microscopic traffic flow model

Also known as time-continuous models, all car-following models have in common that they are defined by ordinary differential equations describing the complete dynamics of the vehicles' positions $x\_\backslash alpha$ and velocities $v\_\backslash alpha$. It is assumed that the input stimuli of the drivers are restricted to their own velocity $v\_\backslash alpha$, the net distance (bumper-to-bumper distance) $s\_\backslash alpha\; =\; x\_\{\backslash alpha-1\}\; -\; x\_\backslash alpha\; -\; \backslash ell\_\{\backslash alpha-1\}$ to the leading vehicle $\backslash alpha-1$ (where $\backslash ell\_\{\backslash alpha-1\}$ denotes the vehicle length), and the velocity $v\_\{\backslash alpha-1\}$ of the leading vehicle. The equation of motion of each vehicle is characterized by an acceleration function that depends on those input stimuli:

:$\backslash ddot\{x\}\_\backslash alpha(t)\; =\; \backslash dot\{v\}\_\backslash alpha(t)\; =\; F(v\_\backslash alpha(t),\; s\_\backslash alpha(t),\; v\_\{\backslash alpha-1\}(t),\; s\_\{\backslash alpha-1\}(t))$

In general, the driving behavior of a single driver-vehicle unit $\backslash alpha$ might not merely depend on the immediate leader $\backslash alpha-1$ but on the $n\_a$ vehicles in front. The equation of motion in this more generalized form reads:

:$\backslash dot\{v\}\_\backslash alpha(t)\; =\; f(x\_\backslash alpha(t),\; v\_\backslash alpha(t),\; x\_\{\backslash alpha-1\}(t),\; v\_\{\backslash alpha-1\}(t),\; \backslash ldots,\; x\_\{\backslash alpha-n\_a\}(t),\; v\_\{\backslash alpha-n\_a\}(t))$

• Optimal velocity model (OVM)
• Velocity difference model (VDIFF)
• Wiedemann model (1974)
• Gipps' model (Gipps, 1981){{Cite journal| doi = 10.1016/0191-2615(81)90037-0| issn = 0191-2615| volume = 15| issue = 2| pages = 105–111| last = Gipps| first = P. G.| title = A behavioural car-following model for computer simulation| journal = Transportation Research Part B: Methodological| access-date = 2022-02-17| date = 1981| url = https://dx.doi.org/10.1016/0191-2615%2881%2990037-0}}
• Intelligent driver model (IDM, 1999){{Cite journal| doi = 10.1103/physreve.62.1805| issn = 1063-651X| volume = 62| issue = 2 Pt A| pages = 1805–1824| last1 = Treiber| first1 = null| last2 = Hennecke| first2 = null| last3 = Helbing| first3 = null| title = Congested traffic states in empirical observations and microscopic simulations| journal = Physical Review E| date = August 2000| pmid = 11088643| arxiv = cond-mat/0002177| bibcode = 2000PhRvE..62.1805T| s2cid = 1100293}}
• DNN based anticipatory driving model (DDS, 2021){{Cite conference |doi=10.1109/IV48863.2021.9575314 |conference=2021 IEEE Intelligent Vehicles Symposium (IV) |pages=496–501 |last1=Isha |first1=Most. Kaniz Fatema |last2=Shawon |first2=Md. Nazirul Hasan |last3=Shamim |first3=Md. |last4=Shakib |first4=Md. Nazmus |last5=Hashem |first5=M.M.A. |last6=Kamal |first6=M.A.S. |title=A DNN Based Driving Scheme for Anticipatory Car Following Using Road-Speed Profile |book-title=2021 IEEE Intelligent Vehicles Symposium (IV) |date=July 2021}}

Cellular automaton (CA) models use integer variables to describe the dynamical properties of the system. The road is divided into sections of a certain length $\backslash Delta\; x$ and the time is discretized to steps of $\backslash Delta\; t$. Each road section can either be occupied by a vehicle or empty and the dynamics are given by updated rules of the form:

:$v\_\backslash alpha^\{t+1\}\; =\; f(s\_\backslash alpha^t,\; v\_\backslash alpha^t,\; v\_\{\backslash alpha-1\}^t,\; \backslash ldots)$

:$x\_\backslash alpha^\{t+1\}\; =\; x\_\backslash alpha^t\; +\; v\_\backslash alpha^\{t+1\}\backslash Delta\; t$

(the simulation time $t$ is measured in units of $\backslash Delta\; t$ and the vehicle positions $x\_\backslash alpha$ in units of $\backslash Delta\; x$).

The time scale is typically given by the reaction time of a human driver, $\backslash Delta\; t\; =\; 1\; \backslash text\{s\}$. With $\backslash Delta\; t$ fixed, the length of the road sections determines the granularity of the model. At a complete standstill, the average road length occupied by one vehicle is approximately 7.5 meters. Setting $\backslash Delta\; x$ to this value leads to a model where one vehicle always occupies exactly one section of the road and a velocity of 5 corresponds to $5\; \backslash Delta\; x/\backslash Delta\; t\; =\; 135\; \backslash text\{km/h\}$, which is then set to be the maximum velocity a driver wants to drive at. However, in such a model, the smallest possible acceleration would be $\backslash Delta\; x/(\backslash Delta\; t)^2\; =\; 7.5\; \backslash text\{m\}/\backslash text\{s\}^2$ which is unrealistic. Therefore, many modern CA models use a finer spatial discretization, for example $\backslash Delta\; x\; =\; 1.5\; \backslash text\{m\}$, leading to a smallest possible acceleration of $1.5\; \backslash text\{m\}/\backslash text\{s\}^2$.

Although cellular automaton models lack the accuracy of the time-continuous car-following models, they still have the ability to reproduce a wide range of traffic phenomena. Due to the simplicity of the models, they are numerically very efficient and can be used to simulate large road networks in real-time or even faster.

• Rule 184
• Biham–Middleton–Levine traffic model
• Nagel–Schreckenberg model (NaSch, 1992)

• Microsimulation

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Category:Road traffic management

Category:Mathematical modeling

Category:Traffic flow