informational network traffic model based on fractional calculus and constructive analysis
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Informational Network Traffic Model Based
On Fractional Calculus
and Constructive Analysis
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
Content
1. Introduction2. Informational Network and Open Dynamic
System Concept3. Spatial-Temporal features of packet traffic
3.1 statistical model 3.2 dynamic process
4. Fractional Calculus models 4.1 fractional calculus formalism 4.2 fractal equations 4.3 fractal oscillator
5. Experimental results and constructive analysis6. Conclusion
Keywords:packet traffic, long-range dependence, self-similarity, fractional calculus, fractional differential equations.
Introduction
1.1
• Packet traffic in Information network has the correlation function decays like (fractal features):
R(k)~Ak–b,
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
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
So, the corresponding expression for
the f(t) can be written as
, )t(F)x(nd)(f)t;1x(n)t;x(nt
00
, dt)t(ftt0
.1dt)t(f ;0)t(f0
10 ,)t1(
)t(f 1
Computer network as an Open System
Features: • Dissipation• Selforganization• Selfsimularity • Multiplicative perturbations• Bifurcation
Telecommunication network Information network
Dynamic Feature
xi y
y= xi
1 2 N
N
ij1i
ijN
1ii
Topological Feature
Point-to-point logical structure
Multi connectedlogical structure
Process Features In Informational Network
• Integral character of data flowparameters – bandwidth, number of users ...
• Differential character of connectionparameters – number of packets, delay, buffer
• Scale invariantness of statistical characteristics
• Fractalness of dynamics process
State space of network process
C(kT) = g(k) C(T)
(t) ~ t
[Sec] astronomical time
[ms] nominal bandwidth
( FLAT CHANNEL)
[ms] effective bandwidth
Goals of the Model
• state forecast• throghtput estimation• loss minimizing• QoS control
Model needs to provide:
Uniting micro and macro descriptions of control object
t0
– min packet discovering timet0 – relaxation time
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
.
Generalized Fractal Dimension Dq Multifractal Spectrum f()
Signal: RTT process
Statistical Description
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
Characteristics - Distribution Function
Parameter - Period of Test Signal (ping procedure)
Correlation Structure of Packet Flow
Fig. 3.6. 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
Fig 3.7. 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
Traffic as a Spatial-Temporal Dynamic Process in IP network
Fig 3.8. Packet delay/drop processes in flat channel.
a)End-to-End model
b)Node-to-Node
model
c)Jump model
Fig 3.9. Fine Structure Packet transfer.
The equation of packet migration
The equation of packet migration in a spatial-temporal channel can be presented as
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
t
)x(nx
)t;x(n)]t;x(n[D)1( 0t
.t
1)()1(
t1
)21()1(k
t1n)t;k(n
or
,t)(
1)1(kt)21()1(k
t1n)t;k(n
122
0
122
0
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):
.t1
)32()1(m)22(
1t)1(n
t1
)131()1(m
t1
)121(1)1(n
)t;m(c
2120
131220
The evolution of c(m,t)/n02 with time t is shown in Fig. 3.11
Fig. 3.11.
Fractional Calculus formalism
)A(EA)(f ,1
define new class of parametric signals
E, - Mittag-Leffler function, - key parameter or order of fractional equation
Fig 4.1. Transmission process f(t) in n-nodes (routers with fractal parameter).
,...1,0n ,10 ,~n
d)()x()(1)t(u
t
a1
4.1
4.4
4.5
0 1 d)(f)(u))(f(L 4.2
Virtual channel operator:
...)(u...)(u)(f1nn10
n
4.3
Multiplicative transformation of input signal:
Analytical description of input signal:
Fractional differential equation
,where
0)(Afd
)(fd
A)(f 10 ,0
Dynamic Operator of Network Signal
)At()t(u))t(f(L n
0ii
n
1II
,
1n
Total transformation of signal in n nodes: model with time and space parameters
a)
b)
4.6
Fig. 4.3.
networksignal
f(t)input process
u(t)output process
Fig. 4.2.Input parameters: , A
network parameters: , n
where E, - Mittag-Leffler function,
input process
output process
burst
delay
burstdissemination
Simple Model: Fractal oscillator
0)t(xdt
)t(xd
4.7
where, 1<2, - frequency, t - time.
Common solution
tEtBtEtA)t(x ,1
1,2 4.8
where A and B – constants
Example =2 z/)zsin()z(E),zcos()z(E 22,221,2
)tsin(B)tcos(Atx
)sin(i)cos(tt
Fig. 4.4.
X(t)
21
11
dt)t(Xd
Fig. 4.5.
t
100
X(t)1 where =1.5
2 where =1.95
Basic solution
The common solution: input ,A,B, output F(t)
tEtBtEtA)t(F ,1
1,2 4.9
Identification formula: input F(t), output F
FFF
FFFF
F tEtBtEtA)t(F ,1
1,2
Modeling example
4.10
2/3ktcos1t
where , 0, +<1, k - whole number then
Fig. 4.6.
X(t)
11
dt)t(Xd
k=4 , =0, = 0,95 and t(0,6).
Phase Plane
Fig. 4.7.
X(t)
11
dt)t(Xd
Fig. 4.8.
X(t)
t
1
2
60
k=4 , =0, = 0,75 and t(0,6).
Model with Biffurcation
2/3))t(xcos(kcos1t
If
Then
Fig. 4.9а
X(t)
11
dt)t(Xd
Fig. 4.9b
X(t)
11
dt)t(Xd
Fig. 4.9c
X(t)
t
12
7
Parameters Identification Model(Detailed chaos)
Identification process formulas
))t/t((E)t/t(C/)t/t(C 0,1000
4.11
а)
b)
c)
d)
Fig. 4.10.
C(t)/C(0)
(0)(t)
(1)(t)
(2)(t)
Experimental results and constructive analysis
Fig. 5.1.
RTTInput
processOutputprocess
PPS
delay: RTT integral characteristic
traffic:PPS differential characteristic
MiniMax Description
Fig. 5.2.
Basic Idea:
• Natural Basis of the Signal
• Constructive Spectr of the Signal
Fig. 5.4.
Constructive Components of the Source Process
blocks sequence
source process
time
Constructive Analysis of RTT Process
Fig. 5.5.
RTT process
sec
number of “max” in each block
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.
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