Applying Mean-Field Approximation to Continuous Time Markov Chains

  • Anna Kolesnichenko
  • Valerio Senni
  • Alireza Pourranjabar
  • Anne Remke
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8453)


The mean-field analysis technique is used to perform analysis of a system with a large number of components to determine the emergent deterministic behaviour and how this behaviour modifies when its parameters are perturbed. The computer science performance modelling and analysis community has found the mean-field method useful for modelling large-scale computer and communication networks. Applying mean-field analysis from the computer science perspective requires the following major steps: (1) describing how the agent populations evolve by means of a system of differential equations, (2) finding the emergent deterministic behaviour of the system by solving such differential equations, and (3) analysing properties of this behaviour. Depending on the system under analysis, performing these steps may become challenging. Often, modifications of the general idea are needed. In this tutorial we consider illustrating examples to discuss how the mean-field method is used in different application areas. Starting from the application of the classical technique, moving to cases where additional steps have to be used, such as systems with local communication. Finally, we illustrate the application of existing model checking analysis techniques.


Model Check Local Model Mobile Agent Goal State Continuous Time Markov Chain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Baccelli, F., Karpelevich, F.I., Kelbert, M.Y., Puhalskii, A.A., Rybko, A.N., Suhov, Y.M.: A mean-field limit for a class of queueing networks. Journal of Statistical Physics 66, 803–825 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Baier, C., Haverkort, B.R., Hermanns, H., Katoen, J.P.: Model-checking algorithms for continuous-time Markov chains. IEEE Trans. Softw. Eng. 29(7), 524–541 (2003)CrossRefzbMATHGoogle Scholar
  3. 3.
    Benaïm, M., Le Boudec, J.Y.: A class of mean field interaction models for computer and communication systems. Perform. Eval. 65(11-12), 823–838 (2008)CrossRefGoogle Scholar
  4. 4.
    Benaïm, M., Weibull, J.W.: Deterministic approximation of stochastic evolution in games. Econometrica 71(3), 873–903 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Billingsley, P.: Probability and Measure, 3rd edn. Wiley-Interscience (1995)Google Scholar
  6. 6.
    Bobbio, A., Gribaudo, M., Telek, M.: Analysis of large scale interacting systems by mean field method. In: QEST, pp. 215–224 (2008)Google Scholar
  7. 7.
    Bortolussi, L.: Hybrid limits of continuous time Markov chains. In: QEST, pp. 3–12. IEEE Computer Society (2011)Google Scholar
  8. 8.
    Bortolussi, L., Hillston, J.: Fluid model checking. In: Koutny, M., Ulidowski, I. (eds.) CONCUR 2012. LNCS, vol. 7454, pp. 333–347. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  9. 9.
    Bortolussi, L., Hillston, J., Latella, D., Massink, M.: Continuous approximation of collective systems behaviour: A tutorial. Performance Evaluation 70(5), 317–349 (2013)CrossRefGoogle Scholar
  10. 10.
  11. 11.
    Chaintreau, A., Le Boudec, J.Y., Ristanovic, N.: The age of gossip: spatial mean field regime. In: SIGMETRICS/Performance, pp. 109–120. ACM (2009)Google Scholar
  12. 12.
    Ciocchetta, F., Hillston, J.: Bio-pepa: A framework for the modelling and analysis of biological systems. Theoretical Computer Science 410(33-34), 3065–3084 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Darling, R.W.R., Norris, J.R.: Differential equation approximations for Markov chains. Probability Surveys 5, 37–79 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Deavours, D.D., Clark, G., Courtney, T., Daly, D., Derisavi, S., Doyle, J.M., Sanders, W.H., Webster, P.G.: The Mobius framework and its implementation. IEEE Transactions on Software Engineering 28(10), 956–969 (2002)CrossRefGoogle Scholar
  15. 15.
    Gast, N., Gaujal, B.: A mean field model of work stealing in large-scale systems. In: SIGMETRICS, pp. 13–24. ACM (2010)Google Scholar
  16. 16.
    Gillespie, C.S.: Moment closure approximations for mass-action models. IET Systems Biology 3, 52–58 (2009)CrossRefGoogle Scholar
  17. 17.
    Gillespie, D.T.: Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81(25), 2340–2361 (1977)CrossRefGoogle Scholar
  18. 18.
    Hayden, R., Stefanek, A., Bradley, J.T.: Fluid computation of passage time distributions in large Markov models. Theoretical Computer Science 413(1), 106–141 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hayden, R.A., Bradley, J.T.: A fluid analysis framework for a markovian process algebra. Theoretical Computer Science 411(22-24), 2260–2297 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hillston, J.: A compositional approach to performance modelling. Cambridge University Press (1996)Google Scholar
  21. 21.
    Hillston, J.: Fluid flow approximation of pepa models. In: QEST, pp. 33–43. IEEE Computer Society (2005)Google Scholar
  22. 22.
    Hillston, J., Tribastone, M., Gilmore, S.: Stochastic process algebras: From individuals to populations. The Computer Journal (2011)Google Scholar
  23. 23.
    Kadanoff, L.P.: More is the Same; Phase Transitions and Mean Field Theories. Journal of Statistical Physics 137, 777–797 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kleczkowski, A., Grenfell, B.T.: Mean-field-type equations for spread of epidemics: the small world model. Physica A: Statistical Mechanics and its Applications 274(12), 355–360 (1999)CrossRefGoogle Scholar
  25. 25.
    Kolesnichenko, A., de Boer, P.T., Remke, A.K.I., Haverkort, B.R.: A logic for model-checking mean-field models. In: DSN/PDF, pp. 1–12. IEEE Computer Society (2013)Google Scholar
  26. 26.
    Kolesnichenko, A., Remke, A., de Boer, P.-T., Haverkort, B.R.: Comparison of the mean-field approach and simulation in a peer-to-peer botnet case study. In: Thomas, N. (ed.) EPEW 2011. LNCS, vol. 6977, pp. 133–147. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  27. 27.
    Kurtz, T.G.: Solutions of ordinary differential equations as limits of pure jump Markov processes. Journal of Applied Probability 7(1), 49–58 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Kurtz, T.G.: Approximation of population processes, vol. 36. Society for Industrial Mathematics (1981)Google Scholar
  29. 29.
    Latella, D., Loreti, M., Massink, M.: On-the-fly Fast Mean-Field Model-Checking: Extended Version. Technical report (2013)Google Scholar
  30. 30.
    Le Boudec, J.Y., McDonald, D., Mundinger, J.: A generic mean field convergence result for systems of interacting objects. In: QEST, pp. 3–18. IEEE Computer Society (2007)Google Scholar
  31. 31.
    McComb, W.D.: Renormalization Methods: A Guide For Beginners. OUP, Oxford (2004)Google Scholar
  32. 32.
    Mitzenmacher, M.: The power of two choices in randomized load balancing. IEEE Trans. Parallel Distrib. Syst. 12(10), 1094–1104 (2001)CrossRefGoogle Scholar
  33. 33.
    Pourranjbar, A., Hillston, J., Bortolussi, L.: Dont Just Go with the Flow: Cautionary Tales of Fluid Flow Approximation. In: Tribastone, M., Gilmore, S. (eds.) EPEW/UKPEW 2012. LNCS, vol. 7587, pp. 156–171. Springer, Heidelberg (2013)Google Scholar
  34. 34.
    Silva, M., Recalde, L.: On fluidification of petri nets: from discrete to hybrid and continuous models. Annual Reviews in Control 28(2), 253–266 (2004)CrossRefGoogle Scholar
  35. 35.
    Tribastone, M.: Relating layered queueing networks and process algebra models. In: WOSP/SIPEW, pp. 183–194 (2010)Google Scholar
  36. 36.
    Tribastone, M., Gilmore, S., Hillston, J.: Scalable differential analysis of process algebra models. IEEE Trans. Software Eng. 38(1), 205–219 (2012)CrossRefGoogle Scholar
  37. 37.
    Van Kampen, N.G.: Stochastic Processes in Physics and Chemistry. North-Holland Personal Library. Elsevier Science (2011)Google Scholar
  38. 38.
    van Ruitenbeek, E., Sanders, W.H.: Modeling peer-to-peer botnets. In: QEST, pp. 307–316. IEEE CS Press (2008)Google Scholar
  39. 39.

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Anna Kolesnichenko
    • 1
  • Valerio Senni
    • 3
  • Alireza Pourranjabar
    • 2
  • Anne Remke
    • 1
  1. 1.DACSUniversity of TwenteThe Netherlands
  2. 2.LFCSUniversity of EdinburghUK
  3. 3.IMT Institute for Advanced StudiesLuccaItaly

Personalised recommendations