交通流模型的研究
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摘要
非平衡态统计力学常被用来研究发生在物理、化学、生物，甚至社会和经济过程中的宏观行为。非平衡态现象出现在系统向平衡态弛豫的过程中以及受驱动系统(系统在外力或自身驱动力驱动下维持在偏离平衡态的状态)过程中。自驱动多粒子系统是我们这篇文章的主要内容，一般情况下不能用平衡态统计力学来描述。这些系统一般会演化到一个非平衡的稳态。和已经充分理解的平衡稳态相比，远离平衡态的稳态的研究才刚刚开始。
自驱动多粒子系统的驱动力是每个粒子自身产生的而不是由外部提供的。作为典型的自驱动远离平衡态多粒子体系，交通流的理论研究可以促进统计物理、非线性动力学、应用数学、流体力学及交通工程学等学科的交叉和发展，可以加深对远离平衡态多体相互作用粒子演化规律的认识。因此，进行交通流研究不仅仅具有工程上的意义，而且还有其深远的科学意义。
交通动力学系统是非线性相互作用复杂系统的典型例子，通常用确定性轨迹来描述。但了解涨落现象的形成原因，以及涨落对全局交通的重要性是非常必要的。描述交通流的模型主要包括：宏观连续模型，介观气体动理论模型，微观跟驰模型、元胞自动机模型，还包括几率交通流理论。
本论文主要工作内容如下：
元胞自动机模型是方便于计算机模拟的模型。然而，由于有限尺寸效应和数值噪音，解析研究也是重要的，它提供了对数值计算结果的检验。我们采用以相邻车辆距离为变量的全局动力学演化的方法，计算各不同长度车距之间的跃迁概率，运用统计力学中的稳态条件，解析研究了Fukui-Ishibashi(FI)加速规则的Nagel-Schreckenberg(NS)元胞自动机模型，获得了平均速度和车流量与车辆密度在不同随机延迟概率下的关系，和数值模拟结果符合的很好。我们得到的一维交通流元胞自动机模型的平均场方程，为交通流复杂系统的自组织临界性和相变行为提供了基本的物理理解。与此相关的研究结果发表于：Chaos，Solitons & Fractals，31，Issue 3，772-776(2007)。具体内容见第三章第二节。
多个体资讯的分析及经验的反馈——判断——再适应是一个多个体复杂适应系统的本质性质，真实的交通行为是与环境信息紧密相关的。从广义角度看，我们每天都会遇到各种各样的问题，需要我们依据有限的信息进行合理的决策。城市的交通状况，就是由多个个体的决策所产生的宏观效应。有趣的是，尽管大多数人所考虑的是自己的利益，很少会有人故意走一条可以让别人省时的路，但很多时候多个体分别争取对自己最有利的安排时，却也很好地利用了整体的资源。通过合理地利用反馈信息，道路的使用效率会变得更高。可以看到多个体资讯的分析及经验的反馈、判断、再适应，最后合理地利用反馈信息是多体复杂适应系统的本质。交通流明显带有以上特性，研究复杂自适应系统与交通流之间的关系和相似之处，是智能交通研究的一个重要分支。我们利用双通道决策模型，研究了智能决策的重要性。数值模拟的结果表明，如果机械地利用反馈信息反而使得道路上的车辆出现了人们不希望看到的不稳定现象，即道路的拥挤程度随时间而震荡，致使系统的利用效率下降，而合理地利用反馈信息使得系统效率得到明显的提高。因此，应在尽可能地提供好的反馈信息的同时合理地利用它们。研究结果有助于对智能决策的重要性有更好的了解。与此相关的研究结果发表于：物理学报，vol．55，No．8，4032-4038(2006)。具体内容见第四章第一节。
更进一步，我们利用平均场近似的方法对依据信息反馈进行决策的双通道交通流问题进行了解析研究，理论结果和数值模拟近似符合，使我们更清晰的了解了信息反馈在决策过程中所起的作用。有助于我们制定好的交通管理策略。具体内容见第四章第二节。
实际交通的加速减速不对称的特征是避免碰撞的重要因素，据此在前人工作的基础上提出了一个不对称全速度差交通流模型，该模型给出的车辆运动延迟时间和车辆启动波速和实测数据符合的很好，而且可以描述微扰下交通逐渐失稳并最终形成时走时停交通的相变，给出了更复杂的迟滞效应。具体内容见第五章。
The statistical mechanics of nonequilibrium systems is required for understanding the macroscopic behaviors of processes occurring throughout physics, chemistry, biology and even sociology and economics. Nonequilibrium phenomena are encountered whenever systems are relaxing towards an equilibrium steady state and also whenever systems are driven i. e., maintained away from equilibrium by external or self-driven forces. Systems of the self-driven kind, which are the main focus of this work, cannot be described by equilibrium statistical mechanics in general. These systems evolve to a nonequilibrium steady state. Though the statistical mechanics of equilibrium steady states are well understood, analogous general principles to guide the study of steady states far from equilibrium are just beginning.
In self-driven many-particle systems, the driving force is not of exerted from outside, but associated with each single particle and self-produced. The traffic system is a typical self-driven system which is far from equilibrium. The study of traffic theory may help to promote the development and cross of such subjects as statistical physics, nonlinear dynamics, applied mathematics, fluid mechanics, traffic engineering and so on, and to better understand the evolution laws of many-particle systems which are far from equilibrium. Therefore, it is not only important for engineering application but also of scientifically significant to study the traffic flow theory.
Vehicular traffic dynamics, usually described by deterministic trajectories, is a representative example of nonlinear complex systems. However, we believe it is necessary how fluctuating phenomena arise and how important they are. The various kinds of traffic flow models can be classified into macroscopic continuum traffic models, gas-kinetic traffic models, microscopic models including car-following models and cellular automata models, and probabilistic description of traffic flow. The contents of the paper are as follows.
Cellular automata (CA), by design, ideal for computer simulations. However, one can not deny the importance of exact analytical results in providing a testing ground for the computer codes with the finite-size effects and "numerical noise" produced by computer simulations. Choosing the length of inter-car spacing as the dynamical variables and study the global evolution of these spacings. We study a one-dimensional traffic flow cellular automaton model of high-speed vehicles with the Fukui - Ishibashi-type (FI) acceleration rule for all cars, and the Nagel - Schreckenberg-type (NS) stochastic delay mechanism. We obtain analytically the fundamental diagrams of the average speed and vehicle flux depending on the vehicle density and stochastic delay probability.
Intelligent decision-making based on the information feedback in a two route traffic flow model is studied. When compared to the mechanical decision-making which induces an oscillation and low efficiency of system, intelligent decision-making will contribute to an improvement of efficiency. Our results suggest that one should not only devote to obtainment of optimal information feedback, but also make an intelligent decision based on information feedback at hand.
Furthermore, we obtain analytical results upon the decision dynamics in a two route traffic flow model by using Mean Field Approximation method. Our results are closely consistent with the simulation ones and clearly characterize the effect of the information feedback on the decision-making process. The result is useful for establishing traffic management strategies.
The characteristic of asymmetry between deceleration and acceleration in realistic traffic is one important factor to prevent collisions. Based on the earlier works, we propose an asymmetric full velocity difference traffic model. Delay times and the kinematic wave speed of car motions from our model are well consistent with the empirical data. Furthermore, under initial disturbance, our model can describe the phase transition from stable traffic to an unstable one which corresponds to stop-and-go traffic. Additionally, more complex hysteresis effect occurs in our model.
自驱动多粒子系统的驱动力是每个粒子自身产生的而不是由外部提供的。作为典型的自驱动远离平衡态多粒子体系，交通流的理论研究可以促进统计物理、非线性动力学、应用数学、流体力学及交通工程学等学科的交叉和发展，可以加深对远离平衡态多体相互作用粒子演化规律的认识。因此，进行交通流研究不仅仅具有工程上的意义，而且还有其深远的科学意义。
交通动力学系统是非线性相互作用复杂系统的典型例子，通常用确定性轨迹来描述。但了解涨落现象的形成原因，以及涨落对全局交通的重要性是非常必要的。描述交通流的模型主要包括：宏观连续模型，介观气体动理论模型，微观跟驰模型、元胞自动机模型，还包括几率交通流理论。
本论文主要工作内容如下：
元胞自动机模型是方便于计算机模拟的模型。然而，由于有限尺寸效应和数值噪音，解析研究也是重要的，它提供了对数值计算结果的检验。我们采用以相邻车辆距离为变量的全局动力学演化的方法，计算各不同长度车距之间的跃迁概率，运用统计力学中的稳态条件，解析研究了Fukui-Ishibashi(FI)加速规则的Nagel-Schreckenberg(NS)元胞自动机模型，获得了平均速度和车流量与车辆密度在不同随机延迟概率下的关系，和数值模拟结果符合的很好。我们得到的一维交通流元胞自动机模型的平均场方程，为交通流复杂系统的自组织临界性和相变行为提供了基本的物理理解。与此相关的研究结果发表于：Chaos，Solitons & Fractals，31，Issue 3，772-776(2007)。具体内容见第三章第二节。
多个体资讯的分析及经验的反馈——判断——再适应是一个多个体复杂适应系统的本质性质，真实的交通行为是与环境信息紧密相关的。从广义角度看，我们每天都会遇到各种各样的问题，需要我们依据有限的信息进行合理的决策。城市的交通状况，就是由多个个体的决策所产生的宏观效应。有趣的是，尽管大多数人所考虑的是自己的利益，很少会有人故意走一条可以让别人省时的路，但很多时候多个体分别争取对自己最有利的安排时，却也很好地利用了整体的资源。通过合理地利用反馈信息，道路的使用效率会变得更高。可以看到多个体资讯的分析及经验的反馈、判断、再适应，最后合理地利用反馈信息是多体复杂适应系统的本质。交通流明显带有以上特性，研究复杂自适应系统与交通流之间的关系和相似之处，是智能交通研究的一个重要分支。我们利用双通道决策模型，研究了智能决策的重要性。数值模拟的结果表明，如果机械地利用反馈信息反而使得道路上的车辆出现了人们不希望看到的不稳定现象，即道路的拥挤程度随时间而震荡，致使系统的利用效率下降，而合理地利用反馈信息使得系统效率得到明显的提高。因此，应在尽可能地提供好的反馈信息的同时合理地利用它们。研究结果有助于对智能决策的重要性有更好的了解。与此相关的研究结果发表于：物理学报，vol．55，No．8，4032-4038(2006)。具体内容见第四章第一节。
更进一步，我们利用平均场近似的方法对依据信息反馈进行决策的双通道交通流问题进行了解析研究，理论结果和数值模拟近似符合，使我们更清晰的了解了信息反馈在决策过程中所起的作用。有助于我们制定好的交通管理策略。具体内容见第四章第二节。
实际交通的加速减速不对称的特征是避免碰撞的重要因素，据此在前人工作的基础上提出了一个不对称全速度差交通流模型，该模型给出的车辆运动延迟时间和车辆启动波速和实测数据符合的很好，而且可以描述微扰下交通逐渐失稳并最终形成时走时停交通的相变，给出了更复杂的迟滞效应。具体内容见第五章。
The statistical mechanics of nonequilibrium systems is required for understanding the macroscopic behaviors of processes occurring throughout physics, chemistry, biology and even sociology and economics. Nonequilibrium phenomena are encountered whenever systems are relaxing towards an equilibrium steady state and also whenever systems are driven i. e., maintained away from equilibrium by external or self-driven forces. Systems of the self-driven kind, which are the main focus of this work, cannot be described by equilibrium statistical mechanics in general. These systems evolve to a nonequilibrium steady state. Though the statistical mechanics of equilibrium steady states are well understood, analogous general principles to guide the study of steady states far from equilibrium are just beginning.
In self-driven many-particle systems, the driving force is not of exerted from outside, but associated with each single particle and self-produced. The traffic system is a typical self-driven system which is far from equilibrium. The study of traffic theory may help to promote the development and cross of such subjects as statistical physics, nonlinear dynamics, applied mathematics, fluid mechanics, traffic engineering and so on, and to better understand the evolution laws of many-particle systems which are far from equilibrium. Therefore, it is not only important for engineering application but also of scientifically significant to study the traffic flow theory.
Vehicular traffic dynamics, usually described by deterministic trajectories, is a representative example of nonlinear complex systems. However, we believe it is necessary how fluctuating phenomena arise and how important they are. The various kinds of traffic flow models can be classified into macroscopic continuum traffic models, gas-kinetic traffic models, microscopic models including car-following models and cellular automata models, and probabilistic description of traffic flow. The contents of the paper are as follows.
Cellular automata (CA), by design, ideal for computer simulations. However, one can not deny the importance of exact analytical results in providing a testing ground for the computer codes with the finite-size effects and "numerical noise" produced by computer simulations. Choosing the length of inter-car spacing as the dynamical variables and study the global evolution of these spacings. We study a one-dimensional traffic flow cellular automaton model of high-speed vehicles with the Fukui - Ishibashi-type (FI) acceleration rule for all cars, and the Nagel - Schreckenberg-type (NS) stochastic delay mechanism. We obtain analytically the fundamental diagrams of the average speed and vehicle flux depending on the vehicle density and stochastic delay probability.
Intelligent decision-making based on the information feedback in a two route traffic flow model is studied. When compared to the mechanical decision-making which induces an oscillation and low efficiency of system, intelligent decision-making will contribute to an improvement of efficiency. Our results suggest that one should not only devote to obtainment of optimal information feedback, but also make an intelligent decision based on information feedback at hand.
Furthermore, we obtain analytical results upon the decision dynamics in a two route traffic flow model by using Mean Field Approximation method. Our results are closely consistent with the simulation ones and clearly characterize the effect of the information feedback on the decision-making process. The result is useful for establishing traffic management strategies.
The characteristic of asymmetry between deceleration and acceleration in realistic traffic is one important factor to prevent collisions. Based on the earlier works, we propose an asymmetric full velocity difference traffic model. Delay times and the kinematic wave speed of car motions from our model are well consistent with the empirical data. Furthermore, under initial disturbance, our model can describe the phase transition from stable traffic to an unstable one which corresponds to stop-and-go traffic. Additionally, more complex hysteresis effect occurs in our model.
引文
[1] B. S. Kemer, H. Rehborn, Experimential properties of phase transitions in traffic flow. Phys.Rev.Lett. 79, 4030～4033(1997).
[2] 唐孝威，张训生，陆坤权，交通流与颗粒流，浙江大学出版社 (2004)．
[3] W. Knospe, L. Santen, A. Schadschneider, and M. Schreckenberg, Empirical test for cellular automaton models of traffic flow. Phys.Rev.E 70, 016115(2004).
[4] M. Bando, K. Hasebe. A. Nakayama, A. Shibata, and Y.Sugiyama, Dynamical model of traffic congestion and numerical simulation. Phys. Rev. E 51, 1035 (1995).
[5] B. S. Kerner, H. Rehborn, Expermental features and characteristics of traffic jam. Phys.Rev.E 53, R 1297～R1300(1996).
[6] B. S. Kerner, H. Rehbom, Experimential properties of complexity in traffic flow. Phys.Rev.E 53, R4275～R4278(1996).
[7] 姜锐，交通流复杂动态特性的微观和宏观模式研究，中国科学技术大学博士学位论文 (2002)．
[8] L. Neubert, L. Santen, A. Schadschneider, M. Schreckenberg, Single-vehicle data of highway traffic: A statistical analysis. Phys. Rev. E 60, 6480(1999).
[9] B. S. Kerner, Experimental features of self-organization in traffic flow. Phys.Rev.Lett. 81, 3797～3800(1997).
[10] R.Mahnke, J.Kaupuzs, I.Lubashevsky, Probabilistic description of traffic flow. Phys.Rep.408, 1-130(2005).
[11] D.Chowdhury, L.Santen, A.Schadschneider, Statistical physics of vehicular traffic and some related system. Phys.Rep.329, 199-329(2000).
[12] M. Schonhof, D.Helbing, Empirical features of congested traffic states and their implications for traffic modeling. Cond-mat/0408138.
[13] D.Helbing, Master: macroscopic traffic simulation based on a gas-kinetic, non-local traffic model. Transport Res B 35, 180～211(2001).
[14] D.Helbing, Jams, waves and clusters. Science 282, 2001～2003(1998).
[15] T. Nagatani, Density wave in traffic flow. Phys.Rev.E 60, 6395(1999).
[16] M.Muramatsu and T. Nagatani, Soliton and kind jams in traffic flow with open boundariws. Phys.Rev. E 60, 180～187(1999).
[17] M.J.Lighthill, G.B.Whitham, On kinematic waves: Ⅱ. A theory of traffic flow on long crowed roads. Proceedings of the Royal Society, London, Ser. A 229, 317～345(1955).
[18] P.I.Richards, Shock waves on the highway. Oper. Res. 4, 42～51 (1956).
[19] H.J.Payne, Models of freeway traffic and control. In: Bekey, G.A.(Ed.), Mathematical Models of Public Systems, Vol. 1, 51～61(Simulation Council, La Jolla, 1971).
[20] C.F.Daganzo, Requiem for second-order fluid approximation of traffic flow. Transpn. Res. B 29, 277～286(1995).
[21] R.Jiang, Q.S.Wu, Z.J.Zhu, A new continuum model for traffic flow and numerical tests. Transpn. Res. B 36, 405～419(2002).
[22] R.Jiang, Q.S.Wu, Z.J.Zhu, A new dynamics model for traffic flow. Chinese Sci. Bull. 46, 345～349(2001).
[23] I.Prigogine, R.Herman, Kinetic theory of vehicular traffic. (American Elsevier, New York, 1971).
[24] S.L.Paveri-Fontana, On Boltzmann-like treatments for traffic flow. A critical review of the basic model and an alternative proposal for dilute traffic analysis. Transpn. Res. 9, 225～235(1975).
[25] D.Helbing, Derivation and empirical validation of a refined traffic flow model. Physica A 233, 253～282(1996).
[26] M.Treiber, A.Hennecke, D.Helbing, Derivation, properties, and simulation of a gas-kinetic-based, nonlocal traffic model. Phys. Rev. E 59, 239～253(1999).
[27] L.A.Pipes, An operational analysis of traffic dynamics. J.Appl.Phys. 24, 274～287(1953).
[28] G.F.Newell, Nonlinear effects in the dynamics of car following. Oper. Res. 9, 209～229 (1961).
[29] L.C.Edie, Car following and steady state theory for non-congested traffic. Oper. Res. 9, 66～76 (1962).
[30] M.Bando, Dynamical model of traffic congestion and simulation. Phys.Rev.E 51, 1035(1995).
[31] D.Helbing, B.Tilch, Generalized force model of traffic dynamics. Phys.Rev.E 58, 133～138(1998).
[32] E.F.Codd, Cellular Automata, Acadenic Press, New York, (1968).
[33] M.Gardner, The fantastic bombination of John Conway's solitaire game life, Sci. Am 220(4), 120(1970).
[34] S.Wolfram, Statistical mechanics of cellular automata. Rev. Mod. Phys 55, 601～644(1983). S.Wolfram, Theory and Applications of Cellular Automata, World Scientific, Singapore,(1986). S.Wolfram, Cellular automaton fluids I: basic theory, J. Stat. Phys 45, 471～526(1986). Cellular Automata and Complexity, Addison-Wesley, Reading, MA, (1994).
[35] D. Stauffer, Computer-simulations of cellular automata, J.Phys. A 24, 909～927(1991).
[36] M.Cremer, J.Ludwig, A fast simulation model for traffic flow on the basis of Boolean operations, J.Math. Comp. Simul 28, 297～303(1986).
[37] Traffic and Granular flow '99, Edited by D.Helbing, M.Herrmann, M.Schreckenberg, and D.E.Wolf. (Springer, Berlin, 2000).
[38] A.Hennecke, et al., Macroscopic simulation of open systems and macro-micro link. In[37], 383～388.
[39] 汪秉宏，王雷，陈龙康，Fukui-Ishibashi 交通流基本图的解析研究，非线性动力学报 4，374(1997)。
[40] L. wang, B. H. Wang and P. M. Hui, Strict Derivation of Mean Field Equation for One-Dimensional Traffic Flow Model. Acta Physica Sinica(Overseas Edition) 6, 829(1997).
[41] B. H. Wang, L. Wang and P. M. Hui, One Dimensional Fukui-Ishibashi Traffic Flow Model. Journal of the Physical Society of Japan 66, 3683 (1997).
[42] 汪秉宏，王雷，许伯铭，胡斑比，福井-石桥交通流模型高密度区平均场方程，应用科学学报 Vol．17 No．2 142(1999)
[43] B. H. Wang, L. Wang, P. M. Hui and B.Hu, Analytical Results for the Steady State of Traffic Flow Models with Stochastic Delay, Physical Review E 55, 2876(1998)
[44] B. H. Wand, L. Wang, E M. Hui and B. Hu, Statistical Mechanical Approach to Phase Transition in Traffic Flow Model, 《STATPHYS 20》 The 20th IUPAP International Conference on Statistical Physics, (Paris, July 20-24, 1998) Oral contribution: Short Communications, Book of Abstracts, A Gervois, M Gingold, D Iagolnitzer, ed., Topic 2(Nonequilibrium Systems) T0791; PO02/145.
[45] 王雷，汪秉宏，决定论性逐步加速交通流模型的渐进稳态行为，物理学报 vol.48, No.5, 808(1999).
[46] B. H. Wang, L. Wang, P. M. Hui and B. Hu, Steady state traffic flow in a model with gradual acceleration and stochastic delay, 《ETOPIM5》 The 5th Inernational Conference on Electrical Transport and Optical Properties of Inhomogeneous Media (Hong Kong, June 21-25, 1999) Proceedings p.81.
[47] B. H. Wang, L. Wang, P. M. Hui and B. Hu, The asymptotic steady states of deterministic one-dimensional traffic flow models, Physica B 279, 237(2000).
[48] 王雷，一维交通流元胞自动机模型中自组织临界性及相变行为研究，中国科学技术大学博士学位论文(2000)。
[49] Chuan-Ji Fu, Bing-Hong Wang, Chuan-Yang Yin, Tao Zhou, Bo Hu, Kun Gao, P.M. Hui and Chin-Kun Hu, Analytical studies on a modified Nagel-Schreckenberg model with the Fukui-Ishibashi acceleration rule, Chaos, Solitons & Fractals, 31, Issue 3, 772-776 (2007)。
[50] 付传技，汪秉宏，殷传洋，高坤，利用智能决策的双通道交通流，物理学报 vol．55，No．8，4032-4038(2006)．
[51] 付传技，汪秉宏，殷传洋，高坤，陆玉风，双通道决策交通流的解析研究，全国博士生学术论坛(物理学)会议论文(2006)。
[52] 付传技，汪秉宏，高坤，龚华鑫，季莉，不对称全速度差模型，CCAST 复杂系统研究论坛会议论文(2005)。
[53] G.B.Whitham, Linear and Nonlinear Waves(Wiley, New York, 1974).
[54] T.Musha and H.Higuchi, Japanese Journal of Applied Physics 17, 811(1978).
[55] B.D.Greenshields, in Proceedings of the Fourteenth Annual Meeting of the Highway Research Board, edited by R.W.Crum(National Research Council, Washington, DC, 1935), Part Ⅰ.
[56] J.M.Burgers, Adv. Appl. Mech. 1, 171(1948).
[57] H. M. Zhang, A theory of nonequilibrium traffic flow. Transport Res B, 32, 485～498(1998).
[58] H. M. Zhang, A nonequilibrium traffic model devoid of gas-like behavior. Transport Res B, 36, 275～290(2002).
[59] I. Prigogine, A Boltzmann-like approach for traffic flow. Operat Res, 8, (1960).
[60] H.J.Payne, FREFLO. A macroscopic simulation model of freeway traffic. TPRR, 772, 68～75(1979).
[61] M.Papageorgiou, Applications of Automatic Control Concepts to Traffic Flow Modeling and Control(Springer, Heidelberg, Germany, 1983).
[62] R.D.Kuhne, in Proceeding of the 9th International Symposium on Transportation and Traffic Theory, edited by I.Volmuller and R.Hamerslag (VNU Science Press, Utrecht, The Netherlands, 1984).
[63] P.S.Babcock IV, D.M.Auslander, M.Tomizuka, and A.D.May, Transportation Research Record 971, 80(1984).
[64] A.K.Rathi, E.B.Lieberman, and M.Yedlin, Transportation Research Record 1112, 61(1987).
[65] P.Ross, Transportation Research B 22, 421 (1988).
[66] C.J.Leo, R.L.Pretty, Transportation Research B 26, 207(1992).
[67] M.Cremer and A.D.May, An Extended Traffic Model for Freeway Control (Research Report UCB-ITS-RR-85-7, Institute of Transportation Studies, University of California, Berkeley).
[68] H.J.Payne, in Research Directions in Computer Control of Urban Traffic Systems, edited by W.S.Levine, E.Lieberman, and J.J.Fearnsides (American Society of Civil Engineers, New York, 1979).
[69] H.J.Payne, in Mathematical Models of Public Systems, edited by G.A.Bekey(Simulation Council, La Jolla, CA, 1971), Vol. 1.
[70] R.D.Kuhne, in Proceedings of the 10th International Symposium on Transportation and Traffic Theory, edited by N.H.Garter and N.H.M.Wilson (Elsevier, New York, 1987).
[71] W.F.Phillips, Kinetic Model for Traffic Flow (Report No. DOT/RSPD/DPB/50-77/17, National Technical Information Service, Springfield, Virginia 22161, 1977).
[72] W.F.Phillips, Transportation Planning and Technology 5, 131(1979).
[73] R.D.Kuhne, Transportation Research Record 1320, 251(1991).
[74] B.S.Kerner and P.Konhauser, Cluster effect in initially homogeneous traffic flow. Phys.Rev.E 48, R2335～R2338(1993).
[75] B.S.Kerner and P.Konhauser, Structure and parameters of clusters in traffic flow. Phys.Rev.E 50, 54(1994).
[76] D.Helbing, Improved fluid-dynamic model for vehicular trffic. Phys.Rev.E 51, 3164(1995).
[77] R.D.Kuhne, Non-linearity stochastics of unstable traffic flow. In: Daganzo, C F(ED). Transportation and Traffic Flow Theory. Elsevier Science Publishers, 367～386(1994).
[78] B.S.Kerner, Determinstic spontaneous appearance of traffic jams in slightly inhomogenoous traffic flow. Phys.Rev.E51, R6243～R6246(1995).
[79] P.Berg, Optimal-velocity models of motorway traffic. PHD thesis, University of Bristol, (2001).
[80] D.A.Kurtz, and D.C.Hong, Traffic jam, granular flow and soliton selection. Phys.Rev.E52, 5574(1993).
[81] T.Komatasu, and S.Sasa, Kink solution charactering traffic congestion. Phys.Rev.E55, 5574～5581(1995).
[82] 戴世强，薛郁，交通流的建模和仿真，66～125(2004)．In[2]．
[83] D.Helbing, Gas-kinetic derivation of Navier-Stokes-like trffic equations. Phys.Rev.E53, 2366(1996).
[84] P.Nelson, Transport Theory and Statistical Physics 24, 383(1995).
[85] J.Keizer, Statistical Thermodynamics of Nonequilibrium Processes (Springer, New York, 1987).
[86] I.Prigogine, in Theory of Traffic Flow, edited by R.Herman (Elservier, Amsterdam, 1961).
[87] D.Helbing, Traffic and related self-driven many-particle systems. Rev Mod Phys, 73, (2001).
[88] R.E.Chandler, R.Herman and E.W.Montroll, Traffic dynamics: studies in car following. Oper Res, 6, 165～184(1958).
[89] R.Herman, R.B.Potts, Single lane traffic theory and experiment. In: Proceedings Symposium on theory of traffic flow. (Elservier, New York, 1961).
[90] M.Bando, K.Hasebe, A.Nakayama, et al, Analysis of optimal velocity model with explicit delay. Phys.Rev.E 58, 5429～5435(1998).
[91] M.J.Del Castillo, F.G.Benitez, On functional form of the speed-density relationship-Ⅰ: general theory, Ⅱ: empirical investigation. Transpn, Res, B, 29, 373～406(1995).
[92] R.Jiang, Q.S.Wu, Z.J.Zhu, Full velocity difference model for car-following theory. Phys.Rev.E 64, 017101(2001).
[93] B.Chopard, M.Droz, Cellular Automata Modelling of Physical Systems, (Cambridge University Press, Cambridge, 1998).
[94] 贾斌，交通流瓶颈处车流复杂动态特性的元胞自动机模拟，中国科学技术大学博士学位论文(2002)．
[95] K.Nagel, M.Schreckenberg, A cellular automaton model for freeway traffic. J Phys I France, 2, 2221～2233(1992).
[96] M.Fukui, and Y.Ishibashi, Traffic flow in 1D cellular automaton model including cars moving with high speed. J Phys Soc Japan, 65, 1868～1870(1996).
[97] O.Biham, A.Middleton, and D.Levine. Self-organization & a dynamic transition in traffic flow models. Phys. Rev.A, 46, R6124～R6127(1992).
[98] M.Takayasu, H.Takayasu, 1/f noise in a traffic model, Fractals 1, 860～866(1993).
[99] S.C.Benjamin, N.F.Johnson, P.M.Hui, Cellular automaton models of traffic flow along a highway containing a junction, J.Phys. A 29, 3119～3127(1996).
[100] R.Barlovic, L.Santen, A.Schreckenburg, Metastable states in cellular automata for traffic flow, Eur.Phys.J.B. 5, 793～800(1998).
[101] X.B.Li, Q.S.Wu, R.Jiang, Cellular automaton model considering the velocity effect of a car on the successive car, Phys.Rev.E 64, 066128(2001).
[102] W.Knospe, L.Santen, A.Schadscheider, et al., Towards a realistis microscopic description of highway traffic, J.Phys.A 33, L477～L485(1996).
[103] A.Schadschneider, M.Schrenkenberg, Cellular-automaton models and traffic flow, J.Phys.A 26, L679～L683(1996).
[104] A. Schadschneider, M.Schrenkenberg, Car-oriented mean-field theory for traffic flow models, J.Phys.A 30, L69～L75(1996).
[105] M.Schrenkenberg, A.Schadschneider, K.Nagel, et al., Discrete stochastic models for traffic flow, Phys.Rev.E 51, 2939～2949(1995).
[106] K.Nagel, D.E.Wolf, P.Wagner, et al., Two-lane traffic rules for cellular automata: A systematic approach, Phys.Rev.E 58, 1425～1437(1998).
[107] W.Knospe, L.Santen, A.Schadschneider, et al., A realistic two-lane traffic model for highway traffic, J.Phys.A 35, 3369～3388(2002).
[108] W.Knospe, L.Santen, A.Schadschneider, et al., Disorder effects in cellular automata for two-lane traffic, Physica A 265, 614～633(1999).
[109] P.Wagner, K.Nagel, D.E.Wolf, Realistic multi-lane traffic rules for cellular automata, Physica A 234, 687～698(1997).
[110] M.Rickert, K.Nagel, M.Schreckenberg, et al., Two lane traffic simulations using cellular automata, Physica A 231, 534～550(1996).
[111] K.Nagel, M.Rickert, Parallel implementation of the TRANSIMS micro-simulation, Parallel Comput 27, 1611～1639(2001).
[112] O.Biham, A.A.Middleton, D.A.Levine, Self-organization and a dynamical transition in traffic flow models, Phys.Rev.A 46, R6124～R6127(1992).
[113] T.Nagatani, Effect of traffic accident on jamming transition in traffic flow model, J.Phys.A 26, L1015～L1020(1993).
[114] T.Nagatani, Jamming transition in the traffic flow model with two-level crossings, Phys.Rev.E48, 3290～3294(1998).
[115] B.H.Wang, Y.F.Woo, P.M.Hui, Improved mean-field theory of two-dimensional traffic flow models, J.Phys.A 29, L31～L35(1996).
[116] B.H.Wang, Y.F.Woo, P.M.Hui, Mean field theory of traffic flow problems with overpasses and asymmetric distributions of cars, J.Physs.Soc.Jpn 65, 2345～2348(1996).
[117] G.Q.Gu, K.H.Chung, P.M.Hui, Two-dimensional traffic flow problems in inhomogeneous lattice, Physica A 217, 339～347(1995).
[118] K.H.Chung, P.M.Hui, G.Q.Gu, Two-dimensional traffic problems with faulty traffic lights, Phys.Rev.E 56, 772～774(1995).
[119] T.Nagatani, Effect of traffic accident on jamming transition in traffic flow model, J.Phys.A 26, 1015～1020(1993).
[120] T.Horiguchi, T.Sakakibara, Numerical simulations for traffic flow models on a decorated square lattice, Physica A 252, 388～404(1998).
[121] 方兆本，缪柏其，随机过程，中国科学技术大学出版社(2002)．
[122] L.Wang, B.H.Wang, Bambi Hu, Cellular automaton traffic flow model between the Fuikui-Ishibashi and Nagel-Schreckenberg models, Phys.Rev.E 63, 056117(2001).
[123] J.Wahle, A.L.C.Bazzan, EKlugl, M.Schreckenberg, Decision dynamics in a traffic scenario, Physica A 287, 669～681(2000).
[124] T.Nagatani, Self-organization and phase-transition in traffic-flow model of a 2-lane roadway, J.Phys.A 26, L781-L787(1993).
[125] T.Nagatani, Dynamical jamming transition induced by a car accident in traffic-flow of a 2-lane roadway, Physica A 202, 449-458(1994).
[126] M.Rickert, K.Nagel, D.E.Wolf, Realistic multi-lane traffic rules for cellular automata, Physica A 231, 534-550(1996).
[127] D.Chowdhury, D.E.wolf, M.Schreckenberg, Particle hopping models for two-lane traffic with two kinds of vehicles: effects of lane changing rules, Physica A 235, 417-439(1997).
[128] M.Rickert, Two Lane Traffic Simulation Using Cellular Automata, Report No.95, 213, (Center for Parallel Computing, University of Cologne, Germany, 1995).
[129] M.Herrmann, B.S.Kener, Local cluster effect in different traffic flow models. Physica A 255, 163-188(1998).
[2] 唐孝威，张训生，陆坤权，交通流与颗粒流，浙江大学出版社 (2004)．
[3] W. Knospe, L. Santen, A. Schadschneider, and M. Schreckenberg, Empirical test for cellular automaton models of traffic flow. Phys.Rev.E 70, 016115(2004).
[4] M. Bando, K. Hasebe. A. Nakayama, A. Shibata, and Y.Sugiyama, Dynamical model of traffic congestion and numerical simulation. Phys. Rev. E 51, 1035 (1995).
[5] B. S. Kerner, H. Rehborn, Expermental features and characteristics of traffic jam. Phys.Rev.E 53, R 1297～R1300(1996).
[6] B. S. Kerner, H. Rehbom, Experimential properties of complexity in traffic flow. Phys.Rev.E 53, R4275～R4278(1996).
[7] 姜锐，交通流复杂动态特性的微观和宏观模式研究，中国科学技术大学博士学位论文 (2002)．
[8] L. Neubert, L. Santen, A. Schadschneider, M. Schreckenberg, Single-vehicle data of highway traffic: A statistical analysis. Phys. Rev. E 60, 6480(1999).
[9] B. S. Kerner, Experimental features of self-organization in traffic flow. Phys.Rev.Lett. 81, 3797～3800(1997).
[10] R.Mahnke, J.Kaupuzs, I.Lubashevsky, Probabilistic description of traffic flow. Phys.Rep.408, 1-130(2005).
[11] D.Chowdhury, L.Santen, A.Schadschneider, Statistical physics of vehicular traffic and some related system. Phys.Rep.329, 199-329(2000).
[12] M. Schonhof, D.Helbing, Empirical features of congested traffic states and their implications for traffic modeling. Cond-mat/0408138.
[13] D.Helbing, Master: macroscopic traffic simulation based on a gas-kinetic, non-local traffic model. Transport Res B 35, 180～211(2001).
[14] D.Helbing, Jams, waves and clusters. Science 282, 2001～2003(1998).
[15] T. Nagatani, Density wave in traffic flow. Phys.Rev.E 60, 6395(1999).
[16] M.Muramatsu and T. Nagatani, Soliton and kind jams in traffic flow with open boundariws. Phys.Rev. E 60, 180～187(1999).
[17] M.J.Lighthill, G.B.Whitham, On kinematic waves: Ⅱ. A theory of traffic flow on long crowed roads. Proceedings of the Royal Society, London, Ser. A 229, 317～345(1955).
[18] P.I.Richards, Shock waves on the highway. Oper. Res. 4, 42～51 (1956).
[19] H.J.Payne, Models of freeway traffic and control. In: Bekey, G.A.(Ed.), Mathematical Models of Public Systems, Vol. 1, 51～61(Simulation Council, La Jolla, 1971).
[20] C.F.Daganzo, Requiem for second-order fluid approximation of traffic flow. Transpn. Res. B 29, 277～286(1995).
[21] R.Jiang, Q.S.Wu, Z.J.Zhu, A new continuum model for traffic flow and numerical tests. Transpn. Res. B 36, 405～419(2002).
[22] R.Jiang, Q.S.Wu, Z.J.Zhu, A new dynamics model for traffic flow. Chinese Sci. Bull. 46, 345～349(2001).
[23] I.Prigogine, R.Herman, Kinetic theory of vehicular traffic. (American Elsevier, New York, 1971).
[24] S.L.Paveri-Fontana, On Boltzmann-like treatments for traffic flow. A critical review of the basic model and an alternative proposal for dilute traffic analysis. Transpn. Res. 9, 225～235(1975).
[25] D.Helbing, Derivation and empirical validation of a refined traffic flow model. Physica A 233, 253～282(1996).
[26] M.Treiber, A.Hennecke, D.Helbing, Derivation, properties, and simulation of a gas-kinetic-based, nonlocal traffic model. Phys. Rev. E 59, 239～253(1999).
[27] L.A.Pipes, An operational analysis of traffic dynamics. J.Appl.Phys. 24, 274～287(1953).
[28] G.F.Newell, Nonlinear effects in the dynamics of car following. Oper. Res. 9, 209～229 (1961).
[29] L.C.Edie, Car following and steady state theory for non-congested traffic. Oper. Res. 9, 66～76 (1962).
[30] M.Bando, Dynamical model of traffic congestion and simulation. Phys.Rev.E 51, 1035(1995).
[31] D.Helbing, B.Tilch, Generalized force model of traffic dynamics. Phys.Rev.E 58, 133～138(1998).
[32] E.F.Codd, Cellular Automata, Acadenic Press, New York, (1968).
[33] M.Gardner, The fantastic bombination of John Conway's solitaire game life, Sci. Am 220(4), 120(1970).
[34] S.Wolfram, Statistical mechanics of cellular automata. Rev. Mod. Phys 55, 601～644(1983). S.Wolfram, Theory and Applications of Cellular Automata, World Scientific, Singapore,(1986). S.Wolfram, Cellular automaton fluids I: basic theory, J. Stat. Phys 45, 471～526(1986). Cellular Automata and Complexity, Addison-Wesley, Reading, MA, (1994).
[35] D. Stauffer, Computer-simulations of cellular automata, J.Phys. A 24, 909～927(1991).
[36] M.Cremer, J.Ludwig, A fast simulation model for traffic flow on the basis of Boolean operations, J.Math. Comp. Simul 28, 297～303(1986).
[37] Traffic and Granular flow '99, Edited by D.Helbing, M.Herrmann, M.Schreckenberg, and D.E.Wolf. (Springer, Berlin, 2000).
[38] A.Hennecke, et al., Macroscopic simulation of open systems and macro-micro link. In[37], 383～388.
[39] 汪秉宏，王雷，陈龙康，Fukui-Ishibashi 交通流基本图的解析研究，非线性动力学报 4，374(1997)。
[40] L. wang, B. H. Wang and P. M. Hui, Strict Derivation of Mean Field Equation for One-Dimensional Traffic Flow Model. Acta Physica Sinica(Overseas Edition) 6, 829(1997).
[41] B. H. Wang, L. Wang and P. M. Hui, One Dimensional Fukui-Ishibashi Traffic Flow Model. Journal of the Physical Society of Japan 66, 3683 (1997).
[42] 汪秉宏，王雷，许伯铭，胡斑比，福井-石桥交通流模型高密度区平均场方程，应用科学学报 Vol．17 No．2 142(1999)
[43] B. H. Wang, L. Wang, P. M. Hui and B.Hu, Analytical Results for the Steady State of Traffic Flow Models with Stochastic Delay, Physical Review E 55, 2876(1998)
[44] B. H. Wand, L. Wang, E M. Hui and B. Hu, Statistical Mechanical Approach to Phase Transition in Traffic Flow Model, 《STATPHYS 20》 The 20th IUPAP International Conference on Statistical Physics, (Paris, July 20-24, 1998) Oral contribution: Short Communications, Book of Abstracts, A Gervois, M Gingold, D Iagolnitzer, ed., Topic 2(Nonequilibrium Systems) T0791; PO02/145.
[45] 王雷，汪秉宏，决定论性逐步加速交通流模型的渐进稳态行为，物理学报 vol.48, No.5, 808(1999).
[46] B. H. Wang, L. Wang, P. M. Hui and B. Hu, Steady state traffic flow in a model with gradual acceleration and stochastic delay, 《ETOPIM5》 The 5th Inernational Conference on Electrical Transport and Optical Properties of Inhomogeneous Media (Hong Kong, June 21-25, 1999) Proceedings p.81.
[47] B. H. Wang, L. Wang, P. M. Hui and B. Hu, The asymptotic steady states of deterministic one-dimensional traffic flow models, Physica B 279, 237(2000).
[48] 王雷，一维交通流元胞自动机模型中自组织临界性及相变行为研究，中国科学技术大学博士学位论文(2000)。
[49] Chuan-Ji Fu, Bing-Hong Wang, Chuan-Yang Yin, Tao Zhou, Bo Hu, Kun Gao, P.M. Hui and Chin-Kun Hu, Analytical studies on a modified Nagel-Schreckenberg model with the Fukui-Ishibashi acceleration rule, Chaos, Solitons & Fractals, 31, Issue 3, 772-776 (2007)。
[50] 付传技，汪秉宏，殷传洋，高坤，利用智能决策的双通道交通流，物理学报 vol．55，No．8，4032-4038(2006)．
[51] 付传技，汪秉宏，殷传洋，高坤，陆玉风，双通道决策交通流的解析研究，全国博士生学术论坛(物理学)会议论文(2006)。
[52] 付传技，汪秉宏，高坤，龚华鑫，季莉，不对称全速度差模型，CCAST 复杂系统研究论坛会议论文(2005)。
[53] G.B.Whitham, Linear and Nonlinear Waves(Wiley, New York, 1974).
[54] T.Musha and H.Higuchi, Japanese Journal of Applied Physics 17, 811(1978).
[55] B.D.Greenshields, in Proceedings of the Fourteenth Annual Meeting of the Highway Research Board, edited by R.W.Crum(National Research Council, Washington, DC, 1935), Part Ⅰ.
[56] J.M.Burgers, Adv. Appl. Mech. 1, 171(1948).
[57] H. M. Zhang, A theory of nonequilibrium traffic flow. Transport Res B, 32, 485～498(1998).
[58] H. M. Zhang, A nonequilibrium traffic model devoid of gas-like behavior. Transport Res B, 36, 275～290(2002).
[59] I. Prigogine, A Boltzmann-like approach for traffic flow. Operat Res, 8, (1960).
[60] H.J.Payne, FREFLO. A macroscopic simulation model of freeway traffic. TPRR, 772, 68～75(1979).
[61] M.Papageorgiou, Applications of Automatic Control Concepts to Traffic Flow Modeling and Control(Springer, Heidelberg, Germany, 1983).
[62] R.D.Kuhne, in Proceeding of the 9th International Symposium on Transportation and Traffic Theory, edited by I.Volmuller and R.Hamerslag (VNU Science Press, Utrecht, The Netherlands, 1984).
[63] P.S.Babcock IV, D.M.Auslander, M.Tomizuka, and A.D.May, Transportation Research Record 971, 80(1984).
[64] A.K.Rathi, E.B.Lieberman, and M.Yedlin, Transportation Research Record 1112, 61(1987).
[65] P.Ross, Transportation Research B 22, 421 (1988).
[66] C.J.Leo, R.L.Pretty, Transportation Research B 26, 207(1992).
[67] M.Cremer and A.D.May, An Extended Traffic Model for Freeway Control (Research Report UCB-ITS-RR-85-7, Institute of Transportation Studies, University of California, Berkeley).
[68] H.J.Payne, in Research Directions in Computer Control of Urban Traffic Systems, edited by W.S.Levine, E.Lieberman, and J.J.Fearnsides (American Society of Civil Engineers, New York, 1979).
[69] H.J.Payne, in Mathematical Models of Public Systems, edited by G.A.Bekey(Simulation Council, La Jolla, CA, 1971), Vol. 1.
[70] R.D.Kuhne, in Proceedings of the 10th International Symposium on Transportation and Traffic Theory, edited by N.H.Garter and N.H.M.Wilson (Elsevier, New York, 1987).
[71] W.F.Phillips, Kinetic Model for Traffic Flow (Report No. DOT/RSPD/DPB/50-77/17, National Technical Information Service, Springfield, Virginia 22161, 1977).
[72] W.F.Phillips, Transportation Planning and Technology 5, 131(1979).
[73] R.D.Kuhne, Transportation Research Record 1320, 251(1991).
[74] B.S.Kerner and P.Konhauser, Cluster effect in initially homogeneous traffic flow. Phys.Rev.E 48, R2335～R2338(1993).
[75] B.S.Kerner and P.Konhauser, Structure and parameters of clusters in traffic flow. Phys.Rev.E 50, 54(1994).
[76] D.Helbing, Improved fluid-dynamic model for vehicular trffic. Phys.Rev.E 51, 3164(1995).
[77] R.D.Kuhne, Non-linearity stochastics of unstable traffic flow. In: Daganzo, C F(ED). Transportation and Traffic Flow Theory. Elsevier Science Publishers, 367～386(1994).
[78] B.S.Kerner, Determinstic spontaneous appearance of traffic jams in slightly inhomogenoous traffic flow. Phys.Rev.E51, R6243～R6246(1995).
[79] P.Berg, Optimal-velocity models of motorway traffic. PHD thesis, University of Bristol, (2001).
[80] D.A.Kurtz, and D.C.Hong, Traffic jam, granular flow and soliton selection. Phys.Rev.E52, 5574(1993).
[81] T.Komatasu, and S.Sasa, Kink solution charactering traffic congestion. Phys.Rev.E55, 5574～5581(1995).
[82] 戴世强，薛郁，交通流的建模和仿真，66～125(2004)．In[2]．
[83] D.Helbing, Gas-kinetic derivation of Navier-Stokes-like trffic equations. Phys.Rev.E53, 2366(1996).
[84] P.Nelson, Transport Theory and Statistical Physics 24, 383(1995).
[85] J.Keizer, Statistical Thermodynamics of Nonequilibrium Processes (Springer, New York, 1987).
[86] I.Prigogine, in Theory of Traffic Flow, edited by R.Herman (Elservier, Amsterdam, 1961).
[87] D.Helbing, Traffic and related self-driven many-particle systems. Rev Mod Phys, 73, (2001).
[88] R.E.Chandler, R.Herman and E.W.Montroll, Traffic dynamics: studies in car following. Oper Res, 6, 165～184(1958).
[89] R.Herman, R.B.Potts, Single lane traffic theory and experiment. In: Proceedings Symposium on theory of traffic flow. (Elservier, New York, 1961).
[90] M.Bando, K.Hasebe, A.Nakayama, et al, Analysis of optimal velocity model with explicit delay. Phys.Rev.E 58, 5429～5435(1998).
[91] M.J.Del Castillo, F.G.Benitez, On functional form of the speed-density relationship-Ⅰ: general theory, Ⅱ: empirical investigation. Transpn, Res, B, 29, 373～406(1995).
[92] R.Jiang, Q.S.Wu, Z.J.Zhu, Full velocity difference model for car-following theory. Phys.Rev.E 64, 017101(2001).
[93] B.Chopard, M.Droz, Cellular Automata Modelling of Physical Systems, (Cambridge University Press, Cambridge, 1998).
[94] 贾斌，交通流瓶颈处车流复杂动态特性的元胞自动机模拟，中国科学技术大学博士学位论文(2002)．
[95] K.Nagel, M.Schreckenberg, A cellular automaton model for freeway traffic. J Phys I France, 2, 2221～2233(1992).
[96] M.Fukui, and Y.Ishibashi, Traffic flow in 1D cellular automaton model including cars moving with high speed. J Phys Soc Japan, 65, 1868～1870(1996).
[97] O.Biham, A.Middleton, and D.Levine. Self-organization & a dynamic transition in traffic flow models. Phys. Rev.A, 46, R6124～R6127(1992).
[98] M.Takayasu, H.Takayasu, 1/f noise in a traffic model, Fractals 1, 860～866(1993).
[99] S.C.Benjamin, N.F.Johnson, P.M.Hui, Cellular automaton models of traffic flow along a highway containing a junction, J.Phys. A 29, 3119～3127(1996).
[100] R.Barlovic, L.Santen, A.Schreckenburg, Metastable states in cellular automata for traffic flow, Eur.Phys.J.B. 5, 793～800(1998).
[101] X.B.Li, Q.S.Wu, R.Jiang, Cellular automaton model considering the velocity effect of a car on the successive car, Phys.Rev.E 64, 066128(2001).
[102] W.Knospe, L.Santen, A.Schadscheider, et al., Towards a realistis microscopic description of highway traffic, J.Phys.A 33, L477～L485(1996).
[103] A.Schadschneider, M.Schrenkenberg, Cellular-automaton models and traffic flow, J.Phys.A 26, L679～L683(1996).
[104] A. Schadschneider, M.Schrenkenberg, Car-oriented mean-field theory for traffic flow models, J.Phys.A 30, L69～L75(1996).
[105] M.Schrenkenberg, A.Schadschneider, K.Nagel, et al., Discrete stochastic models for traffic flow, Phys.Rev.E 51, 2939～2949(1995).
[106] K.Nagel, D.E.Wolf, P.Wagner, et al., Two-lane traffic rules for cellular automata: A systematic approach, Phys.Rev.E 58, 1425～1437(1998).
[107] W.Knospe, L.Santen, A.Schadschneider, et al., A realistic two-lane traffic model for highway traffic, J.Phys.A 35, 3369～3388(2002).
[108] W.Knospe, L.Santen, A.Schadschneider, et al., Disorder effects in cellular automata for two-lane traffic, Physica A 265, 614～633(1999).
[109] P.Wagner, K.Nagel, D.E.Wolf, Realistic multi-lane traffic rules for cellular automata, Physica A 234, 687～698(1997).
[110] M.Rickert, K.Nagel, M.Schreckenberg, et al., Two lane traffic simulations using cellular automata, Physica A 231, 534～550(1996).
[111] K.Nagel, M.Rickert, Parallel implementation of the TRANSIMS micro-simulation, Parallel Comput 27, 1611～1639(2001).
[112] O.Biham, A.A.Middleton, D.A.Levine, Self-organization and a dynamical transition in traffic flow models, Phys.Rev.A 46, R6124～R6127(1992).
[113] T.Nagatani, Effect of traffic accident on jamming transition in traffic flow model, J.Phys.A 26, L1015～L1020(1993).
[114] T.Nagatani, Jamming transition in the traffic flow model with two-level crossings, Phys.Rev.E48, 3290～3294(1998).
[115] B.H.Wang, Y.F.Woo, P.M.Hui, Improved mean-field theory of two-dimensional traffic flow models, J.Phys.A 29, L31～L35(1996).
[116] B.H.Wang, Y.F.Woo, P.M.Hui, Mean field theory of traffic flow problems with overpasses and asymmetric distributions of cars, J.Physs.Soc.Jpn 65, 2345～2348(1996).
[117] G.Q.Gu, K.H.Chung, P.M.Hui, Two-dimensional traffic flow problems in inhomogeneous lattice, Physica A 217, 339～347(1995).
[118] K.H.Chung, P.M.Hui, G.Q.Gu, Two-dimensional traffic problems with faulty traffic lights, Phys.Rev.E 56, 772～774(1995).
[119] T.Nagatani, Effect of traffic accident on jamming transition in traffic flow model, J.Phys.A 26, 1015～1020(1993).
[120] T.Horiguchi, T.Sakakibara, Numerical simulations for traffic flow models on a decorated square lattice, Physica A 252, 388～404(1998).
[121] 方兆本，缪柏其，随机过程，中国科学技术大学出版社(2002)．
[122] L.Wang, B.H.Wang, Bambi Hu, Cellular automaton traffic flow model between the Fuikui-Ishibashi and Nagel-Schreckenberg models, Phys.Rev.E 63, 056117(2001).
[123] J.Wahle, A.L.C.Bazzan, EKlugl, M.Schreckenberg, Decision dynamics in a traffic scenario, Physica A 287, 669～681(2000).
[124] T.Nagatani, Self-organization and phase-transition in traffic-flow model of a 2-lane roadway, J.Phys.A 26, L781-L787(1993).
[125] T.Nagatani, Dynamical jamming transition induced by a car accident in traffic-flow of a 2-lane roadway, Physica A 202, 449-458(1994).
[126] M.Rickert, K.Nagel, D.E.Wolf, Realistic multi-lane traffic rules for cellular automata, Physica A 231, 534-550(1996).
[127] D.Chowdhury, D.E.wolf, M.Schreckenberg, Particle hopping models for two-lane traffic with two kinds of vehicles: effects of lane changing rules, Physica A 235, 417-439(1997).
[128] M.Rickert, Two Lane Traffic Simulation Using Cellular Automata, Report No.95, 213, (Center for Parallel Computing, University of Cologne, Germany, 1995).
[129] M.Herrmann, B.S.Kener, Local cluster effect in different traffic flow models. Physica A 255, 163-188(1998).