If you drive like me, you have no patience for bumper to bumper traffic. There is gotta be a way to beat it somehow, right? Do you sneak into an opening in a neighboring lane if it is moving faster? Do you set goals like: “when I get in front of that van, I’ll switch back?” It doesn’t always seem to work. A lane that was zooming by you comes to a dead stop when you switch into it. If the motion of each lane is random, is there a way to switch lanes and move faster than a car that stays in lane?
It turns out there is a way to beat the traffic. To show this we will use a simple model of traffic flow introduced by Nagel and Schrekenberg (see the previous post). The model consists of a circular track with consecutive slots which can be empty of occupied by cars. Cars have an integer velocity between 0 and vmax. As we saw in the previous post, simple rules for updating the positions and velocities of the cars can reproduce the traffic jam phenomenon thereby a dense region forms in which the cars are at a standstill for a few turns and then, as the jam clears in front of them, the cars accelerate and zoom around the track only to be stuck in the jam again. The jam itself moves in the direction opposite to that of the cars.
Now imagine that we put two of the circular tracks (or lanes) side by side. For starters, let’s require all cars except one to stay in their respective lanes. One rogue car can switch lanes. Can the rogue with the right lane switching strategy move faster than the rest of the cars on average? The answer is most certainly yes although finding the best lane switching strategy is a difficult computational problem. What we are going to do here is compare two lane switching strategies that at first sight seem equally good. What we will discover is that it the lane changing strategy matters. As you might have suspected, if you don’t do it right, you might actually move slower than the rest of the traffic!
Here are the two simple strategies we will compare (I suggest you read the previous post for the description of the model):
1) “Stop-switch:” if the slot directly ahead is occupied, switch if the space in the other lane directly across is not occupied.
2) “Faster-switch:” if the car directly ahead in the neighboring lane is moving faster, switch if there is space available.
The graph above compares the two strategies. It shows the percent improvement of the rogue’s average speed compared to the average speed of the rest of the cars as a function of the car density. When density is low and traffic jams are rare, switching lanes has almost no effect on your average speed for both strategies. When the density is high and traffic jams are abound, switching can make you go slower than the rest of the traffic. The reason is that when a space in the neighboring lane opens up, it is likely to be at the tail end of a jam whereas the jam in the lane you just switched out of might be already partially cleared. The final remark is that the “Stop-switch” strategy is significantly better improving the speed by as much as 35% whereas the best “Faster-switch” can do is a 15% improvement.
Finally let me mention that if all cars switch lanes and use the same strategy, nobody wins. All cars move with the same speed on average. That average speed could be smaller or larger (depending on the car density and the switching strategy) than in the case when everybody says in lane. The graph below explains why everyone is so keen on the advice “Stay in lane!” It turns out that if everyone uses the “Faster-switch” strategy, the average speed is drastically lower for everyone than if everyone stays in lane! The reason for this dramatic result is that when you change lanes, the car behind is likely to slam on the brakes which slows everyone down.
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!