ ǫ. By multiplying by ̺ˆ > 0 on both sides, we obtain the following sufficient condition for the chance constraint Pr {F (x, r) > 0} ≤ ǫ to hold: Ψ(x, ̺ˆ) − ̺ˆǫ ≤ 0.

(A.2)

In fact, condition (A.2) is equivalent to (A.1). Thus, the latter provides a conservative approximation of the chance constraint.

Appendix B Ergodic MIMO Capacity and Moment Generating Function Theorem 5. (cf. [31]) Consider the capacity of MIMO channels given by γ † HH , C = log2 det I + LT where γ is the average SNR per receive antenna, and H ∈ CLR ×LT is the channel matrix with LT and LR denoting the number of transmit antennas and receive antennas, respectively. The ergodic capacity and the moment generating function of C can be written as E{C} = Lf

L min X l=1

det

Z

0

∞

xLmax −Lmin +j+i−2 · e−x

γ · Ui,j log2 1 + x dx , LT i,j=1,··· ,Lmin (Z lnz2 ) ∞ γ dx , x E{ezC } = Lf det xLmax −Lmin +j+i−2 · e−x · 1 + LT 0

i,j=1,··· ,Lmin

155

Appendix B. Ergodic MIMO Capacity and Moment Generating Function

where Lmin = min{LT , LR }, Lmax = max{LT , LR }, and Lf =

L min Y i=1

Ul,j (x) =

  x,  1,

(Lmin − i)!(Lmax − i)! if

l = j,

if

l 6= j.

!−1

,

156

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