# The Network Calculus Book The Network Calculus book by Jean-Yves Le Boudec and Patrick Thiran is available for free download:

Have a look at the Table of Contents. The book is also available as Springer-Verlag Lecture Notes on Computer Science number 2050.

### What is Network Calculus ?

Network Calculus is a collection of results based primarily on min-plus algebra, which applies to deterministic queuing systems found in communication networks. It can be used to understand: • what is a T-SPEC or arrival curve constraint;
• how to compute delay and backlog bounds in deterministic or time sensitive networks;
• why re-shaping delays can be ignored in shapers or spacer-controllers;
• a common model for schedulers;
• and much more.

### Tutorials

• "A Short Introduction to Network Calculus", pdf, IMAG, Grenoble, Nov 12, 2019,
• "A Long Introduction to Network Calculus", pdf, video, Dagstuhl Seminar on Analysis, Design, and Control of Predictable Interconnected Systems, March 4 and EPFL April 5, 2019,
• A Short Course (2 to 3 hours) ppt
• EPFL Doctoral School click here
• Pisa July 2003 click here
• Other Network Calculus Tutorials (powerpoint + zipped): Sigmetrics June 2002, ENS May 2001 (systems theory aspects)

### Table of Contents

Part I A First Course in Network Calculus

1 Network Calculus
1.1 Models for Data Flows
1.2 Arrival Curves
1.3 Service Curves
1.4 Network Calculus Basics
1.5 Greedy Shapers
1.6 Maximum Service Curve, Variable and Fixed Delay
1.7 Handling Variable Length Packets
1.8 Lossless Effective Bandwidth and Equivalent Capacity
1.9 Proof of Theorem\nobreakspace {
1.10 Bibliographic Notes
1.11 Exercises

2 Application of Network Calculus to the Internet
2.1 GPS and Guaranteed Rate Schedulers
2.2 The Integrated Services Model of the IETF
2.3 Schedulability
2.4 Application to Differentiated Services
2.5 Exercises

Part II Mathematical Background

3 Basic Min-plus and Max-plus Calculus
3.1 Min-Plus Calculus
3.2 Max-Plus Calculus
3.3 Exercises

4 Min-plus and Max-Plus System Theory
4.1 Min-Plus and Max-Plus Operators
4.2 Closure of an Operator
4.3 Fixed Point Equation (Space Method)
4.4 Fixed Point Equation (Time Method)
4.5 Conclusion

Part III A Second Course in Network Calculus

5 Optimal Multimedia Smoothing
5.1 Problem Setting
5.2 Constraints Imposed by Lossless Smoothing
5.3 Minimal Requirements on Delays and Playback Buffer
5.4 Optimal Smoothing Strategies
5.5 Optimal Constant Rate Smoothing
5.6 Optimal Smoothing versus Greedy Shaping
5.7 Comparison with Delay Equalization
5.8 Lossless Smoothing over Two Networks
5.9 Bibliographic Notes

6 FIFO Systems and Aggregate Scheduling
6.1 Introduction
6.2 General Bounds for Aggregate Scheduling
6.3 Stability of a Network with Aggregate Scheduling
6.4 Bounds for a FIFO Service Curve Element
6.5 Bounds for a Network of FIFO CBR Servers
6.6 Bibliographic Notes
6.7 Exercises

7 Adaptive and Packet Scale Rate Guarantees
7.1 Introduction
7.2 Adaptive Guarantee
7.3 Application to the Internet: Packet Scale Rate Guarantee
7.4 Bibliographic Notes
7.5 Exercises

8 Time Varying Shapers
8.1 Introduction
8.2 Time Varying Shapers
8.3 Time Invariant Shaper with Non-zero Initial Conditions
8.4 Time Varying Leaky-Bucket Shaper
8.5 Bibliographic Notes

9 Systems with Losses
9.1 A Representation Formula for Losses
9.2 Application 1: Bound on Loss Rate
9.3 Application 2: Bound on Losses in Complex Systems
9.4 Solution to Skohorkhod's Reflection Problem with Two Boundaries
9.5 Bibliographic Notes

Bibliography

Index