Stability of Congestion Control Algorithms Using Control Theory

Stability of Congestion Control Algorithms Using Control Theory with an application to XCP Ioannis Papadimitriou (jpg@stanford. edu) George Mavromatis (gmavr@stanford. edu)

Outline • Previous Work § Motivation behind applying control theory on congestion control protocols • e. Xplicit Control Protocol (XCP) § Stability proof for users with common RTT • Stability § Stability conditions for heterogeneous users • Simulations § NS-2 implementation of XCP and tests

Previous Work • In 1998, F. P. Kelly proposes a fluid-flow description of a network and proves stability • Soon, conditions for stability of this model are established for homo/heterogeneous users • Application of these results to TCP and AQM protocols § TCP unstable for long RTTs and high capacities § RED tradeoffs § Guidelines for AQM implementations § Proposal of new AQM protocols

Survey – Open Issues • Under all these assumptions, are systems really locally stable? • Does local stability imply network stability? • Can we find new fair/efficient algorithms with known stability behavior? • This is a hot research area

XCP – Main Features • Descriptive feedback of congestion levels • Decoupling between efficiency control and fairness control • Congestion header carried by each packet • Stability proof for a single link and N users having the same RTT • Simulations with varying traffic requests and RTTs

XCP stability for different RTTs • Standard assumptions § Constant number of users § One bottleneck link § Local stability around equilibrium point § Negligible queuing delays • Under these assumptions § Average RTT becomes constant (d) § Positive and negative feedback is equally divided among the users around equilibrium § Dynamics become linearized • Our proof: XCP stability conditions for heterogeneous users

Stability Proof • New linearized differential equations with • • arbitrary delay for each user Transform to A • x = 0 System stable when all roots of det[A] = 0 have negative real part We describe the stability conditions that must be satisfied for N users. Now the problem purely algebraic although difficult

Our solution for N = 2 • Padé approximation for exp(-d·s) factors • Code in Matlab to find the roots of det[A] = 0 for different values of parameters a, b and different delays. • Plot of the stability region

Stability Region Plot for N = 2

Simulink Model

XCP simulation • We have implemented XCP in ns-2 • Study XCP behavior under adversarial network events: § Large differences in RTT § Number of users variable in time • Try different values for XCP parameters

Questions ?