Mean-Field Approximation and Quasi-Equilibrium Reduction of Markov Population Models

  • Luca Bortolussi
  • Rytis Paškauskas
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8657)


Markov Population Model is a commonly used framework to describe stochastic systems. Their exact analysis is unfeasible in most cases because of the state space explosion. Approximations are usually sought, often with the goal of reducing the number of variables. Among them, the mean field limit and the quasi-equilibrium approximations stand out. We view them as techniques that are rooted in independent basic principles. At the basis of the mean field limit is the law of large numbers. The principle of the quasi-equilibrium reduction is the separation of temporal scales. It is common practice to apply both limits to an MPM yielding a fully reduced model. Although the two limits should be viewed as completely independent options, they are applied almost invariably in a fixed sequence: MF limit first, QE reduction second. We present a framework that makes explicit the distinction of the two reductions, and allows an arbitrary order of their application. By inverting the sequence, we show that the double limit does not commute in general: the mean field limit of a time-scale reduced model is not the same as the time-scale reduced limit of a mean field model. An example is provided to demonstrate this phenomenon. Sufficient conditions for the two operations to be freely exchangeable are also provided.


Model Check Singular Perturbation Steady State Distribution Continuous Time Markov Chain Multiple Time Scale 
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  1. 1.
    Benäim, M., Le Boudec, J.-Y.: A class of mean field interaction models for computer and communication systems. Perf. Eval. 65(11), 823–838 (2008)CrossRefGoogle Scholar
  2. 2.
    Benäim, M., Le Boudec, J.-Y.: On mean field convergence and stationary regime. CoRR, abs/1111.5710 (2011)Google Scholar
  3. 3.
    Benäim, M., Weibull, J.W.: Deterministic approximation of stochastic evolution in games. Econometrica 71(3), 873–903 (2003)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Bortolussi, L.: Limit behavior of the hybrid approximation of stochastic process algebras. In: Al-Begain, K., Fiems, D., Knottenbelt, W.J. (eds.) ASMTA 2010. LNCS, vol. 6148, pp. 367–381. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  5. 5.
    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
  6. 6.
    Bortolussi, L., Paškauskas, R.: Multiscale reductions of mean field and stochastic models. Technical Report TR-QC-04-2014, QUANTICOL Tech. Rep. (2014)Google Scholar
  7. 7.
    Bortolussi, L., Paškauskas, R.: Mean-Field approximation and Quasi-Equilibrium reduction of Markov Population Models (2014),
  8. 8.
    Bortolussi, L., Hayden, R.: Bounds on the deviation of discrete-time Markov chains from their mean-field model. Perf. Eval. 70(10), 736–749 (2013)CrossRefGoogle Scholar
  9. 9.
    Bortolussi, L., Hillston, J., Latella, D., Massink, M.: Continuous approximation of collective systems behaviour: a tutorial. Perf. Eval. 70, 317–349 (2013)CrossRefGoogle Scholar
  10. 10.
    Bortolussi, L., Lanciani, R.: Model checking markov population models by central limit approximation. In: Joshi, K., Siegle, M., Stoelinga, M., D’Argenio, P.R. (eds.) QEST 2013. LNCS, vol. 8054, pp. 123–138. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  11. 11.
    Cao, Y., Gillespie, D.T., Petzold, L.R.: The slow-scale stochastic simulation algorithm. The Journal of Chemical Physics 122(1), 014116 (2005)Google Scholar
  12. 12.
    Ethier, S.N., Kurtz, T.G.: Markov processes: characterization and convergence. series in probability and statistics. Wiley Interscience. Wiley (2005)Google Scholar
  13. 13.
    Gardiner, C.W.: Handbook of stochastic methods, vol. 3. Springer (1985)Google Scholar
  14. 14.
    Gardner, T.S., Cantor, C.R., Collins, J.J.: Construction of a genetic toggle switch in escherichia coli. Nature 403(6767), 339–342 (2000)CrossRefGoogle Scholar
  15. 15.
    Gillespie, D.T.: Exact stochastic simulation of coupled chemical reactions. The Journal of Physical Chemistry 81(25), 2340–2361 (1977)CrossRefGoogle Scholar
  16. 16.
    Hayden, R., Bradley, J., Clark, A.: Performance specification and evaluation with unified stochastic probes and fluid analysis. IEEE TSE 39(1), 97–118 (2013)Google Scholar
  17. 17.
    Henzinger, T., Jobstmann, B., Wolf, V.: Formalisms for specifying Markovian population models. International Journal of Foundations of Computer Science 22(04), 823–841 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Johnson, K.A., Goody, R.S.: The original Michaelis constant: Translation of the 1913 Michaelis–Menten paper. Biochemistry 50(39), 8264–8269 (2011)CrossRefGoogle Scholar
  19. 19.
    Lam, S.H., Goussis, D.A.: The CSP method for simplifying kinetics. International Journal of Chemical Kinetics 26(4), 461–486 (1994)CrossRefGoogle Scholar
  20. 20.
    Le Boudec, J.-Y.: The stationary behaviour of fluid limits of reversible processes is concentrated on stationary points. NHM 8(2), 529–540 (2013)CrossRefzbMATHGoogle Scholar
  21. 21.
    Marquez-Lago, T.T., Stelling, J.: Counter-intuitive stochastic behavior of simple gene circuits with negative feedback. Biophysical Journal 98(9), 1742–1750 (2010)CrossRefGoogle Scholar
  22. 22.
    Maas, U., Pope, S.B.: Simplifying chemical kinetics: intrinsic low-dimensional manifolds in composition space. Combustion and Flame 88(3), 239–264 (1992)CrossRefGoogle Scholar
  23. 23.
    Mastny, E.A., Haseltine, E.L., Rawlings, J.B.: Two classes of quasi-steady-state model reductions for stochastic kinetics. The Journal of Chemical Physics 127(9), 094106 (2007)Google Scholar
  24. 24.
    Rao, C.V., Arkin, A.P.: Stochastic chemical kinetics and the quasi-steady-state assumption: Application to the Gillespie algorithm. Journal of Chemical Physics 118(11), 4999–5010 (2003)CrossRefGoogle Scholar
  25. 25.
    Schnakenberg, J.: Network theory of microscopic and macroscopic behavior of master equation systems. Reviews of Modern Physics 48(4), 571–585 (1976)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Shwartz, A., Weiss, A.: Large Deviations for Performance Analysis. C&H (1995)Google Scholar
  27. 27.
    Stewart, G.W., Sun, J.: Matrix perturbation theory. Computer Science and Scientific Computing. Academic Press (1990)Google Scholar
  28. 28.
    Verhulst, F.: Methods and applications of singular perturbations: boundary layers and multiple timescale dynamics. Springer (2005)Google Scholar
  29. 29.
    Weinan, E., Liu, D., Vanden-Eijnden, E.: Nested stochastic simulation algorithms for chemical kinetic systems with multiple time scales. Journ. of Comp. Phys. 221(1), 158–180 (2007)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Luca Bortolussi
    • 1
    • 2
  • Rytis Paškauskas
    • 1
  1. 1.DMGUniversity of TriesteItaly
  2. 2.ISTI Area della Ricerca CNRPisaItaly

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