# Informational Network Traffic Model Based On Fractional Calculus and Constructive Analysis - PowerPoint PPT Presentation

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Informational Network Traffic Model Based On Fractional Calculus and Constructive Analysis. Vladimir Zaborovsky, Technical University, Robotics Institute, Saint-Petersburg, Russia e-mail vlad@neva.ru Ruslan Meylanov, Academic Research Center, Makhachkala, Russia e-mail lan_rus@dgu.ru.

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Informational Network Traffic Model Based On Fractional Calculus and Constructive Analysis

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## Informational Network Traffic Model Based On Fractional Calculus and Constructive Analysis

Ruslan Meylanov, Academic Research Center, Makhachkala, Russiae-mail lan_rus@dgu.ru

### Content

1.Introduction

2.Informational Network and Open Dynamic System Concept

3.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 analysis

6.Conclusion

• Keywords:

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

Introduction

• 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

1.1

Computer network as an Open System

• Features:

•         Dissipation

•         Selforganization

•         Selfsimularity

•         Multiplicative perturbations

•         Bifurcation

Telecommunication networkInformation network

Dynamic Feature

1 2 N 

xi

y

y= xi

Topological Feature

Point-to-point

logical structure

Multi connected

logical structure

Process Features In Informational Network

[Sec] astronomical time

[ms] effective bandwidth

[ms] nominal bandwidth

( FLAT CHANNEL)

• Integral character of data flow

• parameters – bandwidth, number of users ...

• Differential character of connection

• parameters – 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

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 time

t0 – 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

Signal: RTT process

Generalized Fractal Dimension Dq

Multifractal Spectrum f()

.

### Statistical Description

Characteristics - Distribution Function

Parameter - Period of Test Signal (ping procedure)

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

### Correlation Structure of Packet Flow

Input signal: ICMP packets

Analysing Structure: Autocorrelation function of number of packets

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

Main Feature: Power Low of Statistical Moments

### Correlation Structure of Time Series

Input: ICMP packets

Analysing Structure: Autocorrelation function of time interval

between packets

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

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):

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

Fig. 3.11.

Virtual channel operator:

4.2

Multiplicative transformation of input signal:

4.3

Analytical description of input signal:

Fractional differential equation

,where

### Fractional Calculus formalism

4.1

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

4.4

define new class of parametric signals

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

4.5

a)

burst

b)

### Dynamic Operator of Network Signal

network

signal

f(t)

input process

u(t)

output process

Fig. 4.2.

Input parameters: , A

network parameters: , n

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

4.6

where E, - Mittag-Leffler function,

input process

delay

output process

burst

dissemination

Fig. 4.3.

X(t)

t

0

10

### Simple Model: Fractal oscillator

4.7

where, 1<2,  - frequency, t -time.

Common solution

4.8

where AandB – constants

Example=2

X(t)

1

2

Fig. 4.4.

1 where =1.5

2 where =1.95

Fig. 4.5.

X(t)

Fig. 4.6.

### Basic solution

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

4.9

Identification formula: input F(t), output F

4.10

Modeling example

where , 0, +<1, k - whole number then

k=4 , =0, = 0,95 and t(0,6).

X(t)

1

2

t

0

6

Fig. 4.8.

X(t)

Fig. 4.7.

### Phase Plane

k=4 , =0, = 0,75 and t(0,6).

X(t)

Fig. 4.9b

X(t)

1

2

t

7

Fig. 4.9c

If

Then

X(t)

Fig. 4.9а

а)

b)

c)

d)

### Parameters Identification Model(Detailed chaos)

Identification process formulas

4.11

C(t)/C(0)

(0)(t)

(1)(t)

(2)(t)

Fig. 4.10.

Experimental results and constructive analysis

PPS

delay:

RTT  integral characteristic

traffic:

PPS  differential characteristic

RTT

Input

process

Output

process

Fig. 5.1.

MiniMax Description

• Basic Idea:

• Natural Basis of the Signal

• Constructive Spectr of the Signal

Fig. 5.2.

blocks sequence

Constructive Components of the Source Process

source process

time

Fig. 5.4.

### Constructive Analysis of RTT Process

RTT process

sec

number of “max” in each block

Fig. 5.5.

Fig. 5.6.

Fig. 5.7.

Fig. 5.8.

### Multilevel Forecasting Procedure

Fig. 5.9.

Conclusion

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

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

• 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.