Abstract
In this paper we consider a Boltzmann type equation arising in the kinetic vehicle traffic flow model with an acceleration variable. The latter model is improved within the framework of the previously developed approach by introducing a set of random parameters. This enables us to take into account different types of interacting vehicles, as well as various parameters describing skills and behavior of particular drivers. We develop new Monte Carlo algorithms to evaluate probabilistic moments of linear functionals of the solution to the considered equation.
Funding: The work was supported by the budget project 0315-2019-0002 for ICM&MG SB RAS and was also partly supported by the Russian Foundation for Basic Research (project No. 18-01-00356).
References
[1] A. V. Burmistrov and M. A. Korotchenko, Algorithms of Monte Carlo method for simulating the vehicular traffic flow with random parameters. In: Marchuk Scientific Readings 2019: Proc. Int. Conf. ‘Advanced Mathematics, Computations and Applications, AMCA-2019’. NSU PPC, Novosibirsk. 2019, pp. 72–79 (in Russian).Search in Google Scholar
[2] A. V. Burmistrov and M.A. Korotchenko, Application of kinetic models and simulation of multi-particle systems in some fields of science. Mathematical Modeling, Computer and Field Experiment in Natural Sciences1 (2019), 3–12.Search in Google Scholar
[3] A. V. Burmistrov and M. A. Korotchenko, Application of statistical methods for the study of kinetic model of traffic flow with separated accelerations. Russ. J. Numer. Anal. Math. Modelling26 (2011), No. 3, 275–293.10.1515/rjnamm.2011.015Search in Google Scholar
[4] A. Burmistrov and M. Korotchenko, Monte Carlo algorithm for simulation of the vehicular traffic flow within the kinetic model with velocity dependent thresholds. Springer Proc. Math. Stat. 114 (2014), 109–117.10.1007/978-1-4939-2104-1_10Search in Google Scholar
[5] A. Burmistrov and M. Korotchenko, Weight Monte Carlo method applied to acceleration oriented traffic flow model. Springer Proc. Math. Stat. 23 (2012), 297–311.10.1007/978-3-642-27440-4_14Search in Google Scholar
[6] B. Coifman, Empirical flow-density and speed-spacing relationships: evidence of vehicle length dependency. Transportation Research Part B78 (2015), 54–65.10.1016/j.trb.2015.04.006Search in Google Scholar
[7] M. S. Ivanov and S. V. Rogasinsky, The Direct Statistical Simulation Method in Dilute Gas Dynamics. CC SB USSR AS, Novosibirsk, 1988 (in Russian).Search in Google Scholar
[8] M. A. Korotchenko and A. V. Burmistrov, The Monte Carlo algorithms for solving the Smoluchowski equation with random coefficients. In: Mathematical and Computer Modeling of Natural Scientific and Social Problems. 2019, pp. 35–40 (in Russian).Search in Google Scholar
[9] M. A. Korotchenko, G. A. Mikhailov, and S. V. Rogazinskii, Value modifications of weighted statistical modelling for solving nonlinear kinetic equations. Russ. J. Numer. Anal. Math. Modelling22 (2007), No. 5, 471–486.10.1515/rnam.2007.023Search in Google Scholar
[10] M. A. Marchenko and G. A. Mikhailov, Distributed computing by the Monte Carlo method. Automation and Remote Control68 (2007), No. 5, 888–900.10.1134/S0005117907050141Search in Google Scholar
[11] G. A. Mikhailov, Efficient Monte Carlo algorithms for evaluating the correlation characteristics of conditional mathematical expectations. USSR Comp. Math. Math. Phys. 17 (1977) No. 1, 244–247 (in Russian).10.1016/0041-5553(77)90088-XSearch in Google Scholar
[12] G. A. Mikhailov, Randomized Monte Carlo algorithms for problems with random parameters (‘double randomization’ method). Numer. Anal. Appl. 12 (2019), No. 2, 155–165.10.1134/S1995423919020058Search in Google Scholar
[13] G. A. Mikhailov and S. V. Rogazinskii, Weighted Monte Carlo methods for approximate solution of a nonlinear Boltzmann equation. Sib. Math. J. 43 (2002) No. 3, 496–503.10.1023/A:1015467719806Search in Google Scholar
[14] G. A. Mikhailov and A. V. Voitishek, Numerical Statistical Modelling. Monte Carlo Methods. Akademia, Moscow, 2006 (in Russian).Search in Google Scholar
[15] S. V. Rogazinskii, Weighted statistical modelling algorithms with branching and extension of a model ensemble of interacting particles. Russ. J. Numer. Anal. Math. Modelling31 (2016) No. 4, 207–215.10.1515/rnam-2016-0021Search in Google Scholar
[16] E. Roidl, B. Frehse, and R. Höger, Emotional states of drivers and the impact on speed, acceleration and traffic violations – a simulator study. Accident Analysis and Prevention70 (2014), 282–292.10.1016/j.aap.2014.04.010Search in Google Scholar
[17] E. I. Vlahogianni, M. G. Karlaftis, and J. C. Golias, Short-term traffic forecasting: where we are and where we are going. Transportation Research Part C43 (2014) No. 1, 3–19.10.1016/j.trc.2014.01.005Search in Google Scholar
[18] K. T. Waldeer, The direct simulation Monte Carlo method applied to a Boltzmann-like vehicular traffic flow model. Computer Physics Communications156 (2003) No. 1, 1–12.10.1016/S0010-4655(03)00368-0Search in Google Scholar
[19] K. T. Waldeer, A vehicular traffic flow model based on a stochastic acceleration process. Transport Theory and Statistical Physics33 (2004) No. 1, 7–30.10.1081/TT-120030341Search in Google Scholar
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