Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Ph.D. advisor: Prof. Jean-Yves Le Boudec EPFL, Lausanne, July 17, 2003 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 2 Part I Equation-based Rate Control 3 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 4 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 5 Problem (cont’d) Equation-based rate control: basic control principles given: a TCP throughput formula p = 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) 6 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 instant Tn = 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 7 send rate Equation-based rate control: the basic control law n L ... Tn L ... ... Tn 3 n 3 n 2 n 1 Tn 2 Tn 1 Tn ... Tn 1 Tn = instant of a loss-event n = a loss-event interval Additional control laws ignored in this slide 8 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 9 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 10 Sketch of the Proof Palm inversion: Throughput: May make the control conservative ? ! 11 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 12 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 13 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 14 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 15 Numerical example: Is the basic control conservative ? SQRT: PFTK-simplified: loss-event intervals: i.i.d., generalized exponential density 16 ns-2 and lab: Is TFRC conservative ? ns-2 lab PFTK-simplified PFTK-standard 16 8 4 L=8 L=2 Setup: a RED link shared by TFRC and TCP connections The same qualitative behavior as observed on the previous slide 17 We turn to check: is TFRC TCP-friendly First check: is Internet, LAN to LAN, EPFL sender negative or slightly positive Internet, LAN to a cable-modem at EPFL Lab 18 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 19 Check: is TFRC TCP-friendly TCP-friendly ? - no, not always although, it is mostly conservative ! 20 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 21 Check the factors separately ! Does TCP conform to its formula ? No Does TFRC see no better loss-event rate than TCP ? No Does TFRC see no better loss-event rate than TCP ? No when a few connections compete, none of the conditions hold 22 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 23 Part II Expedited Forwarding 24 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 25 Problem Obtain 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 26 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 27 Assumptions A1, A2, …, AI stochastically independent Ai is constrained with an arrival curve Ai is such that There exists a finite Note that an EF flow is allowed to be any stochastic process as long as it obeys to the given set of the assumptions s.t. 28 One Result: a Bound on Probability of the Buffer Overflow Assume: fix: all I Then, for Q(t) (= number of bits in the node at an instant t), 29 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 30 Numerical example 31 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 32 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 33 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) 34 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 Use one of our single-node bounds: must be > 0, for the bound to be < 1 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 n bn(t)= rn(t-e)+ Combine the last two to retrieve the asserted d’ 35 Numerical Example Example networks rn = all n 36 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 37 Part III Input-queued Switch 38 Problem at any time slot, connectivity restricted to permutation matrices Switch scheduling problem: schedule crossbar connectivity with guarantees on the rate and latency 39 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 M can be decomposed as a permutation matrix a positive real: 2) Schedule: schedule the permutation matrices with objective to offer a ”smooth” schedule 40 Rate-Latency Service Curve * 41 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 ? 42 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 43 Numerical Example worst-case deterministic w.p. 0.99 44 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 45 Random-distortion Periodic token processes as with Chang et al, but place each token uniformly at random on the periods By derandomization: 46 A Numerical Example Chang et al Random-distortion periodic Random-phase periodic rate-demand matrices drawn in a random manner 47 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 48