MSR, Cambridge, August 5, 2003

Long-Run Behavior of Equation-Based Rate Control& Rate-Latency of Some Input-Queued Switches

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Outline

Part ILong-run Behavior of

Equation-based Rate ControlPart II

Rate-Latency of Some Input-queued Switches

The talk takes from:

M.V., Ph.D. thesis, July 2003

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Part ILong-Run Behavior of

Equation-Based Rate Control

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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

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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

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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)

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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

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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

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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

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When the basic control is conservative assume: the send rate be a stationary ergodic process

In practice: the conditions are true, or almost the result explains overly conservativeness

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Sketch of the Proof

Palm inversion:

Throughput: May make the control conservative ? !

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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

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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:

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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

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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

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Numerical example: Is the basic control conservative ?

SQRT:

PFTK-simplified:

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

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ns-2 and lab: Is TFRC conservative ?

PFTK-simplified

Setup: a RED link shared by TFRC and TCP connections

L=2

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the same qualitative behavior as observed on the previous slide

PFTK-standard

L=8

ns-2 lab

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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

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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

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Check: is TFRC TCP-friendly

TCP-friendly ? - no, not always although, it is mostly conservative !

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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 RTT than TCP ? Is TFRC conservative ?

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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

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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 protocol adjustment

conservativeness against TCP-friendliness TCP-friendliness is difficult to verify conservativeness

amenable to a formal verification not TCP centric

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Part IIRate-Latency of

Some Input-queued Switches

The work done in part while an intern with Dept. of Mathematics of Networks and Systems, Bell Laboratories, Murray Hill, NJ, Summer 2001

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Problem

at any time slot, connectivity restricted to permutation matrices

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

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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 number:

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Rate-Latency Service Curve

*

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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 token types 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)

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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

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Numerical Example

worst-case deterministic w.p. 99/100

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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:

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Random-distortion Periodic token processes as with Chang et al, but place each token uniformly at random on the

periods

By derandomization:

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A Numerical Example

Chang et al

Random-distortionperiodic

Random-phase periodic

rate-demand matrices drawn in a random manner

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Concluding Remarks for Part II

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 may lead to construction of practical algorithms