Multiscale Network Processes: Fractal and p-Adic analysis

1. 10th International Conference on Telecommunications ICT’2003 Multiscale Network Processes:Fractal and p-Adic analysis Vladimir Zaborovsky, Technical University, Robotics Institute,Saint-Petersburg, Russiae-mail vlad@neva.ru February 2003 Tahiti

2. 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.

3. Introduction computer network and network processes Appl n Appl 1 Appl i 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.

4. Spatial-Temporal features: spectral components trend multiplicative cascades Packet traffic - discrete positive process with a singular internal structure.

5. Basic aspects • Common questions:a) metrics and dimension of state space; • b) statistical or dynamical approaches; • c) predictable or chaotic behaviors of congested periods. • 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

6. Experimental data flows in spectral and statistical domain Spectral domain – 1/f process “tail behavior” frequency Second-order statistics domain real data log{varRTT(m)} <1 “tail behavior” =1 classical normal distribution logm

7. Correlation Structure in power law scale time intervals 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”? ICMP packets. Autocorrelation function of number of packets aggregation periodT=pmL0 ; p–2,3,5,… m = 0,1,2,3 L0 = time scale

8. Network environment and logical structure protocol application Virtual channel: macroscopic processes (IP address, port) virtual grid node 1 node n Channel signal: (MAC frame) 01001101 channel structure node 1 node n Physical signal: (signal and noise value levels) physical network 0 1 microscopic processes

9. Models and features peer-to-peer virtual connection signal propagation tn t2 t1 ti node n(1,t) node n(2,t) node n(x,t) … node n(m,t) number of node n(x,t) – number of packets, at node x, at time t • Fractal process and power low correlation decays: 1.1 1.2 R(k)~Ak–band Basic equation (continuous time approximation): 1.3 number of packets that already exist in the node x new comer packets where 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

10. Spatial-Temporal Microscopic Process nodes: Packet delay/drop processes in virtual channel. a) End-to-End model (discrete time scale) b) Node-to-Node model (real time scale) c) Jump model (fractal time scale) Common and Fine Structure of the packet traffic.

11. Basic model of the packet “dissipation” • Common packets loss condition: each packet can be lost, so 1.4 F(t) – distribution function virtual channel intermediate node x source destination node 1 node n this packet never come to the destination node “t” Functional equation for scale invariant or “stable” distribution function

12. Simple F(t) approximation Take into account expression for can be written as 1.5 • Resume: • For the t>>1 density function f(t) has a scale-invariant property and power low decay like (1.1) • Virtual connection can be characterized by dynamics equation (1.3) and statistical (1.4) condition.

13. State Space of the Network Process X Z 0 [Sec] fractal time scale or network signal time propagation measure virtual channel 2 virtual channel 1 virtual channel 4 virtual channel 3 Y X possible packet loss 1/[ms] effective bandwidth measure microscopic dynamics 1/[ms] nominal channel bit rate measure (real number) macroscopic dynamics one-to-one reflection • Features: • Space measure [1/sec  1/sec  sec] = [1/sec] • Fractal time scale

14. Micro Dynamics of packets (network signal) network signal wavelet approximation wavelet image: Curve of Embedding Dimension: n=58 (fractal structure) raw signal: Curve of Embedding Dimension: n >> 1 (white nose) RTT signal

15. Fractal measure Generalized Fractal Dimension Dq Multifractal Spectrum f() Network signal (RTT signal) and its: • 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.

16. Fractal Model of Network Signal (packet flow) The fractional equation of packet flow: (spatial-temporal virtual channel) 4.1 where –fractional derivative of function n(x;t),– Gamma function, n(x; t)– number of packets in node number x at time t; –parameter of density function (1.5) • Why fractional derivative? • Operator - take into account a possible loss of the packets;

17. number of node Equation (4.1) has solution 4.2 The dependence of packets number n(k,100)/n0 for different values of  parameter at the time moment t=100

18. Spatial-temporal co-variation function Initial conditions n(0;t)=n0(t): 4.3 The time evolution of c(m,t)/n02

19. 2-Adic Wavelet Decomposition а) network traffic b) Wavelet coefficients and their maxima/minima lines

20. 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.

21. 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 identificationmapping 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.

22. P-adic field structure cluster , where {0} …p2Zp pZp Zp p-1Zp …Qp , The wavelet basis inL2(R+) is 2-adic multiscale basis

23. 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

24. Constructive analysis: hidden periods and spectrum virtual channel Input process Output process RTT PPS PPS  differential characteristic • Experimental data: • RTT  spatial-temporal integral characteristic t, sec packets per second Location:

25. 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

26. Constructive Components of the Analyzing Process blocks sequence analyzing process: packet-per-second curve time

27. Network Process: Constructive Spectrum Source RTT process and its constructive components: sec number of “max” in each block

28. 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)

29. Quasi Turbulence Network Structure Source signal: number of time interval Filtered signal: block length=5 number of time interval detailed structure

30. Multiscale Forecasting Algorithm: application aspect

31. 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.