1. Acknowledgments S. Athuraliya, D. Lapsley, V. Li, Q. Yin (UMelb) S. Adlakha (UCLA), J. Doyle (Caltech), K. Kim (SNU/Caltech), F. Paganini (UCLA), J. Wang (Caltech) L. Peterson, L. Wang (Princeton)

2. Copyright, 1996 © Dale Carnegie & Associates, Inc. Equilibrium & Dynamics of TCP/AQM Steven Low CS & EE, Caltech netlab.caltech.edu March 2002

3. Congestion Control Heavy tail  Mice-elephants Elephant Internet Mice Congestion control efficient & fair sharing small delay queueing + propagation CDN

4. TCP x i (t)

5. TCP & AQM x i (t) p l (t) Example congestion measure p l (t) Loss (Reno) Queueing delay (Vegas)

6. TCP & AQM x i (t) p l (t) TCP: Reno Vegas AQM: DropTail RED REM

7. TCP & AQM x i (t) p l (t) TCP: Reno Vegas AQM: DropTail RED REM

8. Outline Protocol (Reno, Vegas, RED, REM/PI…) Equilibrium Performance Throughput, loss, delay Fairness Utility Dynamics Local stability Cost of stabilization

9. TCP Reno (Jacobson 1990) SS time window CA SS: Slow Start CA: Congestion Avoidance Fast retransmission/fast recovery

10. TCP Vegas (Brakmo & Peterson 1994) SS time window CA Converges, no retransmission … provided buffer is large enough

11. Model structure F1F1 FNFN G1G1 GLGL R f (s) R b ’ (s) TCP Network AQM Multi-link multi-source network x y q p

12. Reno: F for every ack (ca) {W += 1/W } for every loss {W := W/2 } x(t+1) = F( x(t), q(t) ) p(t+1) = G( p(t), y(t) ) Reno, Vegas DropTail, RED, REM

13. Reno: F for every ack (ca) {W += 1/W } for every loss {W := W/2 } x(t+1) = F( x(t), q(t) ) p(t+1) = G( p(t), y(t) ) Reno, Vegas DropTail, RED, REM

14. Reno: F for every ack (ca) {W += 1/W } for every loss {W := W/2 } x(t+1) = F( x(t), q(t) ) p(t+1) = G( p(t), y(t) ) Reno, Vegas DropTail, RED, REM

15. RED (Floyd & Jacobson 1993) Compute average queue length b l (t+1) = [b l (t) + y l (t) - c l ] + r l (t+1) = (1-  ) r l (t) +  b l (t) Mark with a probability r p l (r) 1 x(t+1) = F( x(t), q(t) ) p(t+1) = G( p(t), y(t) ) Reno, VegasDropTail, RED, REM

16. Summary: Reno/RED F1F1 FNFN G1G1 GLGL R f (s) R b ’ (s) TCP Network AQM x y q p

17. Overview Protocol (Reno, Vegas, RED, REM/PI…) Equilibrium Performance Throughput, loss, delay Fairness Utility Dynamics Local stability Cost of stabilization

18. Model c1c1 c2c2 Network Links l of capacities c l Sources s L(s) - links used by source s U s (x s ) - utility if source rate = x s x1x1 x2x2 x3x3

19. Flow control problem Assumptions Strictly concave increasing U s Unique optimal rates x s exist Direct solution impractical

20. Duality Approach Primal-dual algorithm:

21. Duality Model of TCP/AQM Primal-dual algorithm: Reno, VegasDropTail, RED, REM Source algorithm iterates on rates Link algorithm iterates on prices With different utility functions

22. Duality Model of TCP/AQM Primal-dual algorithm: Reno, VegasDropTail, RED, REM (x*,p*) primal-dual optimal if and only if

23. Duality Model of TCP/AQM Primal-dual algorithm: Reno, VegasDropTail, RED, REM Any link algorithm that stabilizes queue generates Lagrange multipliers solves dual problem

24. Summary: Reno Equilibrium characterization Congestion measure p = loss Implications Reno equalizes window w i =  i x i inversely proportional to delay  i dependence for small p queue length increases as load increases

25. Queue DropTail queue = 94% RED min_th = 10 pkts max_th = 40 pkts max_p = 0.1 p = Lagrange multiplier! p increasing in queue! REM queue = 1.5 pkts utilization = 92%  = 0.05,  = 0.4,  = 1.15 p decoupled from queue

26. Summary: Vegas Delay Congestion measures = end to end queueing delay Sets rate Equilibrium condition: Little’s Law Fairness Utility Weighted proportional fairness Loss No loss if buffers are sufficiently large Otherwise: equilibrium not attainable, loss unavoidable (revert to Reno)

27. Summary: equilibrium Flow control problem TCP/AQM Maximize aggregate source utility With different utility functions Primal-dual algorithm Reno, Vegas DropTail, RED, REM

28. Outline Protocol (Reno, Vegas, RED, REM/PI…) Equilibrium Performance Throughput, loss, delay Fairness Utility Dynamics Local stability Cost of stabilization

29. Dynamics Small effect on queue AIMD Mice traffic Heterogeneity Big effect on queue Stability!

30. Stable: 20ms delay Window Ns-2 simulations, 50 identical FTP sources, single link 9 pkts/ms, RED marking

31. Queue Stable: 20ms delay Window Ns-2 simulations, 50 identical FTP sources, single link 9 pkts/ms, RED marking

32. Unstable: 200ms delay Window Ns-2 simulations, 50 identical FTP sources, single link 9 pkts/ms, RED marking

33. Unstable: 200ms delay Queue Window Ns-2 simulations, 50 identical FTP sources, single link 9 pkts/ms, RED marking

34. Other effects on queue 20ms 200ms 50% noise avg delay 16ms avg delay 208ms

35. Effect of instability Larger jitters in throughput Lower utilization No noise

36. Linearized system F1F1 FNFN G1G1 GLGL R f (s) R b ’ (s) TCP Network AQM x y q p

37. Linearized system F1F1 FNFN G1G1 GLGL R f (s) R b ’ (s) x y q p TCPREDqueuedelay

38. Stability condition Theorem TCP/RED stable if 

39. Stability condition Theorem TCP/RED stable if Small  Slow response Large delay

40. Stability condition Theorem TCP/RED stable if

41. Validation

43. Stability region Unstable for Large delay Large capacity Small load

44. Role of AQM (Sufficient) stability condition

45. Role of AQM (Sufficient) stability condition TCP: AQM: scale down K & shape H

46. Role of AQM TCP: RED:REM/PI: Queue dynamics (windowing) Problem!!

47. Role of AQM (Kim 2002) To stabilize a given TCP F i with least cost Linearized F, single link, identical sources Any AQM solves optimal control with appropriate Q and R

48. Role of AQM (Kim 2002) To stabilize a given TCP F i with least cost Linearized F, single link, identical sources k 1 =0 Nonzero buffer k 3 =0 Slower decay

49. Role of AQM (Kim 2002) To stabilize a given TCP F i with least cost Linearized F, single link, identical sources

50. Papers Scalable laws for stable network congestion control (CDC2001) A duality model of TCP flow controls (ITC, Sept 2000) Optimization flow control, I: basic algorithm & convergence (ToN, 7(6), Dec 1999) Understanding Vegas: a duality model (J. ACM, 2002) REM: active queue management (Network, May/June 2001) netlab.caltech.edu