Fundamentals of Resource Allocation in Wireless Networks - Theory and Algorithms

Preface

Contents

List of Figures

List of Symbols

Part I Mathematical Framework

On the Perron Root of Irreducible Matrices

Some Basic Definitions

Some Bounds on the Perron Root and their Applications

Concavity of the Perron Root on Some Subsets of Irreducible Matrices

Kullback--Leibler Divergence Characterization

A Rate Function Representation for Large Deviations of Finite Dimensional Markov Chains

Some Extended Perron Root Characterizations

Collatz--Wielandt-Type Characterization of the Perron Root

Convexity of the Perron Root

Some Definitions

Sufficient Conditions

Convexity of the Feasibility Set

Necessary Conditions

Special Classes of Matrices

Symmetric Matrices

Symmetric Positive Semidefinite Matrices

The Perron Root under the Linear Mapping

Some Bounds

Disproof of the Conjecture

The Perron Root under Exponential Mapping

A Necessary and Sufficient Condition on Strict Convexity of the Feasibility Set

Graph-theoretic Interpretation

Generalizations to Arbitrary Nonnegative Matrices

Log-Convexity of the Spectral Radius

Characterization of the Spectral Radius

Existence of Positive Eigenvectors

Collatz--Wielandt-Type Characterization of the Spectral Radius

Bibliographical Notes

On the Positive Solution to a Linear System with Nonnegative Coefficients

Basic Concepts and Definitions

Feasibility Sets

Convexity Results

Log-Convexity of the Positive Solution

Convexity of the Feasibility Set

Strict Log-Convexity

Strict Convexity of the Feasibility Sets

The Linear Case

Part II Principles of Resource Allocation in Wireless Networks

Introduction

Network Model

Basic Definitions

Medium Access Control

Wireless Communication Channel

Signal-to-Interference Ratio

Different Receiver Structures

Power Constraints

Data Rate Model

Examples

Resource Allocation Problem in Communications Networks

End-to-End Rate Control in Wired Networks

Fairness Criteria

Algorithms

Problem Formulation for Wireless Networks

Joint Power Control and Link Scheduling

Feasible Rate Region

End-to-End Window-Based Rate Control

MAC Layer Fair Rate Control

Utility-Based Power Control

Efficiency-Fairness Trade-Off

Kuhn--Tucker Conditions

Interpretation in the QoS Domain

Remarks on Joint Power Control and Link Scheduling

Optimal Joint Power Control and Link Scheduling

High SIR Regime

Low SIR Regime

Wireless Links with Self-Interference

QoS-based Power Control

Some Definitions

Axiomatic Interference Functions

QoS-Based Power Control Algorithms

Max-Min SIR Balancing Power Control

Some Preliminary Observations

Characterization under Sum Power Constraints

General Power Constraints

Some Consequences and Applications

Utility-based Power Control with QoS Support

Hard QoS Support

Soft QoS Support

Utility-Based Joint Power and Receiver Control

Problem Statement

Perfect Synchronization

Decentralized Alternating Computation

Max-Min SIR Balancing

Additional Results for a Noiseless Case

The Efficiency--Fairness Trade-off

Existence and Uniqueness of Log-SIR Fair Power Allocation

Proofs

Part III Algorithms

Power Control Algorithms

Introduction

Some Basic Definitions

Convex Statement of the Problem

Strong Convexity Conditions

Gradient Projection Algorithm

Global Convergence

Rate of Convergence

Diagonal Scaling

Projection on a Closed Convex Set

Distributed Implementation

Local and Global Parts of the Gradient Vector

Adjoint Network

Distributed Handshake Protocol

Some Comparative Remarks

Noisy Measurements

Incorporation of QoS Requirements

Hard QoS Support

Soft QoS Support

Primal-Dual Algorithms

Improving Efficiency by Primal-Dual Methods

Generalized Lagrangian

Primal-Dual Algorithms

Decentralized Implementation

Min-max Optimization Framework

Simulation Results

Part IV Appendices

Some Concepts and Results from Matrix Analysis

Vectors and Vector Norms

Matrices and Matrix Norms

Square Matrices and Eigenvalues

Matrix Spectrum, Spectral Radius and Neumann Series

Orthogonal, Symmetric and Positive Semidefinite Matrices

Perron--Frobenius Theory

Perron--Frobenius Theorem for Irreducible Matrices

Perron--Frobenius Theorem for Primitive Matrices

Some Extensions to Reducible Matrices

The Existence of a Positive Solution p to(I-X)p=b

Some Concepts and Results from Convex Analysis

Sets and Functions

Convex Sets and Functions

Strong Convexity

Majorization and Schur-Convexity

Log-Convex Functions

Inverse Functions of Monotonic Log-Convex Functions

Basics of Optimization Theory

Characterization of Numerical Convergence

Convergence of Gradient Projection Algorithms

Basics of Lagrangian Optimization Theory

Saddle Points, Saddle Functions, Min-Max Functions

References

Index