Peer-to-peer fractal models: a new approach to describe multiscale network process Vladimir Zaborovsky, Technical University, Robotics Institute, Saint-Petersburg, Russia Ruslan Meylanov, Academic Research Center, Makhachkala, Russia June 2002

Content 1.Introduction 2.Basic questions 3.Spatial-Temporal features and network ultrametrics 4.Fractional Calculus models and constructive analysis 5.Forecasting procedure 6.Conclusion Keywords: packet traffic, long-range dependence, self-similarity, fractional calculus, fractional differential equations.

Introduction Subject of research: computer network and network processes Appl 1 Appl 2 Appl n Appl i characteristics: number of nodes and links link bit speed (bps – bit per second) and virtual channel capacity (pps – packet per second) applications, protocols, etc. feature: fractal behavior 1/f  spectrum heavy-tailed correlation structure self similarity etc.

Basic questions 1.Computer or packet-switch telecommunication network, what does it means from: theoretical – metrics or ultrametrics spacetime surface pragmatics – statistical/dynamical or discrete (synchronous)/ quantum (asynchronous) application – predictable or chaotic behaviors Point of view 2.Peer-to-peer model of network process, what are relationship between: -line bit speed and packets throughput capacity -subjective/packet and microscopic/physical signal -notion of packets jump in virtual channel and network fractal properties (1/f – noise, heavy-tailed statistics of the link propagation delay, etc.)

Correlation Structure of Packet Flow 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

Correlation Structure of Time Series 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

Internal structure of network virtual channel virtual grid logical domain physical network (IP address, port) node nnode 1 Virtual path: node nnode 1 Physical channel: (MAC frame) Digital signal: (signal and noise value levels) 1 0

Fractal-like process has power low correlation decays: R(k)~Ak –b, 1.1 as a concequence scale-invariant feature 1.2 where k = 0, 1, 2,..., is a discrete time variable; A - scale parameter, b – fractal parameter. Packet flow in each virtual peer-to-peer channel at each time and nodes Basic idea: The most probable number of packets n(x; t) at node number x at the time moment t given by the simple spatial-temporal integral expression where n 0 (x) is the number of packets at site x before the packet's arrival from site x-1; F(t) – distribution function; density of distribution function f(t) 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 RTT – propagation delay {t i } – set of packets delay new comer packetsnumber of packets that already exist in the node x

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) Fine Structure Packet transfer. Traffic as a Spatial-Temporal Dynamic Process

The possible packets loss in virtual connection or event when packet never leaves intermediate node can be count up by the following condition f(t) – density of distribution function. 1.4 source node x destination node 1 node n “t”

Take into account common requirements the corresponding expression for the such f(t) 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 in the packet switched network is a spatial-temporal object which internal features can be characterized by dynamics (1.3) and statistical (1.4) equations.

Channel logical structure virtual or logical structure (Internet) point-to-point physical channel (modem connection) group of point-to-point channels (telephone network) Corresponding topology separate point two connected points in metrics space d(i,j) ≤ d(i,k) + d(k,j) d – real number three structure in ultrametrics space Network Topology Formalism and Channel Structure sourcedestination source destination d(i,j) ≤ max {d(i,k),d(k,j)}

Common parameters bandwidth, propagation delay, trough put... Differential parameters number of packets, delay, buffers capacity Scale invariantness or fractal like integral characteristics Fractalness of network dynamics and dissipation C(p k T) =p k C(T) parameters – scale function p k, power low of C(T) function Measure of space dimension [1/sec  1/sec  sec] = [1/sec] Fractal time – not any time moments have equal influence to the state of process 3D spacetime: network virtual processes (2D or FLAT CHANNEL) [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 packet loss virtual channel 2 Y X 0 Z Network Process Characteristics and State Space

RTT signal (blue) and its wavelet filtering image (black). RTT signal: Curve of Embedding Dimension: n >> 1 (white nose) network signal filtering image Filtering image: Curve of Embedding Dimension: n=5  8 (fractal structure) Internal Dynamics packets flow (network signal)

Resume: Internal dynamics of network process can be characterized by interval (n=5  8) of embedding dimension parameter. Fractal traffic feature can be characterized by D q parameter. Fine scaling structure can be characterized by multifractal spectrum f(  ) parameter. Generalized Fractal Dimension D q Multifractal Spectrum f(  ) Network signal (RTT signal) and its: Fine Structure and Fractal Features of Network Signal

The fractional equation of packet flow: in the spatial-temporal channel Left part of the equation is the 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 Resume: Operator - take into account possible loss of the packets; For the initial conditions: n0(0) = n0 and n0(k) = 0, k = 1,2, …, The fractal model of network signal (packet flow)

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

The co-variation function for the (4.2) solution for the initial conditions n(0;t)=n 0  (t): The time evolution of c(m,t)/n Spatial-temporal co-variation function

Features: Each node in virtual network is a router with  i fractal parameter; In each node packet loss has a non zero probability. Transformation model of input signal f(t) in peer-to-peer channel. can be used to characterize a multiplicative virtual channel operator: Analytical expression for output signal Fractional Calculus formalism and virtual channel model Source Destination f(t)  x a  x a  x a n - intermediate node Virtual channel model (signal time scale) u 1 (t) u 2 (t) u(t) Buffer 1Buffer 2Buffer n

This equation define new class of parametric signals E ,  - Mittag-Leffler function,  - order of fractional equation (fractal channel measure) Fractional differential equation of one physical peer-to-peer channel network process f(t) input signal u(t) output signal Input-output fractal network model Input parameters: , A network parameters: , n Common Description of peer-to-peer network process

Total transformation of network signal in n nodes of virtual channel: model with time (  ) (real number) and space(h) parameters (real number) Network process in fractal network environment. where E ,  - Mittag-Leffler function, a) b) input process output process burst delay burst dissemination Dynamic Operator of peer-to-peer channel Delay and burst dissemination

Identification formula а) b)b) c)c) d)d) Fine structure of chaos signal C(t)/C(0)  (0) (t)  (1) (t)  (2) (t) Identification process Real RTT process 1-st iteration or detalized level 2-ed iteration 3-ed iteration Identification Process: Iterative estimation of fractal parameters

Input process Output process PPS virtual channel RTT Experimental data: delay: round trip time process RTT  spatial-temporal integral characteristic of virtual channel traffic: packets-per-second process PPS  differential characteristic of virtual channel Location: packets per second t, sec Constructive spectral analysis

MiniMax signal dependence and p-adic fractal structure. Basic Idea of constructive analysis approach: Natural Basis of the Signal is define by Signal itself Constructive Specter of the Signal is based on natural Basis and consist of blocks with different numbers of minimax values MiniMax Process Description PPS p-adic time scale

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

Constructive p-adic time scale specter has 1/f  Fourier transform countrpart Source RTT process and its constructive components: sec number of “max” in each block Network Process: Constructive Specter Analysis

Dynamic Reflection diagram RTT(t)/RTT(t+1) RTT delay process: Transitive curve: block length=4 to block length=6 RTT(t) RTT(t+1) Dynamic Analysis of hidden period: Reflection diagram of Transitive Points

Hidden periods fine or detailed structure Source signal: Filtered signal: block length=5 number of time interval number of time interval detailed structure Network Traffic and its Quasi Turbulence Structure

Forecasting procedure based on block length=3 curve block length=3 curve forecasting value Forecast RTT values Forecasting Procedure: Constructive algorithm

Multilevel Forecasting Algorithm

1The features of peer-to-peer processes in computer networks correspond to the chaotic dynamic systems process and can be described by equations in fractional derivatives. 2Fractional equations formalism is the adequate description of network processes on physical and logical levels. 3Concept of ultrametricity in computer network emerges as a possible renormalized distance measure between nodes of virtual channel that means absence of intermediates on the corresponding network level and feasible way to find common description of multiscale network process. 4Using of constructive analysis of network process allows correctly described the traffic dynamic in model with minimum numbers of parameters. Conclusion