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On the Long-Run Behavior of Equation-Based Rate Control

Milan Vojnovićand

Jean-Yves Le Boudec

ACM SIGCOMM 2002, Pittsburgh, PA, August 19-23, 2002

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Ln 3n 2n1n

The Control We Study

)p̂(fX nn

nT1nT 2nT 3nT LnT

)t(X rate send The

t

...

n

n ˆ1

p̂

L

1llnln wˆ

We call this the basic control

The loss events:

The loss intervals:

Function f is typically TCP loss-throughput

function

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The Control We Study (cont’d)

We call this the comprehensive control

3n 2n1n

)p̂(fX nn

nT1nT 2nT 3nT LnT

)t(X rate send The

t

...Ln

Additional rule: If the number of bits sent since the lastloss event included in the loss-event estimator increases its value, then use it in computation of the send rate

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Why do we Study This?

Several send rate controls are proposed for media streaming in the Internet.

We often take TFRC (Floyd et al, 2000) as a recurring example.

The send rate should be smoother than with TCP, but still responsive to congestion.

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What do we Study?

In the long-run, is the control TCP-friendly ?

i.e.: (P) EBRC Throughput TCP Throughput ?

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We split the problem into 3 Sub-Problems

P1) Is Rate Control Conservative ? )p(f)]t(X[E

protocol this of ratio event-loss the is p here

P2) Is our loss no better than TCP’s ? TCPpp

P3) Does TCP conform to function f ? )p(f)]t(X[E TCPTCP

Ob. If P1, P2, and P3 are positive, then the control is TCP-friendly

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)x/1(f

x

Some Functions fSQRT:

Note: c1, c2, c3 are some positive-valued constantsr is the round-trip timeq is TCP retransmit timeout (typically, q=4r)

prc1

)p(f1

PFTK-standard:

)p32p](pc,1min[qprc1

)p(f 321

PFTK-simplified:

)p32p(qcprc1

)p(f 2/72/321

SQRT

PFTK-

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Throughput ExpressionsBasic Control:

])ˆ/1(f

[E

][E)]t(X[E

0

00T

00T

Comprehensive Control (PFTK-simplified):

]1V[E])ˆ/1(f

[E

][E)]t(X[E

01ˆV0

0T

0

00T

00T

where

])ˆ/1(f

1)ˆˆ()ˆˆ(qc

52

)ˆˆ(qc2)ˆˆ(rc2[w1

V

n

n1n2/5

n2/5

1n3

2/1n

2/11n2

2/1n

2/11n1

1n

Ob. Knowing the joint law of one would beable to compute the throughput

L,,10,

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(P1) When is the Control Conservative?

Sufficient Conditions:

]ˆ,[cov)p(fp)p('f

1

1)p(f)]t(X[E :Moreover

000T

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(F1) 1/f(1/x) is convex with x

0]ˆ,[cov nn0T (C1)

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(F1) is true for SQRT and PFTK-simplified

)x/1(f1

PFTK-SQRT

x

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(F1) is true for SQRT and PFTK-simplified

(F1) is almost true for PFTK-standard

If f(1/x) deviates from convexity by the ratio r, and (C1) holds, then the control cannot overshoot by more than the factor r

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When the Conditions are Met?

It follows: (C1) is true for i.i.d. nn)(

],[covw]ˆ,[cov lnn0T

L

1llnn

0T

It is the autocorrelation of what matters! nn)(

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Claim 1Assume:

and are negatively or lightly correlated consider f in the region where takes its values

n n̂

n̂

1) The more convex 1/f(1/x) is, the more conservative the control is

n̂2) The more variable is, the more conservative the control is

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Numerical Example for Claim 1

is i.i.d. with the distribution:nn)(

0x ,xx )),xx(exp()x( 000

000T x][E

variation) of (coeff. x

][cv0

00T

(skewness) 2][S 00T

(kurtosis) 6][K 00T

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Numerical Example for Claim 1 (Cont’d)

Ob. The larger is, the more convex 1/f(1/x) is, and hence the more conservative the control is

p

SQRT PFTK-simplified

Ob 2. PFTK is more convex than SQRT, effect is more pronounced

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Numerical Example for Claim 1 (Cont’d)

Ob. The more variable is, the more conservative the control is

n̂

SQRT PFTK-simplified

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ns-2 Example for Claim 1Single RED bottleneck

shared with equal

number of TFRC and TCP flowsPFTK-simplified:

(likewise for SQRT and PFTK-standard)

)p(f)]0(X[E

pOb. The larger is, the more convex

1/f(1/x) is, and hence the more conservative the control is

p

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Another Set of Conditions

Then, the control is conservative!

Then, the control is non-conservative!

If(F2) f(1/x) is concave with x(C2) 0]S,X[cov nn

0T

If(F2’) f(1/x) is convex with x(C2’) (V) is not fixed to some constant

n̂0]S,X[cov nn

0T

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When is the Control Non-Conservative?

SQRT and PFTK formulas are such that f(1/x) is concave, except

both PFTK formulas are such that f(1/x) is convex for small x

We may have non-conservativeness in this region!EXAMPLE: Some rate controls keep the packet

send rate fixed, but vary packet size

Non-conservativeness!

If packets are dropped at a router independently of the packet length, then

Xn=Ln r

0]S,X[cov nn0T

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Claim 2

Assume Xn and Sn are negatively or lightly correlated

1) If f(1/x) is concave in the region where takes its values, then the controls tends to be conservative

n̂

Assume Xn and Sn are postively or lightly correlated

2) If f(1/x) is convex in the region where takes its values, then the controls tends to be non-conservative

n̂

In both cases: the more variable is, the more pronounced the effect is

n̂

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ns-2 Example for Claim 2Rate control with

fixed packet send rate, but variable packet size through a loss module with fixed packet drop probability

L=4

)p(f)]0(X[E

]ˆ[cv 00T

p

For L=8 (not shown in the slides), we have qualitatively the same effects, but less pronounced (the last part of the claim)

With both PFTK non-conservativeness!Recall: f is convex for

PFTK for largep

With SQRT always conservative!

With trend upwards due to decreasing coeff. of variation of

n̂

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Statistical Bias due to Viewpoint Does Play a Role!

By Palm inversion formula:

]S[E)]0(X[E]S,X[cov

1)]0(X[E)]0(X[E0

0T

0T

000T0

T

4nT 3nT 2nT 1nT nT1nT 2nT

RandomObserver

ObserverSampling atthe Points

Random Observer falls more likely into a large time interval

Random Observer would measure larger average interval than as seen at the points!

This is known as Feller’s Paradox!

If Xn and Sn are positively correlated

Then, the random observer would see larger send rate than as seen at the points

Likewise to Feller’s Paradox!

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(P2) How Do Different Loss-Event Ratios Seen By the Sources

Compare?This is another issue of importance of viewpoint!Different sources may see different loss-event ratios!

Claim 3:PAT ppp

Seen by TCP Seen by Equation-Based Rate Control

Seen by Poisson

Source (non-adaptive)

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How Do Different Loss-Event Ratios Seen By the Sources

Compare? (Cont’d)Suppose there exists a hidden congestion process Z(t)

If at time t, Z(t)=i, then the loss-event ratio is ip

This can be formalized by Palm Calculus

Intuition behind Claim 3: Non-adaptive (Poisson Source) would see time average of the

system loss-event ratio An adaptive source would sample “bad” states less frequently The more adaptive the source is, the smaller loss-event ratio

it would see TCP would be more adaptive than Equation-Based Rate

Control, and hence would see smaller loss-event ratio

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ns-2 Example for Claim 3

p

s)connection of number (total N

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(P3) Does f Match TCP Loss-Throughput Formula,

Actually?Not always!

TCP Sack1:

)]0(X[E

)p(f

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Check the 3 sub-problems separately !

)]0(X[E

A TCP-unfriendly example, even though control conservative and sees larger loss-event ratio!

This is just an artifact of inaccuracy of function f.It is not an intrinsic problem of the control.

Ignoring this might lead the designer to try to“improve” her protocol -- wrongly so

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(P1) We showed when we expect to have either conservative or non-conservative control We explain the throughput-drop encountered

empirically elsewhere We demonstrate a realistic control which would be

non-conservative

(P2) Expect loss-event ratio of equation-based rate control to be larger than TCP would see

(P3) TCP may deviate from PFTK formula It is important to distinguish the three sub-

problems and check them separately.

Conclusion