General motors car-following models (1958-1961)

In the late 50's, American company [[General Motors]] started developing some [[#Car following models]]
They developed several generations of a model. The models were calibrated using real world data.

General Motors car-following models fall under the spectrum of [[stimulus-response model]].

GM car following models

Response

In longitudinal movement, the follower can only react in 2 ways to what the vehicle in front does:

• Accelerate
• Decelerate

The response is therefore measured in terms of follower acceleration:
${\ddot{x}}_{n+1}(t+\mathrm{\Delta }t)$

Stimulus

Sensitivity

measures how attentive you are to traffic.

---

• Response: follower acceleration - ${\ddot{x}}_{n+1}(t+\mathrm{\Delta }t)$ - measured at a time $t+\mathrm{\Delta }t$ (a reaction time after $t$) - If the stimulus happens at $t$, the response always happens $\mathrm{\Delta }t$ after the stimulus
• Stimulus: Relative speed - ${\dot{x}}_n(t)-{\dot{x}}_{n+1}(t)$
• Sensitivity: a proportionality factor (or function) - $\alpha$

The general model is always in the form:
${\ddot{x}}_{n+1}(t+\mathrm{\Delta }t)=\alpha (...)\cdot [{\dot{x}}_n(t)-{\dot{x}}_{n+1}(t)]$
Depending on the model generation then, $\alpha$ is expressed as different functions.

GM model Generations

1st generation - GM model

In the 1st generation of the [[#General motors car-following models (1958-1961)]] the sensitivity $\alpha$ is considered constant:
$\begin{array}{rl}{\ddot{x}}_{n+1}(t+\mathrm{\Delta }t)& =\alpha \cdot [{\dot{x}}_n(t)-{\dot{x}}_{n+1}(t)]\\ \left[\frac{\mathrm{m}}{{\mathrm{s}}^2}\right]& =\left[\frac{1}{s}\right]\cdot \left[\frac{m}{s}\right]\end{array}$
The acceleration of the follower is directly proportional to the relative speed between the leader and the follower.

$\alpha$ and $\mathrm{\Delta }t$ are considered constant parameters to calibrate.

2nd generation - GM model

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GM observed that $\alpha$ had too much variation. So, they decided to use 2 different values of $\alpha$

3rd generation - GM model

Since they didn't know at what point you would get a new value of $\alpha$, GM decided to use a linear function to describe $\alpha$:

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${\ddot{x}}_{n+1}(t+\mathrm{\Delta }t)=\frac{{\alpha }_0}{[x_n(t)-x_{n+1}(t)]}[{\dot{x}}_n(t)-{\dot{x}}_{n+1}(t)]$
where ${\alpha }_0$ is the sensitivity coefficient and is measured in $\left[{\displaystyle \frac{\mathrm{m}}{\mathrm{s}}}\right]$.

So, ${\alpha }_0$ is a speed. It gives the linkage between microscopic and macroscopic traffic flow models.

Relation between 3rd gen micro model and Greenberg macro model

Integrating the 3rd generation model in time, we get (notice that the relative speed is the derivative of the spacing):
${\dot{x}}_{n+1}(t+\mathrm{\Delta }t)={\alpha }_0\ln (x_n(t)-x_{n+1}(t))+C$
In terms of speed:
$V={\alpha }_0\ln \left(\frac{C}{k}\right)$
We can try to find the constant of integration $C$ by the following initial conditions:
$V=0k=k_j$
then:
$0={\alpha }_0\ln \left(\frac{C}{k_j}\right)⟹C=k_j$
and we get:
$V={\alpha }_0\ln \left(\frac{k_j}{k}\right)$
which looks exactly like the [[03 - Fundamentals of traffic flow modeling - OMT#Greenberg k-v model (1959)]] with ${\alpha }_0=v_0$, which is the optimal speed.

4th generation - GM model

In the 4th generation, they proposed that ${\alpha }_0$ is not constant but dependent on the speed of the follower:
${\alpha }_0={\alpha }^{\prime }{\dot{x}}_{n+1}(t)$
the model is then
${\ddot{x}}_{n+1}(t+\mathrm{\Delta }t)=\frac{{\alpha }^{\prime }{\dot{x}}_{n+1}(t)}{[x_n(t)-x_{n+1}(t)]}[{\dot{x}}_n(t)-{\dot{x}}_{n+1}(t)]$
where

• ${\alpha }^{\prime }$ is a-dimensional

This means that how attentive you are, not only depends on the spacing, but also on the traveling speed (more attentive the faster you go)

! We still have 2 parameters.

5th generation - GM model

2 parameters are added, $n$ and $l$. They have no known physical meaning, but they're useful to hav more flexibility in calibrating the model.

${\ddot{x}}_{n+1}(t+\mathrm{\Delta }t)=\frac{{\alpha }^{\prime }[{\dot{x}}_{n+1}(t){]}^n}{[x_n(t)-x_{n+1}(t){]}^l}[{\dot{x}}_n(t)-{\dot{x}}_{n+1}(t)]$

All the [[#General motors car-following models (1958-1961)]] are a particular case of the 5th gen model.

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Pipes car-following model (1953)

Pipes' was one of the first [[#Car following models]] ever proposed.

It comes from the [[California Driving Code]]. The code stated: "Drivers should leave a gap of one vehicle length for every $10\mathrm{m}\mathrm{p}\mathrm{h}$ of traveling speed."

This translates to:

This model works quite well although:

• For low speeds, it yields spacing considerably lower than the measured one
• It only works for heavy traffic (this is true for any car following model)

This model is incredibly simple: in fact it has only 1 parameter: the follower length $l_{n+1}$.