Download Informational Network Traffic Model Based On Fractinal Calculs

Informational Network
Traffic Model Based
On Fractional
Calculus
and
Constructive Analysis
Vladimir Zaborovsky,
Technical University, Robotics Institute,
Saint-Petersburg, Russia
e-mail [email protected]
Ruslan Meylanov,
Academic Research Center,
Makhachkala, Russia
e-mail [email protected]
Content
1.
2.
5.
Introduction
Informational Network and Open Dynamic
System Concept
Spatial-Temporal features of packet traffic
3.1 statistical model
3.2 dynamic process
Fractional Calculus models
4.1 fractional calculus formalism
4.2 fractal equations
4.3 fractal oscillator
Experimental results and constructive analysis
6.
Conclusion
3.
4.
Keywords:
packet traffic, long-range dependence, self-similarity,
fractional calculus, fractional differential equations.
Introduction
•
Packet traffic in Information network has the correlation
function decays like (fractal features):
R(k)~Ak–b,
1.1
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
t
n(x; t)   n(x  1; t  )f()d  n0(x)F(t) ,
0
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
 t 

 t  f(t)dt  
,
0
f(t)  0;

 f(t)dt  1. So, the corresponding expression for
0
the f(t) can
be written as f(t) 

(1  t)
 1
,
0  1
Computer network as an Open System
Features:
•
•
•
•
•
Dissipation
Selforganization
Selfsimularity
Multiplicative perturbations
Bifurcation
Telecommunication network
Information network
Dynamic Feature
1

xi

2
y

y= xi
N
N
N
i1
i1
j i
 i   ij
Topological Feature
Point-to-point
logical structure
Multi connected
logical structure
Process Features In Informational Network
• 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
C(kT) = g(k) C(T)
• Fractalness of dynamics process
(t) ~ t
State space of network process
[Sec] astronomical time
[ms] effective bandwidth
[ms] nominal bandwidth
(
FLAT CHANNEL)
Goals of the Model
• state forecast
• throghtput estimation
• loss minimizing
• QoS control
Model needs to provide:
Uniting micro and macro descriptions of control object

 – min packet discovering time
t0 – relaxation time
t
0
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
Network Traffic: Fine Structure and General Features
Signal: RTT process
Generalized Fractal Dimension Dq
Multifractal Spectrum f()
.
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 Signals
Abscissa – 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
TCP
TCP
Link transmission
delay
Buffering
processing delay
…
…
Packet
drops
node 0
…
node x
node x+1
node M
Fig 3.8. Packet delay/drop processes in flat channel.
Sender
Receiver
Sender
Receiver
i
Sender
Receiver
i+1
tB
tL
RTT
Packet
drop
t
a)
End-to-End model
t
b)
Node-to-Node
model
Fig 3.9. Fine Structure Packet transfer.
Infinite
delay
t
c)
Jump model
The equation of packet migration
The equation of packet migration in a spatial-temporal channel
can be presented as
(1  )Dt [n(x; t)]  
n(x; t) n0(x)

x
t
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
 1

2(1  )
1
n(k; t)  n0   k
 k(1  ) 
,
 t 
(1  2)t2
( )t   1 
or
 1
 2(1  ) 1
(1  )
1  
n(k; t)  n0   k 



 .

2
 1 

(
1

2

)

(


)

 
t
t
 t
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):
c(m; t) 

1
1
(1  )
1 
 n2

(
1


)


m



0
2 1
3 1 

(
1

2


1
)

(
1

3


1
)
t
t


1
(1  ) 1 
12 
 n2

(
1


)
t

m
 .

0

(
2

2

)

(
2

3

)
t 

The evolution of c(m,t)/n02 with time t is shown in Fig. 3.11
Fig. 3.11.
Fractional Calculus formalism
Input
Output
x
f(t)

a
x
x


a
a
n
Virtual channel
1 t
u(t) 
(x  ) 1 ()d 4.1

() a
~
  n   , 0    1, n  0,1,...
Fig 4.1. Transmission process f(t) in n-nodes (routers with  fractal parameter).
Virtual channel operator:
L(f())  u() 


f()
0    
1
d
4.2
Multiplicative transformation of input signal:
0
1
n
n  1
f()  u()  ...  un ()  ...
4.3
Analytical description of input signal:
Fractional differential equation
df()

d
 Af()  0
4.4
f(  )  A ,where   0, 0    1
define new class of parametric signals
f()  A  1E, (A  )
E, - Mittag-Leffler function,
4.5
 - key parameter or order of fractional equation
Dynamic Operator of Network Signal
network
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
n
L(f(t))  un(t)  
 1  I
I1

n
,   i
i 0
where E, - Mittag-Leffler function,
burst
input process
a)
delay
output process
burst
dissemination
b)
Fig. 4.3.
(At )
4.6
Simple Model: Fractal oscillator
d x(t)
dt

  x(t)  0
4.7
where, 1<2,  - frequency, t - time.
Common solution



x(t)  At   2 E,  1  t   Bt  1E,   t 

where A and B – constants
Example =2
E2,1(z2 )  cos(z), E2,2 (z2 )  sin(z) / z
xt   A cos( t)  B sin( t)
 t 
 t  cos()  i sin()
d 1X(t)
dt 1
X(t)
1
2
Fig. 4.4.
X(t)
1 where =1.5
2 where =1.95
t
0
10
Fig. 4.5.
4.8
Basic solution
The common solution: input ,A,B, output F(t)



F(t)  At  2 E, 1  t   Bt  1E,  t 
Identification formula: input F(t), output F




F(t)  At F  2 E F , F 1  t F  Bt F 1 E F , F  t F
Modeling example
t   1      cos kt       3 / 2
where , 0, +<1, k - whole number then
d 1X(t)
dt 1
X(t)
Fig. 4.6.
k=4 , =0,  = 0,95 and t(0,6).
4.9

4.10
Phase Plane
d 1X(t)
dt 1
X(t)
Fig. 4.7.
k=4 , =0,  = 0,75 and t(0,6).
X(t)
1
2
t
0
6
Fig. 4.8.
Model with Biffurcation
If
t   1      cos k cos( x(t))      3 / 2
Then
d 1X(t)
dt 1
X(t)
Fig. 4.9а
d 1X(t)
dt 1
X(t)
Fig. 4.9b
X(t)
1
2
t

7
Fig. 4.9c
Parameters Identification Model
(Detailed chaos)
Identification process formulas
C(t / t 0 ) / C0  (t / t 0 ) 1E, ((t / t 0 ) )
а)
C(t)/C(0)
b)
(0)(t)
c)
(1)(t)
d)
(2)(t)
Fig. 4.10.
4.11
Experimental results and constructive analysis
delay:
RTT  integral characteristic
RTT
Input
process
Output
process
PPS
Fig. 5.1.
traffic:
PPS  differential characteristic
MiniMax Description
Basic Idea:
• Natural Basis of the Signal
• Constructive Spectr of the Signal
Fig. 5.2.
Constructive Components of the Source Process
blocks sequence
source process
time
Fig. 5.4.
Constructive Analysis of RTT Process
RTT process
sec
number of “max” in each block
Fig. 5.5.
Dynamic Reflection
Fig. 5.6.
Network Quasi Turbulence
Fig. 5.7.
Forecasting Procedure
Fig. 5.8.
Multilevel Forecasting Procedure
Fig. 5.9.
Conclusion
1
The features of processes in computer
networks correspond to the open dynamic
systems process.
2
Fractional equations are the adequate
description of micro and macro network
process levels.
3
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.