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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

Vladimir Zaborovsky, Technical University, Robotics Institute,Saint-Petersburg, Russiae-mail [email protected]

Ruslan Meylanov, Academic Research Center, Makhachkala, Russiae-mail [email protected]

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.

- 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 network Information 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

- 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

Generalized Fractal Dimension Dq

Multifractal Spectrum f()

Network Traffic: Fine Structure and General Features.

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.

4.2

Multiplicative transformation of input signal:

4.3

Analytical description of input signal:

Fractional differential equation

,where

Fractional Calculus formalism4.1

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

burst

b)

Dynamic Operator of Network Signalnetwork

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.

t

0

10

Simple Model: Fractal oscillator4.7

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.

Fig. 4.6.

Basic solutionThe 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).

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.

- Basic Idea:
- Natural Basis of the Signal
- Constructive Spectr of the Signal

Fig. 5.2.

Dynamic Reflection

Fig. 5.6.

Network Quasi Turbulence

Fig. 5.7.

Forecasting Procedure

Fig. 5.8.

Multilevel Forecasting Procedure

Fig. 5.9.

- 1 The 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.

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