Interference Calculus: A General Framework For Interference Management And Network Utility Optimization (foundations In Signal Processing, Communications And Networking, Vol. 7)

This book develops a mathematical framework for modeling and optimizing interference-coupled multiuser systems. At the core of this framework is the concept of general interference functions, which provides a simple means of characterizing interdependencies between users. The entire analysis builds on the two core axioms scale-invariance and monotonicity. The proposed network calculus has its roots in power control theory and wireless communications. It adds theoretical tools for analyzing the typical behavior of interference-coupled networks. In this way it complements existing game-theoretic approaches. The framework should also be viewed in conjunction with optimization theory. There is a fruitful interplay between the theory of interference functions and convex optimization theory. By jointly exploiting the properties of interference functions, it is possible to design algorithms that outperform general-purpose techniques that only exploit convexity. The title “network calculus” refers to the fact that the theory of interference functions constitutes a generic theoretical framework for the analysis of interference coupled systems. Certain operations within the framework are “closed”, that is, combinations of interference functions are interference functions again. Also, certain properties are preserved under such operations. This, provides a methodology for analyzing different multiuser performance measures that can be expressed as interference functions or combinations of interference functions. Cover......Page 1 Front matter......Page 2 Introduction......Page 11 Notation......Page 13 Basic Axiomatic Framework of Interference Functions......Page 14 Convex and Concave Interference Functions......Page 15 Log-Convex Interference Functions......Page 16 Examples  Interference in Wireless Networks......Page 17 Linear Interference Function......Page 18 Beamforming......Page 19 Receive Strategies......Page 20 The Spectral Radius  Indicator of Feasibility......Page 21 Min-Max Balancing and Feasible Sets......Page 22 Transmit Strategies and Duality......Page 23 Robust Designs......Page 24 Interference Functions in Other Contexts......Page 25 Systems of Coupled Interference Functions......Page 27 Interference Coupling......Page 28 Asymptotic Coupling Matrix......Page 29 The Dependency Matrix......Page 30 Strict Monotonicity and Strict Log-Convexity......Page 31 Standard Interference Functions......Page 33 Comparison between Standard Interference Functions and General Interference Functions......Page 34 Continuity......Page 36 Continuation on the Boundary......Page 37 Continuity of Standard Interference Functions......Page 38 SIR-Based QoS Sets......Page 39 Comprehensiveness......Page 40 Fixed Point Characterization of Boundary Points......Page 41 Power-Constrained QoS Regions......Page 42 Sum-Power Constraint......Page 43 Individual Power Constraints......Page 44 The QoS Balancing Problem......Page 45 Equivalent Convex Reformulations......Page 46 Equivalent Log-Convex Reformulation......Page 47 The Structure of Interference Functions and Comprehensive Sets......Page 49 Max-Min and Min-Max Representations......Page 50 Majorants and Minorants......Page 51 Representation Theorem based on Level Sets......Page 52 Elementary Sets and Interference Functions......Page 54 Interference Functions and Comprehensive Sets......Page 56 Comprehensive Hull......Page 58 Concave Interference Functions......Page 60 Representation of Concave Interference Functions......Page 61 Properties of the Set $N_0(I)$ ......Page 64 Synthesis of Concave Interference Functions......Page 65 Least Concave Majorant and Alternative Representation......Page 67 Representation of Convex Interference Functions......Page 71 Properties of the set $W_0(I)$ ......Page 73 Synthesis of Convex Interference Functions......Page 74 Greatest Convex Minorant and Alternative Representation......Page 76 Expressing Utility Sets as Sub-/Superlevel Sets of Convex/Concave Interference Functions......Page 79 Basic Building Blocks of Log-Convex Interference Functions......Page 82 Analysis of Log-Convex Interference Functions......Page 85 Synthesis of Log-Convex Interference Functions......Page 87 Connection with the Kullback-Leibler Distance......Page 88 Every Convex Interference Function is a Log-Convex Interference Function......Page 89 Greatest Log-Convex Minorant......Page 90 Application to Standard Interference Functions......Page 92 Convex Weakly Standard Interference Functions......Page 93 Concave Weakly Standard Interference Functions......Page 95 The Structure of Convex and Concave Standard Interference Functions......Page 96 The Structure of Log-Convex Standard Interference Functions......Page 97 Convex/Concave Bounds......Page 99 Least Concave Majorant and Greatest Convex Minorant......Page 101 Comparison of Convex and Log-Convex Minorants......Page 102 Convex and Concave Approximations of SIR Feasible Sets......Page 103 Convex Comprehensive Level Sets......Page 105 The Structure of Interference Functions and Comprehensive Sets......Page 0 Nash Bargaining and Proportional Fairness for Log-Convex Utility Sets......Page 109 Nash Bargaining for Strictly Log-Convex Utility Sets......Page 110 The Conventional Nash Bargaining Solution (NBS)......Page 111 Log-Convex Utility Sets......Page 113 Generalized Nash Bargaining for Compact Log-Convex Sets......Page 114 Non-Symmetric Nash Solution and the Conjugate $f_ I$......Page 118 The SIR Region of Log-Convex InterferenceFunctions......Page 119 Existence of a Fixed Point for Constant W......Page 120 Min-Max and Max-Min Balancing......Page 124 Generalization to Adaptive W......Page 125 Fixed Point Analysis for General Log-Convex Interference Functions......Page 126 Fairness Gap......Page 129 Proportional Fairness - Boundedness, Existence, and Strict Log-Convexity......Page 132 Boundedness of the Cost Function......Page 133 Existence of a Proportionally Fair Optimizer......Page 138 Uniqueness of the Solution......Page 140 Equivalence of Nash Bargaining and Proportional Fairness......Page 145 Weighted Utility and Cost Optimization......Page 147 SINR Region under a Total Power Constraint......Page 148 Individual Power Constraints - Pareto Optimality and Strict Convexity......Page 150 Properties of Boundary Points......Page 151 Analysis of the Pareto Boundary......Page 155 Concept of Strongly Coupled Users......Page 157 Strict Monotonicity......Page 159 Strict Log-Convexity......Page 161 QoS-Constrained Power Minimization......Page 165 Matrix-Based Iteration......Page 167 Feasibility......Page 168 Monotonicity......Page 170 Lipschitz Continuity......Page 171 Global Convergence and Comparison with the Fixed Point Iteration......Page 172 Continuously Dierentiable Interference Functions......Page 174 Generalization to Non-Smooth Interference Functions......Page 175 Invertibility......Page 176 Non-Smooth Versions of Newton's Method......Page 177 Superlinear Convergence......Page 178 Quadratic Convergence for Degree-2 Semi-Continuous Interference Functions......Page 180 Convergence of the Fixed Point Iteration......Page 181 Linear Convergence......Page 182 Geometrical Interpretation for the 2-User Beamforming Case......Page 184 Matrix Iteration for Convex Interference Functions......Page 187 Convergence Analysis......Page 190 Weighted SIR Balancing......Page 193 The Max-Min Optimum......Page 194 Set of Optimal Receive Strategies......Page 195 Existence and Uniqueness of the Optimal Power Allocation......Page 196 Equivalence of Max-Min and Min-Max Balancing......Page 197 Principal Eigenvector (PEV) Iteration......Page 198 Fixed Point Iteration......Page 202 Convergence Behavior of the PEV Iteration......Page 205 Back matter......Page 206