TCP Models Objective Given the loss probability, how fast does TCP send? Deterministic model?
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Transcript of TCP Models Objective Given the loss probability, how fast does TCP send? Deterministic model?
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