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Stochastic description of traffic flow. (English) Zbl 1161.82324
Summary: We propose a traffic model based on microscopic stochastic dynamics. We built a Markov chain equipped with an Arrhenius interaction law. The resulting stochastic process is comprised of both spin-flip and spin-exchange dynamics which models vehicles exiting, entering and interacting in a two-dimensional lattice environment corresponding to a multi-lane highway. The process is further equipped with a novel look-ahead type, anisotropic interaction potential which allows drivers/vehicles to ascertain local fluctuations and advance to new cells forward or sideways. The resulting vehicular traffic model is simulated via kinetic Monte Carlo and examined under both, typical and extreme traffic flow scenarios. The model is shown to correctly predict both qualitative as well as quantitative traffic observables for any highway geometry. Furthermore it also captures interesting multi-scale phenomena in traffic flows after a simulated accident which lead to oscillatory, dissipating, traffic waves with different periods per lane.
MSC:
82C20Dynamic lattice systems and systems on graphs
90B20Traffic problems
82C80Numerical methods of time-dependent statistical mechanics
60J10Markov chains (discrete-time Markov processes on discrete state spaces)
References:
[1]Akcelic: Traffic models-research and software for the transport industry (2007). http://www.sidrasolutions.com , accessed Oct. 10
[2]Athol, P.: Interdependence of certain operational characteristics within a moving traffic stream. In: Highway Research Record, pp. 58–97 (1972)
[3]Aw, A., Rascle, M.: Resurrection of ”second order” models of traffic flow. SIAM J. Appl. Math. 60, 916 (2000) · Zbl 0957.35086 · doi:10.1137/S0036139997332099
[4]Bortz, A.B., Kalos, M.H., Lebowitz, J.L.: A new algorithm for Monte Carlo simulations of Ising spin systems. J. Comput. Phys. 17, 10 (1975) · doi:10.1016/0021-9991(75)90060-1
[5]Daganzo, C.F.: Requiem for second-order fluid approximations of traffic flow. Transp. Res. B 29, 277 (1995) · doi:10.1016/0191-2615(95)00007-Z
[6]Hall, F.L.: Traffic Flow Theory, pp. 2–34. US Federal Highway Administration, Washington (1996)
[7]HCM: Highway capacity manual. Tech. rep., Transportation Research Board (1985)
[8]HCM: Highway capacity manual. Tech. rep., Transportation Research Board, Washington, DC (2000)
[9]Helbing, D.: Gas-kinetic derivation of Navier-Stokes-like traffic equations. Phys. Rev. E 53(3), 2366 (1995) · doi:10.1103/PhysRevE.53.2366
[10]Helbing, D.: Modeling multi-lane traffic flow with queuing effects. cond-mat.stat-mech (1998)
[11]Helbing, D.: Traffic and related self-driven many-particle systems. Rev. Mod. Phys. 73(4), 1067 (2001). cond-mat/0012229 · doi:10.1103/RevModPhys.73.1067
[12]Helbing, D., Hennecke, A., Shvetsov, V., Treiber, M.: Micro and macro simulation of freeway traffic. Math. Comput. Model. 35, 517 (2002) · Zbl 0994.90025 · doi:10.1016/S0895-7177(02)80019-X
[13]Hildebrand, M., Mikhailov, A.S.: J. Phys. Chem. 100, 19089 (1996) · doi:10.1021/jp961668w
[14]Illner, R., Klar, A., Materne, T.: Vlasov-Fokker-Planck models for multilane traffic flow. Commun. Math. Sci. 1, 1 (2003)
[15]Jiang, R., Wu, Q.S.: Cellular automata models for synchronized traffic flow. J. Phys. A 36(2), 281 (2003)
[16]Jin, S., Liu, J.G.: Relaxation and diffusion enhanced dispersive waves. Proc. R. Soc. Lond. A 446, 555–563 (1994) · Zbl 0816.76031 · doi:10.1098/rspa.1994.0120
[17]Kanai, M., Nishinari, K., Tokihiro, T.: Stochastic optimal velocity model and its long-lived metastability. Phys. Rev. E 72 (2005)
[18]Katsoulakis, M., Sopasakis, A., Plechac, P.: Error analysis of coarse-graining for stochastic lattice dynamics. SIAM J. Numer. Anal. (2006)
[19]Kerner, B.S., Klenov, S.L.: A microscopic model for phase transitions in traffic flow. J. Phys. A 35, 31 (2002) · Zbl 1061.90021 · doi:10.1088/0305-4470/35/47/303
[20]Kerner, B.S., Klenov, S.L., Wolf, D.E.: Cellular automata approach to three-phase traffic theory. J. Phys. A: Math. Gen. 35, 9971 (2002) · Zbl 1061.90021 · doi:10.1088/0305-4470/35/47/303
[21]Klar, A., Wegener, R.: A hierarchy of models for multilane vehicular traffic i: modeling. SIAM J. Appl. Math. 59(3), 983–1001 (1995) · Zbl 1009.90019 · doi:10.1137/S0036139997326946
[22]Kurtze, D.A., Hong, D.S.: Traffic jams, granular flow, and soliton selection. Phys. Rev. E 52, 218–221 (1995) · doi:10.1103/PhysRevE.52.218
[23]Liggett, T.M.: Interacting Particle Systems. Springer, Berlin (1985)
[24]Lubashevsky, I.A., Mahnke, R.: Order parameter model for unstable multilane traffic flow. cond-math/9910268 v2 (2000)
[25]Masi, A.D., Orlandi, E., Pressuti, E., Triolo, L.: Proc. R. Soc. Edinb. A 124, 1013 (1994)
[26]McShane, W.R., Roess, R.P.: Traffic Engineering. Prentice-Hall, Englewood Cliff (1990)
[27]Muramatsu, M., Nagatani, T.: Phys. Rev. E 60, 180 (1999) · doi:10.1103/PhysRevE.60.180
[28]Nagatani, T.: Jamming transitions and the modified Korteweg-de Vries equation in a two-lane traffic flow. Physica A 265, 297–310 (1999) · doi:10.1016/S0378-4371(98)00563-9
[29]Nagatani, T.: Stabilization and enhancement of traffic flow by the next-nearest-neighbor interaction. Phys. Rev. E 60, 1535 (1999) · doi:10.1103/PhysRevE.60.1535
[30]Nagatani, T.: The physics of traffic jams. Rep. Prog. Phys. 65, 1331 (2002) · doi:10.1088/0034-4885/65/9/203
[31]Nagel, K., Schreckenberg, M.: A cellular automaton model for freeway traffic. J. Phys. I 2, 2221 (1992) · doi:10.1051/jp1:1992277
[32]Nagel, K., Wolf, D.E., Wagner, P., Simon, P.: Two-lane traffic rules for cellular automata: a systematic approach. Phys. Rev. E 58(2), 1425 (1998) · doi:10.1103/PhysRevE.58.1425
[33]Nelson, P.: Phys. Rev. E 61, 383 (2000) · doi:10.1103/PhysRevE.61.R6052
[34]Nelson, P.: On two-regime flow, fundamental diagrams and kinematic-wave theory. In progress (2004)
[35]Newell, G.F.: Nonlinear effects in theory of car following. Oper. Res. 9, 209–229 (1961) · Zbl 0211.52301 · doi:10.1287/opre.9.2.209
[36]Newell, G.F.: Transp. Res. B 23, 386 (1989) · doi:10.1016/0191-2615(89)90015-5
[37]Phillips, W.F.: Transp. Plann. Technol. 5, 131 (1979) · doi:10.1080/03081067908717157
[38]Rathi, A.K., Lieberman, E.B., Yedlin, M.: Transp. Res. Rec. 61, 1112 (1987)
[39]Ross, P.: Transp. Res. B 22, 421 (1988) · doi:10.1016/0191-2615(88)90023-9
[40]Schadschneider, A.: Traffic flow: a statistical physics point of view. Physica A 312, 153 (2002) · Zbl 0998.90018 · doi:10.1016/S0378-4371(02)01036-1
[41]Schreckenberg, M., Wolf, D.E.: Traffic and Granular Flow. Springer, Singapore (1998)
[42]Sopasakis, A.: Unstable flow theory and modeling. Math. Comput. Model. 35(5–6), 623 (2002)
[43]Sopasakis, A.: Stochastic noise approach to traffic flow modeling. Physica A 342(3-4), 741–754 (2004) · doi:10.1016/j.physa.2004.05.040
[44]Sopasakis, A., Katsoulakis, M.A.: Stochastic modeling and simulation of traffic flow: ASEP with Arrhenius look-ahead dynamics. SIAM J. Appl. Math. 66(3), 921–944 (2006) · Zbl 1141.90014 · doi:10.1137/040617790
[45]Sparmann, U.: Spurwechselvorgänge auf zweispurigen Bab-Richtungsfahrbahnen. In: Forschung Straßenbau und Straßenverkehrstechnik. Bundesminister für Verkehr, Bonn-Bad Godesberg (1978)
[46]Spohn, H.: Large scale dynamics of interacting particles. Springer, Berlin (1991)
[47]Vlachos, D.G., Katsoulakis, M.A.: Derivation and validation of mesoscopic theories for diffusion of interacting molecules. Phys. Rev. Lett. 85(18), 3898 (2000) · doi:10.1103/PhysRevLett.85.3898
[48]Wardrop, J.G.: Some theoretical aspects of road traffic research. Proc. Inst. Civ. Eng. Part II I, 325 (1952)
[49]Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1974)
[50]Wiedemann, R.: Simulation des Straßenverkehrsflusses. Schriftenreihe des Instituts für verkenhrswesen der Universität Karlsruhe, vol. 8, Germany (1974)
[51]Wright, P.H., Dixon, K.: Highway Engineering, 7th edn. New Jersey (2004)