Peer-to-peer fractal models:a new approach to describe multiscale network process

Peer-to-peer fractal models:a new approach to describe multiscale network process

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

June 2002

Content

1. Introduction

2. Basic questions

3. Spatial-Temporal features and network

ultrametrics

4. Fractional Calculus models and constructive

analysis

5. Forecasting procedure   

6. Conclusion

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

Introduction

Subject of research:

computer network and

network processes

Appl 1

Appl 2

Appl n

Appl i

characteristics:• number of nodes and links• link bit speed (bps – bit per second) and virtual channel capacity (pps – packet per second)• applications, protocols, etc.

feature:• fractal behavior• 1/f spectrum• heavy-tailed correlation structure• self similarity• etc.

Basic questions

1. Computer or packet-switch telecommunication network,what does it means from:

theoretical – metrics or ultrametrics spacetime surfacepragmatics – statistical/dynamical or discrete (synchronous)/quantum (asynchronous)application – predictable or chaotic behaviors

Point of view2. Peer-to-peer model of network process,

what are relationship between:- line bit speed and packets throughput

capacity- subjective/packet and

microscopic/physical signal- notion of packets jump in virtual channel

and network fractal properties (1/f – noise, heavy-tailed statistics of the link propagation delay, etc.)

Correlation Structure of Packet Flow

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

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

Internal structure of network virtual channel

virtualgrid

logicaldomain

physicalnetwork

(IP address, port)

node nnode 1

Virtual path:

node nnode 1

Physical channel:

01001101

(MAC frame)

Digital signal:(signal and noise value levels)1

0

• Fractal-like process has power low correlation decays:

R(k)~Ak–b,

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

00

1.1

as a concequence scale-invariant feature

bb kA~

)mk(A~)mk(R 1.2

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

• Packet flow in each virtual peer-to-peer channel at each time and nodes

Basic idea: •The most probable number of packets n(x; t) at node number x at the time moment t given by the simple spatial-temporal integral expression

where n0(x) is the number of packets at site x before the packet's

arrival from site x-1; F(t) – distribution function;

density of distribution function f(t) )t(F

dtd

1.3

Models and features

peer-to-peervirtual connection

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

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

signal propagation

t1 t2 titn

RTT – propagation delay{ti} – set of packets delay

new comer packets number of packets that already exist in the node x

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)

Fine Structure Packet transfer.

Traffic as a Spatial-Temporal Dynamic Process

• The possible packets loss in virtual connection or event when packet never leaves intermediate node can be count up by the following condition

. dt)t(ftt0

f(t) – density of distribution function.

1.4

source node x destination

node 1 node n“t”

10 ,)t1(

)t(f1

Take into account common requirements

the corresponding expression for the such f(t) can be written as

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

1.5

Resume:

1. For the t>>1 density function f(t) has a scale-invariant

property and power low decay like (1.1)

2. Virtual connection in the packet switched network is a

spatial-temporal object which internal features can be

characterized by dynamics (1.3) and statistical (1.4)

equations.

Channel logical structure

virtual orlogical structure

(Internet)

point-to-point physical channel(modem connection)

group of point-to-point channels(telephone network)

Corresponding topology

separate point

two connected pointsin metrics spaced(i,j) ≤ d(i,k) + d(k,j)d – real number

three structure in ultrametrics space

Network Topology Formalism and Channel Structure

source destination

source destination

00 01 10 11

d(i,j) ≤ max {d(i,k),d(k,j)}

•   Common parametersbandwidth, propagation delay, trough put ...

•   Differential parameters number of packets, delay, buffers capacity

•   Scale invariantness or fractal like integral characteristics

• Fractalness of network dynamics and dissipation

C(pkT) =pkC(T)

t ~(t)

parameters – scale function pk, power low of C(T) function

• Measure of space dimension [1/sec 1/sec sec] = [1/sec]• Fractal time – not any time moments have equal influence to the state of process

3D spacetime: network virtual processes

(2D or FLAT CHANNEL)

[Sec] fractal time scale or network signal time propagation measure

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

1/[ms] effective bandwidthmeasure

X

virt

ua

l ch

an

ne

l 1

virt

ua

l ch

an

ne

l 3

virt

ua

l ch

an

ne

l 4

packet loss

virt

ua

l ch

an

ne

l 2

Y

X0

Z

Network Process Characteristics and State Space

RTT signal (blue) and its wavelet filtering image (black).

RTT signal: Curve of Embedding Dimension:

n >> 1 (white nose)

network signal

filtering image

Filtering image: Curve of Embedding Dimension:

n=58(fractal structure)

Internal Dynamics packets flow (network signal)

Resume:• Internal dynamics of network process can be characterized by interval (n=58) of embedding dimension parameter.

• Fractal traffic feature can be characterized by Dq parameter.

• Fine scaling structure can be characterized by multifractal spectrum f() parameter.

Generalized Fractal Dimension Dq Multifractal Spectrum f()

Network signal (RTT signal) and its:

Fine Structure and Fractal Features of Network Signal

The fractional equation of packet flow: in the spatial-temporal channel

Left part of the equation is the 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)

tD

)1(

t

)x(n

x

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

t4.1

Resume:

• Operator - take into account possible loss of the packets;

• For the initial conditions:

n0(0) = n0 and n0(k) = 0, k = 1,2, …,

tD

The fractal model of network signal (packet flow)

The dependence of packets number n(k,100)/n0 for different

values of parameter at the time moment t=100

Equation (4.1) has solution

.t

1

)(

)1(

t

1

)21(

)1(k

t

1n)t;k(n

12

2

0

4.2

number of node

The co-variation function for the (4.2) solution for the initial conditions

n(0;t)=n0(t):

.t

1

)32(

)1(m

)22(

1t)1(n)t;m(c 212

0

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

4.3

Spatial-temporal co-variation function

Features:

• Each node in virtual network is a router with i fractal parameter;

• In each node packet loss has a non zero probability.

Transformation model of input signal f(t) in peer-to-peer channel.

...)(u...)(u)(fn1n10

n

can be used to characterize a multiplicative virtual channel operator:

Analytical expression for output signal

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

n

t

a

1 d)(f)x()(

1)t(u

Fractional Calculus formalism and virtual channel model

Source Destination

f(t)

x

a

x

a

x

a

n - intermediate node

Virtual channel

model (signal time scale)

u1(t) u2(t)u(t)

Buffer 1 Buffer 2 Buffer n

This equation define new class of parametric signals

E, - Mittag-Leffler function,

- order of fractional equation (fractal channel measure)

Fractional differential equation of one physical peer-to-peer channel

0)(Afd

)(fd

10 0, ,A)(fdt

dD where t

)A(EA)(f ,1

networkprocess

f(t)input signal

u(t)output signal

Input-output fractal network modelInput parameters: , A

network parameters: , n

Common Description of peer-to-peer network process

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

0ii

n

1II

,

1

n

Total transformation of network signal in n nodes of virtual channel: model with time () (real number)

and space(h) parameters (real number)

Network process in fractal network environment.

where E, - Mittag-Leffler function,

a)

b)

input process

output process

burst

delay

burstdissemination

Dynamic Operator of peer-to-peer channel

Delay and burst dissemination

Identification formula

))t/t((E)t/t(C/)t/t(C 0,1

000

а)

b)

c)

d)

Fine structure of chaos signal

C(t)/C(0)

(0)(t)

(1)(t)

(2)(t)

Identification process

)t()t()t(C/)t(C )2()1()0(0

Real RTT process

1-st iteration ordetalized level

2-ed iteration

3-ed iteration

Identification Process: Iterative estimation of fractal parameters

Inputprocess

Outputprocess

PPS

virtual channel

RTT

Experimental data:• delay: round trip time processRTT spatial-temporal integral characteristic of virtual channel

• traffic: packets-per-second processPPS differential characteristic of virtual channel

Location:

packets per second

t, sec

Constructive spectral analysis

MiniMax signal dependence and p-adic fractal structure.

Basic Idea of constructive analysis approach:

• Natural Basis of the Signal is define by Signal itself

• Constructive Specter of the Signal is based on natural Basis and consist of blocks with different numbers of minimax values

MiniMax Process Description

PPS

p-adic time scale

blocks sequence

analyzing process: packet-per-second curve

time

Constructive Components of the Analyzing Process

Constructive p-adic time scale specter has 1/f Fourier transform countrpart

Source RTT process

and its constructive components:

sec

number of “max” in each block

Network Process: Constructive Specter Analysis

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

RTT delay process:Transitive curve: block length=4 to block length=6

RTT(t)

RTT(t+1)

Dynamic Analysis of hidden period: Reflection diagram of Transitive Points

Hidden periods fine or detailed structure

Source signal:

Filtered signal: block length=5number of

time interval

number of time interval

detailed structure

Network Traffic and its Quasi Turbulence Structure

Forecasting procedure based on block length=3 curve

block length=3curve

forecasting value

Forecast RTT values

Forecasting Procedure: Constructive algorithm

Multilevel Forecasting Algorithm

1 The features of peer-to-peer processes in computer networks correspond to the chaotic dynamic systems process and can be described by equations in fractional derivatives.

2 Fractional equations formalism is the adequate description of network processes on physical and logical levels.

3 Concept of ultrametricity in computer network emerges as a possible renormalized distance measure between nodes of virtual channel that means absence of intermediates on the corresponding network level and feasible way to find common description of multiscale network process.

4 Using of constructive analysis of network process allows correctly described the traffic dynamic in model with minimum numbers of parameters.

Conclusion