A Markov chain model for traffic equilibrium problems
RAIRO - Operations Research - Recherche Opérationnelle, Tome 36 (2002) no. 3, pp. 209-226.

We consider a stochastic approach in order to define an equilibrium model for a traffic-network problem. In particular, we assume a markovian behaviour of the users in their movements throughout the zones of the traffic area. This assumption turns out to be effective at least in the context of urban traffic, where, in general, the users tend to travel by choosing the path they find more convenient and not necessarily depending on the already travelled part. The developed model is a homogeneous Markov chain, whose stationary distributions (if any) characterize the equilibrium.

DOI : https://doi.org/10.1051/ro:2003003
Mots clés : traffic assignment problems, Markov chains, network flows
@article{RO_2002__36_3_209_0,
     author = {Mastroeni, Giandomenico},
     title = {A Markov chain model for traffic equilibrium problems},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {209--226},
     publisher = {EDP-Sciences},
     volume = {36},
     number = {3},
     year = {2002},
     doi = {10.1051/ro:2003003},
     zbl = {1062.90014},
     mrnumber = {1988277},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ro:2003003/}
}
Mastroeni, Giandomenico. A Markov chain model for traffic equilibrium problems. RAIRO - Operations Research - Recherche Opérationnelle, Tome 36 (2002) no. 3, pp. 209-226. doi : 10.1051/ro:2003003. http://www.numdam.org/articles/10.1051/ro:2003003/

[1] J.A. Barnes and R.L. Disney, Traffic processes in a class of finite Markov queues. Queueing Systems Theory Appl. 9 (1990) 311-326. | MR 1059239 | Zbl 0696.60090

[2] R. Bellman, Dynamic Programming. Princeton University Press, Princeton, New Jersey (1957). | MR 90477 | Zbl 0077.13605

[3] S. Dafermos, Traffic equilibria and variational inequalities. Math. Programming 26 (1980) 40-47. | Zbl 0506.65026

[4] P.T. Harker and J.S. Pang, Finite-Dimensional Variational Inequalities and Nonlinear Complementarity Problem: A Survey of Theory, Algorithms and Appl. Math. Programming 48 (1990) 161-220. | MR 1073707 | Zbl 0734.90098

[5] K.J. Hastings, Introduction to the mathematics of operations research. Dekker, New York (1989). | MR 992197 | Zbl 0709.90074

[6] M. Iosifescu, Finite Markov Processes and Their Applications. John Wiley and Sons (1980). | MR 587116 | Zbl 0436.60001

[7] T. Kurasugi and K. Kobayashi, A Markovian model of coded video traffic which exhibits long-range dependence in statistical analysis. J. Oper. Res. Soc. Japan 42 (1999) 1-17. | MR 1687929 | Zbl 0998.90010

[8] J. Kemeni and J. Snell, Finite Markov Chains. Van Nostrand, Princeton, New Jersey (1960). | MR 115196 | Zbl 0089.13704

[9] W. Leontief, Environmental Repercussions and the Economic Structure: An Input-Output Approach. Rev. Econom. Statist. 52 (1970).

[10] G. Mihoc, On General Properties of Dependent Statistical Variables. Bull. Math. Soc. Roumaine Sci. 37 (1935) 37-82. | JFM 61.1295.03

[11] J.F. Nash, Non-Cooperative games. Ann. Math. 54 (1951) 286-295. | MR 43432 | Zbl 0045.08202

[12] S. Nguyen and S. Pallottino, Equilibrium traffic assignment for large scale transit networks. Eur. J. Oper. Res. 37 (1988) 176-186. | MR 963924 | Zbl 0649.90049

[13] M. Patriksson, Nonlinear Programming and Variational Inequality Problems. Kluwer Academic Publishers, Dordrecht, Boston, London (1999). | MR 1673631 | Zbl 0913.65058

[14] E. Seneta, Non-negative Matrices and Markov Chains. Springer Verlag, New York, Heidelberg, Berlin (1981). | MR 2209438 | Zbl 0471.60001

[15] J.G. Wardrop, Some Theoretical Aspects of Road Traffic Research, in Proc. of the Institute of Civil Engineers, Part II (1952) 325-378.