MSR, Cambridge, August 5, 2003

1. Long-Run Behavior of Equation-Based Rate Control& Rate-Latency of Some Input-Queued Switches MSR, Cambridge, August 5, 2003

2. Outline Part I Long-run Behavior of Equation-based Rate Control Part II Rate-Latency of Some Input-queued Switches The talk takes from: M.V., Ph.D. thesis, July 2003

3. Part I Long-Run Behavior of Equation-Based Rate Control

4. Problem • New transmission control protocols proposed for some packet senders in the Internet • a design goal is to offer a better transport for streaming sources, than offered by TCP • In today’s Internet, TCP is the most used • Axiom: transport protocols other than TCP, should be TCP-friendly—another design goal TCP-friendliness: Throughput <= TCP throughput

5. Problem (cont’d) • Equation-based rate control • a new set of transmission control protocols • an instance: TFRC, IETF proposed standard (Jan 2003) • Past studies of equation-based rate controls mostly restricted to simulations • lack of a formal study • understanding needed before a wide-spread deployment

6. Problem (cont’d) Equation-based rate control: basic control principles • given: a TCP throughput formulap = loss-event rate • p estimated on-line • at an instant t, send rate set as Problem: Is equation-based rate control TCP-friendly ? (TCP throughput formula depends also on other factors, e.g. an event-average of the round-trip time)

7. Where is the Problem ? • The estimators are updated at some special points in time the send rate updated at the special instants(sampling bias)t = an arbitrary instantTn = the nth update of the estimators, a special instant • x->f(x) is non-linear, the estimators are non-fixed values(non-linearity) • Other factors

8. Equation-based rate control: the basic control law send rate = instant of a loss-event = a loss-event interval • additional control laws ignored in this slide

9. We first check: is the control conservative We say a control is conservative iff p = loss-event rate as seen by this protocol • conservativeness is not the same as TCP-friendliness • we come back to TCP-friendliness later

10. When the basic control is conservative • assume: the send rate be a stationary ergodic process In practice: • the conditions are true, or almost • the result explains overly conservativeness

11. Sketch of the Proof Palm inversion: Throughput: May make the control conservative ? !

12. Sketch of the Proof (Cont’d) • 1/f(1/x) is assumed to be convex, thus, it is above its tangents • take the tangent at 1/p • the “overshoot” bounded by a function of p and

13. When 1/f(1/x) is convex Check some typical TCP throughput formulae: SQRT: PFTK-standard almost convex PFTK-standard: PFTK-simplified convex PFTK-simplified: SQRT convex b = number of packets acknowledged by an ack

14. On Covariance of the Estimator and the Next Loss-event Interval • Recall (C1) = a “measure” how well predicts It holds: • if is a bad predictor, that leads to conservativeness • if the loss-event intervals are independent, then (C1) holds with equality

15. Claim • assume: the estimator and the next sample of the loss-event interval are negatively or slightly positive correlated • consider a region where the loss-event interval estimator takes its values • the more convex 1/f(1/x) is in this region => the more conservative • the more variable the is => the more conservative

16. Numerical example: Is the basic control conservative ? SQRT: PFTK-simplified: • loss-event intervals: i.i.d., generalized exponential density

17. ns-2 and lab: Is TFRC conservative ? ns-2 lab PFTK-simplified PFTK-standard 16 8 L=8 4 L=2 Setup: a RED link shared by TFRC and TCP connections • the same qualitative behavior as observed on the previous slide

18. We turn to check: is TFRC TCP-friendly First check: is negative or slightly positive Internet, LAN to LAN, EPFL sender Internet, LAN to a cable-modem at EPFL Lab

19. Check: is TFRC conservative PFTK-standard L=8 • setup: equal number of TCP and TFRC connections (1,2,4,6,8,10), for the experiments (1,2,3,4,5,6) • mostly conservative • slight deviation, anyway

20. Check: is TFRC TCP-friendly TCP-friendly ? - no, not always • although, it is mostly conservative !

21. Conservativeness does not imply TCP-friendliness ! Breakdown TCP-friendliness into: • Does TCP conform to its formula ? • Does TFRC see no better loss-event rate than TCP ? • Does TFRC see no better average RTT than TCP ? • Is TFRC conservative ? • if all conditions hold => TCP-friendliness • if the control is non-TCP-friendly, then at least one condition must not hold • the breakdown is more than a set of sufficient conditions- it tells us about the strength of individual factors

22. Check the factors separately ! Does TFRC see no better loss-event rate than TCP ? Does TCP conform to its formula ? Does TFRC see no better loss-event rate than TCP ? • No • No • No • when a few connections compete, none of the conditions hold

23. Concluding Remarks for Part I • under the conditions we identified,equation-based rate control is conservative • when loss-event rate is large, it is overly conservative • different TCP throughput formulae may yield different bias • breakdown TCP-friendliness problem into sub-problems, check the sub-problems separately ! • the breakdown would reveal a cause of an observed non-TCP-friendliness • an unknown cause may lead a protocol designer to an improper protocol adjustment • conservativeness against TCP-friendliness • TCP-friendliness is difficult to verify • conservativeness • amenable to a formal verification • not TCP centric

24. Part IIRate-Latency of Some Input-queued Switches The work done in part while an intern with Dept. of Mathematics of Networks and Systems, Bell Laboratories, Murray Hill, NJ, Summer 2001

25. Problem • at any time slot, connectivity restricted to permutation matrices switch scheduling problem: schedule crossbar connectivity with guarantees on the rate and latency

26. Problem (Cont’d) Consider: decomposition-based schedulers given:M, a I x I doubly sub-stochastic rate-demand matrix 1) decomposition: decompose M=[mij] into a sequence of permutation matrices, s.t. for an input/output port pair ij, intensity of the offered slots is at least mij • Birkoff/von Neumann: a doubly stochastic matrix Mcan be decomposed as a permutation matrix a positive real number: 2) schedule: schedule the permutation matrices with objective to offer a ”smooth” schedule

27. Rate-Latency Service Curve *

28. Scheduling Permutation Matrices • unique token assigned to a permutation matrix • scheduler by Chang et al can be seen as Known result (Chang et al, 2000) (= subset of permutation matrices that schedule input/output port pair ij) • superposition of point processes on a line marked by the token types • schedule permutation matrices as their tokens appear Scheduler by Chang et al is for deterministic periodic individual token processes Problem: can we have schedules with better bounds on the latency ?

29. Random Permutation • a rate k is an integer multiple of 1/L • L = frame-length Scheduler: • schedule the permutation matrices in a frame, according to a random permutation of the tokens • repeat the frame over time • compare with the worst-case deterministic latency

30. Numerical Example w.p. 99/100 worst-case deterministic

31. Random-phase Periodic • token processes as with Chang et al, but for a token process chose a random phase, independently of other token processes By derandomization: • compare with Chang et al

32. Random-distortion Periodic • token processes as with Chang et al, but place each token uniformly at random on the periods By derandomization:

33. A Numerical Example Chang et al Random-distortionperiodic Random-phase periodic • rate-demand matrices drawn in a random manner

34. Concluding Remarks for Part II • we showed new bounds on the latency for a decomposition-based input-queued switch scheduling • the bounds are in many cases better than previously-known bound by Chang et al • to our knowledge, the approach is novel • conjunction of the superposition of the token processes and probabilistic techniques may lead to new bounds • may lead to construction of practical algorithms