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

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

5 Problem (cont’d) oEquation-based rate control oa new set of transmission control protocols oan instance: TFRC, IETF proposed standard (Jan 2003) oPast studies of equation-based rate controls mostly restricted to simulations olack of a formal study ounderstanding needed before a wide-spread deployment

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

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

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

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

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

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

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

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

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

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

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

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

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

19 Check: is TFRC conservative PFTK-standardL=8 osetup: equal number of TCP and TFRC connections (1,2,4,6,8,10), for the experiments (1,2,3,4,5,6) omostly conservative oslight deviation, anyway

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

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

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

23 Concluding Remarks for Part I ounder the conditions we identified, equation-based rate control is conservative owhen loss-event rate is large, it is overly conservative odifferent TCP throughput formulae may yield different bias obreakdown TCP-friendliness problem into sub-problems, check the sub-problems separately ! othe breakdown would reveal a cause of an observed non-TCP-friendliness oan unknown cause may lead a protocol designer to an improper protocol adjustment oconservativeness against TCP-friendliness oTCP-friendliness is difficult to verify oconservativeness oamenable to a formal verification onot TCP centric

24 Part II Rate-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 oat 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) given: M, a I x I doubly sub-stochastic rate-demand matrix 1) decomposition: decompose M=[m ij ] into a sequence of permutation matrices, s.t. for an input/output port pair ij, intensity of the offered slots is at least m ij –Birkoff/von Neumann: a doubly stochastic matrix M can be decomposed as 2) schedule: schedule the permutation matrices with objective to offer a ”smooth” schedule Consider: decomposition-based schedulers a permutation matrix a positive real number:

27 Rate-Latency Service Curve *

28 Scheduling Permutation Matrices ounique token assigned to a permutation matrix oscheduler by Chang et al can be seen as osuperposition of point processes on a line marked by the token types os chedule 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 ? Known result (Chang et al, 2000) (= subset of permutation matrices that schedule input/output port pair ij)

29 Random Permutation  a rate  k is an integer multiple of 1/L oL = frame-length ocompare with the worst-case deterministic latency Scheduler: oschedule the permutation matrices in a frame, according to a random permutation of the tokens orepeat the frame over time

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

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

32 Random-distortion Periodic otoken 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-distortion periodic Random-phase periodic orate-demand matrices drawn in a random manner

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