The Mean Drift: Tailoring the Mean Field Theory of Markov Processes for Real-World Applications
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.
KeywordsMarkov chains Population processes Mean field approximation Propagation of chaos
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.
- 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.Hillston, J.: Fluid flow approximation of PEPA models. In: QEST 2005, pp. 33–42. IEEE (2005)Google Scholar
- 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.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
- 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)
- 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