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10 th International Conference on Telecommunications ICT’2003. Multiscale Network Processes: Fractal and p-Adic analysis. Vladimir Zaborovsky, Technical University, Robotics Institute, Saint-Petersburg, Russia e-mail vlad@neva.ru February 2003 Tahiti. Content. Introduction

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multiscale network processes fractal and p adic analysis

10th International Conference on Telecommunications

ICT’2003

Multiscale Network Processes:Fractal and p-Adic analysis

Vladimir Zaborovsky, Technical University, Robotics Institute,Saint-Petersburg, Russiae-mail vlad@neva.ru

February 2003

Tahiti

content
Content
  • Introduction
  • Basic questions and experimental background
  • Fractional analysis
  • Wavelet decomposition
  • p-adic and constructive analysis   
  • Conclusion
  • Keywords:
    • packet traffic, long-range dependence, self-similarity, wavelet, p-adic analysis.
introduction
Introduction

computer network and

network processes

Appl n

Appl 1

Appl i

Appl 2

  • characteristics:
    • number of nodes and links
    • performance (bps and pps )
    • applications,
    • control protocols, etc.
  • feature:
    • fractal or 1/fa spectrum
    • heavy-tailed correlation structure
    • self similarity
    • etc.
spatial temporal features
Spatial-Temporal features:

spectral

components

trend

multiplicative

cascades

Packet traffic - discrete positive process with a singular internal structure.

basic aspects
Basic aspects
  • Common questions:a) metrics and dimension of state space;
    • b) statistical or dynamical approaches;
    • c) predictable or chaotic behaviors of congested periods.
  • Relationship between:
    • d) line bit speed and virtual line throughput
    • e) microscopic packet dynamics and heavy-tailed statistical distributions
    • f) fractal properties and QoS issues
experimental data flows in spectral and statistical domain
Experimental data flows in spectral and statistical domain

Spectral domain – 1/f process

“tail

behavior”

frequency

Second-order statistics domain

real

data

log{varRTT(m)}

<1

“tail

behavior”

=1

classical

normal

distribution

logm

correlation structure in power law scale time intervals
Correlation Structure in power law scale time intervals

T= 4ms = 22ms

T= 2ms = 21ms

T= 1ms = 20ms

  • what feature is important

T = 64 ms = 25 ms

T = 8 ms = 23 ms

T = 2 ms = 21 ms

  • which model
  • is “right”?

ICMP packets. Autocorrelation function of number of packets

aggregation periodT=pmL0 ;

p–2,3,5,…

m = 0,1,2,3

L0 = time scale

network environment and logical structure
Network environment and logical structure

protocol

application

Virtual channel:

macroscopic processes

(IP address, port)

virtual

grid

node 1

node n

Channel signal:

(MAC frame)

01001101

channel

structure

node 1

node n

Physical signal:

(signal and noise value levels)

physical

network

0

1

microscopic processes

models and features
Models and features

peer-to-peer

virtual connection

signal propagation

tn

t2

t1

ti

node n(1,t) node n(2,t) node n(x,t) … node n(m,t)

number of node

n(x,t) – number of packets, at node x, at time t

  • Fractal process and power low correlation decays:

1.1

1.2

R(k)~Ak–band

Basic equation (continuous time approximation):

1.3

number of packets that already exist in the node x

new comer packets

where

P(n(x;t)<n0) F(x,t)

n(x;t) – number of packets n(x; t) at node number x at the time moment t

spatial temporal microscopic process
Spatial-Temporal Microscopic Process

nodes:

Packet delay/drop processes in virtual channel.

a)

End-to-End model

(discrete time scale)

b)

Node-to-Node model

(real time scale)

c)

Jump model

(fractal time scale)

Common and Fine Structure of the packet traffic.

basic model of the packet dissipation
Basic model of the packet “dissipation”
  • Common packets loss condition: each packet can be lost, so

1.4

F(t) – distribution function

virtual channel

intermediate

node x

source

destination

node 1

node n

this packet

never come to

the destination node

“t”

Functional equation for scale invariant or

“stable” distribution function

simple f t approximation
Simple F(t) approximation

Take into account

expression for can be written as

1.5

  • Resume:
  • For the t>>1 density function f(t) has a scale-invariant property and power low decay like (1.1)
  • Virtual connection can be characterized by dynamics equation (1.3) and statistical (1.4) condition.
state space of the network process
State Space of the Network Process

X

Z

0

[Sec] fractal time scale or network signal time

propagation measure

virtual channel 2

virtual channel 1

virtual channel 4

virtual channel 3

Y

X

possible

packet loss

1/[ms] effective

bandwidth

measure

microscopic

dynamics

1/[ms] nominal channel bit rate measure (real number)

macroscopic

dynamics

one-to-one reflection

  • Features:
  • Space measure [1/sec  1/sec  sec] = [1/sec]
  • Fractal time scale
micro dynamics of packets network signal
Micro Dynamics of packets (network signal)

network signal

wavelet approximation

wavelet image:

Curve of Embedding Dimension: n=58

(fractal structure)

raw signal:

Curve of Embedding Dimension: n >> 1

(white nose)

RTT signal

fractal measure
Fractal measure

Generalized Fractal Dimension Dq

Multifractal Spectrum f()

Network signal (RTT signal) and its:

  • Resume:
  • Dynamics of network process has limited value (n=58) of embedded dimension parameters (or signal has internal structure).
  • Temporal fractality associated to p-adic time scale, where T=pmL0, L0 – time scale.
fractal model of network signal packet flow
Fractal Model of Network Signal (packet flow)

The fractional equation of packet flow: (spatial-temporal virtual channel)

4.1

where –fractional derivative of function n(x;t),– Gamma function,

n(x; t)– number of packets in node number x at time t;

–parameter of density function (1.5)

  • Why fractional derivative?
    • Operator - take into account a possible loss of the packets;
slide17

number of node

Equation (4.1) has solution

4.2

The dependence of packets number n(k,100)/n0 for different values of  parameter at the time moment t=100

spatial temporal co variation function
Spatial-temporal co-variation function

Initial conditions

n(0;t)=n0(t):

4.3

The time evolution of c(m,t)/n02

2 adic wavelet decomposition
2-Adic Wavelet Decomposition

а) network traffic

b) Wavelet coefficients and their maxima/minima lines

p adic analyze basic ideas
P-adic analyze: Basic ideas
  • p-adic numbers
        • (p is prime: 2,3,5,…)
  • can be regarded as a completion of the rational numbers using norm
        • |x|p = 0 if x = 0
        • |xy|p = |x|p  |y|p
        • |xy|p  max {|xp|, |yp|}  |x|p + |y|p
  • The distance function d(x,y)=|xy|p possesses a general property called ultrametricity
        • d(x,z)  max {d(x,y),d(y,z)}
  • p-Adic decomposition:
        • x and y belong to same class if the distance between x and y satisfies the condition
          • d(x,y) < D
    • Classes form a hierarchical tree.
p adic fractality
p-Adic Fractality

Basic feature:

  • p-adic norm for a sum of p-adic numbers cannot be larger than the maximum of the p-adic norm for the items
  • the canonical identificationmapping p-adics to real
  • i:th structural detail appears in finite region of the fractal structure is:

infinite as a real number

and has finite norm as a p-adic number

This norm – p-adic invariant of the fractal.

p adic field structure
P-adic field structure

cluster ,

where

{0} …p2Zp pZp Zp p-1Zp …Qp ,

The wavelet basis inL2(R+)

is 2-adic multiscale basis

p adic self similar feature of power low function
p-Adic Self-Similar Feature of Power Low Function
  • Power low functions as f(x)=xn are self-similar in p-adic sence:
    • the value of the function at interval (pk,pk+1) determines the function completely
    • function y=x2

p = 2

p = 3

p = 7

p = 11

constructive analysis hidden periods and spectrum
Constructive analysis: hidden periods and spectrum

virtual channel

Input

process

Output

process

RTT

PPS

PPS  differential characteristic

  • Experimental data:
    • RTT  spatial-temporal integral characteristic

t, sec

packets per second

Location:

minimax process decomposition
MiniMax Process Decomposition
  • Basic Idea:
  • Natural Basis of Signal is defined by Signal itself
  • Constructive Spectrum of the Signal consist of blocks with different numbers of minimax values

PPS

time scale

constructive components of the analyzing process
Constructive Components of the Analyzing Process

blocks sequence

analyzing process: packet-per-second curve

time

network process constructive spectrum
Network Process: Constructive Spectrum

Source RTT process

and its constructive components:

sec

number of “max” in each block

2 adic analysis of hidden period
2-Adic Analysis of hidden period:

Transitive curve: block length=4 to block length=8

RTT(t+1)

RTT(t)

Dynamic Reflection diagram RTT(t)/RTT(t+1)

quasi turbulence network structure
Quasi Turbulence Network Structure

Source signal:

number of

time interval

Filtered signal: block length=5

number of

time interval

detailed structure

conclusion
Conclusion
  • The features of processes in computer networks correspond to the multiscale chaotic dynamic systems process .
  • Fractional equations and wavelet decomposition can be used to describe network processes on physical and logical levels.
  • Concept of p-adic ultrametricity in computer network emerges as a possible renormalized distance measure between nodes of virtual channel .
  • Constructive analysis p-adic of network process allows correctly describe the multiscale traffic dynamic with limited numbers of parameters.