1. Informational Network Traffic Model Based On Fractional Calculus and Constructive Analysis Vladimir Zaborovsky, Technical University, Robotics Institute, Saint-Petersburg, Russia e-mail vlad@neva.ru Ruslan Meylanov, Academic Research Center, Makhachkala, Russia e-mail lan_rus@dgu.ru

2. Content 1.Introduction 2.Informational Network and Open Dynamic System Concept 3.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 analysis 6.Conclusion Keywords: packet traffic, long-range dependence, self-similarity, fractional calculus, fractional differential equations.

3. 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 n 0 (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

4. Computer network as an Open System Features: Dissipation Selforganization Selfsimularity Multiplicative perturbations Bifurcation Telecommunication networkInformation network Dynamic Feature  xixi y y=  x i  1  2  N  Topological Feature Point-to-point logical structure Multi connected logical structure

5. 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 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

6. Goals of the Model state forecast throghtput estimation loss minimizing QoS control Model needs to provide: Uniting micro and macro descriptions of control object  t0t0  – min packet discovering time t 0 – relaxation time

7. 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

8. Network Traffic: Fine Structure and General Features. Generalized Fractal Dimension D q Multifractal Spectrum f(  ) Signal: RTT process

9. 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)

10. Correlation Structure of Packet Flow Fig. 3.6. Autocorrelation functions: upper RTT Ping Signals Abscissa – numbers of the packets Main Feature: Power Low of Statistical Moments Input signal: ICMP packets Analysing Structure: Autocorrelation function of number of packets

11. 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

12. 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.

13. 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: n 0 (0) = n 0 and n 0 (k) = 0, k = 1,2, …,. we finally obtain The dependence n(k,100)/n 0 is shown graphically in Fig.3.10. Fig.3.10.

14. Spatial-temporal co-variation function The co-variation function for the obtained solution for the initial conditions n(0;t)=n 0  (t): The evolution of c(m,t)/n 0 2 with time t is shown in Fig. 3.11 Fig. 3.11.

15. Fractional Calculus formalism 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). 4.1 4.4 4.5 4.2 Virtual channel operator: 4.3 Multiplicative transformation of input signal: Analytical description of input signal: Fractional differential equation,where

16. Dynamic Operator of Network Signal Total transformation of signal in n nodes: model with time and space parameters a) b) 4.6 Fig. 4.3. network signal 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 burst dissemination

17. Simple Model: Fractal oscillator 4.7 where, 1<  2,  - frequency, t - time. Common solution 4.8 where A and B – constants Example  =2 Fig. 4.4. X(t) 2 1 Fig. 4.5. t 10  0 X(t) 1 where  =1.5 2 where  =1.95

18. Basic solution The common solution: input ,A,B, output F(t) 4.9 Identification formula: input F(t), output  F Modeling example 4.10 where ,  0,  +  <1, k - whole number then Fig. 4.6. X(t) k=4,  =0,  = 0,95 and t  (0,6  ).

19. Phase Plane Fig. 4.7. X(t) Fig. 4.8. X(t) t 1 2 66 0 k=4,  =0,  = 0,75 and t  (0,6  ).

20. Model with Biffurcation If Then Fig. 4.9а X(t) Fig. 4.9b X(t) Fig. 4.9c X(t) t 1 2 77 

21. Parameters Identification Model (Detailed chaos) Identification process formulas 4.11 а) b)b) c)c) d)d) Fig. 4.10. C(t)/C(0)  (0) (t)  (1) (t)  (2) (t)

22. Experimental results and constructive analysis Fig. 5.1. RTT Input process Output process PPS delay: RTT  integral characteristic traffic: PPS  differential characteristic

23. MiniMax Description Fig. 5.2. Basic Idea: Natural Basis of the Signal Constructive Spectr of the Signal

24. Fig. 5.4. Constructive Components of the Source Process blocks sequence source process time

25. Constructive Analysis of RTT Process Fig. 5.5. RTT process sec number of “max” in each block

26. Dynamic Reflection Fig. 5.6.

27. Network Quasi Turbulence Fig. 5.7.

28. Forecasting Procedure Fig. 5.8.

29. Multilevel Forecasting Procedure Fig. 5.9.

30. Conclusion 1The features of processes in computer networks correspond to the open dynamic systems process. 2Fractional equations are the adequate description of micro and macro network process levels. 3Using 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.