1. Communication Networks
A Second Course
Jean Walrand
Department of EECS
University of California at Berkeley

2. Transport and Optimization
Review of Duality
Examples
Congestion Control
AIMD
Reno
Vegas
Dual Problem
Decomposition
Solving the Dual Problem

3. Review of Duality

4. Review of Duality

5. Review of Duality

6. Example of Duality: Cube

7. Example of Duality: Discrete N(1, s2)

8. Example of Duality: HOT
Highly Optimized Tolerance, John Doyle, Caltech…
Idea: Systems optimized for typical situations
As a consequence, they are vulnerable to extreme situations
We explore this effect for the WWW and other models.

9. Example of Duality: HOT
Heavy Tails
DC = length of codewords
after data compression [exponential]
FF = size of forest fires
[heavy]
WWW = file lengths

10. Example of Duality: HOT
Power laws, Highly Optimized Tolerance and generalized source coding
John Doyle, J.M. Carlson, 2000

11. Congestion Control: AIMDB x D E y
Rates equalize  fair share

12. Congestion Control: AIMDB x C D E y y C
Chiu and Jain, 1988
Limit rates:
x = y x

13. Congestion Control: RenoA B x C D E y
In practice (Reno):
window increases at
rate ~ 1/RTT
Limiting window ~ 1/RTT
But throughput = window/RTT
Limiting throughput
~ 1/RTT2

14. Congestion Control: VegasB x C D E y
Adjust rates based on estimated backlog. Roughly,
X ~ 1/q where q is backlog in router.
Then, one can show that x ~ y. The intuition is that the flows will have similar backlogs.
Two types of proof:
Lyapunov
Show that algorithm is a gradient projection algorithm for a convex problem [ converges if …]

15. Congestion Control: Dual

16. Congestion Control: Decomposition

17. Congestion Control: Decomposition

18. Congestion Control: Solving Dual

19. Congestion Control: Solving Dual

20. Congestion Control: Solving Dual

21. Optimization: Summary
Convex Programming
Primal  Dual; Shadow cost
TCP Vegas == gradient alg. for dual
HOT
Minimize average cost  Events have heavy tail
Congestion control
Dual decomposes