TCP Models

Objective

Given the loss probability, how fast does TCP send?

Deterministic model?

Simple Stationary modeldrops

cwnd

time

wmax

wmax/2

Wmax/2* RTTData rate = cwnd/RTT

Total packet sent=Total area wmax wm ax

212

wm ax

2 3

8wmax

2

so loss probability p 1

3/8wm ax2

or wmax 83

1p

Average window size wmax wm ax

212

34wmax

or average window size 34

83

1p

32

1p

Simple Stationary modeldrops

cwnd

time

wmax

wmax/2

Wmax/2* RTT

Total packets sent wm ax

2wm ax

2 1 2 3 . . . wm ax

2

wm ax

2wm ax

2 wm ax

2wm ax

2 1 1

2

wm ax2

4 wmax

2 18

wmax14

wmax2 3

8as previously shown

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.210

0

101

102

103

probability

log

(Me

an

Cw

nd

)

ObservedLeastSquares, rhat=-0.53, chat=1.05 LeastSquares, r=-0.5, chat=1.18 r=-0.5, c=(3/2) (̂0.5}r=-0.5, c=1.31

10-4

10-3

10-2

10-1

100

100

101

102

103

probability

log

(Me

an

Cw

nd

)ObservedLeastSquares, rhat=-0.53, chat=1.05 LeastSquares, r=-0.5, chat=1.18 r=-0.5, c=(3/2) (̂0.5}r=-0.5, c=1.31

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10-3

101

102

103

probability

log

(Me

an

Cw

nd

)

ObservedLeastSquares, rhat=-0.51, chat=1.15 LeastSquares, r=-0.5, chat=1.27 r=-0.5, c=(3/2) (̂0.5}r=-0.5, c=1.31

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10-3

0

0.5

1

1.5

2

2.5

probability

pe

rce

nt

err

or

LeastSquares, rhat=-0.51, chat=1.15 LeastSquares, r=-0.5, chat=1.27 r=-0.5, c=(3/2) (̂0.5}r=-0.5, c=1.31

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.210

0

101

102

probability

log

(Me

an

Cw

nd

)

ObservedLeastSquares, rhat=-0.54, chat=1.01 LeastSquares, r=-0.5, chat=1.14 r=-0.5, c=(3/2) (̂0.5}r=-0.5, c=1.31

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

5

10

15

20

25

probability

pe

rce

nt

err

or

LeastSquares, rhat=-0.54, chat=1.01 LeastSquares, r=-0.5, chat=1.14 r=-0.5, c=(3/2) (̂0.5}r=-0.5, c=1.31

More complicated model

dW t 1RTTdt 1

2W tdN t

pw,t t 1

RTT

pw,tw w t RTTpw, t 4p2w, t

pw,t t 0 dpw

dw w 4p2w pw .

wpwp 2/11

2/1

The mth moment around the origin scales like -m/2, i.e.,

2/mm

m

C

The median scales = 1.2/1/2

=1

=0.1

=0.05=0.01 =0.005

cwnd

p(cwnd)

wpwp 2/11

2/1

C1 =1.3, C2 = 2.0, C3 = 3.5, C4 = 7.1, …0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

The PDF of the cwnd for a Simplified TCP

10-4

10-3

10-2

10-1

100

100

101

102

103

probability

log

(Me

an

Cw

nd

)ObservedLeastSquares, rhat=-0.53, chat=1.05 LeastSquares, r=-0.5, chat=1.18 r=-0.5, c=(3/2) (̂0.5}r=-0.5, c=1.31

varw 0. 31/

varw 0.285 0.18

Distribution of cwndCwnd is nearly distributed according to the negative binomial distribution

pw N w 1 Nw 1 !

1 qNqw 1

Gamma function is factorial if argument is an integer

q 1 Ew 1

Varwand N 1 q

q Ew 1

q 1

c 1

1 and N 1 q

qc 1

1

Where: Ewm cm

m/2

E(w²)-E(w)²=(γ/δ)

c1~sqrt(3/2)

γ≈0.3

0 50 100 150 200 250 300 3500

0.002

0.004

0.006

0.008

0.01

0.012

=0.00010

observedsqrt(3/2)/1.27/() or observed mean

0 5 10 15 20 25 30 35 400

0.02

0.04

0.06

0.08

0.1

=0.01000

observedsqrt(3/2)/1.27/() or observed mean

1%

0 2 4 6 8 10 12 14 16 180

0.05

0.1

0.15

0.2

0.25

=0.05000

observedsqrt(3/2)/1.27/() or observed mean 5%

0 2 4 6 8 10 12 140

0.05

0.1

0.15

0.2

0.25

0.3

0.35

=0.10000

observedsqrt(3/2)/1.27/() or observed mean

10%

Time-out model

w, wR k maxminw/2 ,w 3 ,0

w 1w 1k

k1 w 1 k

maxminw/2 ,w 3, 0 drops out of the next w 1 packets

Rate of going to time-out =

: E w, |not TO w, ga ,b w ,

I If a flow experiences so many losses that triple duplicate acknowledgements are not received, i.e., if more thanmaxw 2, 1 losses occur in one window.

II In the case of the ns-2 implementation of TCP-SACK, if more than w/2 packets are droppedIII If a retransmitted packet is dropped.

Timeout modelIf a retransmission is not dropped (only if 1 and 2 didn’t apply).

In particular, if less than maxminw/2 ,w 3, 0 packets are dropped out of the next w 1 packets.

w, : wR

1

k maxminw/2 ,w 3 ,0

w 1w 1k

k1 w 1 k .

: E w, 2 1 R

Total rate of entering timeout is: = ’ + ’’

Time-out. Let I₁(t), denote the rate that flows enter timeout at time t

denote the rate that flows enter timeout for this second time with I2 t

The fraction of flows in timeout are t RTO

tI1 d

t 2RTO

tI2 t d

I1 t t t

1 t RTO

tI1 d

t 2RTO

tI2 t d .

#

I2 t I1 t RTO.

In steady state, I1 t and I2 t are constant.

I1 1 I1 RTO 2I2 RTO

I2 I1 .

I1 1 RTO 1 2

I2 .

1 RTO 1 2

Time out prob

T c 1

R 1 PTO MSS.

Dynamics of cwndddtw t 1

R 1

21R t Rw t Rw t

ddtw t 1

R 1

21R t Rw 2 t .

ddtw t 1

R 1

R t REwt Rwt

if proper stochastic calculus is applied, the correct dynamics for the mean are

ddtw t 1

R 1

R t Rw 2 t .Approximately:

ddtw 2 t 2

Rw t 3

41R t REw 3 t .SDE gives

Ew 3 t 83

c 12 0.31

3/2w 2 3/2

Using:

ddtw 2 t 2

Rw t 3

483

c 12 0.31

3/2

1R t R w 2 3/2

.

P TO at time t t RTO

tI1 d

t 2RTO

tI2 d

Models of slow-start are in the works