Small scale analysis of data traffic models B. D’Auria - Eurandom joint work with S. Resnick - Cornell University

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


Telecommunication data traffic

• 1st column

Ethernet Traffic• 2nd column

Poisson Model

Taqqu et al., (1997)


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


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)


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


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.


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.


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


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


α-traffic and β-traffic

from Sarvotham et al. (2001)


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


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


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


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


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


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


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




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




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




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


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


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


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


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


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


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


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


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


Proof

0,1R

0,2R


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


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


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


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


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


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


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


A(0,δ ] and A(kδ,(k+1)δ ]

0, , 1k

pA o a

R

RF


A(iδ,(i+1)δ ] with 0 ≤i ≤k

22, 1 k pA i i m A m o a R

RF


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


Question

The over-sampling asymptotically

implies perfect correlation.

What can we say about

correlation structure for finite δ?


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


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


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


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


Some simulationsRF


Simulation analysisRF


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.

Documents

Small scale analysis of data traffic models B. D’Auria - Eurandom joint work with S. Resnick - Cornell University