Study of Equation-Based Rate Control Long-Run Behavior

1. On the Long-Run Behavior of Equation-Based Rate Control Milan Vojnović and Jean-Yves Le Boudec ACM SIGCOMM 2002, Pittsburgh, PA, August 19-23, 2002

2. The Control We Study Function f is typically TCP loss-throughput function The loss intervals: The loss events: We call this the basic control

3. The Control We Study (cont’d) Additional rule: If the number of bits sent since the last loss event included in the loss-event estimator increases its value, then use it in computation of the send rate We call this the comprehensive control

4. Why do we Study This? • Several send rate controls are proposed for media streaming in the Internet. • We often take TFRC (Floyd et al, 2000) as a recurring example. • The send rate should be smoother than with TCP, but still responsive to congestion.

5. What do we Study? In the long-run, is the control TCP-friendly ? i.e.: (P) EBRC Throughput  TCP Throughput ?

6. We split the problem into 3 Sub-Problems P1) Is Rate Control Conservative ? P2) Is our loss no better than TCP’s ? P3) Does TCP conform to function f ? Ob. If P1, P2, and P3 are positive, then the control is TCP-friendly

7. Some Functions f SQRT: SQRT PFTK-standard: PFTK- PFTK-simplified: Note: c1, c2, c3 are some positive-valued constants r is the round-trip time q is TCP retransmit timeout (typically, q=4r)

8. where Ob. Knowing the joint law of one would be able to compute the throughput Throughput Expressions Basic Control: Comprehensive Control (PFTK-simplified):

9. (C1) (P1) When is the Control Conservative? Sufficient Conditions: (F1)1/f(1/x) is convex with x

10. (F1) is true for SQRT and PFTK-simplified PFTK- SQRT

11. (F1) is true for SQRT and PFTK-simplified (F1) is almost true for PFTK-standard If f(1/x) deviates from convexity by the ratio r, and (C1) holds, then the control cannot overshoot by more than the factor r

12. It is the autocorrelation of what matters! It follows: (C1) is true for i.i.d. When the Conditions are Met?

13. Assume: • and are negatively or lightly correlated • consider f in the region where takes its values 2) The more variable is, the more conservative the control is Claim 1 1) The more convex 1/f(1/x) is, the more conservative the control is

14. is i.i.d. with the distribution: Numerical Example for Claim 1

15. SQRT PFTK-simplified Ob. The larger is, the more convex 1/f(1/x) is, and hence the more conservative the control is Numerical Example for Claim 1 (Cont’d) Ob 2. PFTK is more convex than SQRT, effect is more pronounced

16. SQRT PFTK-simplified Ob. The more variable is, the more conservative the control is Numerical Example for Claim 1 (Cont’d)

17. Ob. The larger is, the more convex 1/f(1/x) is, and hence the more conservative the control is ns-2 Example for Claim 1 Single RED bottleneck shared with equal number of TFRC and TCP flows PFTK-simplified: (likewise for SQRT and PFTK-standard)

18. If (F2)f(1/x) is concave with x (C2) If (F2’)f(1/x) is convex with x (C2’) (V) is not fixed to some constant Another Set of Conditions Then, the control is conservative! Then, the control is non-conservative!

19. When is the Control Non-Conservative? SQRT and PFTK formulas are such that f(1/x) is concave, except both PFTK formulas are such that f(1/x) is convex for small x We may have non-conservativeness in this region! EXAMPLE: Some rate controls keep the packet send rate fixed, but vary packet size Xn=Ln r If packets are dropped at a router independently of the packet length, then Non-conservativeness!

20. Assume Xn and Sn are negatively or lightly correlated 1) If f(1/x) is concave in the region where takes its values, then the controls tends to be conservative Assume Xn and Sn are postively or lightly correlated 2) If f(1/x) is convex in the region where takes its values, then the controls tends to be non-conservative In both cases: the more variable is, the more pronounced the effect is Claim 2

21. With both PFTK non-conservativeness! Recall: f is convex for PFTK for large With SQRT always conservative! With trend upwards due to decreasing coeff. of variation of ns-2 Example for Claim 2 Rate control with fixed packet send rate, but variable packet size through a loss module with fixed packet drop probability L=4 For L=8 (not shown in the slides), we have qualitatively the same effects, but less pronounced (the last part of the claim)

22. By Palm inversion formula: Observer Sampling at the Points Random Observer Statistical Bias due to Viewpoint Does Play a Role! If Xn and Sn are positively correlated Random Observer falls more likely into a large time interval Then, the random observer would see larger send rate than as seen at the points Random Observer would measure larger average interval than as seen at the points! This is known as Feller’s Paradox! Likewise to Feller’s Paradox!

23. (P2) How Do Different Loss-Event Ratios Seen By the Sources Compare? Seen by Equation-Based Rate Control Seen by Poisson Source (non-adaptive) Seen by TCP This is another issue of importance of viewpoint! Different sources may see different loss-event ratios! Claim 3:

24. If at time t, Z(t)=i, then the loss-event ratio is How Do Different Loss-Event Ratios Seen By the Sources Compare? (Cont’d) Suppose there exists a hidden congestion process Z(t) Intuition behind Claim 3: • Non-adaptive (Poisson Source) would see time average of the system loss-event ratio • An adaptive source would sample “bad” states less frequently • The more adaptive the source is, the smaller loss-event ratio it would see • TCP would be more adaptive than Equation-Based Rate Control, and hence would see smaller loss-event ratio This can be formalized by Palm Calculus

25. ns-2 Example for Claim 3

26. (P3) Does f Match TCP Loss-Throughput Formula, Actually? Not always! TCP Sack1:

27. Check the 3 sub-problems separately ! A TCP-unfriendly example, even though control conservative and sees larger loss-event ratio! This is just an artifact of inaccuracy of function f.It is not an intrinsic problem of the control. Ignoring this might lead the designer to try to “improve” her protocol -- wrongly so

28. Conclusion • (P1) We showed when we expect to have either conservative or non-conservative control • We explain the throughput-drop encountered empirically elsewhere • We demonstrate a realistic control which would be non-conservative • (P2) Expect loss-event ratio of equation-based rate control to be larger than TCP would see • (P3) TCP may deviate from PFTK formula • It is important to distinguish the three sub-problems and check them separately.