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NOBEL WP2 Meeting Activity A.2.2

NOBEL WP2 Meeting Activity A.2.2. Krzysztof Wajda, Piotr Żuraniewski, AGH – UST, Department of Telecommunications e-mail: wajda @ kt.agh.edu.pl, zuraniew@wms.mat.agh.edu .pl. WP2 - A2.2 : Subject - outline. Outline: Measurement, characterization and modelling

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NOBEL WP2 Meeting Activity A.2.2

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  1. NOBEL WP2 MeetingActivity A.2.2 Krzysztof Wajda, Piotr Żuraniewski, AGH – UST, Department of Telecommunications e-mail: wajda@kt.agh.edu.pl, zuraniew@wms.mat.agh.edu.pl

  2. WP2 - A2.2: Subject - outline Outline: • Measurement, characterization and modelling • Solutions to optimise utilisation of network resources • New methods for route management (intra- and inter-domain) • Develop accurate statistical models for evaluating the impact on QoS AGH University of Science and Technology

  3. WP2 - A2.2: Motivation for NOBEL AGH University of Science and Technology • Common approach to explain irregularity in measured data: external random noise influences a linear system. • Results ? May be very poor [1] • Traffic traces from ChinaSat, • SARIMA model used for traffic prediction, • normalized mean square error • for uploaded traffic: ~35%, • for downloaded traffic: ~60%

  4. WP2 - A2.2: Motivation for NOBEL AGH University of Science and Technology In some situations linear stochastic models (like ARMA – autoregressive moving average) are good approximation of existing systems… …but it may occur that we can take advantage by using non-linear deterministic description (like chaos theory)… …however it is dubious that in a real situation we will deal with the pure non-linear deterministic or pure linear stochastic system non-linearity chaos reality ? ARMA stochasticity

  5. WP2 - A2.2: Motivation for NOBEL AGH University of Science and Technology • Example of usage non-linear approach in teletraffic: discretedynamic, deterministic,non-linear, chaotic maps can model an ON/OFF source. • Before presenting a model we have to explain the above terms.

  6. WP2 - A2.2: Definitions: AGH University of Science and Technology • Q: What is a dynamic system ? • A: The one evolving in time.

  7. WP2 - A2. 2: Definitions AGH University of Science and Technology • If f and x0 are completely determined (no randomness) then a system is deterministic. • If f is not a linear function then a system is called non-linear • We can plot the evolution of our system or its trajectory – the sequence of x0, x1, x2,…

  8. WP2 - A2. 2: Definitions AGH University of Science and Technology • Deterministic chaos – irregular motion, resulting from non-linear, deterministic system.Note: this only an intuitive description, NOT a formal mathematical definition. • Not all non-linear deterministic systems are chaotic. • One of the features of chaotic system is its sensitivity to initial conditions: two trajectories starting very close to each other may diverge during their evolutionNote: two other are: topological transitivity and denseness of periodic points. Refer to [2] for details

  9. WP2 - A2.2: Model:general description AGH University of Science and Technology • ON/OFF source modelled via chaotic maps ([3])

  10. WP2 - A2.1: Model:general properties AGH University of Science and Technology • ON/OFF source modelled via chaotic maps ([3]) • We can choose different parameters or even different functions f1 and f2 in order to catch the differences between ON and OFF state characteristics. For example we may want to model a source with a heavy-taileddistribution of sojourn time in ON state and a light-taileddistribution of sojourn time in OFF state • If we need to model more than just two states (ON and OFF) we just take the required number of functions f’s

  11. WP2 - A2.n: Model:Intermittency AGH University of Science and Technology • Problem: how to choose the functions f1 and f2 in order to catch a variety of possible source characteristics (light/heavy tails etc.) ? • Possible solution: consider the intermittency maps Intermittency (by Schuster) –occurrence of a signal that alternates randomly between regular (laminar) phases and relatively short irregular bursts zoomed on next slide

  12. WP2 - A2.n: Subject – next slides:Intermittency map AGH University of Science and Technology Evolution of the intermittency map

  13. WP2 - A2.n: Subject – next slides:Intermittency – laminar phase AGH University of Science and Technology • Intermittent signal in laminar phase: trajectory passes through a narrow „channel”. • Many iterations are needed to leave this area and the values differ only slightly (note the plot range: [0.512 , 0.518]). • If for example a trajectory is in the ON region this corresponds to a „packet train” (many subsequent iterations in the ON region = many subsequent packets). g(x)=x

  14. WP2 - A2.2: Macroscopic view AGH University of Science and Technology • Q: What can we say about such system in a macro scale i.e., can we characterize a generated „traffic” ({yn} sequence)? • A: Relying on the statistics of the dynamics we can for example address the problems of: • mean rate of the traffic • distribution of sojourn times of ON and OFF periods • autocovariance function • We have to find out how frequent during system evolution the trajectory visits a given segment ]x,x+dx[. • It will result in obtaining a probability density called invariant density, denoted as ρ(x) • There are several methods to find ρ(x): analitycal, graphical, numerical • Example: mean rate of the traffic

  15. WP2 - A2.2: Function - example: AGH University of Science and Technology • Functions called Fixed Point Double Intermittency Maps (FPDI), described by Hu and Rudnick in 1982, were found useful in considered model due to their analytical tractability [4]. • Note the number of parameters: m1,m2(this can be reduced to single m if we assume some symmetry) and d.

  16. WP2 - A2. 2: Function - example: AGH University of Science and Technology • Note: for the simplicity we consider a symmetric case with m1=m2 = m • It can be proven that for Fixed Point Double Intermittency Maps (FPDI) the following, among others, hold • Sojourn time distribution (k – number of iterations) • P(K > k) ~ k-1/(m-1) (2>m>3/2 – heavy tail) • The output of the aggregates of FPDI maps will tend to self-similar traffic with H=(3m-4)/(2m-2), 3/2<m<2 • The invariant density

  17. WP2 - A2.2: Elastic traffic modelling: AGH University of Science and Technology • A special choice of f1 and f2as Fixed Point Double Intermittency Maps (FPDI), allows not only to mimic an open-loop source and catch a wide range of source characteristic – from geometric to heavy-tailed distributed ON/OFF times – but also take a network feedback under consideration (for example TCP congestion control mechanism)

  18. WP2 - A2.2: Elastic traffic modelling: AGH University of Science and Technology The simplest example: increasing TCP window – slow start • Recall: by f2(x) we denote 2nd iteration of function f i.e., f(f(x)) ; by fk(x) we mean k-th iteration. Every iteration corresponds to one RTT (round trip time) which is fixed • wn denotes sender’s window size, wmax is fixed and represents maximum allowed window size • No packet losses are assumed: window size only increases (more complicated cases were investigated by Erramilli et al. in [4])

  19. WP2 - A2.2: Conclusions AGH University of Science and Technology • Non-linear analysis methods (including chaos theory) can be successfully used in teletraffic modeling. • ON/OFF source can be modeled by the couple of chaotic maps. • The presented model is parsimonious and is able to catch a variety of source behaviors. • Special choice of functions it the model makes the analysis easier. • Analysis of multiplexed sources is possible • Elastic traffic can also be modeled. • Queuing theory for sources modeled by chaotic maps is available (also was not mentioned during a presentation) • Further investigations based on measured data files are planned

  20. WP2 - A2.2: references: AGH University of Science and Technology • [1] TRAJKOVIC L., “Modelling and Characterization of Traffic in Deployed Networks”, International Workshop Nonlinear Dynamics and Complexity in Information and Communication Technology, Bologna 2004 • [2] LASOTA A., MACKEY M.C., “Chaos, Fractals, and Noise – Stochastic Aspects of Dynamics,” second edition, Springer-Verlag, 1994. • [3] Erramilli A., Singh R.P. , and Pruthi P. , “Modeling Packet Traffic with Chaotic Maps,” Royal Institute of Technology, ISRN KTH/IT/R-94/18--SE,Stockholm-Kista, Sweden, Aug. 1994 • [4] ERRAMILLI A., ROUGHAN M.,VEITCH D.,WILLINGER W.,Self-Similar Traffic and Network Dynamics, Special Issue IEEE Proceedings: Applications of nonlinear dynamics to electronics and information engineering Vol 90 No 5 May 2002 pp. 800-815

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