Langevin method for a continuous stochastic car-following model and its stability conditions

Abstract In car-following models, the driver reacts according to his physical and psychological abilities which may change over time. However, most car-following models are deterministic and do not capture the stochastic nature of human perception. It is expected that purely deterministic traffic models may produce unrealistic results due to the stochastic driving behaviors of drivers. This paper is devoted to the development of a distinct car-following model where a stochastic process is adopted to describe the time-varying random acceleration which essentially reflects the random individual perception of driver behavior with respect to the leading vehicle over time. In particular, we apply coupled Langevin equations to model complex human driver behavior. In the proposed model, an extended Cox-Ingersoll-Ross (CIR) stochastic process will be used to describe the stochastic speed of the follower in response to the stimulus of the leader. An important property of the extended CIR process is to enhance the non-negative properties of the stochastic traffic variables (e.g. non-negative speed) for any arbitrary model parameters. Based on stochastic process theories, we derive stochastic linear stability conditions which, for the first time, theoretically capture the effect of the random parameter on traffic instabilities. Our stability results conform to the empirical results that the traffic instability is related to the stochastic nature of traffic flow at the low speed conditions, even when traffic is deemed to be stable from deterministic models.

[1]  Jean-Patrick Lebacque,et al.  A stochastic jump process applied to traffic flow modelling , 2014 .

[2]  S. Ross,et al.  A theory of the term structure of interest rates'', Econometrica 53, 385-407 , 1985 .

[3]  Fangfang Zheng,et al.  Stochastic Lagrangian modeling of traffic dynamics , 2018 .

[4]  Agachai Sumalee,et al.  Stochastic cell transmission model for traffic network with demand and supply uncertainties , 2013 .

[5]  M. Treiber,et al.  Improved 2D Intelligent Driver Model simulating synchronized flow and evolution concavity in traffic flow , 2016, 1603.00264.

[6]  C. Gardiner.  Stochastic Methods: A Handbook for the Natural and Social Sciences , 2009 .

[7]  P. Mazur.  On the theory of brownian motion , 1959 .

[8]  Serge P. Hoogendoorn,et al.  A geometric Brownian motion car-following model: towards a better understanding of capacity drop , 2018, Transportmetrica B: Transport Dynamics.

[9]  W. Y. Szeto,et al.  Stochastic cell transmission model (SCTM): A stochastic dynamic traffic model for traffic state surveillance and assignment , 2011 .

[10]  Henry X. Liu,et al.  A stochastic model of traffic flow: Theoretical foundations , 2011 .

[11]  Martin Treiber,et al.  Traffic Flow Dynamics , 2013 .

[12]  Mehdi Keyvan-Ekbatani,et al.  Integrated deep learning and stochastic car-following model for traffic dynamics on multi-lane freeways , 2019, Transportation Research Part C: Emerging Technologies.

[13]  Xiaobo Qu,et al.  A recurrent neural network based microscopic car following model to predict traffic oscillation , 2017 .

[14]  Daiheng Ni,et al.  Analysis of LWR model with fundamental diagram subject to uncertainties , 2011 .

[15]  C. Voloşencu,et al.  Nonlinear Science and Numerical Simulation 1 A 2 D chaotic path planning for mobile robots accomplishing boundary surveillance missions in adversarial conditions , 2014 .

[16]  Ziyou Gao,et al.  Empirical analysis and simulation of the concave growth pattern of traffic oscillations , 2016 .

[17]  Nakayama,et al.  Dynamical model of traffic congestion and numerical simulation. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[18]  J. B. Walsh,et al.  An introduction to stochastic partial differential equations , 1986 .

[19]  Jorge A. Laval,et al.  A parsimonious model for the formation of oscillations in car-following models , 2014 .

[20]  G. Uhlenbeck,et al.  On the Theory of the Brownian Motion , 1930 .

[21]  Jianfeng Zheng,et al.  A probabilistic stationary speed–density relation based on Newell’s simplified car-following model , 2014 .

[22]  Dirk Helbing,et al.  Understanding widely scattered traffic flows, the capacity drop, and platoons as effects of variance-driven time gaps. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  Henry X. Liu,et al.  A stochastic model of traffic flow: Gaussian approximation and estimation , 2013 .

[24]  Gordon F. Newell,et al.  A simplified car-following theory: a lower order model , 2002 .

[25]  R. Jiang,et al.  Full velocity difference model for a car-following theory. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  Hai Le Vu,et al.  A multiclass microscopic model for heterogeneous platoon with vehicle-to-vehicle communication , 2019 .

[27]  Jorge A. Laval,et al.  The Distribution of Congestion on a Class of Stochastic Kinematic Wave Models , 2014, Transp. Sci..

[28]  Martin Treiber,et al.  The Intelligent Driver Model with Stochasticity -New Insights Into Traffic Flow Oscillations , 2017 .

[29]  Bin Jia,et al.  Experimental and empirical investigations of traffic flow instability , 2018, Transportation Research Part C: Emerging Technologies.

[30]  Dirk Helbing,et al.  Delays, inaccuracies and anticipation in microscopic traffic models , 2006 .

[31]  Alireza Talebpour,et al.  Influence of connected and autonomous vehicles on traffic flow stability and throughput , 2016 .

[32]  H. Kushner.  Introduction to stochastic control , 1971 .

[33]  Ren Wang,et al.  Comparing traffic state estimators for mixed human and automated traffic flows , 2017 .

[34]  Reinhard Mahnke,et al.  Physics of Stochastic Processes: How Randomness Acts in Time , 2009 .

[35]  A. Skabardonis,et al.  Understanding stop-and-go traffic in view of asymmetric traffic theory , 2009 .

[36]  Simon M. J. Lyons.  Introduction to stochastic differential equations , 2011 .

[37]  Dong Ngoduy,et al.  Multiclass first-order traffic model using stochastic fundamental diagrams , 2011 .

[38]  Bin Jia,et al.  Improved 2D intelligent driver model in the framework of three-phase traffic theory simulating synchronized flow and concave growth pattern of traffic oscillations , 2016 .

[39]  X. Mao,et al.  Stochastic Differential Equations and Applications , 1998 .

[40]  Dong Ngoduy,et al.  Instability of cooperative adaptive cruise control traffic flow: A macroscopic approach , 2013, Commun. Nonlinear Sci. Numer. Simul..

[41]  Dong Ngoduy,et al.  Analytical studies on the instabilities of heterogeneous intelligent traffic flow , 2013, Commun. Nonlinear Sci. Numer. Simul..