EPFL, Lausanne, July 17, 2003

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

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: • increase-decrease controls (Chapter 3) • fairness of bandwidth sharing • analysis and synthesis

3. Part I 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 is 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 round-trip times 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 adjustment of a protocol • TCP-friendliness is difficult to verify • we propose the concept of conservativeness • conservativeness is amenable to a formal verification

24. Part IIExpedited Forwarding

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

26. Problem • Obtain performance bounds to dimension EF networksAssumption: EF flows stochastically independent at ingressStep 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 curveStep 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: • a property that holds for one of the node abstractions, holds for a PSRG node

28. Assumptions • A1, A2, …, AI stochastically independent • Ai is constrained with an arrival curve • Ai is such that • There exists a finite s.t. • Note that an EF flow is allowed to be any stochastic process as long as it obeys to the given set of the assumptions

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

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 • Bounds on probability of the buffer overflow • for identical and non-identical arrival curve constraints • in terms of some global knowledge about the arrival curves (for leaky-bucket shapers) • Bounds on probability of the buffer overflow as seen by bit and packet arrivals • Bounds on complementary cdf of a packet delay • Bounds on the arrival bit loss rate

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

34. A dimensioning rule • Given, in addition: Dimensioning rule: fix the buffer lengths such that qn=d’rn, all n • The e2e delay-jitter is bounded by h(d’+e)(delay-from-backlog property of PSRG nodes)

35. Sketch of the Proof • Majorize 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 must be > 0, for the bound to be < 1 • Use one of our single-node bounds: horizontal deviation between an arrival curve of the aggregate EF arrival process to a node n, an(t)=rn(at+b+a(h-1)(d+e))and a service curve offered by the node nbn(t)= rn(t-e)+ Combine the last two to retrieve the asserted d’

36. Numerical Example • Example networks rn = all n

37. Concluding Remarks for Part II • We obtained probabilistic bounds on performance of a PSRG (r,e) node • Our bounds hold in probability • the bounds would be more optimistic, than worst-case deterministic bounds • Our bounds are exact • Network of nodes: we showed probabilistic bounds for a network of PSRG nodes • The bounds are still with a bound on the EF load, likewise to some known worst-case deterministic bounds • With 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 IIIInput-queued Switch

39. Problem • at 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) 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: 2) Schedule: schedule the permutation matrices with objective to offer a ”smooth” schedule

41. Rate-Latency Service Curve *

42. 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 tokens • 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 ?

43. 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

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

45. 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

46. Random-distortion Periodic • token 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-distortionperiodic Random-phase periodic • rate-demand matrices drawn in a random manner

48. Concluding Remarks for Part III • 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 • construction of practical algorithms