Traffic flow is frequently studied because it is an example of a system far from equilibrium. The practical applications are important as well. Many models from crude to sophisticated have been advanced. Massive amounts of data exist and are frequently used to estimate model parameters and make predictions. I am not going to attempt to review the vast field here. My goal is simply to elucidate the physiological limitation of the human mind that causes the driving patters leading to congestion.
Although great progress has been made in modeling traffic as a compressible fluid, a class of models that fall into the category of Cellular Automata are more intuitive and instructive.
Cellular Automata, promoted by Stephen Wolfram of Mathematica fame as the solution to all problems, are indeed quite nifty. It turns out that autonomous agents, walking on a lattice and interacting according a simple set of rules can reproduce a surprising variety of observed macroscopic phenomena. If you want to learn more the Wikipedia article is a good start.
A pioneering work of Nagel and Schreckenberg published in Journal de Physique in 1992 introduced a simple lattice model of traffic which reproduced the traffic jam phenomenon and came to a surprising conclusion that the essential ingredient was infrequent random slowdowns.
You have probably done so yourself, you change the radio station or adjust the rear view mirror, or speak the child in the seat behind you. As you do so, your foot eases off the accelerator ever so slightly irritating the person behind you who has to disengage the cruise control. You and people like you are responsible for the traffic jams when the volume is heavy but there are no obvious obstructions to traffic.
Allow me to reproduce the authors’ description of the model since it is concise and elegant:
“Our computational model is defined on a one-dimensional array of L sites and with open or periodic boundary conditions. Each site may either be occupied by one vehicle, or it may be empty. Each vehicle has an integer velocity with values between zero and vmax. For an arbitrary configuration, one update of the system consists of the following four consecutive steps, which are performed in parallel for all vehicles:
- Acceleration: if the velocity v of a vehicle is lower than vmax and if the distance to the next car ahead is larger than v + 1, the speed is increased by one.
- Slowing down (due to other cars): if a vehicle at site i sees the next vehicle at site i + j (with j < v), it reduces its speed to j.
- Randomization: with probability p, the velocity of each vehicle (if greater than zero) is decreased by one.
- Car motion: each vehicle is advanced v sites.”
Without the randomizing step 3) the motion is deterministic: “every initial configuration of vehicles and corresponding velocities reaches very quickly a stationary pattern which is shifted backwards (i.e. opposite the vehicle motion) one site per time step.”
The model exhibits the congestion phenomenon when the mean spacing between the cars is smaller then vmax.
Below are the links to the simulations of the model for a circular track with 100 lattice sites, the cars are colored circles which move along the track. It helps to follow a particular color car with your eyes to see what’s happening.
The two simulations are done with 15 cars (density lower than critical) and with 23 cars (above the critical density–exhibits congestion). As you probably guessed vmax=5 in these simulation hence 20 cars correspond to the critical density. The probability of random slowing down is 10% per turn.
The second simulation (above the critical density) shows the development of a jam of 5 cars. Cars zoom around the track and then spend 5 turns not moving at all, before the traffic clears ahead of them and they can accelerate to full velocity again.
The moral of the story? People like you and me can be the cause of traffic congestion!