EPFL, Lausanne, July 17, 2003 Ph.D. advisor: Prof. Jean-Yves Le Boudec

2 Outline Part I Equation-based Rate Control Part II Expedited Forwarding Part III Input-queued Switch In the thesis, but not in the slides: oincrease-decrease controls (Chapter 3) ofairness of bandwidth sharing oanalysis and synthesis

3 Part I 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 is 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 Consider 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 round-trip times 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 adjustment of a protocol oTCP-friendliness is difficult to verify owe propose the concept of conservativeness oconservativeness is amenable to a formal verification

24 Part II Expedited Forwarding

25 Problem oExpedited Forwarding (EF): a service of differentiated services Internet - network of nodes - each node offers service to the aggregate EF traffic, not per-EF-flow oEF per-hop-behavior: PSRG, Packet Scale Rate Guarantee with a rate r and a latency e oEF flows: individually shaped at the network ingress

26 Problem oObtain performance bounds to dimension EF networks Assumption: EF flows stochastically independent at ingress Step 1: Find probabilistic bounds on backlog, delay, and loss for a single PSRG node, with stochastically independent EF arrival processes, each constrained with an arrival curve Step 2: Apply the results to a network of PSRG nodes

27 Packet Scale Rate Guarantee with a rate r and a latency e Relations among different node abstractions: oa property that holds for one of the node abstractions, holds for a PSRG node

28 Assumptions oNote that an EF flow is allowed to be any stochastic process as long as it obeys to the given set of the assumptions oA 1, A 2, …, A I stochastically independent oA i is constrained with an arrival curve oA i is such that oThere exists a finite s.t.

29 One Result: a Bound on Probability of the Buffer Overflow Then, for Q(t) (= number of bits in the node at an instant t), oAssume: all I ofix:

30 A Method to Derive Bounds Step 1: containment into a union of the “arrival overflow events” (by def. of a service curve and  ) Step 2: use the union probability bound Step 3: apply Hoeffding’s inequalities key observation: is a sum of I random variables - independent, with bounded support, bounded means - fits the assumptions by Hoeffding (1963) Note: realizing that we can apply Hoeffding’s inequalities, enabled us to obtain new performance bounds

31 Numerical example

32 Our Other Bounds that apply to a PSRG node oBounds on probability of the buffer overflow ofor identical and non-identical arrival curve constraints oin terms of some global knowledge about the arrival curves (for leaky-bucket shapers) oBounds on probability of the buffer overflow as seen by bit and packet arrivals oBounds on complementary cdf of a packet delay oBounds on the arrival bit loss rate

33 Dimensioning an EF network oKnown result: for, a bound on the e2e delay-jitter is oGiven: ( = set of EF flows that traverse the node n) (= maximum number of hops an EF flow can traverse) oProblem: obtain a bound on the e2e delay-jitter

34 A dimensioning rule Dimensioning rule: fix the buffer lengths such that q n =d’r n, all n oThe e2e delay-jitter is bounded by h(d’+e) (delay-from-backlog property of PSRG nodes) oGiven, in addition:

35 Sketch of the Proof oMajorize by the fresh traffic: bits of an EF flow i seen at the node n in (s,t]bits of an EF flow i seen at the network ingress (fresh traffic) = (h-1)(d+e), a bound on the delay-jitter to any node in the network oUse one of our single-node bounds: horizontal deviation between an arrival curve of the aggregate EF arrival process to a node n, a n (t)=r n (at+b+a(h-1)(d+e)) and a service curve offered by the node n b n (t)= r n (t-e) + Combine the last two to retrieve the asserted d’ must be > 0, for the bound to be < 1

36 Numerical Example oExample networks r n = all n

37 Concluding Remarks for Part II oWe obtained probabilistic bounds on performance of a PSRG (r,e) node oOur bounds hold in probability othe bounds would be more optimistic, than worst-case deterministic bounds oOur bounds are exact oNetwork of nodes: we showed probabilistic bounds for a network of PSRG nodes oThe bounds are still with a bound on the EF load, likewise to some known worst-case deterministic bounds oWith an additional global parameter, we obtained a bound on the e2e delay-jitter that is more optimistic than a known worst-case deterministic bound

38 Part III Input-queued Switch

39 Problem oat any time slot, connectivity restricted to permutation matrices Switch scheduling problem: schedule crossbar connectivity with guarantees on the rate and latency

40 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:

41 Rate-Latency Service Curve *

42 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 tokens 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)

43 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

44 Numerical Example worst-case deterministic w.p. 0.99

45 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:

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

47 A Numerical Example Chang et al Random-distortion periodic Random-phase periodic orate-demand matrices drawn in a random manner

48 Concluding Remarks for Part III 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 oconstruction of practical algorithms