Advertisement

The Mean Drift: Tailoring the Mean Field Theory of Markov Processes for Real-World Applications

  • Mahmoud TalebiEmail author
  • Jan Friso Groote
  • Jean-Paul M. G. Linnartz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10378)

Abstract

The statement of the mean field approximation theorem in the mean field theory of Markov processes particularly targets the behaviour of population processes with an unbounded number of agents. However, in most real-world engineering applications one faces the problem of analysing middle-sized systems in which the number of agents is bounded. In this paper we build on previous work in this area and introduce the mean drift. We present the concept of population processes and the conditions under which the approximation theorems apply, and then show how the mean drift can be linked to observations which follow from the propagation of chaos. We then use the mean drift to construct a new set of ordinary differential equations which address the analysis of population processes with an arbitrary size.

Keywords

Markov chains Population processes Mean field approximation Propagation of chaos 

Notes

Acknowledgments

The research from DEWI project (www.dewi-project.eu) leading to these results has received funding from the ARTEMIS Joint Undertaking under grant agreement No. 621353.

References

  1. 1.
    Kurtz, T.G.: Approximation of Population Processes. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 36. SIAM, Philadelphia (1981)CrossRefGoogle Scholar
  2. 2.
    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
  3. 3.
    Dietz, K., Heesterbeek, J.A.P.: Daniel Bernoulli’s epidemiological model revisited. Math. Biosci. 180(1), 1–21 (2002)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Darling, R.W.R., Norris, J.R.: Differential equation approximations for Markov chains. Probab. Surv. 5, 37–79 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Kurtz, T.G.: Solutions of ordinary differential equations as limits of pure jump Markov processes. J. Appl. Probab. 7(1), 49–58 (1970)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence. Wiley, Hoboken (2009)zbMATHGoogle Scholar
  7. 7.
    Bianchi, G.: Performance analysis of the IEEE 802.11 distributed coordination function. IEEE J. Sel. Areas Commun. 18(3), 535–547 (2000)CrossRefGoogle Scholar
  8. 8.
    Le Boudec, J.-Y., McDonald, D., Mundinger, J.: A generic mean field convergence result for systems of interacting objects. In: Fourth International Conference on the Quantitative Evaluation of Systems, QEST 2007, pp. 3–18. IEEE (2007)Google Scholar
  9. 9.
    Hillston, J.: Fluid flow approximation of PEPA models. In: QEST 2005, pp. 33–42. IEEE (2005)Google Scholar
  10. 10.
    Hayden, R.A., Bradley, J.T.: A fluid analysis framework for a Markovian process algebra. Theoret. Comput. Sci. 411(22), 2260–2297 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bortolussi, L., Hillston, J., Latella, D., Massink, M.: Continuous approximation of collective system behaviour: a tutorial. Perform. Eval. 70(5), 317–349 (2013)CrossRefGoogle Scholar
  12. 12.
    Pourranjbar, A., Hillston, J., Bortolussi, L.: Don’t just go with the flow: cautionary tales of fluid flow approximation. In: Tribastone, M., Gilmore, S. (eds.) EPEW 2012. LNCS, vol. 7587, pp. 156–171. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-36781-6_11CrossRefGoogle Scholar
  13. 13.
    Beccuti, M., Bibbona, E., Horvath, A., Sirovich, R., Angius, A., Balbo, G.: Analysis of Petri net models through Stochastic Differential Equations. In: Ciardo, G., Kindler, E. (eds.) PETRI NETS 2014. LNCS, vol. 8489, pp. 273–293. Springer, Cham (2014). doi: 10.1007/978-3-319-07734-5_15CrossRefzbMATHGoogle Scholar
  14. 14.
    Bobbio, A., Gribaudo, M., Telek, M.: Analysis of large scale interacting systems by mean field method. In: Fifth International Conference on Quantitative Evaluation of Systems. QEST 2008, pp. 215–224. IEEE (2008)Google Scholar
  15. 15.
    Kurtz, T.G.: Strong approximation theorems for density dependent Markov chains. Stoch. Process. Their Appl. 6(3), 223–240 (1978)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Talebi, M., Groote, J.F., Linnartz, J.-P.M.G.: The mean drift: tailoring the mean-field theory of Markov processes for real-world applications. arXiv preprint math/1703.04327 (2017)
  17. 17.
    Sznitman, A.-S.: Topics in propagation of chaos. In: Hennequin, P.-L. (ed.) Ecole d’Eté de Probabilités de Saint-Flour XIX — 1989. LNM, vol. 1464, pp. 165–251. Springer, Heidelberg (1991). doi: 10.1007/BFb0085169CrossRefGoogle Scholar
  18. 18.
    Vvedenskaya, N.D., Sukhov, Y.M.: Multiuser multiple-access system: stability and metastability. Problemy Peredachi Informatsii 43(3), 105–111 (2007)MathSciNetGoogle Scholar
  19. 19.
    Talebi, M., Groote, J.F., Linnartz, J.-P.M.G.: Continuous approximation of stochastic models for wireless sensor networks. In: 2015 IEEE Symposium on Communications and Vehicular Technology in the Benelux (SCVT), pp. 1–6. IEEE (2015)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Mahmoud Talebi
    • 1
    Email author
  • Jan Friso Groote
    • 1
  • Jean-Paul M. G. Linnartz
    • 2
  1. 1.Department of Mathematics and Computer ScienceEindhoven University of TechnologyEindhovenThe Netherlands
  2. 2.Department of Electrical EngineeringEindhoven University of TechnologyEindhovenThe Netherlands

Personalised recommendations