Estimating Congestion in TCP Traffic

Stephan Bohacek and Boris Rozovskii University of Southern California

Objective: Develop stochastic model of TCPNecessary ingredients: Models of the network. Specifically packet drop probability and roundtrip time. The model parameters are indicators of congestion.

Outline

• A very brief introduction to TCP

• Modeling packet drop probability– Modeling roundtrip time– Dynamics of drop probability parameters

• A diffusion model of roundtrip time– Estimating the model parameters

• Stochastic models of TCP

• Results and insights

An Introduction to TCP

• TCP is acknowledgment based. The sender sends a packet of data. When the receiver receives the packet, it responds by sending an acknowledgment packet to the sender.

• If the sender fails to receive an acknowledgment, it assumes that the packet has been dropped and decreases the sending rate.

• If the sender receives an acknowledgment, it assumes that the network is not congested and increases the sending rate.

• The congestion window, X, defines the maximum number unacknowledged packets.

• When the sender receives an acknowledgment it increases the congestion window by 1/[Xt].

• When a packet drop is detected, the sender divides the congestion window in half.

Some TCP Details

Note: the congestion window is not the sending rate. sending rate = congestion window / roundtrip time.

Number of unacknowledgedpackets equals Cwnd. Send no more packets.

How the Congestion Window Increases

send pkt Asend pkt Bsend pkt Csend pkt D

X=4

ack A

ack B

ack C

ack Dsend pkt Esend pkt Fsend pkt Gsend pkt Hsend pkt F

X=4.25X=4.5X=4.75X=5

time

Packet arrives at receiver.Receiver sends an acknowledgment.

roundtriptime

Time Series of the Congestion Window(simulation)

50 55 60 65 70 75 80 850

5

10

15

20

25

30

35

40

45

50

Seconds

Con

gest

ion

Win

dow

linear increasewhen no drops occur

divide by two when a drop is detected

Time Series of the Congestion Window (simulation)

50 100 150 200 250 300 350 400 450 5000

5

10

15

20

25

30

35

40

45

50

Seconds

Con

gest

ion

Win

dow

Stochastic model of TCP•Stochastic model of packet drops.•Stochastic model of the roundtrip time.

Drop Models

• Ott (1997) considered deterministic drops.• Padhye (1998) assumed drops to be highly correlated over short

time scales, but independent over longer time scales.• Altman (2000) assumes drops are bursty.• Altman (2000) drop events are modeled as renewal processes with

particular examples, deterministic, Poisson, i.i.d., and Markovian.• Savari (1999) drop events are modeled as Poisson where the

intensity depends on the window size of the TCP protocol.

Models for Packet Drop Probability

0 ,,,,, dRSKRSg tttt

ttt

tt

tt

ttt

ttt RScRbSaSaaRSg 112

210,,

Let St be the sending rate at time t.Let Rt be the roundtrip time experienced by a packet sent at time t.Let t be the congestion level at time t.Let g be the probability that a packet is dropped

general model

memoryless

2210, t

tt

tttt RaRaaRg

depends on roundtrip time only

Preliminary work indicates that, for reasonable sending rates, the drop probability mostly depends on roundtrip time.

Since drops are rare, it is difficult to collect data for slow sending rates.

* *

Determining the Conditional Drop Probability

1

111

||

|,||,

ktktktk

ktktktktkktktk

RRpRdP

RRpRRdPRRdp

ktktkt

ikt

n

ii

ktktktikt

n

ii

ktktktkt

ktktktktkktktktkktk

dRRRpRa

dRRRpRa

dRRRpRg

dRRRpRdPdRRRdpRdp

10

10

1

111

|

|

|

|||,|

system of linear equations

time.roundtrip for the iesprobabilitn transitio the- |

timeroundtrip a experiencepacket past given the

dropped ispacket next y that theprobabilit the- |

1

1

1

ktkt

kt

ktk

RRp

R

Rdp

observable

assume the congestion level, , is constant

We assume that given Rtk, dk is independent of Rtk-1

End-to-end model with many queues

q1t q2

t qn-1t qn

tsource destination

queues

queuing delay at time t = D1

t

queuing delay at time t = Dn-1

t

kTk DTD 1

2th is queue second in thepacket k by the dexpereincedelay The

kTD1th is queuefirst in thepacket k by the dexpereincedelay The

prop

kTn D kT

n

kTD kT

D kTD kT kT

D kT kTkT D D D D RTT

...

1 ... 12 1

31

2 1...

The kth is sent at time Tk

propagation delay

Modified Diffusion Approximation for a Single Queue

oo dtdDPdtdF ,|,,

otherwise 0

for ,,2

2

,tdd

t

mtdde

t

mtdddtdF o

o

md

o

om

t

mtdd

xo

o

dxet

mtdd

2

2

2

1

Histogram of Observed RTT Increments (real data)

Gaussian (RTT0=28)Queue empties slowly

Queue empties quickly

Agrees with queuing theory(Diffusion Approximation)

Observed and Smoothed Conditional Drop Probabilities

16 18 20 22 24 26 28 30-0.005

0

0.005

0.01

0.015

0.02

0.025

Roundtrip Time

16 18 20 22 24 26 28 300

0.005

0.01

0.015

0.02

0.025

Roundtrip Time

observed conditional drop probability

smoothed conditional drop probability

night

12 noon

3pm6pm

P(D

rop

| Rou

ndtr

ip T

ime

)P(D

rop

| Rou

ndtr

ip T

ime

)

9am

Drop Model Parameter Variation

1

5

10

15

x 10-3 Drop Model Parameter Variation

X 1

1

0

10

20x 10

-3

X 2

1-20

-10

0

x 10-3

X 3

1-10

-50

5

x 10-3

Monday Tuesday Wednesday Thursday Friday Saturday Sunday

X 4

g(Rt, t) = 0(t) + 1( ) T1(R) + 2( ) T2(R) + …

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1

0

1

2x 10

-7

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-2

0

2

4x 10

-7

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1

0

1

2x 10

-7

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-5

0

5x 10

-7

Time (hours)

Autocorrelation of the Increments of the Drop Probability Parameters

Iit := i(t+1) - i(t) - increments

E(I0t I0t+)

E(I1t I1t+)

E(I2t I2t+)

E(I3t I3t+)

It appears that the increments are uncorrelated.

Transition Probability for Drop Model Parameter

1

1

Transition Probabilities for X10.000

0.0000.007

0.007

0.014

0.014

0.020

0.020

0.027

0.027

0.034

0.034

0.041

0.041

0.047

0.047

0.054

0.054

0.061

0.061

0.068

0.068

It appears that the transition probability is spatially homogenous.

controls the rate at which the transition probability converges to the stationary distribution.

trrpr

trprr

trpt

,,2

,2

22

A Simple Stochastic Model of the Roundtrip Time

tttt dBRdtRdR

2

2

CIR Model (short-term interest rates)

0for on Distributi Stationary 1

xexxp x

Note that the stationary distribution does not depend on .

forward equation

Define Rt to be the queuing delay. Roundtrip time – propagation delay

mean reverting mean = /

similar to gamma

distribution

Estimation of , , and (many approaches)

tttt dBRdtRdR

td

X

XdX

X

X

dP

dP T

t

tTt

t

t 0 2

2

0 2,0,0

,,

2

1

Tt

T

t

Tt

t

T

tT

T

t

Tt

t

dtXdtX

T

dXX

TdtX

X

dtX

dXX

T

002

00

0

0

1

11

1

1

likelihoodratio

maximum likelihoodestimates

note the change in representation

Estimation of , , and

tttt dBRdtRdR 2

2

tt

n

kkt

RR

Rpn

log1loglog

log 1

likelihood logn

1

1

Select a set of observations {Rtk} where tk << tk+1 Hence, {Rtk} are approximately i.i.d.

tt RR loglog'

log

tR

the maximum likelihood estimates

satisfy

mean of Rt

mean of log( Rt )

under the independence assumption

Estimating Diffusion

tttt dBRdtRdR

tttt dBRdtRdR 2

2

2

2

Estimation of , , and

qorder of kindfirst theoffunction Bessel modified

1/2: , ,: ,1/2:

2,,;|

2

2/12/

q

tt

ot

q

qvu

ot

I

qcRvecRuec

uvIu

vceRRp

The transition probability is known

Computationally difficult

n

kktkt

RRp0

1* ,,;|logminarg:

and are found as before and

0 2 4 6 80

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5Fitted Measured

0 2 4 6 8 10 12 140

0.05

0.1

0.15

0.2

0.25Fitted Measured

Roundtrip Time Quasi-stationary DistributionNight Midday

50 55 60 65 70 75 80 85 90 95 1000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

ns-2 simulation

RTT

P(R

TT

)

Observed RTT Dist Fitted RTT Dist

RTTRTT

P(R

TT

)

P(R

TT

)

Basic Stochastic Model of TCP

tt

proptt dN

Xdt

TRdX

2

1

Nt is a Cox process with intensity n(Rt, Xt, t)

tttt dBRdtRdR 2

2

n(Rt, Xt, t) = Xt / Rt · g(Rt, Xt, t)

Sending Rate = Xt / Rt = Packets per RTT / Seconds per RTT = Packets / Second

Drop Probability = g(Rt, Xt, t)

Xt – congestion windowRt + Tprop – roundtrip time

drop probability

Stochastic Model of TCPwith fast recovery

tttt

tpropttWtWt

tt

propttWt

dBRdtRdR

dNTRdtdW

dNX

dtTR

dX

2

11

2

11

2

00

0

After a drop is detected, the congestion window, X, is halved and remains constant until the dropped packet is resent and acknowledged.

• A drop is detected at t=0, so W0=R0

• Wt > 0 for 0 < t < R0

• WRo = 0 and the congestion window

continues to increase until the next

drop occurs.

fast recovery

Stochastic Model of TCPwith fast recovery and time-out

tttt

ttpropttWtWt

tttt

propttWt

dBRdtRdR

dTdNTRdtdW

dTXdNX

dtTR

dX

2

11

12

11

2

0

0

When a time-out occurs, the congestion window is set to 1.

The Forward Equation

trxpxrntrwxpxrn

trwxpw

trwxpxTr

trwxrpr

trwxprr

trwxpt

propTrww

wwprop

,,0,22,21,,,,1

,,,1,,,11

,,,2

1,,,

2,,,

0

00

22

22

tdrdwdxPrwx

trwxp

CRBWAXPtCBAP ttt

,,,,,,Let

,,:,,,Let 2

No Time-outsStationary environment (model parameters are constant)

Assumptions:

Midday RTT and Drop Data

0 5 10 15 20 25 300

0.02

0.04

0.06

0.08

0.1

0.12Marginal Density p(X)

X - Congestion Window

p

15 20 25 300

5

10

15

20

25

30

X -

Co

ng

est

ion

Win

do

w

R - Roundtrip Time

Marginal Density p(X,R)

= 4, = 0.85, = 0.5

0 2 4 6 8 10 12 140

0.05

0.1

0.15

0.2

0.25

RTT

Pro

b(R

TT

)

Fitted Measured

0 2 4 6 8 10 12 140

0.005

0.01

0.015

0.02

0.025

0.03

RTT

drop

pro

b

Fitted Measured

Drop Probability

Roundtrip Time Stationary Distribution

ns-2 (network simulator)topology

Source 1 2 3 4 5 Destination

A B DC

S T U V

Competing TCP Sources

ns-2 Simulation

20 30 40 50 60 700

5

10

15

20

25

30

X -

Con

gest

ion

Win

dow

R - Roundtrip Time

Marginal Density p(X,R)

= 2, = 0.17, = 0.34

0 5 10 15 20 25 30 35 40 45 500

0.01

0.02

0.03

0.04

0.05

0.06

0.07

RTT

P(R

TT

)

Fitted Measured

50 60 70 80 90 100 110-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

RTT

dro

p p

rob

Drop Probability

Roundtrip Time Stationary Distribution

0 10 20 30 40 500

0.01

0.02

0.03

0.04

0.05

0.06

0.07current Simulated

Marginal Density p(X)

The Dependence on Drop Probability

0 5 10 15 20 25 300

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

X - Congestion Window

p

Drop Prob Scale = 0.75Drop Prob Scale = 1.0 Drop Prob Scale = 1.25Drop Prob Scale = 1.5 Drop Prob Scale = 1.75Drop Prob Scale = 2.0

10-3

10-2

10-1

100

101

102

The TCP Friendly Rule

p1/2

RT

T

Se

nd

ing

Ra

te =

Co

ng

es

tion

Win

do

w

Drops are modeled as a Cox process with intensity

n(Rt, Xt, t) = Xt / Rt · g(Rt, Xt, t) • Scale

Dependence on

10-2

101

The TCP Friendly Rule

p1/2

RT

T

Se

nd

ing

Ra

te =

Co

ng

es

tion

Win

do

w

0 5 10 15 20 25 300

0.02

0.04

0.06

0.08

0.1

0.12

X - Congestion Window

p

= 0.069 = 0.138 = 0.275 = 0.413 = 0.551 = 0.688 tttt dBRdtRdR

2

2

15 20 25 300

5

10

15

20

25

X -

Co

ng

est

ion

Win

do

w

R - Roundtrip Time

Marginal Density p(X,R)

= 0.07

recall that controls the rate at which the transition probability converges to the stationary distribution

0 5 10 15 20 25 30

0

0.02

0.04

0.06

0.08

0.1

0.12Conditional Probability Density p(X | RTT)

Sending Rate

p

compatiblesending rate

Application to Variable Bit Rate Video Transmission

TCP Friendly – Send data at a rate similar to the rate that TCP would.

When the video image changes quickly, the bit rate increases and the sending rate must also increase.The “compatibility” with TCP’s sending rate can the judged by examining the probability density function of the congestion window.

Future Work

• Better models: dynamic models that depend on the past sending rate.– For example:

• Suppose that the sending rate is initially high. Then other TCP flows should decrease their sending rate.

• If the sending rate suddenly decreases, then there is temporarily extra capacity and there should be few drops (maybe).

• Time-out and slow start.• Doubly stochastic processes: Allow the parameters to vary

with time.• More accurate roundtrip time models• Experimental verification• More data, do general models for drop probability exist?

Conclusions

• System theoretical (input/output) view point to the Internet is valid.– Drop probability models– Roundtrip time models (queuing theory works!)

• Stochastic models – seem to accurately predict the TCP in complex

networks.– give useful insight into the performance of TCP

(e.g. the dependence on )– might be for other types of congestion control

(e.g. VBR video)