Informational Network Traffic Model Based On Fractional Calculus and Constructive Analysis

1. 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

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 • 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 1.1

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

5. Process Features In Informational Network [Sec] astronomical time [ms] effective bandwidth [ms] nominal bandwidth ( FLAT CHANNEL) • 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

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  t0 • – min packet discovering time t0 – 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. Signal: RTT process Generalized Fractal Dimension Dq Multifractal Spectrum f() Network Traffic: Fine Structure and General Features .

9. 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

10. 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 SignalsAbscissa – numbers of the packets Main Feature: Power Low of Statistical Moments

11. 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

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: n0(0) = n0 and n0(k) = 0, k = 1,2, …,. we finally obtain The dependence n(k,100)/n0 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)=n0(t): The evolution of c(m,t)/n02 with time t is shown in Fig. 3.11 Fig. 3.11.

15. Virtual channel operator: 4.2 Multiplicative transformation of input signal: 4.3 Analytical description of input signal: Fractional differential equation ,where Fractional Calculus formalism 4.1 Fig 4.1. Transmission process f(t) in n-nodes (routers with  fractal parameter). 4.4 define new class of parametric signals E, - Mittag-Leffler function,  - key parameter or order of fractional equation 4.5

16. a) burst b) 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 4.6 where E, - Mittag-Leffler function, input process delay output process burst dissemination Fig. 4.3.

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

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

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

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

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

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

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

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

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

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