Post on 05-Jan-2016
Multiscale Network Processes:
Fractal and p-Adic analysis
Vladimir Zaborovsky, Technical University, Robotics Institute,
Saint-Petersburg, Russiae-mail vlad@neva.ru
February 2003Tahiti
10th International Conference on TelecommunicationsICT’2003
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.
computer network and
network processes
Appl 1
Appl 2
Appl n
Appl i
characteristics:• number of nodes and links• performance (bps and pps )• applications, • control protocols, etc.
feature:• fractal or 1/f spectrum• heavy-tailed correlation structure• self similarity• etc.
Introduction
Packet traffic - discrete positive process with a singular internal structure.
trend
multiplicativecascades
Spatial-Temporal features:
spectralcomponents
1. Common questions:
a) metrics and dimension of state
space;
b) statistical or dynamical
approaches;
c) predictable or chaotic behaviors of
congested periods.
2. 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
Basic aspects
Experimental data flows in spectral and statistical domain
“tailbehavior”
“tailbehavior”
realdata
classicalnormaldistribution
=1
Spectral domain – 1/f process
Second-order statistics domainlog{varRTT(m)}
logm
frequency
<1
Correlation Structure in power law scale time intervals
ICMP packets. Autocorrelation function of number of packets
aggregation period T=pmL0 ;
T = 64 ms = 25 ms
T = 8 ms = 23 ms
T = 2 ms = 21 ms
- which modelis “right”?
T= 4ms = 22msT= 2ms = 21msT= 1ms = 20ms
- what feature is important
p – 2,3,5,…
m = 0,1,2,3
L0 = time scale
virtualgrid
channelstructure
physicalnetwork
(IP address, port)
node nnode 1
Virtual channel:
node nnode 1
Channel signal:
01001101
(MAC frame)
Physical signal:(signal and noise value levels)1
0
Network environment and logical structure
protocol application
macroscopic processes
microscopic processes
R(k)~Ak–b and
• Fractal process and power low correlation decays:
, )t;x(F )x(nd )(f )t; 1x( n)t;x( nt
00
1.1 bkA~)mk(R 1.2
Basic equation (continuous time approximation):
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
new comer packetsnumber of packets that already exist in the node x
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
where
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.
Spatial-Temporal Microscopic Process
nodes:
• Common packets loss condition: each packet can be lost, so
. dt)t(ftt}t{M0
1.4
sourceintermediate
node x destination
node 1 node n“t”
Basic model of the packet “dissipation”
);t(Fdt
d)t(f F(t) – distribution function
virtual channel
Functional equation for scale invariant or “stable” distribution function
2.0 ; 1cc
)c/t(F)c/t(F)t(F
21
21
this packet never come to the destination node
10 ,)t1(
)t(f1
Take into account
expression for 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 can be characterized by
dynamics equation (1.3) and statistical (1.4)
condition.
Simple F(t) approximation
)t(Fdt
d)t(f
Features:• Space measure [1/sec 1/sec sec] = [1/sec]• Fractal time scale
microscopic dynamics
[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
ual c
hann
el 1
virt
ual c
hann
el 3
virt
ual c
hann
el 4
possiblepacket loss
virt
ual c
hann
el 2
Y
X0 Z
State Space of the Network Process
one-to-one reflection
macroscopic dynamics
RTT signal
raw signal: Curve of Embedding Dimension:
n >> 1 (white nose)
network signal
wavelet approximation
wavelet image: Curve of Embedding Dimension:
n=58(fractal structure)
Micro Dynamics of packets (network signal)
Resume:• Dynamics of network process has limited value (n=58) of embedded dimension parameters (or signal has internal structure).
• Temporal fractality associated to p-adic time scale, where T=pmL0, L0 – time scale.
Generalized Fractal Dimension Dq Multifractal Spectrum f()
Network signal (RTT signal) and its:
Fractal measure
The fractional equation of packet flow: (spatial-temporal virtual channel)
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)
tD
)1(
t
)x(n
x
)t;x(n)]t;x(n[D)1( 0
t4.1
Fractal Model of Network Signal (packet flow)
Why fractional derivative?
Operator - take into account a possible loss
of the packets;
tD
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
Initial conditions
n(0;t)=n0(t):
.t
1)32(
)1(m
)22(1
t)1(n)t;m(c 2120
The time evolution of c(m,t)/n02
4.3
Spatial-temporal co-variation function
2-Adic Wavelet Decomposition
0j
2
0njnjnn0
1j
)t(WV)t(x
а) network traffic
b) Wavelet coefficients and their maxima/minima lines
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
|xy|p = |x|p |y|p
|xy|p max {|xp| , |yp|} |x|p + |y|p
The distance function d(x,y)=|xy|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
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 identification
mapping p-adics to real
• i:th structural detail appears in finite region of the fractal structure is:infinite as a real numberand has finite norm as a p-adic number
This norm – p-adic invariant of the fractal.
Nm
mm
Nmp
mm RpxRpxx
The wavelet basis in L2(R+)
is 2-adic multiscale basis
P-adic field structure
)x()x()x(1,
21
21
,0
pn
np ZpZ
)...)pZa(pa(Z pkik,j,i
p
cluster ,
where
{0} …p2Zp pZp Zp p-1Zp …Qp ,
Zn Z, ),nx2(2 2n
)R(LVU
...VVVV...
2jZj
2101
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 = 11p = 7
Inputprocess
Outputprocess
PPS
virtual channel
RTT
Experimental data:RTT spatial-temporal integral characteristic
Location:
packets per second
t, sec
Constructive analysis: hidden periods and spectrum
PPS differential characteristic
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
MiniMax Process Decomposition
PPS
time scale
blocks sequence
analyzing process: packet-per-second curve
time
Constructive Components of the Analyzing Process
Source RTT process
and its constructive components:
sec
number of “max” in each block
Network Process: Constructive Spectrum
Dynamic Reflection diagram RTT(t)/RTT(t+1)
Transitive curve: block length=4 to block length=8
RTT(t)
RTT(t+1)
2-Adic Analysis of hidden period:
Source signal:
Filtered signal: block length=5
number of time interval
number of time interval
detailed structure
Quasi Turbulence Network Structure
Multiscale Forecasting Algorithm: application aspect
• 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.
Conclusion