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Modeling and SimulatingModeling and SimulatingNetworking Systems Networking Systems

with with Markov ProcessesMarkov Processes

Tools and Methods of Wide Tools and Methods of Wide Applicability ?Applicability ?

Jean-Yves Le BoudecJean-Yves Le BoudecEPFL/I&C/ISC-LCA-2EPFL/I&C/ISC-LCA-2

jean-yves.leboudec@epfl.chjean-yves.leboudec@epfl.ch

ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE

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Examples of Research in my Group (I&C/ISC/LCA2) Understanding simulation of mobility models

Theoretical understanding of the model explains simulation artifacts Involves Palm calculus and Harris chains

J.-Y. Le Boudec and M. Vojnovic, Perfect Simulation and Stationarity of a Class of Mobility Models, IEEE INFOCOM 2005; tools available at http://ica1www.epfl.ch/RandomTrip

Evaluate best design for ultra-wide band communication

R. Merz, J.-Y. Le Boudec and S. Vijayakumaran “Effect on Network Performance of Common versus Private Acquisition Sequences for Impulse Radio UWB Networks” IEEE International Conference on Ultra-Wideband (ICUWB 2006), 2006

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Methods for Performance Evaluation

Communication systems require modelling in the design phase for validation / tuning

Simulation (discrete event)Most often usedBut does not apply to the large scale

AnalysisOften very hard to use / obtain proven results / re-usableSometimes too late

Fast simulation is also often an alternativeBased on hybrid of analytical results and detailed simulation

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We Need Methods / Tools for The Domain Expert

Domain experts cannot spend a PhD on learning one method

We need theories of general applicability Like e.g. product form queuing network / max-plus algebra

We need methods that can be implemented in a mechanical way / in tools

An exploration track:What can the maths of natural sciences provide us with ?Methods for large markov processes

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Example of Large Scale Model

[ELS-2006] A. El Fawal, J.-Y. Le Boudec, K. Salamatian. Performance Analysis of Self-Limiting Epidemic Forwarding. Technical report LCA-REPORT-2006-127.

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Markov Model for Epidemic Forwarding

The model is complex, O(AN^2) statesN: nb nodes A: a fixed integer

Can we use simple approximations ? What is the corresponding fluid model ?

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Fluid Model is Often Derived Heuristically[KYBR-2006] R. Kumar, D. Yao, A. Bagchi, K.W. Ross, D. Rubenstein, Fluid

Modeling of Pollution Proliferation in P2P Networks, ACM Sigmetrics 2006, St. Malo, France, 2006

Original (micro-) model is continuous time markov process on finite (but huge) state space

Found too large, replaced by a fluid model Step from micro to fluid is ad-hoc, based on informal reasoning Q1: Is there a formal (mechanical) way to derive the fluid model

from the microscopic description ?

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A Similar Step is Common Place in Chemistry/Biology[L-2006] Jean-Yves Le Boudec, Modelling The Immune SystemToolbox:

Stochastic Reaction Models, infoscience.epfl.ch, doc id: LCA-TEACHING-2007-001

Q2: What is the link between the micro quantities and fluid ones ?

Is the fluid quantity the expectation of a microscopic quantity ? Or a re-scaled approximation ?

Micro modelMarkov process

Fluid model

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The Maths of Physics, Chemistry and Biology Help Us

Infinitesimal generator (drift of f)

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Examples of Forward Equations

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Fluid model

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A Fluid Limit Theorem

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Towards a Mechanical Derivation of Fluid Model1. Define the state variable2. Pick functions of interest of the state variable3. Define the transitions jumps r and rates hr(x)4. Compute the generator and write the ODE

1. Define the state variable2. Pick functions of interest of the state variable3. Define the transitions jumps r and rates hr(x)4. Compute the generator and write the ODE

What do we obtain from the fluid model ?• transients• stable points

Implemented for models of the type below in the TSED tool at

http://ica1www.epfl.ch/IS/tsed/index.html

Implemented for models of the type below in the TSED tool at

http://ica1www.epfl.ch/IS/tsed/index.html

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Application to Self-Limiting Epidemic Forwarding

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Application to Self-Limiting Epidemic Forwarding

There is description complexity, but no modelling complexity

A: Age of packet sent by node in middle

ODE

simulation

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Other Results That Are Candidate For Automatic Generation of Solution

Hybrid simulationFast transitions simulated as deterministic fluid, slow transitions as stochastic processExample: mobility + message transmission

Mobility modeled as fluidChange in mobility state changes the rate of the process of packet transmission

“Hybrid Simulation Method” based on representation (martingale approach)

Approximation by SDE

Mean Field, Pairwise approximation Other scaling limits derived from generator approach

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Conclusion

It seems possible to define classes of models thatHave enough generality for networking and computer systemsCan be analyzed approximately in an automatic way

Example:Jump process for which fluid limit is well defined

Many issues remain to explore, many potential applications !