12.2. Traffic Jams¶
What causes traffic jams? Sometimes there is an obvious cause, like an accident, a speed trap, or something else that disturbs the flow of traffic. But other times traffic jams appear for no apparent reason.
Agent-based models can help explain spontaneous traffic jams. As an example, we implement a highway simulation based on a model in Mitchell Resnick’s book, Turtles, Termites and Traffic Jams.
Here’s the class that represents the “highway”:
class Highway: def __init__(self, n=10, length=1000, eps=0): self.length = length self.eps = eps # create the drivers locs = np.linspace(0, length, n, endpoint=False) self.drivers = [Driver(loc) for loc in locs] # and link them up for i in range(n): j = (i+1) % n self.drivers[i].next = self.drivers[j]
n is the number of cars,
length is the length of the highway, and
eps is the amount of random noise we’ll add to the system.
locs contains the locations of the drivers; the NumPy function
linspace creates an array of
n locations equally spaced between
drivers attribute is a list of
Driver objects. The for loop links them so each
Driver contains a reference to the next. The highway is circular, so the last
Driver contains a reference to the first.
During each time step, the
Highway moves each of the
# Highway def step(self): for driver in self.drivers: self.move(driver)
The move method lets the
Driver choose its acceleration. Then
move computes the updated
speed and position. Here’s the implementation:
# Highway def move(self, driver): dist = self.distance(driver) # let the driver choose acceleration acc = driver.choose_acceleration(dist) acc = min(acc, self.max_acc) acc = max(acc, self.min_acc) speed = driver.speed + acc # add random noise to speed speed *= np.random.uniform(1-self.eps, 1+self.eps) # keep it nonnegative and under the speed limit speed = max(speed, 0) speed = min(speed, self.speed_limit) # if current speed would collide, stop if speed > dist: speed = 0 # update speed and loc driver.speed = speed driver.loc += speed
dist is the distance between
driver and the next
driver ahead. This distance is passed to
choose_acceleration, which specifies the behavior of the driver. This is the only decision the driver gets to make; everything else is determined by the “physics” of the simulation.
accis acceleration, which is bounded by
max_acc. In the given implementation, cars can accelerate with
max_acc=1and brake with
speedis the old speed plus the requested acceleration, but then we make some adjustments. First, we add random noise to
speed, because the world is not perfect.
epsdetermines the magnitude of the relative error; for example, if
speedis multiplied by a random factor between 0.98 and 1.02.
Speed is bounded between 0 and
speed_limit, which is 40 in the given implementation, so cars are not allowed to go backward or speed.
If the requested speed would cause a collision with the next car,
speedis set to 0.
Finally, we update the
Here’s the definition for the
class Driver: def __init__(self, loc, speed=0): self.loc = loc self.speed = speed def choose_acceleration(self, dist): return 1
speed are the location and speed of the driver.
This implementation of
choose_acceleration is simple: it always accelerates at the maximum rate.
Since the cars start out equally spaced, we expect them all to accelerate until they reach the speed limit, or until their speed exceeds the space between them. At that point, at least one “collision” will occur, causing some cars to stop.
Figure 12.1 shows a few steps in this process, starting with 30 cars and
eps=0.02. On the left is the configuration after 16 time steps, with the highway mapped to a circle. Because of random noise, some cars are going faster than others, and the spacing has become uneven.
During the next time step (middle) there are two collisions, indicated by the triangles.
During the next time step (right) two cars collide with the stopped cars, and we can see the initial formation of a traffic jam. Once a jam forms, it tends to persist, with additional cars approaching from behind and colliding, and with cars in the front accelerating away.
Under some conditions, the jam itself propagates backwards, as you can see if you watch the animations in the notebook for this chapter.