Mathematics of Internet Congestion Control

• 1 Introduction.- 2 Resource Allocation.- 2.1 Resource allocation as an optimization problem.- 2.2 A general class of utility functions.- 2.3 Appendix: Convex optimization.- 3 Congestion Control: A decentralized solution.- 3.1 Primal algorithm.- 3.2 Dual algorithm.- 3.3 Exact penalty functions.- 3.4 Primal-dual approach.- 3.5 Other variations in the primal approach.- 3.6 REM: A one-bit marking scheme.- 3.7 Multipath routing.- 3.8 Multirate multicast congestion control.- 3.9 A pricing interpretation of proportional fairness.- 3.10 Appendix: Lyapunov stability.- 4 Relationship to Current Internet Protocols.- 4.1 Window flow control.- 4.2 Jacobson’s adaptive window flow control algorithm.- 4.3 TCP-Vegas.- 4.4 Random Early Detection (RED).- 4.5 Explicit Congestion Notification (ECN).- 4.6 High-throughput TCP.- 5 Linear Analysis with Delay: The single link case.- 5.1 Single TCP-Reno source with droptail.- 5.2 Multiple TCP sources with identical RTTs.- 5.3 TCP-Reno and RED.- 5.4 Proportionally-fair controller.- 5.5 High-throughput TCP.- 5.6 Dual algorithm.- 5.7 Primal-dual algorithm.- 5.8 Appendix: The Nyquist criterion.- 6 Linear Analysis with Delay: The network case.- 6.1 Primal controllers.- 6.2 Dual algorithm.- 6.3 Primal-dual algorithm.- 6.4 Appendix: Multivariable Nyquist criterion.- 7 Global Stability for a Single Link and a Single Flow.- 7.1 Proportionally-fair controller over a single link.- 8 Stochastic Models and their Deterministic Limits.- 8.1 Deterministic limit for proportionally-fair controllers.- 8.2 Individual source dynamics.- 8.3 Price feedback.- 8.4 Queue-length-based marking.- 8.5 TCP-type congestion controllers.- 8.6 Appendix: The weak law of large numbers.- 9 Connection-level Models.- 9.1 Stability of weighted proportionally-fair controllers.- 9.2Priority resource allocation.- 10 Real-time Sources and Distributed Admission Control.- 10.1 Resource sharing between elastic and inelastic users.- 10.2 Probing and distributed admission control.- 10.3 A simple model for queueing at the link buffer.- 10.4 Appendix: Diffusion approximation.- 11 Conclusions.- References.

"This first book on the mathematics of congestion control [is] very useful for both students and researchers, but also for system engineers developing new Internet protocols. This is an elementary book for graduate students that uses a rich pallet of mathematical tools: convex optimization and control theory (Lyapunov stability and Nyquist criteria and stochastic processes). The tools are summarized well in appendices which makes the book self-contained and easy to read. The book is very well written and will undoubtedly remain a major reference on the topic for years to come."   —Mathematical Reviews"This well-written book presents an overview of the mathematics behind internet congestion control. Focusing on the well-established TCP protocol, it starts with a gentle introduction on TCP and the packet/acknowledgement structure of the internet, before proceeding to congestion and the principles of congestion control…. Overall, this book presents in an easy-to-understand way the principles of internet congestion control with a focus on the mathematical concepts behind them. The book concludes with a useful index and a rich source of bibliographical references."   —Zentralblatt Math"The investigation of fairness, efficiency, and dynamics of TCP led to the development of a new fast-growing research domain of mathematical models for congestion control. The Mathematics of Internet Congestion Control by R. Srikant provides a valuable comprehensive introduction to this new exciting research area. …Using mathematical models, Srikant successfully explains the principal ideas behind Internet congestion control. The publication of the book is very timely. Currently, there are no other books on mathematical models for congestion control. …All material in the book is accessible to graduate or even undergraduate students of mathematical and electrical engineeringdisciplines. Many chapters are supplied with appendices which review the necessary background material and make the book self-contained. In conclusion, this book can be equally recommended for beginners as well as for expert researchers working in the domain of congestion control. For beginners, the book will be a good starting point from which to explore a vast and rapidly growing body of literature on the subject of Internet congestion control. For expert researchers, the book will definitely help to place congestion control in perspective and will point to new avenues in this exciting research domain." —SIAM Book Reviews