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

• eXplicit 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

1 1 2 1

2 1 2 2

( )2

( )2

0.1 0.1

2 2 20.1 0.1

2 2 21 1

r r r

r r r

d s d d s d s

d d s d s d s

a a bs e e e

d d da a b

A e s e ed d d

s

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

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