Abstract

We study the limiting behaviour of stochastic models of populations of interacting agents, as the number of agents goes to infinity. Classical mean-field results have established that this limiting behaviour is described by an ordinary differential equation (ODE) under two conditions: (1) that the dynamics is smooth; and (2) that the population is composed of a finite number of homogeneous sub-populations, each containing a large number of agents. This paper reviews recent work showing what happens if these conditions do not hold. In these cases, it is still possible to exhibit a limiting regime at the price of replacing the ODE by a more complex dynamical system. In the case of non-smooth or uncertain dynamics, the limiting regime is given by a differential inclusion. In the case of multiple population scales, the ODE is replaced by a stochastic hybrid automaton.

Keywords

Population models Markov chain Mean-field limits Differential inclusions Hybrid systems 

Notes

Acknowledgements

L.B. and N.G. acknowledge partial support from the EU-FET project QUANTICOL (nr. 600708).

References

  1. 1.
    Andersson, H., Britton, T.: Stochastic Epidemic Models and Their Statistical Analysis. Springer, Heidelberg (2000)CrossRefMATHGoogle Scholar
  2. 2.
    Aubin, J., Cellina, A.: Differential Inclusions. Springer, Heidelberg (1984)CrossRefMATHGoogle Scholar
  3. 3.
    Baier, C., et al.: Model-checking algorithms for continuous-time Markov chains. IEEE Trans. Softw. Eng. 29(6), 524–541 (2003). http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1205180 CrossRefGoogle Scholar
  4. 4.
    Benaim, M., Le Boudec, J.-Y.: A class of mean field interaction models for computer and communication systems. Perform. Eval. 65(11), 823–838 (2008)CrossRefGoogle Scholar
  5. 5.
    Billingsley, P.: Probability and Measure. English. Wiley, Hoboken (2012). ISBN: 9781118122372 1118122372MATHGoogle Scholar
  6. 6.
    Bortolussi, L., Gast, N.: Mean field approximation of imprecise population processes. QUANTICOL Technical report TR-QC-07-2015 (2015)Google Scholar
  7. 7.
    Bortolussi, L., et al.: Continuous approximation of collective systems behaviour: a tutorial. Perform. Eval. 70(5), 317–349 (2013). ISSN: 0166-5316, doi:10.1016/j.peva.2013.01.001, http://www.sciencedirect.com/science/article/pii/S0166531613000023
  8. 8.
    Bortolussi, L.: Hybrid behaviour of Markov population models. In: Information and Computation (2015)Google Scholar
  9. 9.
    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). http://link.springer.com/chapter/10.1007/978-3-642-13568-2_26. Accessed 11 June 2015CrossRefGoogle Scholar
  10. 10.
    Bortolussi, L., Hillston, J.: Model checking single agent behaviours by uid approximation. Inf. Comput. 242, 183–226 (2015). ISSN: 0890-5401, doi:10.1016/j.ic.2015.03.002
  11. 11.
    Bortolussi, L., Lanciani, R.: Fluid model checking of timed properties. In: Sankaranarayanan, S., Vicario, E. (eds.) FORMATS 2015. LNCS, vol. 9268, pp. 172–188. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  12. 12.
    Bortolussi, L., Lanciani, R.: Model checking Markov population models by central limit approximation. In: Joshi, K., Siegle, M., Stoelinga, M., DArgenio, P.R. (eds.) QEST 2013. LNCS, vol. 8054, pp. 123–138. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  13. 13.
    Bortolussi, L., Lanciani, R.: Stochastic approximation of global reachability probabilities of Markov population models. In: Horvath, A., Wolter, K. (eds.) EPEW 2014. LNCS, vol. 8721, pp. 224–239. Springer, Heidelberg (2014)Google Scholar
  14. 14.
    Bortolussi, L., Policriti, A.: Dynamical systems and stochastic programming: to ordinary differential equations and back. In: Priami, C., Back, R.-J., Petre, I. (eds.) Transactions on Computational Systems Biology XI. LNCS, vol. 5750, pp. 216–267. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  15. 15.
    Bortolussi, L., Policriti, A.: (Hybrid) automata, (stochastic) programs: the hybrid automata lattice of a stochastic program. J. Logic Comput. 23, 761–798 (2013). http://dx.doi.org/10.1093/logcom/exr045 MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Bortolussi, L., Policriti, A.: Hybrid dynamics of stochastic programs. Theor. Comput. Sci. 411(20), 2052–2077 (2010). ISSN: 0304-3975MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Chaintreau, A., Le Boudec, J.-Y., Ristanovic, N.: The age of gossip: spatial mean field regime. In: Proceedings of the ACM SIGMETRICS, vol. 37, issue 1, pp. 109–120. ACM (2009)Google Scholar
  18. 18.
    Ciocchetta, F., Hillston, J.: Bio-PEPA: a framework for the modelling and analysis of biological systems. Theor. Comput. Sci. 410(33), 00185, 3065–3084 (2009). http://www.sciencedirect.com/science/article/pii/S0304397509001662. Accessed 25 Nov 2013
  19. 19.
    Crudu, A., et al.: Convergence of stochastic gene networks to hybrid piecewise deterministic processes. Ann. Appl. Probab. 22(5), 00015, 1822–1859 (2012). http://projecteuclid.org/euclid.aoap/1350067987. Accessed 05 Nov 2013
  20. 20.
    Darling, R., Norris, J.R., et al.: Differential equation approximations for Markov chains. Probab. Surv. 5, 37–79 (2008)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Davis, M.H.A.: Markov Models and Optimization. Chapman & Hall, London (1993)CrossRefMATHGoogle Scholar
  22. 22.
    Doncel, J., Gast, N., Gaujal, B.: Mean-Field Games with Explicit Interactions. Working paper or preprint, February 2016. https://hal.inria.fr/hal-01277098
  23. 23.
    Durrett, R.: Essentials of Stochastic Processes. Springer, Heidelberg (2012). ISBN: 9781461436157CrossRefMATHGoogle Scholar
  24. 24.
    Fricker, C., Gast, N.: Incentives and redistribution in homogeneous bike-sharing systems with stations of finite capacity. EURO J. Trans. Logistics, 1–31 (2014)Google Scholar
  25. 25.
    Fricker, C., Gast, N., Mohamed, H.: Mean field analysis for inhomogeneous bike sharing systems. DMTCS Proc. 01, 365–376 (2012)MathSciNetMATHGoogle Scholar
  26. 26.
    Galpin, V.: Spatial representations, analysis techniques. In: SFM (2016)Google Scholar
  27. 27.
    Galpin, V., Bortolussi, L., Hillston, J.: HYPE: hybrid modelling by composition of flows. Formal Aspects Comput. 25(4), 503–541 (2013)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Gast, N., Gaujal, B.: Markov chains with discontinuous drifts have differential inclusion limits. Perform. Eval. 69(12), 623–642 (2012)CrossRefGoogle Scholar
  29. 29.
    Gast, N., Gaujal, B.: Mean field limit of non-smooth systems and differential inclusions. ACM SIGMETRICS Perform. Eval. Rev. 38(2), 30–32 (2010)CrossRefGoogle Scholar
  30. 30.
    Gast, N., Le Boudec, J.-Y., Tomozei, D.-C.: Impact of demand-response on the efficiency, prices in real-time electricity markets. In: Proceedings of the 5th International Conference on Future Energy Systems, pp. 171–182. ACM (2014)Google Scholar
  31. 31.
    Gast, N., Van Houdt, B.: Transient and steady-state regime of a family of list-based cache replacement algorithms. In: ACM SIGMETRICS 2015 (2015)Google Scholar
  32. 32.
    Hasenauer, J., et al.: Method of conditional moments (MCM) for the chemical master equation: a unified framework for the method of moments and hybrid stochastic-deterministic models. J. Math. Biol. 69, 687–735 (2013). ISSN: 0303-6812, 1432–1416, doi:10.1007/s00285-013-0711-5, http://link.springer.com/10.1007/s00285-013-0711-5. Accessed 31 July 2014
  33. 33.
    Henzinger, T., Jobstmann, B., Wolf, V.: Formalisms for specifying Markovian population models. Int. J. Found. Comput. Sci. 22(04), 823–841 (2011). http://www.worldscience.com/doi/abs/10.1142/S0129054111008441 MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Hu, L., Le Boudec, J.-Y., Vojnoviae, M.: Optimal channel choice for collaborative ad-hoc dissemination. In: 2010 Proceedings of the IEEE INFOCOM, pp. 1–9. IEEE (2010)Google Scholar
  35. 35.
    Huang, M., Malhame, R.P., Caines, P.E., et al.: Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst. 6(3), 221–252 (2006)MathSciNetMATHGoogle Scholar
  36. 36.
    Katoen, J.-P., Khattri, M., Zapreevt, I.S.: A Markov reward model checker. In: Second International Conference on the Quantitative Evaluation of Systems, pp. 243–244 (2005). Accessed 18 Jan 2014Google Scholar
  37. 37.
    Kurtz, T.: Solutions of ordinary differential equations as limits of pure jump Markov processes. J. Appl. Probab. 7, 49–58 (1970)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Kwiatkowska, M., Norman, G., Parker, D.: PRISM 4.0: verification of probabilistic real-time systems. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 585–591. Springer, Heidelberg (2011). http://link.springer.com/chapter/10.1007/978- 3-642-22110-1_47. Accessed 18 Jan 2014CrossRefGoogle Scholar
  39. 39.
    Krn, M., et al.: Stochasticity in gene expression: from theories to phenotypes. Nat. Rev. Genet. 6(6), 451–464 (2005). ISSN: 1471-0056, 1471–0064, doi:10.1038/nrg1615, http://www.nature.com/doifinder/10.1038/nrg1615. Accessed 09 Feb 2016
  40. 40.
    Lasry, J.-M., Lions, P.-L.: Mean field games. Jpn. J. Math. 2(1), 229–260 (2007)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Le Boudec, J.-Y.: Performance Evaluation of Computer and Communication Systems. EPFL Press, Lausanne (2010)MATHGoogle Scholar
  42. 42.
    Loreti, M.: Modeling and analysis of collective adaptive systems with CARMA and its tools. In: SFM (2016)Google Scholar
  43. 43.
    Mitzenmacher, M.: The power of two choices in randomized load balancing. IEEE Trans. Parallel Distrib. Syst. 12(10), 1094–1104 (2001)CrossRefGoogle Scholar
  44. 44.
    Norris, J.R.: Markov Chains. English. Cambridge University Press, Cambridge (1998). ISBN: 978-0-511-81063-3 0-511-81063-6Google Scholar
  45. 45.
    Pahle, J.: Biochemical simulations: stochastic, approximate stochastic and hybrid approaches. Briefings Bioinform. 10(1), 53–64 (2008). ISSN: 1467-5463, 1477–4054, doi:10.1093/bib/bbn050, http://bib.oxfordjournals.org/cgi/doi/10./bib/bbn050. Accessed 14 July 2014
  46. 46.
    Todorov, E.: Optimal control theory. In: Bayesian Brain: Probabilistic Approaches to Neural Coding, pp. 269–298 (2006)Google Scholar
  47. 47.
    Tribastone, M., Gilmore, S., Hillston, J.: Scalable differential analysis of process algebra models. IEEE Trans. Softw. Eng. 38(1), 205–219 (2012). http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=5567115. Accessed 24 Nov 2013CrossRefGoogle Scholar
  48. 48.
    Tschaikowski, M., Tribastone, M.: Approximate reduction of heterogenous nonlinear models with differential hulls. IEEE Trans. Autom. Control 61(4), 1099–1104 (2016). doi:10.1109/TAC.2015.2457172 CrossRefGoogle Scholar
  49. 49.
    Tsitsiklis, J.N., Xu, K., et al.: On the power of (even a little) resource pooling. Stochast. Syst. 2(1), 1–66 (2012)MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    Van Houdt, B.: A mean field model for a class of garbage collection algorithms in flash-based solid state drives. In: Proceedings of the ACM SIGMETRICS, SIGMETRICS 2013, Pittsburgh, PA, USA, pp. 191–202. ACM (2013). ISBN: 978-1-4503-1900-3, doi:10.1145/2465529.2465543, http://doi.acm.org/10.1145/2465529.2465543
  51. 51.
    Wilkinson, D.: Stochastic Modelling for Systems Biology. Chapman & Hall, Florida (2006)MATHGoogle Scholar
  52. 52.
    Yang, T., Mehta, P.G., Meyn, S.P.: A mean-field control-oriented approach to particle filtering. In: American Control Conference (ACC), pp. 2037–2043. IEEE (2011)Google Scholar
  53. 53.
    Ying, L.: On the rate of convergence of mean-field models: Stein’s method meets the perturbation theory. arXiv preprint arXiv:1510.00761 (2015)

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.DMGUniversity of TriesteTriesteItaly
  2. 2.MOSISaarland UniversitySaarbrückenGermany
  3. 3.CNR-ISTIPisaItaly
  4. 4.Inria, University of Grenoble Alpes, CNRS, LIGGrenobleFrance

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