small scale analysis of data traffic models b. d’auria - eurandom joint work with s. resnick -...
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Small scale analysis of data traffic models
B. D’Auria - Eurandom
joint work with
S. Resnick - Cornell University
14/03/2006 B. D'Auria - EURANDOM
Content
• Data traffic - stylized facts:Heavy-Tails, LRD, self-similarity,
Burstiness
• M/G/∞ input modelsSmall-scale asymptotic results
• Conclusions
14/03/2006 B. D'Auria - EURANDOM
Telecommunication data traffic
• 1st column
Ethernet Traffic• 2nd column
Poisson Model
Taqqu et al., (1997)
14/03/2006 B. D'Auria - EURANDOM
Stylized facts
• Heavy tails for distributions of such quantities as– File sizes– Transmission rates– Transmission durations
• Long Range Dependence (LRD)• Self-similarity (s-s)• Burstiness
14/03/2006 B. D'Auria - EURANDOM
Heavy tails
We use the class of regular varying distributions
( ) ( ) 1 2P X x L x x
where L(x) is a slowly varying function at ∞
• File sizes– Arlitt and Williamson (1996)
– Resnick and Rootzén (2000)
• Transmission durations– Maulik et al. (2002)
– Resnick (2003)
• Number of packets per slot– Leland et al. (1993)
– Willinger et al. (1995)
14/03/2006 B. D'Auria - EURANDOM
Long Range Dependence (LRD)
In our context, we define LRD as the
non-summability of the covariance
function, i.e. a stationary stochastic
process is Long Range
Dependent if
, ov ,k
k k Y n k Y n
Z
C
n
Y nZ
14/03/2006 B. D'Auria - EURANDOM
Self-similarity (s-s)
A stationary process is
strictly self-similar (ss-s) if
n
Y nZ
1 1
1
1,
n mm
j n m
mHdn n
m Y n Y jm
Y m Y
N
0<H<1 is called Hurst parameter.
14/03/2006 B. D'Auria - EURANDOM
Self-similarity (s-s)
A ss-s process has
covariance function
n
Y nZ
2 22
2 2 a
01 2 1
~ s
2
0 2 1
H HH
H
k k k k
H H k k
When H>1/2 self-similarity implies
LRD.
14/03/2006 B. D'Auria - EURANDOM
Let be a stationary process
Weak self-similarity
Exact 2nd order self-similarity
(es-s)
Asymptotic 2nd order self-similarity
(as-s)
n
Y nZ
2 220 1 2 1H HHk k k k
2 22lim 0 1 2 1
ov ,
H Hm H
m
m m m
k k k k
k Y n k Y n
C
14/03/2006 B. D'Auria - EURANDOM
Burstiness
Following Sarvotham et al. (2005),
data traffic can be spitted in two parts:
α-traffic - large files at very high rate
β-traffic - the rest.
• The α-component is a small fraction of the total
workload but is entirely responsible for burstiness
• The β-component is responsible for the
dependence structure
• At high levels of aggregation traffic appears to be
Gaussian
14/03/2006 B. D'Auria - EURANDOM
α-traffic and β-traffic
from Sarvotham et al. (2001)
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M/G/∞ Input Model
• Transmitting sources arrive according to a Poisson Process with rate λ
• The generic transmission k has associated 4 parameters:– the arrival time
– the transmission rate
– the file size
– the transmission length
, , ,k k k kR L FkkR
kL
kF
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M/G/∞ Input Model
0
0
Γ
A(0,δ]
δ 2δ-δ
A(δ,2δ] A(2δ,3δ] A(3δ,4δ] A(4δ,5δ] A(5δ,6δ]A(-δ,0]A(-2δ,δ]
3δ 4δ 5δ 6δ
t
L0
A(δ)
F0=R0·L0R0
Γ1 Γ2 Γ3-1 Γ0Γ-2Γ-3
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Two Models
• Model RF where k kR F
• Model RL where k kR L
1 , 2R
L
R RR L
L L
F r P R r r L r
F l P L l l L l
1 , 2R
F
R RR F
F
F r P R r r L r
G u P F u u L u
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M/G/∞ Input ModelGiven the relation
F R L
FL L FF l R G l E
by Breiman’s theorem we have the following:• Model RF
R
L
R R L
L L R
L F uG u
R F u
E
E
• Model RL
14/03/2006 B. D'Auria - EURANDOM
Poisson Random Measure
The counting function, N, of the pointsis a Poisson
Random measure on , given by
, , ,k k k kR L F
with mean measure
, , ,k k k kR L Fk
N 30, R
# , , , , , , ,ds dr dl du ds P R L F dr dl du
14/03/2006 B. D'Auria - EURANDOM
The data process A(δ)
Fixed the window size, δ, we consider the discrete time process
, 1 ,A k k k A Z
where represents the total amount of work inputted to the system in the k-th time slot
We assume that the arrival rate is function of δ
, 1A k k
, 1k k
011/ RF
14/03/2006 B. D'Auria - EURANDOM
Decomposition of A(0,δ]
It is possible to decompose A(δ)=A(0,δ ] in 4 independent parts related to the following 4 regions
0,0,1 2 0,1 0,20, A A AA A A
0,2
0,1
0
1
,2
0,
, , , :
, , , : 0
0 ,0 ,
, , , : ,0 ,
, , , : 0, .
,,0s r l u s
s r l u s s l
s r l u s s l
s r l u s
s l
s l
R
R
R
R
14/03/2006 B. D'Auria - EURANDOM
Decomposition of A(0,δ]
It is possible to decompose A(δ)=A(0,δ ] in 4 independent parts related to the following 4 regions
0,0,1 1 ,22 0, 00, AA AA A A
0,1
0,2
0,
0,1
2 , , , :
, , , : 0
, , , : ,0 ,
, , , : 0, .
0 0
,
,
0 ,
,
s r l u s
s r l u s s l
s r
s
l u s s l
l
s r l u s s l
R
R
R
R
14/03/2006 B. D'Auria - EURANDOM
Decomposition of A(0,δ]
It is possible to decompose A(δ)=A(0,δ ] in 4 independent parts related to the following 4 regions
0,1 0,2 0 2, ,1 00, AA AA A A
0,
0,1
,1
2
0
0,2
, ,
, ,
, ,
,
, : ,0
, : 0,
: 0 ,0 ,
,
.
, , : 0 ,0 ,
,
s r l u s s l
s r
s r l u s s
s r
l u
l u s s l
s s l
l
R
R
R
R
14/03/2006 B. D'Auria - EURANDOM
Decomposition of A(0,δ]
It is possible to decompose A(δ)=A(0,δ ] in 4 independent parts related to the following 4 regions
0,1 0, 0,2 1 20,0, .A A A A A A
0,
0,1
0,2
2
0,1
, , , : 0 ,0 ,
, , , : 0 ,0 ,
, , , : ,0
, ,
,
, .: 0,
s r l u s s l
s r
s r l u s s
l u s s l
s r
l
l u s s l
R
R
R
R
14/03/2006 B. D'Auria - EURANDOM
Random Measure decomposition
The restriction of the Poisson measure to the 4 regions give 4 independent Poisson processes
0 ,1
0 , 2
0 ,1
0 , 2
0,1
0,2
0,1
0,2
, , ,
, , ,
, , ,
, , ,
1
1
1
1
k k k k
k k k k
k k k k
k k k k
k kk
k kk
k k kk
kk
R L F
R L F
R L F
R L F
A R L
A R
A R L
A R
R
R
R
R
0,1 0,2 0,1 0,2, , , .N N N N R R R R
and
14/03/2006 B. D'Auria - EURANDOM
A>0,1(0,δ ]
Contributions from sessions starting in (0,δ ] and terminating before δ.
0,1
0,1 0,1
1
P
kk
A F
Where P >0,1(δ) is Poisson distributed with parameter # 0,1 R
14/03/2006 B. D'Auria - EURANDOM
Region
Proposition. If
with P >0,1(0) Poisson distributed with parameter and
0,1R 0RF E
11 R
R F E
1
asR
R
R
Fi
R F
F F xP R x x G x x
F
E
E
0,1 00
0,1 0,1 0,1 0,1
1
0 where P
kk
A X X R
RF
14/03/2006 B. D'Auria - EURANDOM
We have
that implies
Having that
and using
we finally get the
result.
Proof
0,1 1
R
R
FP
EE
0,1 0,1 1 11
R
R
P P F x P R F F x
E E
0,1 0,1 0P P
11/ RF
RF
14/03/2006 B. D'Auria - EURANDOM
A>0,2(0,δ ]
Contributions from sessions starting in (0,δ ] and terminating after δ.
0,2
0,2 0,2 0,2
1
P
k kk
A R
Where P >0,2(δ) is Poisson distributed with
parameter # 0,2 R
14/03/2006 B. D'Auria - EURANDOM
Region
Proposition.
X>0,2 is infinitely divisible with Lévy measure with density
0,2R
1
1RR
R
x G x
1 0
0,2 0,2 0,2
0
A s ds X
0,20
.
0,2 1 0,20
v
r s
ds G s r dr ds ds
1
1
R
R
F dr
Fdr
where
RF
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We have that
then we prove that
Then having convergence of the r.h.s of
the following relation we get the result
Proof
0,2 0,2
0exp 1i A i s
se e ds
E
.
0,2 0,20 0on ,
v
ds ds
10,2 0,2 0,2
0 1
1 0,2
0
exp exp 1
exp 1
i s
s
i s
s
i A s ds e ds
e i s ds
E
RF
14/03/2006 B. D'Auria - EURANDOM
A<0,1(0,δ ]
Contributions from sessions starting before 0 and terminating in (0,δ ].
0,1
0,1 0,1 0,1 0,1
1
P
k k kk
A R L
Where P <0,1(δ) is Poisson distributed with
parameter # 0,1 R
Proposition. 0,1 0,2
d
A A
14/03/2006 B. D'Auria - EURANDOM
Proof
0,1R
0,2R
14/03/2006 B. D'Auria - EURANDOM
A<0,2(0,δ ]
Contributions from sessions starting before 0 and terminating after δ.
0,2
0,2 0,2
1
P
kk
A R
Where P <0,2(δ) is Poisson distributed with parameter # 0,2 R
14/03/2006 B. D'Auria - EURANDOM
Region
Proposition.
where
0,2R
0,2 00,2 0,1
A mX N
a
1
01
1;R
R r
F dr
Fdr G r G u du
F
E
1
0
0
12
0
0
;
.
r
r
m F G r dr
a F rG r dr
E
E
RF
14/03/2006 B. D'Auria - EURANDOM
and we prove that
Proof
1
1
0,21
00
0,2
1
exp exp 1
exp 1
exp
ia r
ia r
FA mi e i r G r dr
a a r
e dr
EE
We have that
2
20
RF
14/03/2006 B. D'Auria - EURANDOM
Comparing the models
0,1 0,2A A
Model RF Model RL
0,20,2
R
A nY
b
10,2 0,2 0,2
0R F
A s ds X
0,1 0,1
R FA X
0,2
0,1A m
Na
0,1 0,2A A
1
0,2 0,2 0,2
0R
A s ds Y
0,1 0A
10,2
0
2
0,1
A s ds m
Na
10,2
0,20
2
R
A s ds n
Yb
14/03/2006 B. D'Auria - EURANDOM
Normal contribution for the RL Model
0,2 0,2 1
0,2 0,2 1
0 r
r
R R
R R
0,2
0,1A n
Nb
0,2
0,2A nY
b
n n n
b o b
RL
14/03/2006 B. D'Auria - EURANDOM
Dependence structure
Proposition.
where
0
0, 01
,2 111
, 1 1
A X
A Xm
a
A k k X k
1 11
0
0 0
12
0
0
2 , ;
.
r v
r
m vG v r dr F G r dr
a F rG r dr
E
E
0,1X i N , 1ov X i X j C
and
RF
14/03/2006 B. D'Auria - EURANDOM
A(0,δ ] and A(kδ,(k+1)δ ]
22
22
0
0,1
kA m
a
X N
R
R
1
0
0
1 0askm F G k r dr m E
RF
14/03/2006 B. D'Auria - EURANDOM
A(0,δ ] and A(kδ,(k+1)δ ]
0, , 1k
pA o a
R
RF
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A(iδ,(i+1)δ ] with 0 ≤i ≤k
22, 1 k pA i i m A m o a R
RF
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Dependence structure
Proposition.
where is infinite divisible with Lévy measure with density and
0
0, 1
,2 11
, 1 1
A Y
A Yn
b
A k k Y
10,2
0
1/ 1
2 ,
.R
n s ds n
b
Y
RL
1RRL x E
14/03/2006 B. D'Auria - EURANDOM
Question
The over-sampling asymptotically
implies perfect correlation.
What can we say about
correlation structure for finite δ?
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LRD for δ>0
Proposition.
For fixed δ>0, as k→∞,
0
1
0, , , 1
F
F
ov A A k k const G k
const k L k
C
RF
14/03/2006 B. D'Auria - EURANDOM
LRD for δ>0
22
21 12
12
11
0
0
0
0
0
1Var A k rG k r dr
const G k
Var A k Var A k
Var A k o G k
Var A k o G k
R
R R
R
R
RF
14/03/2006 B. D'Auria - EURANDOM
with
LRD for fixed t>0
Proposition.
For fixed t>0, as δ→0,
1
2
0,1,
A m N N
A t t m N Na
RF
1 2N N N
21 2 0,
0,
d
N N N t
N N t
and
2
0
1 t
t cG t
14/03/2006 B. D'Auria - EURANDOM
with
LRD for fixed t>0
Proposition.
For fixed t>0, as δ→0,
1
2
0,1
,
A n Y Y
A t t n Y Yb
RL
1 2Y Y Y
11 2 0
10
R
R
R
R
dx dx L L t x dx
dx L L t x dx
E
E
14/03/2006 B. D'Auria - EURANDOM
Some simulationsRF
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Simulation analysisRF
14/03/2006 B. D'Auria - EURANDOM
Conclusions The Model RF seems to well represent real data traces as
it show gaussianity in the limit.
The Model RL instead converges marginally to heavy-tailed infinite divisible limit. That implies difficulty in handling with correlation structure and LRD.