1. Stability of Congestion Control Algorithms Using Control Theory with an application to XCP
Ioannis Papadimitriou
George Mavromatis

2. Outline Previous Work eXplicit Control Protocol (XCP) Stability
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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. Stability Region Plot for N = 2

10. Simulink Model

12. 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

13. Questions ?