Английская Википедия:Boris Kerner

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Шаблон:Short description Шаблон:Infobox scientist Boris S. Kerner (born 1947) is a German physicist and civil engineer who created three phase traffic theory.[1][2][3][4][5][6] The three phase traffic theory is the framework for the description of empirical vehicular traffic states in three traffic phases: (i) free traffic flow (F), (ii) synchronized traffic flow (S), and (iii) wide moving jam (J). The synchronized traffic flow and wide moving jam phases belong to congested traffic.

Biography

Kerner is an engineer and physicist. He was born in Moscow, Soviet Union in 1947 and graduated from the Moscow Technical University MIREA in 1972. Boris Kerner was received Ph.D. and Sc.D. (Doctor of Sciences) degrees in the Academy of Sciences of the Soviet Union, respectively, in 1979 and 1986. Between 1972 and 1992, his major interests include the physics of semiconductors, plasma and solid state physics. During this time, Boris Kerner together with V.V. Osipov developed a theory of Autosolitons – solitary intrinsic states, which form in a broad class of physical, chemical and biological dissipative systems.[7]

After emigration from Russia to Germany in 1992, Boris Kerner worked for the Daimler company in Stuttgart. His major interest since then was the understanding of vehicular traffic.[8][9][10][11][12][13][14] Boris Kerner was awarded with Daimler Research Award 1994.[15] The empirical nucleation nature of traffic breakdown at highway bottlenecks understood by Boris Kerner is the basis for Kerner's three phase traffic theory, which he introduced and developed in 1996–2002.[16][17][18][19][20][21][22][23]

Between 2000 and 2013 Boris Kerner was a head of a scientific research field Traffic at the Daimler company. In 2011 Boris Kerner was awarded with the degree Professor at the University of Duisburg-Essen in Germany.[24] After his retirement from the Daimler company on 31 January 2013 Prof. Kerner works at the University Duisburg-Essen.[25]

Scientific work

Three phase traffic theory

In Kerner's three phase traffic theory, in addition to the free flow traffic phase (F), there are two traffic phases in congested traffic: the synchronized flow traffic phase (S) and the wide moving jam phase (J). One of the main results of Kerner's theory is that traffic breakdown at a highway bottleneck is a random (probabilistic) phase transition from free flow to synchronized flow (F → S transition) that occurs in a metastable state of free flow at a highway bottleneck. This means that traffic breakdown (F → S transition) exhibits the nucleation nature.[26][27][28][29][30][31][32][33][34][35][36][37][38] The main reason for the Kerner's three-phase theory is the explanation of the empirical nucleation nature of traffic breakdown (F → S transition) at highway bottlenecks observed in real field traffic data.

The prediction of the Kerner's three-phase theory is that this metastability of free flow with respect to the F → S phase transition is governed by the nucleation nature of an instability of synchronized flow with respect to the growth of a large enough local increase in speed in synchronized flow (called a S → F instability). The S → F instability is a growing speed wave of a local increase in speed in synchronized flow at the bottleneck. The development of Kerner's S → F instability leads to a local phase transition from synchronized flow to free flow at the bottleneck (S → F transition).[16][17][18]

In 2011–2014, Boris Kerner has expanded three phase traffic theory, which he developed initially for highway traffic, for the description of city traffic.[39][40][41]

Synchronized traffic flow

At the end of 1990's Kerner introduced a new traffic phase, called synchronized flow whose basic feature leads to the nucleation nature of the F → S transition at a highway bottleneck.[16][17][18][42][43] Therefore, Kerner's synchronized flow traffic phase can be used synonymously with the term three-phase traffic theory.

In 1998 Kerner found that the well-known empirical phenomenon moving jam "without obvious reason" occurs due to a sequence of F → S → J transitions.[26] This study was conducted using empirical traffic data. The explanation for the sequence of F → S → J transitions is as follows: in the three-phase traffic theory it is assumed that the probability of a F → S transition in metastable free flow is considerably larger than the probability of a F → J transition.[16]

In Kerner's three-phase traffic theory any phase transition between the three traffic phases exhibits the nucleation nature, as in accordance to the results of empirical observations.[16][17][18]

In 2011 Kerner introduced the breakdown minimization principle that is devoted to control and optimization of traffic and transportation networks while keeping the minimum of the probability of the occurrence of traffic congestion in a network.[44] Rather than an explicit minimization of travel time that is the objective of System Optimum and User Equilibrium, the BM principle minimizes the probability of the occurrence of congestion in a traffic network.[45]

Mathematical models in the framework of three-phase traffic theory

Rather than a mathematical model of traffic flow, Kerner's three-phase traffic theory is a qualitative traffic flow theory that consists of several hypotheses. The first mathematical model of traffic flow in the framework of Kerner's three-phase traffic theory that mathematical simulations can show and explain traffic breakdown by an F → S phase transition in the metastable free flow at the bottleneck was the Kerner-Klenov stochastic microscopic traffic flow model introduced in 2002.[46] Some months later, Kerner, Klenov, and Wolf developed a cellular automaton (CA) traffic flow model in the framework of Kerner's three-phase traffic theory.[47] The Kerner-Klenov stochastic traffic flow model in the framework of Kerner's theory has further been developed for different applications, in particular to simulate on-ramp metering, speed limit control, dynamic traffic assignment in traffic and transportation networks, traffic at heavy bottlenecks and on moving bottlenecks, features of heterogeneous traffic flow consisting of different vehicles and drivers, jam warning methods, vehicle-to-vehicle (V2V) communication for cooperative driving, the performance of self-driving vehicles in mixture traffic flow, traffic breakdown at traffic signals in city traffic, over-saturated city traffic, vehicle fuel consumption in traffic networks.[48][49][50][51][52][53][54][55][56][57][58][59][60][39][40][41][61]

Intelligent transportation systems in the framework of three-phase traffic theory

ASDA/FOTO methods for reconstruction of congested traffic patterns

Three phase traffic theory is a theoretical basis for applications in transportation engineering.[16][17] One of the first applications of the three-phase traffic theory is ASDA/FOTO methods that are used in on-line applications for spatiotemporal reconstruction of congested traffic patterns in highway networks.[62][63]

Congested pattern control approach

In 2004 Kerner introduced congested pattern control approach.[16][64][65] Contrarily to standard traffic control at a network bottleneck in which a controller (for example, through the use of on-ramp metering, speed limit, or other traffic control strategies) tries to maintain free flow conditions at the maximum possible flow rate at the bottleneck, in congested pattern control approach no control of traffic flow at the bottleneck is realized as long as free flow is realized at the bottleneck. Only when an F → S transition (traffic breakdown) has occurred at the bottleneck, the controller starts to work trying to return free flow at the bottleneck. Congested pattern control approach is consistent with the empirical nucleation nature of traffic breakdown. Due to the congested pattern control approach, either free flow recovers at the bottleneck or traffic congestion is localized at the bottleneck.[66][67]

In 2004 Kerner introduced a concept of an autonomous driving vehicle in the framework of the three-phase traffic theory. The autonomous driving vehicle in the framework of the three-phase traffic theory is a self-driving vehicle for which there is no fixed time headway to the preceding vehicle.[68][69][70]

Work after 2015

In 2015 Kerner found that before traffic breakdown occurs at a highway bottleneck, there can be a random sequence of F → S → F transitions at the bottleneck<: The development of a F → S transition is interrupted by a S → F instability that leads to synchronized flow dissolution resulting in a S → F transition at the bottleneck. The effect of Kerner's F → S → F transitions is as follows: The F → S → F transitions determine a random time delay of traffic breakdown at the bottleneck.[71]

Kerner argues there is a new paradigm of traffic and transportation science following from the empirical nucleation nature of traffic breakdown (F → S transition) and that three-phase traffic theory changes the meaning of stochastic highway capacity as follows. At any time instant there is a range of highway capacity values between a minimum and a maximum highway capacity, which are themselves stochastic values. When the flow rate at a bottleneck is inside this capacity range related to this time instant, traffic breakdown can occur at the bottleneck only with some probability, i.e., in some cases traffic breakdown occurs, in other cases it does not occur.[16][17][18][72]Шаблон:Page needed

In 2016 Kerner developed an application of the breakdown minimization principle called network throughput maximization approach. Kerner's network throughput maximization approach is devoted to the maximization of the network throughput while keeping free flow conditions in the whole network.[73]

In 2016 Kerner introduced a measure (or "metric") of a traffic or transportation network called network capacity.[73][20]

In 2019 Kerner found that there is a spatiotemporal competition between S → F and S → J instabilities.[38]

See also

References

Шаблон:Reflist

Sources

Шаблон:Authority control

Партнерские ресурсы
Криптовалюты
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Разное

  1. ↑ The article in "The New York Times" titled "Stuck in Traffic? Consult a Physicist" on Webpage
  2. ↑ Science News Online, Volume 156, Number 1 (July 3, 1999). Stop-and-Go Science. By better understanding traffic flow, researchers hope to keep down highway congestion
  3. ↑ Article by Davis in "APS News" titled "Physicists and traffic flow"
  4. ↑ The Economist: Traffic jams – Adapting to road conditions – Jul 1st 2004 – From The Economist print edition
  5. ↑ Physics Today – November 2005 by Henry Lieu (Federal Highway Administration, McLean, Virginia), Reviewer of the book "The Physics of Traffic: Empirical Freeway Pattern Features, Engineering Applications, and Theory" by Boris S. KernerШаблон:Dead link
  6. ↑ Article "Curing Congestion" in Discover Magazine, 1999
  7. ↑ B.S. Kerner, V.V. Osipov, Autosolitons: A New Approach to Problems of Self-Organization and Turbulence (Fundamental Theories of Physics), Kluwer, Dordrecht, 1994
  8. ↑ Boris S. Kerner, Peter Konhäuser, "Cluster effect in initially homogeneous traffic flow" Phys. Rev. E 48, 2335–2338 (1993). doi: 10.1103/PhysRevE.48.R2335 ]
  9. ↑ Boris S. Kerner, Peter Konhäuser, "Structure and parameters of clusters in traffic flow" Phys. Rev. E 50, 54–83 (1994). doi: 10.1103/PhysRevE.50.54
  10. ↑ Boris S. Kerner, Peter Konhäuser, Martin Schilke, "Deterministic spontaneous appearance of traffic jams in slightly inhomogeneous traffic flow" Phys. Rev. E 51, 6243–6246 (1995). doi: 10.1103/PhysRevE.51.6243
  11. ↑ Boris S. Kerner, Hubert Rehborn, "Experimental features and characteristics of traffic jams" Phys. Rev. E 53, R1297-R1300 (1996). doi: 10.1103/PhysRevE.53.R1297
  12. ↑ Boris S. Kerner, Hubert Rehborn, "Experimental properties of complexity in traffic flow" Phys. Rev. E 53, R4275-R4278 (1996). doi: 10.1103/PhysRevE.53.R4275
  13. ↑ Boris S. Kerner, Hubert Rehborn, "Experimental Properties of Phase Transitions in Traffic Flow" Physical Review Letters 79, 4030–4033 (1997). doi: 10.1103/PhysRevLett.79.4030
  14. ↑ Boris S. Kerner, Sergey L. Klenov, Peter Konhäuser, "Asymptotic theory of traffic jams" Phys. Rev. E 56, 4200–4216 (1997). doi: 10.1103/PhysRevE.56.4200
  15. ↑ "Daimler-Benz, das Geschäftsjahr 1994", page 41
  16. ↑ 16,0 16,1 16,2 16,3 16,4 16,5 16,6 16,7 Boris S. Kerner, The Physics of Traffic: Empirical Freeway Pattern Features, Engineering Applications, and Theory, Springer, Berlin, Heidelberg, New York 2004
  17. ↑ 17,0 17,1 17,2 17,3 17,4 17,5 Boris S. Kerner, Introduction to Modern Traffic Flow Theory and Control: The Long Road to Three-Phase Traffic Theory, Springer, Heidelberg, Dordrecht, London, New York, 2009
  18. ↑ 18,0 18,1 18,2 18,3 18,4 Boris S. Kerner, Breakdown in Traffic Networks: Fundamentals of Transportation Science, Springer, Berlin, 2017
  19. ↑ Boris S. Kerner, "Failure of classical traffic flow theories: Stochastic highway capacity and automatic driving", Physica A: Statistical Mechanics and its Applications 450, 700–747 (2016). doi.org/10.1016/j.physa.2016.01.034
  20. ↑ 20,0 20,1 Boris S. Kerner, "Breakdown minimization principle versus Wardrop's equilibria for dynamic traffic assignment and control in traffic and transportation networks: A critical mini-review", Physica A: Statistical Mechanics and its Applications 466, 626–662 (2017)
  21. ↑ Boris S. Kerner, "Criticism of generally accepted fundamentals and methodologies of traffic and transportation theory: A brief review", Physica A: Statistical Mechanics and its Applications 392, 5261–5282 (2013). doi: 10.1016/j.physa.2013.06.004
  22. ↑ Boris S. Kerner, "Failure of classical traffic flow theories: a critical review", Elektrotech. Inftech. 132, 417–433 (2015). doi: 10.1007/s00502-015-0340-3
  23. ↑ Boris S. Kerner (Ed.), Complex Dynamics of Traffic Management, Encyclopedia of Complexity and Systems Science Series, Springer, New York, 2019
  24. ↑ Pressemitteilung der Universität Duisburg-Essen: UDE verleiht Verkehrsforscher außerplanmäßige Professur. Von Daimler zum Campus
  25. ↑ Fakultät der Physik der Universität Duisburg-Essen, Physik von Transport und Verkehr: Mitglieder der Arbeitsgruppe
  26. ↑ 26,0 26,1 Boris S. Kerner, "Experimental Properties of Self-Organization in Traffic Flow" Physical Review Letters 81, 3797–3800 (1998). doi: 10.1103/PhysRevLett.81.3797
  27. ↑ Boris S. Kerner, "Congested Traffic Flow: Observations and Theory" Transportation Research Record, 1678, 160–167 (1999). doi: 10.3141/1678-20
  28. ↑ Boris S. Kerner, "The Physics of Traffic" Physics World 12, No. 8, 25–30 (August 1999). doi: 10.1088/2058-7058/12/8/30
  29. ↑ Boris S. Kerner, "Experimental features of the emergence of moving jams in free traffic flow" J. Physics A: Math. Gen. 33, L221-L228 (2000). doi: 10.1088/0305-4470/33/26/101
  30. ↑ Boris S. Kerner, "Theory of Breakdown Phenomenon at Highway Bottlenecks" Transportation Research Record, 1710, 136–144 (2000). doi: 10.3141/1710-16
  31. ↑ Boris S. Kerner, "Complexity of Synchronized Flow and Related Problems for Basic Assumptions of Traffic Flow Theories" Networks and Spatial Economics. 1, 35–76 (2001). doi: 10.1023/A:1011577010852
  32. ↑ Boris S. Kerner, "Synchronized Flow as a New Traffic Phase and related Problems for Traffic Flow Modelling" Mathematical and Computer Modelling. 35, 481–508 (2002). doi: 10.1016/S0895-7177(02)80017-6
  33. ↑ Boris S. Kerner, "Empirical Features of Congested Patterns at Highway Bottlenecks" Transportation Research Record, 1802, 145–154 (2002). doi: 10.3141/1802-17
  34. ↑ Boris S. Kerner, "Empirical macroscopic features of spatial-temporal traffic patterns at highway bottlenecks" Phys. Rev. E. 65, 046138 (2002). doi: 10.1103/PhysRevE.65.046138
  35. ↑ Boris S. Kerner, "Three-phase traffic theory and highway capacity" Physica A, 333, 379–440 (2004). doi: 10.1016/j.physa.2003.10.017
  36. ↑ Boris S. Kerner, "A theory of traffic congestion at heavy bottlenecks" J. Phys. A: Math. Gen. 41, 215101 (2008). doi: 10.1088/1751-8113/41/21/215101
  37. ↑ Boris S. Kerner, "Complexity of Spatiotemporal Traffic Phenomena in Flow of Identical Drivers: Explanation based on Fundamental Hypothesis of Three-Phase Theory", Phys. Rev. E 85, 036110 (2012). doi: 10.1103/PhysRevE.84.045102
  38. ↑ 38,0 38,1 Boris S. Kerner, "Statistical Physics of Synchronized Traffic Flow: Spatiotemporal Competition between S → F and S → J Instabilities", Phys. Rev. E 100, 012303 (2019). doi: 10.1103/PhysRevE.100.012303
  39. ↑ 39,0 39,1 Boris S. Kerner, "Physics of traffic gridlock in a city", Phys. Rev. E 84, 045102(R) (2011). doi:10.1103/PhysRevE.84.045102
  40. ↑ 40,0 40,1 Boris S. Kerner, "The physics of green-wave breakdown in a city " Europhysics Letters 102, 28010 (2013). doi:10.1209/0295-5075/102/28010
  41. ↑ 41,0 41,1 Boris S. Kerner, "Three-phase theory of city traffic: Moving synchronized flow patterns in under-saturated city traffic at signals", Physica A: Statistical Mechanics and its Applications 397, 76–110 (2014). doi:10.1016/j.physa.2013.11.009
  42. ↑ Boris S. Kerner, Micha Koller, Sergey L. Klenov, Hubert Rehborn, Michael Leibel, "The physics of empirical nuclei for spontaneous traffic breakdown in free flow at highway bottlenecks" Physica A 438 365–397 (2015). doi: 10.1016/j.physa.2015.05.102
  43. ↑ Boris S. Kerner, Peter Hemmerle, Micha Koller, Gerhard Hermanns, Sergey L. Klenov, Hubert Rehborn, and Michael Schreckenberg, "Empirical synchronized flow in oversaturated city traffic" Phys. Rev. E 90, 032810 (2014). doi: 10.1103/PhysRevE.90.032810
  44. ↑ Шаблон:Cite journal
  45. ↑ Minimizing the probability of the occurrence of traffic congestion in a traffic network
  46. ↑ Boris S. Kerner, Sergey L. Klenov, "A microscopic model for phase transitions in traffic flow" J. Phys. A: Math. Gen. 35, L31-L43 (2002). doi: 10.1088/0305-4470/35/3/102
  47. ↑ Boris S. Kerner, Sergey L. Klenov, Dietrich E Wolf, "Cellular automata approach to three-phase traffic theory" J. Phys. A: Math. Gen. 35, 9971–10013 (2002). doi: 10.1088/0305-4470/35/47/303
  48. ↑ Boris S. Kerner, Sergey L. Klenov, "Microscopic theory of spatio-temporal congested traffic patterns at highway bottlenecks" Phys. Rev. E 68, 036130 (2003). doi: 10.1103/PhysRevE.68.036130
  49. ↑ Boris S. Kerner, Sergey L. Klenov, "Spatiotemporal patterns in heterogeneous traffic flow with a variety of driver behavioural characteristics and parameters" J. Phys. A: Math. Gen. 37, 8753–8788 (2004). doi: 10.1088/0305-4470/37/37/001
  50. ↑ Boris S. Kerner, Sergey L. Klenov, "Deterministic microscopic three-phase traffic flow models" J. Phys. A: Math. Gen. 39, 1775–1809 (2006). doi: 10.1088/0305-4470/39/8/002
  51. ↑ Boris S. Kerner, Sergey L. Klenov, "Phase transitions in traffic flow on multilane roads" Phys. Rev. E 80, 056101 (2009). doi: 10.1103/PhysRevE.80.056101
  52. ↑ Boris S. Kerner, Sergey L. Klenov, "A study of phase transitions on multilane roads in the framework of three-phase traffic theory", Transportation Research Record, 2124, 67–77 (2009). doi: 10.3141/2124-07
  53. ↑ Boris S. Kerner, Sergey L. Klenov, "A Theory of Traffic Congestion at Moving Bottlenecks" J. Phys. A: Math. Gen. 43, 425101 (2010). doi: 10.1088/1751-8113/43/42/425101
  54. ↑ Boris S. Kerner, Sergey L. Klenov, and Michael Schreckenberg, "Simple cellular automaton model for traffic breakdown, highway capacity, and synchronized flow" Phys. Rev. E 84, 046110 (2011). doi: 10.1103/PhysRevE.84.046110
  55. ↑ Boris S. Kerner, Sergey L. Klenov, Gerhard Hermanns, and Michael Schreckenberg, "Effect of driver over-acceleration on traffic breakdown in three-phase cellular automaton traffic flow models" Physica A 392, 4083–4105 (2013). doi: 10.1016/j.physa.2013.04.035
  56. ↑ Boris S. Kerner, Sergey L. Klenov, and Michael Schreckenberg, "Probabilistic physical characteristics of phase transitions at highway bottlenecks: Incommensurability of three-phase and two-phase traffic-flow theories" Phys. Rev. E 89, 052807 (2014). doi: 10.1103/PhysRevE.89.052807
  57. ↑ Boris S. Kerner, Sergey L. Klenov, Andreas Hiller, "Criterion for traffic phases in single vehicle data and empirical test of a microscopic three-phase traffic theory" J. Phys. A: Math. Gen. 39, 2001–2020 (2006). doi: 10.1088/0305-4470/39/9/002
  58. ↑ Boris S. Kerner, Sergey L. Klenov, Hubert Rehborn, and Andreas Hiller, "Microscopic features of moving traffic jams" Phys. Rev. E 73, 046107 (2006). doi: 10.1103/PhysRevE.73.046107
  59. ↑ Boris S. Kerner, Sergey L. Klenov, Andreas Hiller, "Empirical test of a microscopic three-phase traffic theory" Nonlinear Dynamics, 49, 525–553 (2007). doi: 10.1007/s11071-006-9113-1
  60. ↑ Boris S. Kerner, Sergey L. Klenov, Gerhard Hermanns, Peter Hemmerle, Hubert Rehborn, and Michael Schreckenberg "Synchronized flow in oversaturated city traffic", Phys. Rev. E 88, 054801 (2013). doi: 10.1103/PhysRevE.88.054801
  61. ↑ Boris S. Kerner, Sergey L. Klenov, and Michael Schreckenberg, "Traffic breakdown at a signal: classical theory versus the three-phase theory of city traffic" Journal of Statistical Mechanics: Theory and Experiment, P03001 (2014). doi: 10.1088/1742-5468/2014/03/p03001
  62. ↑ Boris S. Kerner, Hubert Rehborn, Mario Aleksic, Andreas Haug "Recognition and tracking of spatial-temporal congested traffic patterns on freeways", Transportation Research Part C: Emerging Technologies, 12, 369–400 (2004). doi: 10.1016/j.trc.2004.07.015
  63. ↑ Hubert Rehborn, Micha Koller, Stefan Kaufmann, Data-Driven Traffic Engineering: Understanding of Traffic and Applications Based on Three-Phase Traffic Theory, Elsevier, Amsterdam, 2020
  64. ↑ Boris S. Kerner, "Control of spatiotemporal congested traffic patterns at highway bottlenecks", Physica A, 355, 565–601 (2005). doi: 10.1016/j.physa.2005.04.025
  65. ↑ Boris S. Kerner, "Control of Spatiotemporal Congested Traffic Patterns at Highway Bottlenecks", IEEE Transactions on Intelligent Transportation Systems 8, 308–320 (2007). doi: 10.1109/TITS.2007.894192
  66. ↑ Boris S. Kerner, "Study of freeway speed limit control based on three-phase traffic theory", Transportation Research Record, 1999, 30–39 (2007). doi: 10.3141/1999-04
  67. ↑ Boris S. Kerner, "On-Ramp Metering Based on Three-Phase Traffic Theory: Downstream Off-Ramp and Upstream On-Ramp Bottlenecks", Transportation Research Record, 2088, 80–89 (2008). doi: 10.3141/2088-09
  68. ↑ Boris S. Kerner, "Physics of automated driving in framework of three-phase traffic theory" Phys. Rev. E, 97, 042303 (2018). doi: 10.1103/PhysRevE.97.042303
  69. ↑ Boris S. Kerner, "Autonomous Driving in the Framework of Three-Phase Traffic Theory". In: "Complex Dynamics of Traffic Management", Encyclopedia of Complexity and Systems Science Series, 2nd ed., edited by Boris S. Kerner (Springer, New York, 2019), pp. 343–385. doi: 10.1007/978-1-4939-8763-4_724
  70. ↑ Boris S. Kerner, "Effect of Autonomous Driving on Traffic Breakdown in Mixed Traffic Flow: A Comparison of Classical ACC with Three-Traffic-Phase-ACC (TPACC)". Physica A: Statistical Mechanics and its Applications, 562, 125315 (2021). doi: 10.1016/j.physa.2020.125315
  71. ↑ Boris S. Kerner, "Microscopic theory of traffic-flow instability governing traffic breakdown at highway bottlenecks: Growing wave of increase in speed in synchronized flow", Phys. Rev. E, 92, 062827 (2015). doi: 10.1103/PhysRevE.92.062827
  72. ↑ Boris S. Kerner, Understanding Real Traffic: Paradigm Shift in Transportation Science, Springer, Berlin, Heidelberg, New York 2021
  73. ↑ 73,0 73,1 Boris S. Kerner, "The maximization of the network throughput ensuring free flow conditions in traffic and transportation networks: Breakdown minimization (BM) principle versus Wardrop's equilibria", Eur. Phys. B J., 89, 199 (2016). doi: 10.1140/epjb/e2016-70395-8
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