Deterministic Network Calculus

Abstract

7.1. MIMO servers and aggregation of flows. Cover; Half-Title Page; Title Page; Copyright Page; Contents; Acknowledgments; Introduction; I.1. Organization of the book; I.2. How to read this book; I.3. Network calculus in four pages; 1. Basic Model: Single Server, Single Flow; 1.1. Modeling principles; 1.2. Constant rate server; 1.3. Flow model; 1.4. Server model; 1.5. Delay and memory usage models; 1.6. Summary; PART 1: (min, plus) Functions and Algorithms; 2. The (min, plus) Functions Semi-ring; 2.1. The (min, plus)-based dioids; 2.1.1. Dioids; 2.1.2. The (min, plus) dioid; 2.1.3. The dioid of (min, plus) functions. 2.2. Sub-additive closure2.2.1. Kleene star operator; 2.2.2. Sub-additive closure; 2.3. Deconvolution; 2.3.1. Residuation theory; 2.3.2. (min, plus) deconvolution as a residuation operator; 2.4. Link with (max, plus) dioid; 2.5. Summary; 3. Sub-classes of Functions; 3.1. Usual functions; 3.1.1. Convolution and deconvolution of the usual classes of functions; 3.1.2. Horizontal and vertical deviations for the usual classes of functions; 3.2. Non-negative and non-decreasing functions; 3.2.1. Pseudo-inverse of a function; 3.2.2. Convolution and continuity; 3.3. Concave and convex functions. 3.3.1. Concave functions3.3.2. Convex functions; 3.4. Summary; 4. Efficient Computations for (min, plus) Operators; 4.1. Classes of functions with finite representations; 4.2. Piecewise linear concave/convex functions; 4.2.1. Representation of piecewise linear concave and convex functions; 4.2.2. (min, plus)-convolution of convex and concave functions; 4.3. A stable class of functions; 4.3.1. Examples of instability of some classes of functions; 4.3.2. Class of plain piecewise linear ultimately pseudo-periodic functions; 4.3.3. Functions with discrete domain. 4.4. Containers of (min, plus) functions4.4.1. Notations and context; 4.4.2. The object container; 4.4.3. Inclusion functions for containers; 4.5. Implementations; PART 2: Network Calculus: Local Analysis; 5. Network Calculus Basics: a Server Crossed by a Single Flow; 5.1. Arrival curve; 5.2. Service curves; 5.2.1. Min-plus minimal service curve; 5.2.2. Strict minimal service curve; 5.2.3. Comparison of min-plus and strict minimal service curves; 5.2.4. Maximal service curve; 5.3. From curves to performance guarantees; 5.3.1. Backlog and delay bounds. 5.3.2. Arrival curve for the departure cumulative function5.3.3. Complements on the performance operators; 5.4. Bibliographic and historic notes; 5.5. Summary; 6. Single Flow Crossing Several Servers; 6.1. Servers in tandem; 6.1.1. Concatenation and service convolution; 6.1.2. The pay burst only once phenomenon; 6.1.3. Composition of strict service curves; 6.2. Control design; 6.2.1. Tandem control; 6.2.2. Feedback control; 6.3. Essential use cases; 6.3.1. Essential services curves; 6.3.2. Essential arrival curves; 6.4. Summary; 7. Multiple Flows Crossing One Server.