# Car Following Models in Traffic Engineering

Car following models in traffic engineering are mathematical formulations used to describe the behavior of a vehicle following another vehicle in a traffic stream. These models help in understanding traffic flow dynamics, designing traffic control systems, and simulating traffic scenarios. Here is a summary of some major car following models:

1. Pipes Car-Following Model (1953):

Developed by L. A. Pipes, this model is based on the idea that drivers maintain a constant time headway (t) - the time it takes for the following vehicle to reach the position of the leading vehicle. In this model, the acceleration (a) of the following vehicle is determined by the difference in speed (Î”v) between the two vehicles and the time headway:

a = k(Î”v/t)

where k is a sensitivity factor that represents the responsiveness of the driver.

1. General Motors (GM) Model (1961):

The Gazis-Herman-Rothery (GHR) model, often called the GM model, assumes that drivers react proportionally to both the speed difference (Î”v) and the time headway (t). The acceleration of the following vehicle is given by:

a = k1(Î”v) + k2(Î”x/t^2)

where k1 and k2 are sensitivity factors, and Î”x is the distance between the two vehicles.

1. Linear (Optimal Velocity) Model (1971):

This model, proposed by B. S. Kerner and P. KonhÃ¤user, introduces the concept of optimal velocity (v_opt), which is the desired speed of a driver given the distance to the leading vehicle. The acceleration of the following vehicle depends on the difference between its current speed (v) and the optimal velocity:

a = k(v_opt - v)

where k is a sensitivity factor.

1. Gipps' Model (1981):

Developed by P. G. Gipps, this model considers the speeds, positions, and acceleration rates of both the leading (v1, x1, a1) and following vehicles (v2, x2, a2). It also accounts for driver reaction time (Ï„) and the physical constraints of the vehicle. The model calculates the following vehicle's maximum safe acceleration:

a2_max = [v2(2a1 + (v1^2 - v2^2)/(2s_max)) + (x1 - x2 - l)(2a1 - (v1^2)/(2s_max))]/[(v1 + v2)(v1 + v2 + 2Ï„a1)]

where l is the vehicle length and s_max is the maximum braking distance.

1. Intelligent Driver Model (IDM) (2000):

Developed by Martin Treiber and Ansgar Hennecke, IDM is a more advanced car-following model that adapts to different traffic scenarios. The model calculates the acceleration of the following vehicle based on several parameters:

a = a_max [1 - (v/v0)^Î´ - (s*(v,Î”v)/s0)^2]

where a_max is the maximum acceleration, v0 is the desired speed, Î´ is the speed exponent,          s*(v,Î”v) is the desired distance, and s0 is the minimum distance.

1. Adaptive Cruise Control (ACC) Models:

These models focus on the behavior of vehicles equipped with ACC systems. ACC uses sensors and control algorithms to automatically adjust vehicle speed and maintain a safe distance from the leading vehicle. Various ACC models have been proposed, incorporating different levels of detail and assumptions about human-driven and ACC-equipped vehicles' interactions.

1. Cellular Automaton (CA) Models:

In these models, the road is divided into discrete cells, and vehicles move from one cell to another based on simple rules and predefined states. The Nagel-Schreckenberg model is a well-known example of a CA-based car-following model. It uses the following rules:

a.      Acceleration:

If a vehicle's speed is lower than its maximum speed and there is enough space in front, it accelerates by one unit.

b.      Deceleration:

If there is not enough space in front, the vehicle decelerates to avoid collision, reducing its speed to match the distance to the vehicle ahead.

c.      Randomization:

With a certain probability, the vehicle's speed is reduced by one unit to account for random fluctuations in driver behavior.

d.      Movement:

The vehicle moves forward according to its updated speed.

CA models are computationally efficient and well-suited for large-scale traffic simulations. However, their discrete nature can sometimes lead to unrealistic representations of traffic flow.

These car-following models, while varying in complexity and assumptions, all aim to capture the behavior of vehicles in a traffic stream. They are useful for understanding traffic flow dynamics, designing traffic control systems, and simulating various traffic scenarios. Each model has its strengths and limitations, and the choice of model depends on the specific goals and requirements of a given study or application.