1. 1 Modeling and Simulating Networking Systems with Markov Processes Tools and Methods of Wide Applicability ? Jean-Yves Le Boudec EPFL/I&C/ISC-LCA-2jean-yves.leboudec@epfl.ch

2. 2 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/RandomTriphttp://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

3. 3 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

4. 4 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

5. 5 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.

6. 6 Markov Model for Epidemic Forwarding  The model is complex, O(A N^2 ) states N: nb nodes A: a fixed integer  Can we use simple approximations ? What is the corresponding fluid model ?

7. 7 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 ?

8. 8 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 model Markov process Fluid model

9. 9 The Maths of Physics, Chemistry and Biology Help Us Infinitesimal generator (drift of f)

10. 10 Examples of Forward Equations

11. 11 Fluid model

12. 12 A Fluid Limit Theorem

13. 13 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 h r (x) 4.Compute the generator and write the ODE 1.Define the state variable 2.Pick functions of interest of the state variable 3.Define the transitions jumps  r and rates h r (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 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 http://ica1www.epfl.ch/IS/tsed/index.html

14. 14 Application to Self-Limiting Epidemic Forwarding

15. 15 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

16. 16 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

17. 17 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 !