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EPFL, Lausanne, July 17, 2003

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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
- analysis and synthesis

Part I

Equation-based Rate Control

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

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:

Throughput:

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:

SQRT:

PFTK-standard

almost convex

PFTK-standard:

PFTK-simplified

convex

PFTK-simplified:

SQRT

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

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

Numerical example: Is the basic control conservative ?

SQRT:

PFTK-simplified:

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

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

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

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

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

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

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

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

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

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

Numerical example

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

Problem

- 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

Rate-Latency Service Curve

*

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

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

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-distortionperiodic

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