Multiscale Network Processes: Fractal and p-Adic analysis Vladimir Zaborovsky, Technical University, Robotics Institute, Saint-Petersburg, Russia February 2003 Tahiti 10 th International Conference on Telecommunications ICT’2003

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.

computer network and network processes Appl 1 Appl 2 Appl n Appl i characteristics: number of nodes and links performance (bps and pps ) applications, control protocols, etc. feature: fractal or 1/f  spectrum heavy-tailed correlation structure self similarity etc. Introduction

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

1.Common questions: a)metrics and dimension of state space; b)statistical or dynamical approaches; c)predictable or chaotic behaviors of congested periods. 2.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 Basic aspects

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

Correlation Structure in power law scale time intervals ICMP packets. Autocorrelation function of number of packets aggregation period T=p m L 0 ; T = 64 ms = 2 5 ms T = 8 ms = 2 3 ms T = 2 ms = 2 1 ms - which model is “right”? T= 4ms = 2 2 ms T= 2ms = 2 1 ms T= 1ms = 2 0 ms - what feature is important p – 2,3,5,… m = 0,1,2,3 L 0 = time scale

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

R(k)~Ak –b and Fractal process and power low correlation decays: Basic equation (continuous time approximation): 1.3 Models and features peer-to-peer virtual connection 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 signal propagation t1t1 t2t2 titi tntn new comer packets number of packets that already exist in the node x P(n(x;t)<n 0 )  F(x,t) n(x;t) – number of packets n(x; t) at node number x at the time moment t where

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. Spatial-Temporal Microscopic Process nodes:

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

Take into account expression for can be written as 1.5 Resume: 1.For the t>>1 density function f(t) has a scale- invariant property and power low decay like (1.1) 2.Virtual connection can be characterized by dynamics equation (1.3) and statistical (1.4) condition. Simple F(t) approximation

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

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

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=p m L 0, L 0 – time scale. Generalized Fractal Dimension D q Multifractal Spectrum f(  ) Network signal (RTT signal) and its: Fractal measure

The fractional equation of packet flow: (spatial-temporal virtual channel) 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) 4.1 Fractal Model of Network Signal (packet flow) Why fractional derivative? Operator - take into account a possible loss of the packets;

The dependence of packets number n(k,100)/n 0 for different values of  parameter at the time moment t=100 Equation (4.1) has solution 4.2 number of node

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

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 |x  y| p = |x| p  |y| p |x  y| p  max {|x p |, |y p |}  |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.

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 identification mapping 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.

The wavelet basis in L 2 (R + ) is 2-adic multiscale basis P-adic field structure cluster, where {0}  …p 2 Z p  pZ p  Z p  p -1 Z p  …Q p,

p-Adic Self-Similar Feature of Power Low Function Power low functions as f(x)=x n are self-similar in p-adic sence: the value of the function at interval (p k,p k+1 ) determines the function completely function y=x 2 p = 2p = 3 p = 11p = 7

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

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 MiniMax Process Decomposition PPS time scale

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

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

Dynamic Reflection diagram RTT(t)/RTT(t+1) Transitive curve: block length=4 to block length=8 RTT(t) RTT(t+1) 2-Adic Analysis of hidden period:

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

Multiscale Forecasting Algorithm: application aspect

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. Conclusion