# Optimal Velocity Model

## About the Model

Optimal Velocity (OV) model is a traffic flow model proposed by Bando et al.[1,2].
The governing equation is as follows,

dx_{i}/dt = v_{i}
dv_{i}/dt = a [V(x_{i+1}-x_{i})-v_{i}],

with positions x_{i} and velocities v_{i} of cars.
The parameter "a" is sensitivity denoting the speed of the response.
This model involves the phase transition
from the uniform flow phase to the traffic jam.
The function V(b) denotes the optimal velocity
determined by the intervehicle distance.
Here, we take the following tanh type function

V(b) = tanh(b-2) + tanh(2)

as V(x). The function form is shown in the following figure.

### Model

ă€€This OV model describes the following features,

- A car will keep the maximum speed with enough the distance to the next car.
- A car tries to run with optimal velocity determined by the distance to the next car.

### Traffic Jam

In this model, there are two free parameters,
the density "L/N" and the sensitivity "a".
The phase diagram obtained from the linear stability analysis
is shown in the following diagram.

The uniform flow becomes unstable when a < 2 V'(L/N).

## Flash Demonstration

There are 20 cars (the red circles) running on
the circuit with the size L=40.
The green circles denote the distance between two cars (dx = x_{i+1}-x_{i})
and velocities (v_{i}) for each cars.

While the cars are running uniformly at the beginning of the simulation,
some clusters will be formed, and finally, the system will have traffic jam.
Click the flash to restart the simulator.

## Source Files of OV Model Simulator

Here is the sample code to simulate OV model using
the 4th order Runge-Kutta scheme.
There are 10 cars in the system with size L=20.
The sensitivity a is set to be 1.

Source Files (HTML)

ov.tar.gz (1.9KB)

Usage:

% tar xvzf ov.tar.gz
% make
% ./ov > result.dat

The executable file will put the spatiotemporal diagram.
You can see the diagram using gnuplot or other plotters.
% gnuplot
gnuplot> p "result.dat" 0 0

## Result

Java Applet is available.

At the beginning, the cars run uniformly.
However, the uniform flow becomes unstable and
finally the traffic jam occurs.

## References

- M. Bando, K. Hasebe, A. Nakayama, A. Shibata, and Y. Sugiyama, Jpn. J. Ind. Appl. Math.
**11**, 203 (1994).
- M. Bando, K. Hasebe, A. Nakayama, A. Shibata, and Y. Sugiyama, Phys. Rev. E
**51**, 1035 (1995).
- T. S. Komatsu, S. Sasa, Phys. Rev. E 52 5574 (1995).

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