{{Multiple issues| {{original research|date=June 2017}} {{More citations needed|date=October 2021}} }}
'''Microscopic traffic flow models''' are a class of scientific models of vehicular traffic dynamics.
In contrast, to macroscopic models, microscopic traffic flow models simulate single vehicle-driver units, so the dynamic variables of the models represent microscopic properties like the position and velocity of single vehicles.
==Car-following models== 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 <math>x_\alpha</math> and velocities <math>v_\alpha</math>. It is assumed that the input stimuli of the drivers are restricted to their own velocity <math>v_\alpha</math>, the net distance (bumper-to-bumper distance) <math>s_\alpha = x_{\alpha-1} - x_\alpha - \ell_{\alpha-1}</math> to the leading vehicle <math>\alpha-1</math> (where <math>\ell_{\alpha-1}</math> denotes the vehicle length), and the velocity <math>v_{\alpha-1}</math> of the leading vehicle. The equation of motion of each vehicle is characterized by an acceleration function that depends on those input stimuli:
:<math>\ddot{x}_\alpha(t) = \dot{v}_\alpha(t) = F(v_\alpha(t), s_\alpha(t), v_{\alpha-1}(t), s_{\alpha-1}(t))</math>
In general, the driving behavior of a single driver-vehicle unit <math>\alpha</math> might not merely depend on the immediate leader <math>\alpha-1</math> but on the <math>n_a</math> vehicles in front. The equation of motion in this more generalized form reads:
:<math>\dot{v}_\alpha(t) = f(x_\alpha(t), v_\alpha(t), x_{\alpha-1}(t), v_{\alpha-1}(t), \ldots, x_{\alpha-n_a}(t), v_{\alpha-n_a}(t))</math>
===Examples of car-following models=== * Optimal velocity model (OVM) * Velocity difference model (VDIFF) * Wiedemann model (1974) * Gipps' model (Gipps, 1981)<ref>{{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| url-access = subscription}}</ref> * Intelligent driver model (IDM, 1999)<ref>{{Cite journal| doi = 10.1103/physreve.62.1805| issn = 1063-651X| volume = 62| issue = 2 Pt A| pages = 1805–1824| last1 = Treiber| first1 = M.| last2 = Hennecke| first2 = A.| last3 = Helbing| first3 = D.| 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}}</ref> * DNN based anticipatory driving model (DDS, 2021)<ref>{{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}}</ref> * Rakha-Pasumarthy-Adjerid model (RPA model)<ref>{{Cite journal |last=Rakha |first=Hesham |last2=Pasumarthy |first2=Praveen |last3=Adjerid |first3=Slimane |date=April 2009 |title=A simplified behavioral vehicle longitudinal motion model |url=http://www.tandfonline.com/doi/full/10.3328/TL.2009.01.02.95-110 |journal=Transportation Letters |language=en |volume=1 |issue=2 |pages=95–110 |doi=10.3328/TL.2009.01.02.95-110 |issn=1942-7867|url-access=subscription }}</ref> * Fadhloun-Rakha model (FR model)<ref>{{Cite journal |last=Fadhloun |first=Karim |last2=Rakha |first2=Hesham |date=March 2020 |title=A novel vehicle dynamics and human behavior car-following model: Model development and preliminary testing |url=https://linkinghub.elsevier.com/retrieve/pii/S2046043018301631 |journal=International Journal of Transportation Science and Technology |language=en |volume=9 |issue=1 |pages=14–28 |doi=10.1016/j.ijtst.2019.05.004}}</ref>
==Cellular automaton models== 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 <math>\Delta x</math> and the time is discretized to steps of <math>\Delta t</math>. Each road section can either be occupied by a vehicle or empty and the dynamics are given by updated rules of the form:
:<math>v_\alpha^{t+1} = f(s_\alpha^t, v_\alpha^t, v_{\alpha-1}^t, \ldots)</math> :<math>x_\alpha^{t+1} = x_\alpha^t + v_\alpha^{t+1}\Delta t</math>
(the simulation time <math>t</math> is measured in units of <math>\Delta t</math> and the vehicle positions <math>x_\alpha</math> in units of <math>\Delta x</math>).
The time scale is typically given by the reaction time of a human driver, <math>\Delta t = 1 \text{s}</math>. With <math>\Delta t</math> 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 <math>\Delta x</math> 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 <math>5 \Delta x/\Delta t = 135 \text{km/h}</math>, 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 <math>\Delta x/(\Delta t)^2 = 7.5 \text{m}/\text{s}^2</math> which is unrealistic. Therefore, many modern CA models use a finer spatial discretization, for example <math>\Delta x = 1.5 \text{m}</math>, leading to a smallest possible acceleration of <math>1.5 \text{m}/\text{s}^2</math>.
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.
=== Examples of cellular automaton models === * Rule 184 * Biham–Middleton–Levine traffic model * Nagel–Schreckenberg model (NaSch, 1992)
==See also== *Microsimulation
==References== {{Reflist}}
{{DEFAULTSORT:Microscopic Traffic Flow Model}} Category:Road traffic management Category:Mathematical modeling Category:Traffic flow