Stochastic Ordering for Internet Congestion Control and its Applications

1. Stochastic Ordering for Internet Congestion Control and its Applications Sangtae Ha, Injong Rhee Dept. of CS NC State Univ. Lisong Xu Dept. of CSE Univ. of Nebraska Han Cai, Do Young Eun Dept. of ECE NC State Univ.

2. Motivation • Key factor deciding the congestion control protocol performance? • Congestion signal: loss, delay, hybrid • Inc/Dec behavior: Is there any key factor? • Explore the key factor for better future design • Fluid method: simple, powerful, widely used • Stochastic method: possibility for better system design by unclosing details covered by average behavior analysis, is it possible to find good and intuitive guidance for design?

3. Introduction • Background: • Stochastic method • High speed TCP protocol • Main work • Stationary loss case • Non-stationary loss case • Shape the increasing profile in changing environment • Simulation & Experiment results

4. High-Speed TCP • Why High-Speed TCP? • For high-speed network (Reno-type TCP: not scalable) • Variants of loss based High-Speed TCP: HSTCP, STCP, HTCP, BIC, CUBIC, etc. • Key factor deciding the performance for better future design? • Growth function

5. Example of Growth Functions (1) Growth function: define the protocol’s behavior when there is no congestion, e.g., linear for AIMD Polynomial Exponential

6. Example of Growth Functions (2) Square Mix: Log. and Exp. Cubic

7. Latency variation? Growth function Stability? Time BIC Other stochastic behavior? HTCP HSTCP STCP CUBIC Our work Our work

8. Stochastic vs. Fluid method (1) Random vector (e.g. sending rate) Stochastic recursion Stochastic Recursion Driving sequence AIMD Equivalent representation

9. Fluid version of Stochastic vs. Fluid method (2) Random vector (e.g. sending rate) Stochastic recursion Stochastic Recursion Driving sequence average behavior Fluid recursion Fluid Recursion

10. Stochastic vs. Fluid method (3) First-order behavior Fixed point Fairness Responsiveness

11. Stochastic vs. Fluid method (4) • What is higher-order behavior? • Sending rate distribution • Rate variance • Why higher-order behavior? • Average behavior analysis losses too much details

12. Stochastic vs. Fluid method (5) Direct method: calculate the distribution Possible, but both the method and the result are complicated Our method: use stochastic ordering without calculating the distribution • Simple method • Good intuition on the effect of the shape of growth function

13. Related work • [Altman05] uses the tool of stochastic ordering • Background: congestion control protocol • Ordering relationship: due to the difference in upper bound • Our work • Ordering relationship: due to the difference in the convexity of growth function • Another conclusion of our work: importance of stochastic method

14. Model Time T1 Tn-2 Tn-1 • Inter-loss interval: Ti • Increasing profile (growth function): f

15. Stationary version of Main result: stationary loss Theorem 1: Under conditions 1 and 2, we have Condition 1: The Inter-loss intervals Tn are i.i.d. • Widely observed by many Internet measurement studies [Zhang01, Altman00, Padhye98]. • Allow dependence among congestion events over different RTTs. Condition 2: There exists only one root for

16. Stationary loss (2) Intuition: one function is more concave than the other • Classes of growth functions satisfying the assumption: • Concave vs. Convex (e.g., STCP): monotone • Polynomial vs. Polynomial (e.g., HSTCP): monotone • Concave-convex (e.g., CUBIC) vs. Convex: non-monotone

17. Stationary loss (3) Convex function • More concave growth function: smaller rate fluctuation

18. Stationary loss (4) • Same mean behavior vs. different stochastic behavior

19. Proposition 1: If the rescaled increasing profiles with same mean only have one intersection, the more concave one has less variation. Main result: non-stationary loss

20. Non-stationary loss result Intuition: the concave-convex type profile makes the pdf more concentrated around the mean.

21. Shape the increasing profile (1)

22. Shape the increasing profile (2)

23. NS2 simulation result • Simulation set-up: • Dumbbell, bottleneck 250Mbps, RTT 100ms,100% BDP buffer size • Loss generated by predefined models or by using background traffic (20% of total link bandwidth, long/short flows) • Five pseudo protocols simulated: Root (concave), Linear, Power/Square (convex), Exponential (convex), Concave-Convex • Conclusion: Same as predicted by our theoretical result

24. Experiment result -- Testbed Setup • Bottleneck 400Mbps, RTT for background traffic (recent measurement study [Aikat03]) • Long lived flows (Iperf) and short lived flows (Surge) • Convex: HSTCP (Polynomial), HTCP (Power), and STCP (Exponential) • Concave-Convex functions: CUBIC and BIC-TCP

25. CoV and Link utilization • Buffer size (1MB), four HS flows with the same RTT (40ms – 320ms)

26. Conclusion • Importance of stochastic method • Intuition: provide more details than fluid method • Conclusion from our result: indistinguishable protocol from average sense may be actually very different • Example: comparing stochastic performance of protocols different in the convexity of growth function to guide future design • Use of stochastic method • Can also be simple and intuitive

27. Thanks!

28. Reference (I) • [1]S. Floyd, “HighSpeed TCP for large congestion windows,” RFC 3649, December 2003. • [2] T. Kelly, “Scalable TCP: Improving performance in highspeed wide area networks,” ACM SIGCOMM Computer Communication Review, vol. 33, no. 2, pp. 83–91, April 2003. • [3] R. N. Shorten and D. J. Leith, “H-TCP: TCP for high-speed and longdistance networks,” in Proceedings of the Second PFLDNet Workshop, Argonne, Illinois, February 2004. • [4] L. Xu, K. Harfoush, and I. Rhee, “Binary increase congestion control for fast long-distance networks,” in Proceedings of IEEE INFOCOM, Hong Kong, March 2004. • [5] I. Rhee and L. Xu, “CUBIC: A new TCP-friendly high-speed TCP variant,”in Proceedings of the third PFLDNet Workshop, France, February 2005.

29. Reference (II) • [6] E. Altman, A. A. Kherani, K. Avratchenkov, and B. J. Prabhu, “Performance analysis and stochastic stability of congestion control protocols,” in Proceedings of IEEE INFOCOM, Miami, FL, March 2005. • [7] Y. Zhang, N. Duffield, V. Paxson, and S. Shenker, “On the constancy of Internet path properties,” ACM SIGCOMM IMW 2001 • [8] E. Altman, K. Avrachenkov, and C. Barakat, “A Stochastic Model of TCP/IP with Stationary Random Loss,” ACM SIGCOMM 2000 • [9] J. Padhye, V. Firoiu, D. Towsley, and J. Kurose, “Modeling TCP Throughput: a Simple Model and its Empirical Validation,” ACM SIGCOMM 1998 • [10] J. Aikat, J. Kaur, F. D. Smith, and K. Jeffay, Variability in TCP round-trip times, IMC 2003