2. 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

3. 2 The Control We Study We call this the basic control The loss events: The loss intervals: Function f is typically TCP loss-throughput function

4. 3 The Control We Study (contd) We call this the comprehensive control 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

5. 4 Why do we Study This? oSeveral send rate controls are proposed for media streaming in the Internet. oWe often take TFRC (Floyd et al, 2000) as a recurring example. oThe send rate should be smoother than with TCP, but still responsive to congestion.

6. 5 What do we Study? In the long-run, is the control TCP- friendly ? i.e.: (P) EBRC Throughput TCP Throughput ?

7. 6 We split the problem into 3 Sub-Problems P1) Is Rate Control Conservative ? P2) Is our loss no better than TCPs ? P3) Does TCP conform to function f ? Ob. If P1, P2, and P3 are positive, then the control is TCP-friendly

8. 7 Some Functions f SQRT: Note: c1, c2, c3 are some positive-valued constants r is the round-trip time q is TCP retransmit timeout (typically, q=4r) PFTK-standard: PFTK-simplified: SQRT PFTK-

9. 8 Throughput Expressions Basic Control: Comprehensive Control (PFTK-simplified): where Ob. Knowing the joint law of one would be able to compute the throughput

10. 9 (P1) When is the Control Conservative? Sufficient Conditions: (F1) 1/f(1/x) is convex with x (C1)

11. 10 (F1) is true for SQRT and PFTK- simplified PFTK- SQRT

12. 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

13. 12 When the Conditions are Met? It follows: (C1) is true for i.i.d. It is the autocorrelation of what matters!

14. 13 Claim 1 Assume: o and are negatively or lightly correlated o consider f in the region where takes its values 1) The more convex 1/f(1/x) is, the more conservative the control is 2) The more variable is, the more conservative the control is

15. 14 Numerical Example for Claim 1 is i.i.d. with the distribution:

16. 15 Numerical Example for Claim 1 (Contd) Ob. The larger is, the more convex 1/f(1/x) is, and hence the more conservative the control is SQRTPFTK-simplified Ob 2. PFTK is more convex than SQRT, effect is more pronounced

17. 16 Numerical Example for Claim 1 (Contd) Ob. The more variable is, the more conservative the control is SQRTPFTK-simplified

18. 17 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) Ob. The larger is, the more convex 1/f(1/x) is, and hence the more conservative the control is

19. 18 Another Set of Conditions Then, the control is conservative! Then, the control is non-conservative! 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

20. 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 Non-conservativeness! If packets are dropped at a router independently of the packet length, then X n =L n r

21. 20 Claim 2 Assume X n and S n 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 X n and S n 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

22. 21 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) 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

23. 22 Statistical Bias due to Viewpoint Does Play a Role! By Palm inversion formula: Random Observer Sampling at the Points Random Observer falls more likely into a large time interval Random Observer would measure larger average interval than as seen at the points! This is known as Fellers Paradox! If X n and S n are positively correlated Then, the random observer would see larger send rate than as seen at the points Likewise to Fellers Paradox!

24. 23 (P2) How Do Different Loss-Event Ratios Seen By the Sources Compare? This is another issue of importance of viewpoint! Different sources may see different loss-event ratios! Claim 3: Seen by TCP Seen by Equation- Based Rate Control Seen by Poisson Source (non- adaptive)

25. 24 How Do Different Loss-Event Ratios Seen By the Sources Compare? (Contd) Suppose there exists a hidden congestion process Z(t) If at time t, Z(t)=i, then the loss-event ratio is This can be formalized by Palm Calculus Intuition behind Claim 3: oNon-adaptive (Poisson Source) would see time average of the system loss-event ratio oAn adaptive source would sample bad states less frequently oThe more adaptive the source is, the smaller loss-event ratio it would see oTCP would be more adaptive than Equation-Based Rate Control, and hence would see smaller loss-event ratio

26. 25 ns-2 Example for Claim 3

27. 26 (P3) Does f Match TCP Loss- Throughput Formula, Actually? Not always! TCP Sack1:

28. 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

29. 28 o(P1) We showed when we expect to have either conservative or non-conservative control oWe explain the throughput-drop encountered empirically elsewhere oWe demonstrate a realistic control which would be non- conservative o(P2) Expect loss-event ratio of equation-based rate control to be larger than TCP would see o(P3) TCP may deviate from PFTK formula oIt is important to distinguish the three sub- problems and check them separately. Conclusion