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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

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Epfl lausanne july 17 2003

Ph.D. advisor: Prof. Jean-Yves Le Boudec

EPFL, Lausanne, July 17, 2003


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


Epfl lausanne july 17 2003

Part I

Equation-based Rate Control


Epfl lausanne july 17 2003

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


Epfl lausanne july 17 2003

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


Epfl lausanne july 17 2003

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)


Epfl lausanne july 17 2003

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


Epfl lausanne july 17 2003

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


Epfl lausanne july 17 2003

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


Epfl lausanne july 17 2003

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


Epfl lausanne july 17 2003

Sketch of the Proof

Palm inversion:

Throughput:

May make the control conservative ? !


Epfl lausanne july 17 2003

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


Epfl lausanne july 17 2003

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


Epfl lausanne july 17 2003

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


Epfl lausanne july 17 2003

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


Epfl lausanne july 17 2003

Numerical example: Is the basic control conservative ?

SQRT:

PFTK-simplified:

  • loss-event intervals: i.i.d., generalized exponential density


Epfl lausanne july 17 2003

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


Epfl lausanne july 17 2003

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


Epfl lausanne july 17 2003

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


Epfl lausanne july 17 2003

Check: is TFRC TCP-friendly

TCP-friendly ? - no, not always

  • although, it is mostly conservative !


Epfl lausanne july 17 2003

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


Epfl lausanne july 17 2003

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


Epfl lausanne july 17 2003

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


Epfl lausanne july 17 2003

Part IIExpedited Forwarding


Epfl lausanne july 17 2003

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


Epfl lausanne july 17 2003

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


Epfl lausanne july 17 2003

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


Epfl lausanne july 17 2003

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


Epfl lausanne july 17 2003

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),


Epfl lausanne july 17 2003

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


Epfl lausanne july 17 2003

Numerical example


Epfl lausanne july 17 2003

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


Epfl lausanne july 17 2003

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


Epfl lausanne july 17 2003

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)


Epfl lausanne july 17 2003

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’


Epfl lausanne july 17 2003

Numerical Example

  • Example networks

rn = all n


Epfl lausanne july 17 2003

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


Epfl lausanne july 17 2003

Part IIIInput-queued Switch


Epfl lausanne july 17 2003

Problem

  • at any time slot, connectivity restricted to permutation matrices

Switch scheduling problem: schedule crossbar connectivity with guarantees on the rate and latency


Epfl lausanne july 17 2003

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


Epfl lausanne july 17 2003

Rate-Latency Service Curve

*


Epfl lausanne july 17 2003

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 ?


Epfl lausanne july 17 2003

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


Epfl lausanne july 17 2003

Numerical Example

w.p. 0.99

worst-case deterministic


Epfl lausanne july 17 2003

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


Epfl lausanne july 17 2003

Random-distortion Periodic

  • token processes as with Chang et al, but place each token uniformly at random on the periods

By derandomization:


Epfl lausanne july 17 2003

A Numerical Example

Chang et al

Random-distortionperiodic

Random-phase periodic

  • rate-demand matrices drawn in a random manner


Epfl lausanne july 17 2003

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


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