Download 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, Russia
e-mail [email protected]
February 2003
Tahiti
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
Appl n
Appl 1
computer network and
network processes
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.
Appl i
Spatial-Temporal features:
spectral
components
trend
multiplicative
cascades
Packet traffic - discrete positive process with a singular
internal structure.
Basic aspects
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
Experimental data flows in spectral
and statistical domain
Spectral domain – 1/fa process
“tail
behavior”
frequency
Second-order statistics domain
log{varRTT(m)}
real
data
a<1
a=1
“tail
behavior”
classical
normal
distribution
logm
Correlation Structure in power law scale time
intervals
ICMP packets. Autocorrelation function of number of packets
aggregation period T=pmL0 ; p – 2,3,5,…
m = 0,1,2,3
L0 = time scale
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”?
Network environment and logical structure
protocol
application
NETWORK ENVIRONMENT
TCP
TCP
physical link bit
speed
buffer
packet
…
…
Packet drops direct virtual channel
node 0
…
…
node x+1
node x
node M
…
…
feedback virtual channel
Virtual channel:
macroscopic processes
(IP address, port)
virtual
grid
node 1
node n
Channel signal:
channel
structure
(MAC frame)
01001101
node 1
node n
Physical signal:
physical
network
microscopic processes
0
1
(signal and
noise value
levels)
Models and features
• Fractal process and power low correlation decays:
R(k)~Ak–b
1.1
~ b
R(mk )  Ak
and
1.2
peer-to-peer
virtual connection
signal propagation
t1
node n(1,t)
t2
node n(2,t)
tn
ti
node n(x,t)
…
node n(m,t)
number of node
n(x,t) – number of packets, at node x, at time t
Basic equation (continuous time approximation):
t
n ( x; t )   n ( x  1; t  ) f () d  n0 ( x) F( x; t ) ,
1.3
0
new comer packets
where
number 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
Spatial-Temporal Microscopic Process
nodes:
TCP
TCP
Link bit speed
Buffering
processing delay
…
…
Packet
drops
…
node 0
n(x;t)
node x
n(x+1;t)
node x+1
node M
Packet delay/drop processes in virtual channel.
Sender
Receiver
Sender
Receiver
i
Sender
Receiver
i+1
tB
tL
RTT
Packet
drop
a)
End-to-End model
(discrete time scale)
t
b)
Node-to-Node model
(real time scale)
t
Infinite
delay
c)
Jump model
(fractal time scale)
t
Common and Fine Structure of the packet traffic.
Basic model of the packet “dissipation”
• Common packets loss condition:
each packet can be lost, so

M{t}  t   t  f ( t )dt   .
d
f ( t )  F( t );
dt
1.4
0
F(t) – distribution function
virtual channel
source
node 1
intermediate
node x
“t”
destination
node n
this packet
never come to
the destination node
Functional equation for scale invariant or
“stable” distribution function
F( t )  F( t / c1)  F( t / c 2 )
c1a  c a2  1; 0  a  2.
Simple F(t) approximation
Take into account
f ( t )  0;

 f ( t )dt  1.
0
expression for
f (t) 
d
f ( t )  F( t )
dt

(1  t ) 1
can be written as
, 0   1
1.5
Resume:
1. For the t>>1 density function f(t) has a scaleinvariant property and power low decay like (1.1)
2. Virtual connection can be characterized by
dynamics equation (1.3) and statistical (1.4)
condition.
State Space of the Network Process
virtual channel 4
virtual channel 3
virtual channel 2
virtual channel 1
[Sec] fractal time scale or network signal time
propagation measure
macroscopic
dynamics
X
0 Z
X possible
packet loss
one-to-one reflection
Y
1/[ms]
effective
bandwidth
measure
microscopic
dynamics
1/[ms] nominal channel bit rate measure (real number)
Features:
• Space measure [1/sec  1/sec  sec] = [1/sec]
• Fractal time scale
Micro Dynamics of packets (network signal)
network signal
wavelet approximation
RTT signal
raw signal:
Curve of Embedding Dimension:
n >> 1
(white nose)
wavelet image:
Curve of Embedding Dimension:
n=58
(fractal structure)
Fractal measure
Network signal (RTT signal) and its:
Generalized Fractal Dimension Dq
Multifractal Spectrum f(a)
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)
The fractional equation of packet flow:
(spatial-temporal virtual channel)
(1 
where
 )Dt [n( x; t )]
n( x; t ) n0 ( x)

 
x
t
D t – fractional derivative
(1  ) – Gamma function,
n(x; t)

4.1
of function n(x;t),
– number of packets in node number x at time t;
– parameter of density function (1.5)
Why fractional derivative?
Operator D - take into account a possible loss
t
of the packets;
Equation (4.1) has solution

  2 (1   ) 1
(1   ) 1  
1

n(k; t )  n0   k 



 .

2



1
(   ) t

 (1  2 ) t
 
t

4.2
number of node
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
Initial conditions
n(0;t)=n0(t):
1
(1   ) 1 
2
1 2 
c(m; t )  n0(1   )t
 (2  2 )  m (2  3 )   .

The time evolution of c(m,t)/n02
t 
4.3
2-Adic Wavelet Decomposition
 2 j 1
x( t )  V0n    Wjn  jn ( t )
j 0 n 0
L packet rate
40
30
20
10
0
50
100
150
200
250
300
350
40 0
450
а) network traffic
b) Wavelet coefficients and their maxima/minima lines
сек
500
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
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
x
 xmp
m N
m
 Rp 
 xmp
m
R
mN
mapping 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

cluster
Z p   pn Z p
n 
,
Zp   (ai  p(ak  pZ p )...)
where
i, j,k
{0} …p2Zp  pZp  Zp  p-1Zp  …Qp ,
The wavelet basis in L2(R+)
( x )   
1 (x)   1  (x)
0, 2 
 2,1
is 2-adic multiscale basis
n 



2
2 ( 2 x
 n),   Z, n  Z
...  V 1  V 0  V1  V 2  ...
UjZ V j  L2 (R)
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
Input
process
virtual channel
RTT
Output
process
PPS
Experimental data:
RTT  spatial-temporal
integral characteristic
PPS  differential
characteristic
t, sec
packets per second
Location:
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
blocks sequence
analyzing process: packet-per-second curve
time
Network Process: Constructive Spectrum
Source RTT process
and its constructive components:
sec
number of “max” in each block
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
Source signal:
Filtered signal: block length=5
number of
time interval
number of
time interval
detailed structure
Multiscale Forecasting Algorithm:
application aspect
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.