Modeling and Simulating Networking Systems with Markov Processes

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Modeling and SimulatingNetworking Systems with
Markov Processes
Tools and Methods of Wide Applicability
? Jean-Yves Le Boudec EPFL/IC/ISC-LCA-2 jean-yv
es.leboudec_at_epfl.ch
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Examples of Research in my Group (IC/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 used
• But does not apply to the large scale
• Analysis
• Often very hard to use / obtain proven results /
  re-usable
• Sometimes too late
• Fast simulation is also often an alternative
• Based 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(AN2) 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 ?

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The Maths of Physics, Chemistry and Biology Help
Us
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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 Model

1. Define the state variable
2. Pick functions of interest of the state variable
3. Define the transitions jumps ?r and rates hr(x)
4. Compute the generator and write the ODE

• Implemented for models of the type below in the
  TSED tool at http//ica1www.epfl.ch/IS/tsed/inde
  x.html

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

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Application to Self-Limiting Epidemic Forwarding
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Application to Self-Limiting Epidemic Forwarding
A Age of packet sent by node in middle
ODE
simulation

• There is description complexity, but no modelling
  complexity

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Other Results That Are Candidate For Automatic
Generation of Solution

• Hybrid simulation
• Fast transitions simulated as deterministic
  fluid, slow transitions as stochastic process
• Example mobility message transmission
• Mobility modeled as fluid
• Change 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
  that
• Have enough generality for networking and
  computer systems
• Can 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 !