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## Multiscale Network Processes: Fractal and p-Adic analysis

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**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**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.**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.**Spatial-Temporal features:**spectral components trend multiplicative cascades Packet traffic - discrete positive process with a singular internal structure.**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**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**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**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**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**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.**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**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.**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**Micro Dynamics of packets (network signal)**network signal wavelet approximation wavelet image: Curve of Embedding Dimension: n=58 (fractal structure) raw signal: Curve of Embedding Dimension: n >> 1 (white nose) RTT signal**Fractal measure**Generalized Fractal Dimension Dq Multifractal Spectrum f() Network signal (RTT signal) and its: • Resume: • Dynamics of network process has limited value (n=58) of embedded dimension parameters (or signal has internal structure). • Temporal fractality associated to p-adic time scale, where T=pmL0, L0 – time scale.**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;**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**Spatial-temporal co-variation function**Initial conditions n(0;t)=n0(t): 4.3 The time evolution of c(m,t)/n02**2-Adic Wavelet Decomposition**а) network traffic b) Wavelet coefficients and their maxima/minima lines**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 • |xy|p = |x|p |y|p • |xy|p max {|xp|, |yp|} |x|p + |y|p • The distance function d(x,y)=|xy|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.**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.**P-adic field structure**cluster , where {0} …p2Zp pZp Zp p-1Zp …Qp , The wavelet basis inL2(R+) is 2-adic multiscale basis**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**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:**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**Constructive Components of the Analyzing Process**blocks sequence analyzing process: packet-per-second curve time**Network Process: Constructive Spectrum**Source RTT process and its constructive components: sec number of “max” in each block**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)**Quasi Turbulence Network Structure**Source signal: number of time interval Filtered signal: block length=5 number of time interval detailed structure**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.