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doi: 10.3934/dcdsb.2022100
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Model of vehicle interactions with autonomous cars and its properties

1. 

Institute of Geometry and Applied Mathematics, RWTH Aachen University, Templergraben 55, 52062 Aachen, Germany

2. 

Department of Mathematics "G. Castelnuovo", Sapienza University of Rome, P.le Aldo Moro 5, 00185 Roma, Italy

*Corresponding author: Giuseppe Visconti

Received  October 2021 Revised  February 2022 Early access June 2022

We study a hierarchy of models based on kinetic equations for the descriptions of traffic flow in presence of autonomous and human–driven vehicles. The autonomous cars considered in this paper are thought of as vehicles endowed with some degree of autonomous driving which decreases the stochasticity of the drivers' behavior. Compared to the existing literature, we do not model autonomous cars as externally controlled vehicles. We investigate whether this feature is enough to provide a stabilization of traffic instabilities such as stop and go waves. We propose two indicators to quantify traffic instability and we find, with analytical and numerical tools, that traffic instabilities are damped as the penetration rate of the autonomous vehicles increases.

Citation: Michael Herty, Gabriella Puppo, Giuseppe Visconti. Model of vehicle interactions with autonomous cars and its properties. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022100
References:
[1]

G. Albi and L. Pareschi, Binary interaction algorithms for the simulation of flocking and swarming dynamics, Multiscale Model. Simul., 11 (2013), 1-29.  doi: 10.1137/120868748.

[2]

A. AwA. KlarT. Materne and M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader models, SIAM J. Appl. Math., 63 (2002), 259-278.  doi: 10.1137/S0036139900380955.

[3]

A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916–938 (electronic). doi: 10.1137/S0036139997332099.

[4]

P. L. BhatnagarE. P. Gross and M. Krook, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94 (1954), 511-525.  doi: 10.1103/PhysRev.94.511.

[5]

A. V. Bobylev and K. Nanbu, Theory of collision algorithms for gases and plasmas based on the Boltzmann equation and the Landau-Fokker-Planck equation, Phys. Rev. E, 61 (2000), 4576-86.  doi: 10.1103/PhysRevE.61.4576.

[6]

D. Borra and T. Lorenzi, Asymptotic analysis of continuous opinion dynamics models under bounded confidence, Commun. Pur. Appl. Anal., 12 (2013), 1487-1499.  doi: 10.3934/cpaa.2013.12.1487.

[7]

R. Borsche and A. Klar, A nonlinear discrete velocity relaxation model for traffic flow, SIAM J. Appl. Math., 78 (2018), 2891-2917.  doi: 10.1137/17M1152681.

[8]

C. Canuto, F. Fagnani and P. Tilli, An Eulerian approach to the analysis of Krause's consensus models, SIAM J. Control Optim., 50 (2012), 243–265. doi: 10.1137/100793177.

[9]

G.-Q. ChenC. D. Levermore and T.-P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math, 47 (1992), 787-830.  doi: 10.1002/cpa.3160470602.

[10]

J. Chow, Informed Urban Transport Systems: Classic and Emerging Mobility Methods Toward Smart Cities, Elsevier, 2018.

[11]

A. Corli and H. Fan, Hysteresis and stop-and-go waves in traffic flows, Mathematical Models and Methods in Applied Sciences, 29 (2019), 2637-2678.  doi: 10.1142/S0218202519500568.

[12]

V. CosciaM. Delitala and P. Frasca, On the mathematical theory of vehicular traffic flow. II. Discrete velocity kinetic models, Internat. J. Non-Linear Mech., 42 (2007), 411-421.  doi: 10.1016/j.ijnonlinmec.2006.02.008.

[13]

C. F. Daganzo, Requiem for second-order fluid approximation to traffic flow, Transport. Res. B-Meth., 29 (1995), 277-286.  doi: 10.1016/0191-2615(95)00007-Z.

[14]

M. Delitala and A. Tosin, Mathematical modeling of vehicular traffic: A discrete kinetic theory approach, Math. Models Methods Appl. Sci., 17 (2007), 901-932.  doi: 10.1142/S0218202507002157.

[15]

M. L. Delle Monache, T. Liard, A. Rat, R. Stern, R. Bhadani, B. Seibold, J. Sprinkle, D. B. Work and B. Piccoli, Feedback control algorithms for the dissipation of traffic waves with autonomous vehicles, Springer International Publishing, 2019,275–299. doi: 10.1007/978-3-030-25446-9_12.

[16]

D. Farooq and J. Juhasz, An investigation of speed variance effect on lane-changing for driving logic "cautious" on highways, Advances in Transportation Studies: An international Journal. Section B 51, 109–120.

[17]

L. Fermo and A. Tosin, A fully-discrete-state kinetic theory approach to modeling vehicular traffic, SIAM J. Appl. Math., 73 (2013), 1533-1556.  doi: 10.1137/120897110.

[18]

D. Helbing, Video of traffic waves, Website, http://trafficforum.org/.

[19]

M. HertyA. Klar and L. Pareschi, General kinetic models for vehicular traffic flows and Monte-Carlo methods, Comput. Methods Appl. Math., 5 (2005), 155-169.  doi: 10.2478/cmam-2005-0008.

[20]

M. HertyG. PuppoS. Roncoroni and G. Visconti, The BGK approximation of kinetic models for traffic, Kinet. Relat. Models, 13 (2020), 279-307.  doi: 10.3934/krm.2020010.

[21]

M. Herty and G. Visconti, Analysis of risk levels for traffic on a multi-lane highway, IFAC-PapersOnLine, 51 (2018), 43-48.  doi: 10.1016/j.ifacol.2018.07.008.

[22]

S. Jin and Z. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math, 48 (1995), 235-276.  doi: 10.1002/cpa.3160480303.

[23]

B. S. Kerner, Experimental features of self-organization in traffic flow, Phys. Rev. Lett., 81 (1998), 3797-3800.  doi: 10.1103/PhysRevLett.81.3797.

[24]

A. Klar and R. Wegener, Enskog-like kinetic models for vehicular traffic, J. Stat. Phys., 87 (1997), 91-114.  doi: 10.1007/BF02181481.

[25]

Y. KuangX. Qu and S. Wang, A tree-structured crash surrogate measure for freeways, Accid. Anal. Prev., 77 (2015), 137-148.  doi: 10.1016/j.aap.2015.02.007.

[26]

J. A. Laval and L. Leclercq, A mechanism to describe the formation and propagation of stop-and-go waves in congested freeway traffic, Philosophical Transactions: Mathematical, Physical and Engineering Sciences, 368 (2010), 4519-4541.  doi: 10.1098/rsta.2010.0138.

[27]

M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London. Ser. A., 229 (1955), 317-345.  doi: 10.1098/rspa.1955.0089.

[28]

P. Nelson and A. Sopasakis, The Chapman-Enskog expansion: A novel approach to hierarchical extensions of Lighthill-Whitham models, in Transportation and Traffic Theory. Proceedings of the 14th International Symposium on Transportation and Traffic Theory, Jerusalem, Israel, July, 1999 (ed. A. Ceder), Pergamon Press Ltd., 1999, 51–79, 14th International Symposium on Transportation and Traffic Theory (ISTTT14 1999).

[29]

W. H. Organization, Global Status Report on Road Safety, Technical Report, 2015.

[30]

L. Pareschi and G. Toscani, Interacting Multiagent Systems. Kinetic equations and Monte Carlo methods, Oxford University Press, 2013.

[31]

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, Transport. Res., 9 (1975), 225-235.  doi: 10.1016/0041-1647(75)90063-5.

[32]

H. J. Payne, Models of freeway traffic and control, Math. Models Publ. Sys., Simulation Council Proc. 28, 1 (1971), 51-61. 

[33]

B. Piccoli and A. Tosin, Vehicular traffic: A review of continuum mathematical models, in Encyclopedia of Complexity and Systems Science (ed. R. A. Meyers), vol. 22, Springer, New York, 2009, 9727–9749. doi: 10.1007/978-0-387-30440-3_576.

[34]

B. Piccoli, A. Tosin and M. Zanella, Model-based assessment of the impact of driver-assist vehicles using kinetic theory, Zeitschrift fur Angewandte Mathematik und Physik, 71, (2020), Paper No. 152, 25 pp. doi: 10.1007/s00033-020-01383-9.

[35]

I. Prigogine and R. Herman, Kinetic Theory of Vehicular Traffic, American Elsevier Publishing Co., New York, 1971.

[36]

G. PuppoM. SempliceA. Tosin and G. Visconti, Analysis of a multi-population kinetic model for traffic flow, Commun. Math. Sci., 15 (2017), 379-412.  doi: 10.4310/CMS.2017.v15.n2.a5.

[37]

G. PuppoM. SempliceA. Tosin and G. Visconti, Kinetic models for traffic flow resulting in a reduced space of microscopic velocities, Kinet. Relat. Mod., 10 (2017), 823-854.  doi: 10.3934/krm.2017033.

[38]

R. Ramadan, R. Rosales and B. Seibold, Structural properties of the stability of jamitons, in Mathematical Descriptions of Traffic Flow: Micro, Macro and Kinetic Models (eds. G. Puppo and A. Tosin), vol. 12 of ICIAM2019 SEMA SIMAI Springer Series, Springer International Publishing, 2021, 35–62. doi: 10.1007/978-3-030-66560-9_3.

[39]

P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42.

[40]

B. SeiboldM. R. FlynnA. R. Kasimov and R. R. Rosales, Constructing set-valued fundamental diagrams from jamiton solutions in second order traffic models, Netw. Heterog. Media, 8 (2013), 745-772.  doi: 10.3934/nhm.2013.8.745.

[41]

S. Singh and B. Singh Saini, Autonomous cars: Recent developments, challenges, and possible solutions, IOP Conference Series: Materials Science and Engineering, 1022 (2021), 012028. doi: 10.1088/1757-899X/1022/1/012028.

[42]

A. Sopasakis, Formal asymptotic models of vehicular traffic model closures, SIAM J. Appl. Math., 63 (2003), 1561-1584.  doi: 10.1137/S0036139902403020.

[43]

R. E. SternS. CuiM. L. Delle MonacheR. BhadaniM. BuntingM. ChurchillN. HamiltonR. HaulcyH. PohlmannF. WuB. PiccoliB. SeiboldJ. Sprinkle and D. B. Work, Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments, Transport. Res. C-Emer., 89 (2018), 205-221.  doi: 10.1016/j.trc.2018.02.005.

[44]

A. Tosin and M. Zanella, Uncertainty damping in kinetic traffic models by driver-assist controls, Math. Control Relat. Fields, 11 (2021), 681-713.  doi: 10.3934/mcrf.2021018.

[45]

A. Vadeby and A. Forsman, Speed distribution and traffic safety measures, in Presented at the Transport Research Arena (TRA), 2014.

[46]

R. Wegener and A. Klar, A kinetic model for vehicular traffic derived from a stochastic microscopic model, Transport. Theor. Stat., 25 (1996), 785-798.  doi: 10.1080/00411459608203547.

[47]

G. B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, New York, 1974.

[48]

E. YurtseverJ. LambertA. Carballo and K. Takeda, A survey of autonomous driving: Common practices and emerging technologies, IEEE Access, 8 (2020), 58443-58469.  doi: 10.1109/ACCESS.2020.2983149.

[49]

H. M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transport. Res. B-Meth., 36 (2002), 275-290.  doi: 10.1016/S0191-2615(00)00050-3.

show all references

References:
[1]

G. Albi and L. Pareschi, Binary interaction algorithms for the simulation of flocking and swarming dynamics, Multiscale Model. Simul., 11 (2013), 1-29.  doi: 10.1137/120868748.

[2]

A. AwA. KlarT. Materne and M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader models, SIAM J. Appl. Math., 63 (2002), 259-278.  doi: 10.1137/S0036139900380955.

[3]

A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916–938 (electronic). doi: 10.1137/S0036139997332099.

[4]

P. L. BhatnagarE. P. Gross and M. Krook, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94 (1954), 511-525.  doi: 10.1103/PhysRev.94.511.

[5]

A. V. Bobylev and K. Nanbu, Theory of collision algorithms for gases and plasmas based on the Boltzmann equation and the Landau-Fokker-Planck equation, Phys. Rev. E, 61 (2000), 4576-86.  doi: 10.1103/PhysRevE.61.4576.

[6]

D. Borra and T. Lorenzi, Asymptotic analysis of continuous opinion dynamics models under bounded confidence, Commun. Pur. Appl. Anal., 12 (2013), 1487-1499.  doi: 10.3934/cpaa.2013.12.1487.

[7]

R. Borsche and A. Klar, A nonlinear discrete velocity relaxation model for traffic flow, SIAM J. Appl. Math., 78 (2018), 2891-2917.  doi: 10.1137/17M1152681.

[8]

C. Canuto, F. Fagnani and P. Tilli, An Eulerian approach to the analysis of Krause's consensus models, SIAM J. Control Optim., 50 (2012), 243–265. doi: 10.1137/100793177.

[9]

G.-Q. ChenC. D. Levermore and T.-P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math, 47 (1992), 787-830.  doi: 10.1002/cpa.3160470602.

[10]

J. Chow, Informed Urban Transport Systems: Classic and Emerging Mobility Methods Toward Smart Cities, Elsevier, 2018.

[11]

A. Corli and H. Fan, Hysteresis and stop-and-go waves in traffic flows, Mathematical Models and Methods in Applied Sciences, 29 (2019), 2637-2678.  doi: 10.1142/S0218202519500568.

[12]

V. CosciaM. Delitala and P. Frasca, On the mathematical theory of vehicular traffic flow. II. Discrete velocity kinetic models, Internat. J. Non-Linear Mech., 42 (2007), 411-421.  doi: 10.1016/j.ijnonlinmec.2006.02.008.

[13]

C. F. Daganzo, Requiem for second-order fluid approximation to traffic flow, Transport. Res. B-Meth., 29 (1995), 277-286.  doi: 10.1016/0191-2615(95)00007-Z.

[14]

M. Delitala and A. Tosin, Mathematical modeling of vehicular traffic: A discrete kinetic theory approach, Math. Models Methods Appl. Sci., 17 (2007), 901-932.  doi: 10.1142/S0218202507002157.

[15]

M. L. Delle Monache, T. Liard, A. Rat, R. Stern, R. Bhadani, B. Seibold, J. Sprinkle, D. B. Work and B. Piccoli, Feedback control algorithms for the dissipation of traffic waves with autonomous vehicles, Springer International Publishing, 2019,275–299. doi: 10.1007/978-3-030-25446-9_12.

[16]

D. Farooq and J. Juhasz, An investigation of speed variance effect on lane-changing for driving logic "cautious" on highways, Advances in Transportation Studies: An international Journal. Section B 51, 109–120.

[17]

L. Fermo and A. Tosin, A fully-discrete-state kinetic theory approach to modeling vehicular traffic, SIAM J. Appl. Math., 73 (2013), 1533-1556.  doi: 10.1137/120897110.

[18]

D. Helbing, Video of traffic waves, Website, http://trafficforum.org/.

[19]

M. HertyA. Klar and L. Pareschi, General kinetic models for vehicular traffic flows and Monte-Carlo methods, Comput. Methods Appl. Math., 5 (2005), 155-169.  doi: 10.2478/cmam-2005-0008.

[20]

M. HertyG. PuppoS. Roncoroni and G. Visconti, The BGK approximation of kinetic models for traffic, Kinet. Relat. Models, 13 (2020), 279-307.  doi: 10.3934/krm.2020010.

[21]

M. Herty and G. Visconti, Analysis of risk levels for traffic on a multi-lane highway, IFAC-PapersOnLine, 51 (2018), 43-48.  doi: 10.1016/j.ifacol.2018.07.008.

[22]

S. Jin and Z. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math, 48 (1995), 235-276.  doi: 10.1002/cpa.3160480303.

[23]

B. S. Kerner, Experimental features of self-organization in traffic flow, Phys. Rev. Lett., 81 (1998), 3797-3800.  doi: 10.1103/PhysRevLett.81.3797.

[24]

A. Klar and R. Wegener, Enskog-like kinetic models for vehicular traffic, J. Stat. Phys., 87 (1997), 91-114.  doi: 10.1007/BF02181481.

[25]

Y. KuangX. Qu and S. Wang, A tree-structured crash surrogate measure for freeways, Accid. Anal. Prev., 77 (2015), 137-148.  doi: 10.1016/j.aap.2015.02.007.

[26]

J. A. Laval and L. Leclercq, A mechanism to describe the formation and propagation of stop-and-go waves in congested freeway traffic, Philosophical Transactions: Mathematical, Physical and Engineering Sciences, 368 (2010), 4519-4541.  doi: 10.1098/rsta.2010.0138.

[27]

M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London. Ser. A., 229 (1955), 317-345.  doi: 10.1098/rspa.1955.0089.

[28]

P. Nelson and A. Sopasakis, The Chapman-Enskog expansion: A novel approach to hierarchical extensions of Lighthill-Whitham models, in Transportation and Traffic Theory. Proceedings of the 14th International Symposium on Transportation and Traffic Theory, Jerusalem, Israel, July, 1999 (ed. A. Ceder), Pergamon Press Ltd., 1999, 51–79, 14th International Symposium on Transportation and Traffic Theory (ISTTT14 1999).

[29]

W. H. Organization, Global Status Report on Road Safety, Technical Report, 2015.

[30]

L. Pareschi and G. Toscani, Interacting Multiagent Systems. Kinetic equations and Monte Carlo methods, Oxford University Press, 2013.

[31]

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, Transport. Res., 9 (1975), 225-235.  doi: 10.1016/0041-1647(75)90063-5.

[32]

H. J. Payne, Models of freeway traffic and control, Math. Models Publ. Sys., Simulation Council Proc. 28, 1 (1971), 51-61. 

[33]

B. Piccoli and A. Tosin, Vehicular traffic: A review of continuum mathematical models, in Encyclopedia of Complexity and Systems Science (ed. R. A. Meyers), vol. 22, Springer, New York, 2009, 9727–9749. doi: 10.1007/978-0-387-30440-3_576.

[34]

B. Piccoli, A. Tosin and M. Zanella, Model-based assessment of the impact of driver-assist vehicles using kinetic theory, Zeitschrift fur Angewandte Mathematik und Physik, 71, (2020), Paper No. 152, 25 pp. doi: 10.1007/s00033-020-01383-9.

[35]

I. Prigogine and R. Herman, Kinetic Theory of Vehicular Traffic, American Elsevier Publishing Co., New York, 1971.

[36]

G. PuppoM. SempliceA. Tosin and G. Visconti, Analysis of a multi-population kinetic model for traffic flow, Commun. Math. Sci., 15 (2017), 379-412.  doi: 10.4310/CMS.2017.v15.n2.a5.

[37]

G. PuppoM. SempliceA. Tosin and G. Visconti, Kinetic models for traffic flow resulting in a reduced space of microscopic velocities, Kinet. Relat. Mod., 10 (2017), 823-854.  doi: 10.3934/krm.2017033.

[38]

R. Ramadan, R. Rosales and B. Seibold, Structural properties of the stability of jamitons, in Mathematical Descriptions of Traffic Flow: Micro, Macro and Kinetic Models (eds. G. Puppo and A. Tosin), vol. 12 of ICIAM2019 SEMA SIMAI Springer Series, Springer International Publishing, 2021, 35–62. doi: 10.1007/978-3-030-66560-9_3.

[39]

P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42.

[40]

B. SeiboldM. R. FlynnA. R. Kasimov and R. R. Rosales, Constructing set-valued fundamental diagrams from jamiton solutions in second order traffic models, Netw. Heterog. Media, 8 (2013), 745-772.  doi: 10.3934/nhm.2013.8.745.

[41]

S. Singh and B. Singh Saini, Autonomous cars: Recent developments, challenges, and possible solutions, IOP Conference Series: Materials Science and Engineering, 1022 (2021), 012028. doi: 10.1088/1757-899X/1022/1/012028.

[42]

A. Sopasakis, Formal asymptotic models of vehicular traffic model closures, SIAM J. Appl. Math., 63 (2003), 1561-1584.  doi: 10.1137/S0036139902403020.

[43]

R. E. SternS. CuiM. L. Delle MonacheR. BhadaniM. BuntingM. ChurchillN. HamiltonR. HaulcyH. PohlmannF. WuB. PiccoliB. SeiboldJ. Sprinkle and D. B. Work, Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments, Transport. Res. C-Emer., 89 (2018), 205-221.  doi: 10.1016/j.trc.2018.02.005.

[44]

A. Tosin and M. Zanella, Uncertainty damping in kinetic traffic models by driver-assist controls, Math. Control Relat. Fields, 11 (2021), 681-713.  doi: 10.3934/mcrf.2021018.

[45]

A. Vadeby and A. Forsman, Speed distribution and traffic safety measures, in Presented at the Transport Research Arena (TRA), 2014.

[46]

R. Wegener and A. Klar, A kinetic model for vehicular traffic derived from a stochastic microscopic model, Transport. Theor. Stat., 25 (1996), 785-798.  doi: 10.1080/00411459608203547.

[47]

G. B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, New York, 1974.

[48]

E. YurtseverJ. LambertA. Carballo and K. Takeda, A survey of autonomous driving: Common practices and emerging technologies, IEEE Access, 8 (2020), 58443-58469.  doi: 10.1109/ACCESS.2020.2983149.

[49]

H. M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transport. Res. B-Meth., 36 (2002), 275-290.  doi: 10.1016/S0191-2615(00)00050-3.

Figure 1.  Left: flux–density diagram (4). Middle: variance of microscopic speeds at equilibrium (9) as function of the density $ \rho $. Right: diffusion coefficient (13) in the diffusive limit of the BGK model (11) as function of the density $ \rho $. The plots are obtained using the Maxwellian of the spatially homogeneous model (3)–(6) with $ \Delta v = \frac{ v_{\max}}{3} $. Here, $ \rho_{\max} = 1 $ and $ v_{\max} = 1 $
Figure 2.  Schematic representation of the Ansatz (29) for the kinetic distribution with $ \rho = 0.8 \rho_{\max} $ and $ \delta_1 = 1 $ (dotted line), $ \delta_2 = 4 $ (dashed line), $ \delta_3 = 7 $ (solid line). The values $ \overline{f}_i $, $ i = 1,2,3 $, are found in order to satisfy mass conservation
Figure 3.  Fundamental diagrams of the single–distribution model (21) for autonomous and human–driven vehicles for three choices of the parameter $ \bar{\rho} $. The different diagrams refer to different penetration rates, $ p\leq 0.4 $. Here $ v_{\max} = 1 = \rho_{\max} = 1 $, $ \Delta v = \frac13 $ and $ \hat{u}(\rho) = \frac{1}{\rho}\int_{\mathcal{V}} v f(t,v) \mathrm{d} v $
Figure 4.  Variance of microscopic speeds at equilibrium of the single–distribution model (21) for autonomous and human–driven vehicles. Three choices of the parameter $ \bar{\rho} $ and different penetration rates $ p $ are considered. Here $ v_{\max} = 1 = \rho_{\max} = 1 $, $ \Delta v = \frac13 $ and $ \hat{u}(\rho) = \frac{1}{\rho}\int_{\mathcal{V}} v f(t,v) \mathrm{d} v $
Figure 5.  Top row: sign of the diffusion coefficient $ \mu(\rho) $ (13) of the single–distribution model (21) for autonomous and human–driven vehicles. Three choices of the parameter $ \bar{\rho} $ and different penetration rates $ p $ are considered. Here $ v_{\max} = 1 = \rho_{\max} = 1 $, $ \Delta v = \frac13 $ and $ \hat{u}(\rho) = \frac{1}{\rho}\int_{\mathcal{V}} v f(t,v) \mathrm{d} v $. Bottom row: left boundary $ \alpha $ and right boundary $ \beta $ of the interval of instability, and its amplitude $ \Gamma $, as function of the penetration rate $ p $
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