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

EPFL, Lausanne, July 17, 2003


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

Part I

Equation-based Rate Control


  • 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

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

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)

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

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

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

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

Sketch of the Proof

Palm inversion:


May make the control conservative ? !

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

When 1/f(1/x) is convex

Check some typical TCP throughput formulae:



almost convex







b = number of packets acknowledged by an ack

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


  • 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

Numerical example: Is the basic control conservative ?



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

ns-2 and lab: Is TFRC conservative ?










Setup: a RED link shared by TFRC and TCP connections

  • The same qualitative behavior as observed on the previous slide

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


Check is TFRC conservative



  • 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

Check: is TFRC TCP-friendly

TCP-friendly ? - no, not always

  • although, it is mostly conservative !

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

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

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

Part IIExpedited Forwarding


  • 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


  • 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

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


  • 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

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),

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

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

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

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)

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’

Numerical Example

  • Example networks

rn = all n

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

Part IIIInput-queued Switch


  • at any time slot, connectivity restricted to permutation matrices

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

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

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 ?

Random Permutation

  • a rate k is an integer multiple of 1/L

  • L = frame-length


  • 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

Numerical Example

w.p. 0.99

worst-case deterministic

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

Random-distortion Periodic

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

By derandomization:

A Numerical Example

Chang et al


Random-phase periodic

  • rate-demand matrices drawn in a random manner

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