34.3.3. General Motors’ model

The General Motors’ model is the most popular of the car-following theories because of the following reasons:

1. Agreement with field data; the simulation models developed based on General Motors’ car following models shows good correlation to the field data.
2. Mathematical relation to macroscopic model; Greenberg’s logarithmic model for speed-density relationship can be derived from General motors car following model.

The General Motors’ model is widely recognized in traffic flow analysis due to its effectiveness and reliability. It correlates well with real-world data collected from traffic studies, meaning that the predictions made by this model closely match what happens on actual roads. Furthermore, this model has a mathematical connection to macroscopic models, specifically through Greenberg’s logarithmic relationship between speed and density in traffic, allowing for a broader understanding of traffic behaviors under different conditions.

Imagine a chef who relies on a popular recipe that consistently yields great results. The approval from diners (field data) and the ability to tweak the recipe into various cuisines (mathematical relation) make it a trusted choice for the chef. Similarly, the General Motors’ model is a reliable framework traffic engineers can use to predict and analyze vehicle interactions.

In car following models, the motion of individual vehicle is governed by an equation which is analogous to the Newton’s Laws of motion. In Newtonian mechanics, acceleration can be regarded as the response of the particle to the stimulus it receives in the form of force which includes both the external force as well as those arising from the interaction with all other particles in the system. This model is the widely used and will be discussed in detail later.

The General Motors’ model uses principles from Newtonian physics to describe how vehicles behave when following each other. Essentially, just as a ball responds to forces acting on it, like gravity or a push, a vehicle reacts to forces including its speed relative to the car in front, the distance separating them, and other traffic dynamics. This helps in creating a realistic simulation of how drivers adjust their speed and spacing in response to one another.

Think about a line of cars in a drive-thru. If the car in front moves forward quickly, the one behind reacts by moving forward as well. This reaction mimics Newton's laws—just as an object accelerates or decelerates when acted upon by forces, cars adjust their acceleration based on the distance and speed of the car ahead.

Therefore, the governing equations of a traffic flow can be developed as below. Let ∆T be the reaction time, and ∆t is the updation time, the governing equations can be written as, v(t) = v(t − ∆t) + a(t) ∆t x(t) = x(t − ∆t) + v(t) ∆t + ½ a(t) ∆t²

To create a simulation using the General Motors' model, two main equations determine how vehicles behave over time. The first equation calculates the velocity at a given time based on the previous velocity and acceleration. The second equation calculates the position of the vehicle at that same point in time, combining past position, current speed, and the effect of acceleration. This systematic approach enables the simulation to predict how a car’s speed and position change as it interacts with other vehicles.

Consider tracking a runner in a race. Using their previous speed and how much they speed up or slow down at a given moment lets you predict where they'll be in the next few seconds. Just like in the race, the General Motors’ model uses the vehicle's past performance (speed and position) to estimate future movement, allowing for realistic traffic simulations.

--