EPFL, Lausanne, July 17, 2003

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

2

Outline

Part IEquation-based Rate

ControlPart II

Expedited ForwardingPart 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 IEquation-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)

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 ?

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 ?

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

Ln 3n 2n1n

Equation-based rate control: the basic control law

...

nT1nT 3nT LnT

Additional control laws ignored in this slide

2nT ...... ...

send rate

1nT

nT = instant of a loss-event

= a loss-event intervaln

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)

the “overshoot” bounded by a function of p and

1/f(1/x) is assumed to be convex, thus, it is above its tangents take the tangent at 1/p

13

SQRT

PFTK-standard

PFTK-simplified

convex

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

Recall (C1)

It holds:

if is a bad predictor, that leads to conservativeness

if the loss-event intervals are independent, then (C1) holds with equality

= a “measure” how well predicts

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 ?

PFTK-simplified

Setup: a RED link shared by TFRC and TCP connections

L=2

48

16

The same qualitative behavior as observed on the previous slide

PFTK-standard

L=8

ns-2 lab

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-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:

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

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 ?

22

Check the factors separately !

when 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 ?

Does TFRC see no better loss-event rate

than TCP ?

No No No

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 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:

a property that holds for one of the node abstractions, holds for a PSRG node

28

Assumptions

Note that an EF flow is allowed to be any stochastic process as long as it obeys to the given set of the assumptions

A1, A2, …, AI stochastically independent

Ai is constrained with an arrival curve

Ai is such that

There 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),

Assume: all I fix:

30

A Method to Derive BoundsStep 1: containment into a union of the “arrival overflow events”

(by def. of a service curve and )

Step 2: use the union probability boundStep 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

Known result: for , a bound on the e2e delay-jitter is

Given:

( = set of EF flows that traverse the node n)

(= maximum number of hops an EF flow can traverse)

Problem: obtain a bound on the e2e delay-jitter

34

A dimensioning rule

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)

Given, in addition:

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

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’

must be > 0, for the bound to be < 1

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)

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

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 unique token assigned to a permutation matrix scheduler by Chang et al can be seen as

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 ?

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 L = frame-length

compare with the worst-case deterministic latency

Scheduler: schedule the permutation matrices in a frame,

according to a random permutation of the tokens repeat the frame over time

44

Numerical Example

worst-case deterministic w.p. 0.99

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

compare with Chang et al

By derandomization:

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