Informational Network Traffic Model Based

On Fractional Calculus

and Constructive Analysis

Vladimir Zaborovsky, Technical University, Robotics Institute,

Saint-Petersburg, Russiae-mail vlad@neva.ru

Ruslan Meylanov, Academic Research Center,

Makhachkala, Russiae-mail lan_rus@dgu.ru

Content

1. Introduction2. Informational Network and Open Dynamic

System Concept3. Spatial-Temporal features of packet traffic

    3.1 statistical model    3.2 dynamic process

4. Fractional Calculus models    4.1 fractional calculus formalism    4.2 fractal equations     4.3 fractal oscillator

5. Experimental results and constructive analysis6. Conclusion

Keywords:packet traffic, long-range dependence, self-similarity, fractional calculus, fractional differential equations.

Introduction

1.1

• Packet traffic in Information network has the correlation function decays like (fractal features):

R(k)~Ak–b,

where k = 0, 1, 2, . . ., is a discrete time variable; b - scale parameter

• QoS engineering for Internet Information services requires adequate models of each spatial-temporal virtual connection; the most probable number of packets n(x; t) at site х at the moment t given by the expression

where n0(x) is the number of packets at site х before the packet's arrival from site х-1.

• The possible packets loss can be count up by distribution function f(t) in the following condition

So, the corresponding expression for

the f(t) can be written as

, )t(F)x(nd)(f)t;1x(n)t;x(nt

00

, dt)t(ftt0

.1dt)t(f ;0)t(f0

10 ,)t1(

)t(f 1

Computer network as an Open System

Features: •        Dissipation•        Selforganization•        Selfsimularity •        Multiplicative perturbations•        Bifurcation

Telecommunication network Information network

Dynamic Feature

xi y

y= xi

1 2 N

N

ij1i

ijN

1ii

Topological Feature

Point-to-point logical structure

Multi connectedlogical structure

Process Features In Informational Network

•   Integral character of data flowparameters – bandwidth, number of users ...

•   Differential character of connectionparameters – number of packets, delay, buffer

•   Scale invariantness of statistical characteristics

• Fractalness of dynamics process

State space of network process

C(kT) = g(k) C(T)

(t) ~ t

[Sec] astronomical time

[ms] nominal bandwidth

( FLAT CHANNEL)

[ms] effective bandwidth

Goals of the Model

• state forecast• throghtput estimation• loss minimizing• QoS control

Model needs to provide:

Uniting micro and macro descriptions of control object

t0

– min packet discovering timet0 – relaxation time

Spatial-Temporal Features of Traffic

Fig. 3.1. 2 RTT signal – blue and its wavelet filtering image.

Fig. 3.2. Curve of Embedding Dimension: n=6

Fig. 3.3. Curve of Embedding Dimension: n >> 1

Network Traffic: Fine Structure and General Features

.

Generalized Fractal Dimension Dq Multifractal Spectrum f()

Signal: RTT process

Statistical Description

Fig. 3.5. RTT Distribution Function: Ping Signals with intervals T=1 ms green, T=2 ms red, T=5 ms blue

Main Feature: Long-Range Dependence

Characteristics - Distribution Function

Parameter - Period of Test Signal (ping procedure)

Correlation Structure of Packet Flow

Fig. 3.6. Autocorrelation functions: upper RTT Ping SignalsAbscissa – numbers of the packets

Main Feature: Power Low of Statistical Moments

Input signal: ICMP packets

Analysing Structure: Autocorrelation function of number of packets

Correlation Structure of Time Series

Fig 3.7. Autocorrelation function for ping signal T=5 ms, T=10 ms, T=50 ms Abscissa –time between packets

Input: ICMP packets

Analysing Structure: Autocorrelation function of time interval

between packets

Traffic as a Spatial-Temporal Dynamic Process in IP network

Fig 3.8. Packet delay/drop processes in flat channel.

a)End-to-End model

b)Node-to-Node

model

c)Jump model

Fig 3.9. Fine Structure Packet transfer.

The equation of packet migration

The equation of packet migration in a spatial-temporal channel can be presented as

where the left part of equation with an exponent is the fractional derivative of function n(x; t) – number of packets in node number x at time t

For the initial conditions: n0(0) = n0 and n0(k) = 0, k = 1,2, …,. we finally obtain

t

)x(nx

)t;x(n)]t;x(n[D)1( 0t

.t

1)()1(

t1

)21()1(k

t1n)t;k(n

or

,t)(

1)1(kt)21()1(k

t1n)t;k(n

122

0

122

0

The dependence n(k,100)/n0 is shown graphically in Fig.3.10.

Fig.3.10.

Spatial-temporal co-variation function

The co-variation function for the obtained solution for the initial conditions n(0;t)=n0(t):

.t1

)32()1(m)22(

1t)1(n

t1

)131()1(m

t1

)121(1)1(n

)t;m(c

2120

131220

The evolution of c(m,t)/n02 with time t is shown in Fig. 3.11

Fig. 3.11.

Fractional Calculus formalism

)A(EA)(f ,1

define new class of parametric signals

E, - Mittag-Leffler function, - key parameter or order of fractional equation

Fig 4.1. Transmission process f(t) in n-nodes (routers with fractal parameter).

,...1,0n ,10 ,~n

d)()x()(1)t(u

t

a1

4.1

4.4

4.5

0 1 d)(f)(u))(f(L 4.2

Virtual channel operator:

...)(u...)(u)(f1nn10

n

4.3

Multiplicative transformation of input signal:

Analytical description of input signal:

Fractional differential equation

,where

0)(Afd

)(fd

A)(f 10 ,0

Dynamic Operator of Network Signal

)At()t(u))t(f(L n

0ii

n

1II

,

1n

Total transformation of signal in n nodes: model with time and space parameters

a)

b)

4.6

Fig. 4.3.

networksignal

f(t)input process

u(t)output process

Fig. 4.2.Input parameters: , A

network parameters: , n

where E, - Mittag-Leffler function,

input process

output process

burst

delay

burstdissemination

Simple Model: Fractal oscillator

0)t(xdt

)t(xd

4.7

where, 1<2, - frequency, t - time.

Common solution

tEtBtEtA)t(x ,1

1,2 4.8

where A and B – constants

Example =2 z/)zsin()z(E),zcos()z(E 22,221,2

)tsin(B)tcos(Atx

)sin(i)cos(tt

Fig. 4.4.

X(t)

21

11

dt)t(Xd

Fig. 4.5.

t

100

X(t)1 where =1.5

2 where =1.95

Basic solution

The common solution: input ,A,B, output F(t)

tEtBtEtA)t(F ,1

1,2 4.9

Identification formula: input F(t), output F

FFF

FFFF

F tEtBtEtA)t(F ,1

1,2

Modeling example

4.10

2/3ktcos1t

where , 0, +<1, k - whole number then

Fig. 4.6.

X(t)

11

dt)t(Xd

k=4 , =0, = 0,95 and t(0,6).

Phase Plane

Fig. 4.7.

X(t)

11

dt)t(Xd

Fig. 4.8.

X(t)

t

1

2

60

k=4 , =0, = 0,75 and t(0,6).

Model with Biffurcation

2/3))t(xcos(kcos1t

If

Then

Fig. 4.9а

X(t)

11

dt)t(Xd

Fig. 4.9b

X(t)

11

dt)t(Xd

Fig. 4.9c

X(t)

t

12

7

Parameters Identification Model(Detailed chaos)

Identification process formulas

))t/t((E)t/t(C/)t/t(C 0,1000

4.11

а)

b)

c)

d)

Fig. 4.10.

C(t)/C(0)

(0)(t)

(1)(t)

(2)(t)

Experimental results and constructive analysis

Fig. 5.1.

RTTInput

processOutputprocess

PPS

delay: RTT integral characteristic

traffic:PPS differential characteristic

MiniMax Description

Fig. 5.2.

Basic Idea:

• Natural Basis of the Signal

• Constructive Spectr of the Signal

Fig. 5.4.

Constructive Components of the Source Process

blocks sequence

source process

time

Constructive Analysis of RTT Process

Fig. 5.5.

RTT process

sec

number of “max” in each block

Dynamic Reflection

Fig. 5.6.

Network Quasi Turbulence

Fig. 5.7.

Forecasting Procedure

Fig. 5.8.

Multilevel Forecasting Procedure

Fig. 5.9.

Conclusion

1 The features of processes in computer networks correspond to the open dynamic systems process.

2 Fractional equations are the adequate description of micro and macro network process levels.

3 Using of constructive analysis together with identification procedures based on fractional calculus formalism allows correctly described the traffic dynamic in information network or Internet with minimum numbers of parameters.