AUTHORS: Alexander Tatashev, Marina Yashina

Download as PDF

This paper considers a dynamical system of Buslaev contour network type, containing two contours. There are Ni cells in the contour i, i = 1, 2. There is a common point of all contours. This point is called a node. There are M particles in the system. At any time t = 0, 1, 2, . . . , each particle occupies a cell. No cell can be occupied by more than one particle simultaneously. The particles move in a given direction. At any step, each particle moves onto one cell forward if the cell ahead is vacant. If two particles come to the node simultaneously, then a competition of these particles occurs, and only one particle moves. This particle is chosen in accordance with a deterministic or stochastic competition resolution rule. After completing the movement in the contour i, the particle moves in the contour j with probability αij , i, j = 1, 2. We say that the system is in the state of free movement if all particles move without delays at the present moment and in the future. We have obtained the conditions for the system to result in a state of free movement over a time interval with a finite expectation.

KEYWORDS: -Dynamical systems, cellular automata, traffic models, self-organization, contour networks

REFERENCES:

[ 1] M. Schreckenberg, A. Schadschneider, K. Nagel and N. Ito, Discrete stochastic models for traffic flow Phys. Rev. E51, 1995, pp. 2939–249.

[2] V. Belitsky and P.A. Ferrari, Invariant measures and convergence properties for cellular automation 184 and related processes, J. Stat.Phys.118(3), 2005, pp. 589–623.

[3] M. L. Blank, Exact analysis of dynamical systems arising in models of flow traffic, Russian Mathematical Reviews 55(5), 2005, pp. 562–563.

[4] S. Wolfram, Statistical mechanics of cellular automata, Rev. Mod. Phys., 55, 1983, pp. 601–644.

[5] L. Gray and D. Griffeath, The ergodic theory of traffic jams, J. Stat. Phys.105(3/4), 2001, pp. 413— 452

[6] M. Blank, Metric properties of discrete time exclusion type processes in continuum, J. Stat. Phys. 140(1), pp. 170-–197.

[7] O. Biham, A. A. Middleton and D. Levine, Selforganization and a dynamical transition in trafficflow models J. Phys. Rev. 46(10), 1992, R6124– R61278.

[8] T. Austin and I. Benjamini, For what number of cars must self-organization occur in the Biham– Middleton–Levine traffic model from any possible starting configuration? 2006, arXiv.math/0607759

[9] R. M. D’Souza, Coexisting phases and lattice dependence of a cellular automaton model for traffic flow, J. Stat. Phys. 71(6): 066112.

[10] O. Angel, A.E. Horloyd and J.–B. Martin, The jammed phase of the Biham-Middleton-Levine traffic model, Electronic Communication in Probability 10, pp. 167–178.

[11] H. Moradi, A. Zardadi and Z. Heydarbeigi. The number of collisions in Biham–Middelion–Levine on a square lattice with limited number of cars. Applied Mathematics E-Notes, 19, 2019, pp. 243– 249.

[12] A.S. Bugaev, A.P. Buslaev, V.V. Kozlov and M.V. Yashina, Distributed problems of monitoring and modern approaches to traffic modeling,14th International IEEE Conference on Intelligent Transactions Systems (ITSC 2011), Washington, USA, 5-7.10.2011, 2011, pp. 477–481.

[13] A.P. Buslaev and A.G. Tatashev. Exact results for discrete dynamical systems on a pair of contours Math. Meth. Appl. Sci. 41(17), 2018, pp. 7283– 7294.

[14] A.P. Buslaev, A.V. Gasnikov and M.V. Yashina. Selected mathematical problems of traffic flow theory, International Journal of Computer Mathematics 89(3), 2012, pp. 409–432.

[15] V.V. Kozlov, A.P. Buslaev and A.G. Tatashev, On synergy of totally connected flow on chainmails CMMSE-2013Cadis Spain 3, 2013, pp. 861–873.

[16] A.P. Buslaev, M.Yu. Fomina, A.G. Tatashev and M.V. Yashina. On discrete flow networks model spectra: statement, simulation, hypotheses. J. Phys. Conf. Ser.1053, 2018, 012034.

[17] A.P. Buslaev and A.G. Tatashev, Spectra of local cluster flows on open chain of contours European J. of Pure and Applied Math.11(3), 2018,pp. 628–641.

[18] V.V. Kozlov., A.P. Buslaev and A.G. Tatashev, Monotonic walks on a necklace and a coloured dynamic vector Int. J. Comput. Math. 92(9), 2015, pp. 1910-1920.

[19] V.V. Kozlov, A.P. Buslaev, A.G. Tatashev and M.V. Yashina. Dynamical systems on honey-combs Traffic and Granular Flow ’13, Springer, 2015, Part II, pp. 441–452.

[20] M.J. Fomina, D.A. Tolkachev, A.G. Tatashev and M.V. Yashina, Cellular automata as traffic models and spectrum of two-dimensional con-tour networks open chainmail, Proceedings of the 2018 International Conference ”Quality Management, Transport and Information Security, Information Technologies”, IT and QM and IS no. 8525079, 2018, pp. 435–440.

[21] A.G. Tatashev and M.V. Yashina, Spectrum of Elementary Cellular Automata and Closed Chains of Contours Machines 7(2), 2019, 28.

[22] A.P. Buslaev, A.G. Tatashev and M.V. Yashina, Flows spectrum on closed trio of contours European J. of Pure and Applied Math.11(1), 2018, pp 260– 283.

[23] A.G. Tatashev and M.V. Yashina, Spectrum of continuous two-contours system, ITM Web of Conferences 24, 2019, 01014.

[24] A.G. Tatashev and M.V. Yashina, Behavior of continuous two-contours system WSEAS Transactions on Mathematics18, art. #5, 2019, pp. 37–45.

[25] A.G. Tatashev and M.V. Yashina, Spectrum off lows on discrete pair of contours J. Phys. Conf.Ser.1324, 2019, 012001.

[26] A.P. Buslaev, A.G. Tatashev and M.V. Yashina, About synergy of flows on flower Dependability Engineering and Complex Systems. Proceedings of the Eleventh International Conference on Dependability and Complex Systems DepCoS-– RELCOMEX, June 27-–July 1, 2016, Brunow, Poland, vol. 470 of series Advances in Intelligent Systems and Computing Springer, 2016, pp. 75-84.

[27] A.P. Buslaev and A.G. Tatashev, Flows on discrete traffic flower Journal of Mathematical Research 9(1), pp. 98–108.

[28] A. Buslaev, A. Zernov, P. Sokolov and M. Yashina, Computer network traffic models: research, hypotheses, results. In proc. of the2015 International Conference on Pure Mathematics, Applied Mathematics and Computational Methods (PMAMCM 2015), 2015.

[29] A.A. Buchstab, Number Theory, Kniga po Trebovaniyu, Moscow, 2002 (In Russian.)