Cellular Automaton Models in the Framework of Three-Phase Traffic Theory

  • Junfang Tian
  • Chenqiang Zhu
  • Rui JiangEmail author
Reference work entry
Part of the Encyclopedia of Complexity and Systems Science Series book series (ECSSS)


CA models

Cellular automata (CA) models are a class of microscopic traffic flow models. In the CA models, the time and space are discrete, and the evolution is described by update rules.

First-order phase transition

In three-phase traffic theory, the F → S and S → J transitions are claimed to be first-order phase transition, in which the flow rate abrupt decreases.

Fundamental diagram

Fundamental diagram describes the relationship between flow rate and density. In the empirical data, the flow rate and density are usually collected by loop detector and averaged over 1 min. In the simulation on a circular road, usually the global density vs. averaged flow rate is plotted.

Kerner’s three-phase traffic theory

In Kerner’s three-phase theory, the congested flow has been further classified into synchronized flow (S) and wide moving jam (J). Therefore, there are three phases in traffic flow, i.e., the free flow (F), synchronized flow, and wide moving jam. The usually observed traffic...


  1. Ahn S, Cassidy MJ (2007) Freeway traffic oscillations and vehicle lane-change maneuvers. In: Proceedings of the transportation and traffic theory, 2007Google Scholar
  2. Bando M, Hasebe K, Nakayama A, Shibata A, Sugiyama Y (1995) Dynamical model of traffic congestion and numerical simulation. Phys Rev E 51(2):1035CrossRefGoogle Scholar
  3. Barlovic R, Santen L, Schadschneider A, Schreckenberg M (1998) Metastable states in cellular automata for traffic flow. Eur Phys J B-Condens Matter Complex Syst 5(3):793–800CrossRefGoogle Scholar
  4. Benjamin SC, Johnson NF, Hui PM (1996) Cellular automata models of traffic flow along a highway containing a junction. J Phys A Math Gen 29(12):3119zbMATHCrossRefGoogle Scholar
  5. Brackstone M, McDonald M (1999) Car-following: a historical review. Transport Res F: Traffic Psychol Behav 2(4):181–196CrossRefGoogle Scholar
  6. Brilon W, Geistefeldt J, Regler M (2005) Reliability of freeway traffic flow: a stochastic concept of capacity. In: Proceedings of the 16th international symposium on transportation and traffic theoryGoogle Scholar
  7. Chandler RE, Herman R, Montroll EW (1958) Traffic dynamics: studies in car following. Oper Res 6(2):165–184MathSciNetCrossRefGoogle Scholar
  8. Chen D, Laval J, Zheng Z, Ahn S (2012) A behavioral car-following model that captures traffic oscillations. Transp Res B Methodol 46(6):744–761CrossRefGoogle Scholar
  9. Chen D, Ahn S, Laval J, Zheng Z (2014) On the periodicity of traffic oscillations and capacity drop: the role of driver characteristics. Transp Res B Methodol 59:117–136CrossRefGoogle Scholar
  10. Chmura T, Herz B, Knorr F, Pitz T, Schreckenberg M (2014) A simple stochastic cellular automaton for synchronized traffic flow. Physica A: Stat Mech Appl 405:332–337MathSciNetzbMATHCrossRefGoogle Scholar
  11. Chowdhury D, Santen L, Schadschneider A (2000) Statistical physics of vehicular traffic and some related systems. Phys Rep 329(4–6):199–329MathSciNetCrossRefGoogle Scholar
  12. Chung K, Rudjanakanoknad J, Cassidy MJ (2007) Relation between traffic density and capacity drop at three freeway bottlenecks. Transp Res B Methodol 41(1):82–95CrossRefGoogle Scholar
  13. Coifman B, Kim S (2011) Extended bottlenecks, the fundamental relationship, and capacity drop on freeways. Transp Res A Policy Pract 45(9):980–991CrossRefGoogle Scholar
  14. Daganzo C, Daganzo CF (1997) Fundamentals of transportation and traffic operations, vol 30. Pergamon, OxfordzbMATHCrossRefGoogle Scholar
  15. Fukui M, Ishibashi Y (1997) Effect of delay in restarting of stopped cars in a one-dimensional traffic model. J Phys Soc Jpn 66(2):385–387CrossRefGoogle Scholar
  16. Gao K, Jiang R, Hu SX, Wang BH, Wu QS (2007) Cellular-automaton model with velocity adaptation in the framework of Kerner’s three-phase traffic theory. Phys Rev E 76(2):026105CrossRefGoogle Scholar
  17. Gao K, Jiang R, Wang BH, Wu QS (2009) Discontinuous transition from free flow to synchronized flow induced by short-range interaction between vehicles in a three-phase traffic flow model. Phys A Stat Mech Appl 388(15–16):3233–3243CrossRefGoogle Scholar
  18. Gazis DC, Herman R, Potts RB (1959) Car-following theory of steady-state traffic flow. Oper Res 7(4):499–505MathSciNetCrossRefGoogle Scholar
  19. Gazis DC, Herman R, Rothery RW (1961) Nonlinear follow-the-leader models of traffic flow. Oper Res 9(4):545–567MathSciNetzbMATHCrossRefGoogle Scholar
  20. Gipps PG (1981) A behavioural car-following model for computer simulation. Transp Res B Methodol 15(2):105–111CrossRefGoogle Scholar
  21. Greenberg H (1959) An analysis of traffic flow. Oper Res 7(1):79–85MathSciNetCrossRefGoogle Scholar
  22. Greenshields BD, Channing W, Miller H (1935) A study of traffic capacity. In: Highway research board proceedings, vol 1935. National Research Council (USA), Highway Research BoardGoogle Scholar
  23. Hall FL, Agyemang-Duah K (1991) Freeway capacity drop and the definition of capacity. Transp Res Rec 1320:91Google Scholar
  24. Helbing D (2001) Traffic and related self-driven many-particle systems. Rev Mod Phys 73(4):1067MathSciNetCrossRefGoogle Scholar
  25. Jia B, Li XG, Chen T, Jiang R, Gao ZY (2011) Cellular automaton model with time gap dependent randomisation under Kerner’s three-phase traffic theory. Transportmetrica 7(2):127–140CrossRefGoogle Scholar
  26. Jiang R, Wu QS (2003) Cellular automata models for synchronized traffic flow. J Phys A Math Gen 36(2):381MathSciNetzbMATHCrossRefGoogle Scholar
  27. Jiang R, Wu QS (2005) First order phase transition from free flow to synchronized flow in a cellular automata model. Eur Phys J B-Condens Matter Complex Syst 46(4):581–584CrossRefGoogle Scholar
  28. Jiang R, Wu Q, Zhu Z (2001) Full velocity difference model for a car-following theory. Phys Rev E 64(1):017101CrossRefGoogle Scholar
  29. Jiang R, Wu QS, Zhu ZJ (2002) A new continuum model for traffic flow and numerical tests. Transp Res B Methodol 36(5):405–419CrossRefGoogle Scholar
  30. Jiang R, Hu MB, Zhang HM, Gao ZY, Jia B, Wu QS, Wang B, Yang M (2014) Traffic experiment reveals the nature of car-following. PLoS One 9(4):e94351CrossRefGoogle Scholar
  31. Jiang R, Hu MB, Zhang HM, Gao ZY, Jia B, Wu QS (2015) On some experimental features of car-following behavior and how to model them. Transp Res B Methodol 80:338–354CrossRefGoogle Scholar
  32. Jiang R, Jin CJ, Zhang HM, Huang YX, Tian JF, Wang W, Hu MB, Wang H, Jia B (2017) Experimental and empirical investigations of traffic flow instability, Transp Res Part C: Emerg Technol.
  33. Jin CJ, Wang W (2011) The influence of nonmonotonic synchronized flow branch in a cellular automaton traffic flow model. Phys A Stat Mech Appl 390(23–24):4184–4191CrossRefGoogle Scholar
  34. Jin CJ, Wang W, Jiang R, Zhang HM, Wang H, Hu MB (2015) Understanding the structure of hyper-congested traffic from empirical and experimental evidences. Transp Res Part C: Emerg Technol 60:324–338CrossRefGoogle Scholar
  35. Kerner BS (1998) Experimental features of self-organization in traffic flow. Phys Rev Lett 81(17):3797–3800zbMATHCrossRefGoogle Scholar
  36. Kerner BS (1999a) Congested traffic flow: observations and theory. Transp Res Rec J Transp Res Board 1678(1):160–167CrossRefGoogle Scholar
  37. Kerner BS (1999b) Theory of congested traffic flow: self-organization without bottlenecks. In: 14th international symposium on transportation and traffic theoryGoogle Scholar
  38. Kerner B (1999c) Congested traffic flow: observations and theory. Transp Res Rec J Transp Res Board 1678(1):160–167CrossRefGoogle Scholar
  39. Kerner BS (2000) Experimental features of the emergence of moving jams in free traffic flow. J Phys A Math Gen 33(26):L221zbMATHCrossRefGoogle Scholar
  40. Kerner BS (2002) Empirical macroscopic features of spatial-temporal traffic patterns at highway bottlenecks. Phys Rev E 65(4):046138CrossRefGoogle Scholar
  41. Kerner BS (2004) The physics of traffic: empirical freeway pattern features, engineering applications, and theory. Phys Today 58(11):54–56Google Scholar
  42. Kerner BS (2009) Introduction to modern traffic flow theory and control: the long road to three-phase traffic theory. Springer, BerlinzbMATHCrossRefGoogle Scholar
  43. Kerner BS (2013) Criticism of generally accepted fundamentals and methodologies of traffic and transportation theory: a brief review. Phys A Stat Mech Appl 392(21):5261–5282MathSciNetzbMATHCrossRefGoogle Scholar
  44. Kerner BS (2017) Breakdown in traffic networks: fundamentals of transportation science. Springer, HeidelbergzbMATHCrossRefGoogle Scholar
  45. Kerner BS, Klenov SL (2002) A microscopic model for phase transitions in traffic flow. J Phys A Math Gen 35(3):L31MathSciNetzbMATHCrossRefGoogle Scholar
  46. Kerner BS, Rehborn H (1996) Experimental properties of complexity in traffic flow. Phys Rev E 53(5):R4275CrossRefGoogle Scholar
  47. Kerner BS, Rehborn H (1997) Experimental properties of phase transitions in traffic flow. Phys Rev Lett 79(20):4030CrossRefGoogle Scholar
  48. Kerner BS, Klenov SL, Wolf DE (2002) Cellular automata approach to three-phase traffic theory. J Phys A Math Gen 35(47):9971MathSciNetzbMATHCrossRefGoogle Scholar
  49. Kerner BS, Klenov SL, Schreckenberg M (2011) Simple cellular automaton model for traffic breakdown, highway capacity, and synchronized flow. Phys Rev E 84(4):046110CrossRefGoogle Scholar
  50. Knospe W, Santen L, Schadschneider A, Schreckenberg M (2000) Towards a realistic microscopic description of highway traffic. J Phys A Math Gen 33(48):L477MathSciNetzbMATHCrossRefGoogle Scholar
  51. Knospe W, Santen L, Schadschneider A, Schreckenberg M (2004) Empirical test for cellular automaton models of traffic flow. Phys Rev E 70(1):016115CrossRefGoogle Scholar
  52. Kokubo S, Tanimoto J, Hagishima A (2011) A new cellular automata model including a decelerating damping effect to reproduce Kerner’s three-phase theory. Phys A Stat Mech Appl 390(4):561–568CrossRefGoogle Scholar
  53. Lárraga ME, Alvarez-Icaza L (2010) Cellular automaton model for traffic flow based on safe driving policies and human reactions. Phys A Stat Mech Appl 389(23):5425–5438zbMATHCrossRefGoogle Scholar
  54. Laval J (2006) Stochastic processes of moving bottlenecks: approximate formulas for highway capacity. Transp Res Rec: J Transp Res Board 1988:86–91CrossRefGoogle Scholar
  55. Laval JA, Daganzo CF (2006) Lane-changing in traffic streams. Transp Res B Methodol 40(3):251–264CrossRefGoogle Scholar
  56. Laval JA, Leclercq L (2010) A mechanism to describe the formation and propagation of stop-and-go waves in congested freeway traffic. Philos Trans R Soc Lond A: Math, Phy Eng Sci 368(1928):4519–4541MathSciNetzbMATHCrossRefGoogle Scholar
  57. Lee HK, Barlovic R, Schreckenberg M, Kim D (2004) Mechanical restriction versus human overreaction triggering congested traffic states. Phys Rev Lett 92(23):238702CrossRefGoogle Scholar
  58. Li X, Ouyang Y (2011) Characterization of traffic oscillation propagation under nonlinear car-following laws. Transp Res B Methodol 45(9):1346–1361CrossRefGoogle Scholar
  59. Li X, Wu Q, Jiang R (2001) Cellular automaton model considering the velocity effect of a car on the successive car. Phys Rev E 64(6):066128CrossRefGoogle Scholar
  60. Li X, Wang X, Ouyang Y (2012) Prediction and field validation of traffic oscillation propagation under nonlinear car-following laws. Transp Res B Methodol 46(3):409–423CrossRefGoogle Scholar
  61. Li X, Cui J, An S, Parsafard M (2014) Stop-and-go traffic analysis: theoretical properties, environmental impacts and oscillation mitigation. Transp Res B Methodol 70:319–339CrossRefGoogle Scholar
  62. Lighthill MJ, Whitham GB (1955) On kinematic waves II. A theory of traffic flow on long crowded roads. Proc R Soc Lond A 229(1178):317–345. The Royal SocietyMathSciNetzbMATHCrossRefGoogle Scholar
  63. Maerivoet S, De Moor B (2005) Cellular automata models of road traffic. Phys Rep 419(1):1–64MathSciNetCrossRefGoogle Scholar
  64. Mauch M, Cassidy MJ (2002) Freeway traffic oscillations: observations and predictions. In: Proceedings of the 15th international symposium on transportation and traffic theoryGoogle Scholar
  65. May AD (1990) Traffic flow fundamentals. Prentice Hall, Englewood CliffsGoogle Scholar
  66. Nagel K, Schreckenberg M (1992) A cellular automaton model for freeway traffic. J Phys I 2(12):2221–2229Google Scholar
  67. Nishinari K, Takahashi D (2000) Multi-value cellular automaton models and metastable states in a congested phase. J Phys A Math Gen 33(43):7709MathSciNetzbMATHCrossRefGoogle Scholar
  68. Payne HJ (1979) FREFLO: a macroscopic simulation model of freeway traffic. Transp Res Rec 722:68–77Google Scholar
  69. Pipes LA (1967) Car following models and the fundamental diagram of road traffic. Trans Res 1(1):21–29Google Scholar
  70. Richards PI (1956) Shock waves on highway. Oper Res 4:42–51Google Scholar
  71. Saberi M, Mahmassani HS (2013) Empirical characterization and interpretation of hysteresis and capacity drop phenomena in freeway networks. Transp Res Rec: J Transp Res Board. Transportation Research Board of the National Academies, Washington, DCGoogle Scholar
  72. Saifuzzaman M, Zheng Z (2014) Incorporating human-factors in car-following models: a review of recent developments and research needs. Transp Res Part C: Emerg Technol 48:379–403CrossRefGoogle Scholar
  73. Saifuzzaman M, Zheng Z, Haque MM, Washington S (2017) Understanding the mechanism of traffic hysteresis and traffic oscillations through the change in task difficulty level. Transp Res B Methodol 105:523–538CrossRefGoogle Scholar
  74. Schadschneider A, Schreckenberg M (1997) Traffic flow models with ‘slow-to-start’ rules. Ann Phys 509(7):541–551MathSciNetzbMATHCrossRefGoogle Scholar
  75. Schönhof M, Helbing D (2007) Empirical features of congested traffic states and their implications for traffic modeling. Transp Sci 41(2):135–166CrossRefGoogle Scholar
  76. Srivastava A, Geroliminis N (2013) Empirical observations of capacity drop in freeway merges with ramp control and integration in a first-order model. Transp Res Part C: Emerg Technol 30:161–177CrossRefGoogle Scholar
  77. Sugiyama Y, Fukui M, Kikuchi M, Hasebe K, Nakayama A, Nishinari K et al (2008) Traffic jams without bottlenecks – experimental evidence for the physical mechanism of the formation of a jam. New J Phys 10(3):033001CrossRefGoogle Scholar
  78. Takayasu M, Takayasu H (1993) 1/f noise in a traffic model. Fractals 1(04):860–866zbMATHCrossRefGoogle Scholar
  79. Tian JF, Yuan ZZ, Jia B, Fan HQ, Wang T (2012a) Cellular automaton model in the fundamental diagram approach reproducing the synchronized outflow of wide moving jams. Phys Lett A 376(44):2781–2787zbMATHCrossRefGoogle Scholar
  80. Tian JF, Yuan ZZ, Treiber M, Jia B, Zhang WY (2012b) Cellular automaton model within the fundamental-diagram approach reproducing some findings of the three-phase theory. Phys A Stat Mech Appl 391(11):3129–3139CrossRefGoogle Scholar
  81. Tian J, Treiber M, Ma S, Jia B, Zhang W (2015) Microscopic driving theory with oscillatory congested states: model and empirical verification. Transp Res B Methodol 71:138–157CrossRefGoogle Scholar
  82. Tian J, Li G, Treiber M, Jiang R, Jia N, Ma S (2016) Cellular automaton model simulating spatiotemporal patterns, phase transitions and concave growth pattern of oscillations in traffic flow. Trans Res B Methodol 93:560–575CrossRefGoogle Scholar
  83. Tian J, Jia B, Ma S, Zhu C, Jiang R, Ding Y (2017) Cellular automaton model with dynamical 2D speed-gap relation. Trans Sci 51(3):807–822CrossRefGoogle Scholar
  84. Treiber M, Helbing D (2003) Memory effects in microscopic traffic models and wide scattering in flow-density data. Phys Rev E 68(4):046119CrossRefGoogle Scholar
  85. Treiber M, Kesting A (2013) Traffic flow dynamics: data, models and simulation. no. Book, Whole. Springer, Berlin/HeidelbergGoogle Scholar
  86. Treiber M, Hennecke A, Helbing D (2000) Congested traffic states in empirical observations and microscopic simulations. Phys Rev E 62(2):1805zbMATHCrossRefGoogle Scholar
  87. Treiber M, Kesting A, Helbing D (2006) Understanding widely scattered traffic flows, the capacity drop, and platoons as effects of variance-driven time gaps. Phys Rev E 74(1):016123CrossRefGoogle Scholar
  88. Treiterer J, Myers J (1974) The hysteresis phenomenon in traffic flow. Transp Traffic Theory 6:13–38Google Scholar
  89. Wang Z, Ma S, Jiang R, Tian J (2017) A cellular automaton model reproducing realistic propagation speed of downstream front of the moving synchronized pattern. Transportmetrica B: Transp Dyn.
  90. Yeo H, Skabardonis A (2009) Understanding stop-and-go traffic in view of asymmetric traffic theory. In: Transportation and traffic theory 2009: Golden Jubilee. Springer, Boston, pp 99–115Google Scholar
  91. Zhao BH, Hu MB, Jiang R, Wu QS (2009) A realistic cellular automaton model for synchronized traffic flow. Chin Phys Lett 26(11):118902CrossRefGoogle Scholar
  92. Zheng Z (2014) Recent developments and research needs in modeling lane changing. Transp Res B Methodol 60:16–32CrossRefGoogle Scholar
  93. Zheng Z, Ahn S, Chen D, Laval J (2011) Applications of wavelet transform for analysis of freeway traffic: bottlenecks, transient traffic, and traffic oscillations. Transp Res B Methodol 45(2):372–384CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of Systems Engineering, College of Management and EconomicsTianjin UniversityTianjinChina
  2. 2.MOE Key Laboratory for Urban Transportation Complex Systems Theory and TechnologyBeijing Jiaotong UniversityBeijingChina

Personalised recommendations