THE SIMULATIONS

Krauss Model

The velocity update of the Krauss model (28,29) reads as follows:

			(7)
			(8)
			(9)
			(10)
			(11)
			(12)
			(13)
			(14)
			(15)
			(16)
			(17)
			(18)
			(19)

is the speed of the car in front, is the average velocity of the two cars involved, is the maximum allowed velocity, $a$ is the maximum acceleration of the vehicles, their maximum deceleration for , is the noise amplitude, and is a random number in . The meaning of the terms is as follows:

Eq. (7): Calculation of a ``safe'' velocity. This is the maximum velocity that the follower can drive when she wants to be sure to avoid a crash (28,29). The main assumption is that the car ahead will never decelerate faster than , and that the car of the follower can also decelerate with up to .

Eq. (8): The desired velocity is the minimum of: (a) current velocity plus acceleration, (b) safe velocity, (c) maximum velocity (e.g.limit).

Eq. (9): Some randomness is added to the desired velocity.

After the velocities of all vehicles are updated, all vehicles are moved.

The Krauss model has been proven to be free of crashes for numerical time steps smaller than or equal to the reaction time, (22,29). We will use as has conventionally been used for the Krauss model. We further use , , for all simulations.

The model is free of units; this is a property that it has inherited from the cell-based cellular automata models. A reasonable calibration is: time steps correspond to seconds and cells correspond to  meters. The reaction time then is assumed to be  second, and corresponds to m/s or  km/h. corresponds to a maximum acceleration of m/s per second or km/h per second. corresponds to a maximum deceleration of  km/h per second.

All simulations are done in a 1-lane system of length with periodic boundary conditions (i.e. the road is bent into a ring). Let be the number of cars on the road. The (global) density is .

Pictures

Before analysing the Krauss-model numerically, it is instructive to look at the space-time plots in Fig. 1(c). Space-time plots are pictures of the time evolution of the system. In Fig. 1(c), vehicles drive to the right and time points down. Each row of pixels is a ``snapshot'' of the state of the road. In principle, one could reconstruct the trajectory of a particular car by connecting the corresponding pixels. In practice, at the displayed resolution this is close to impossible and one mostly observes the larger scale traffic jam structure. Traffic jams move against the direction of driving. The following refers to each individual case (i)-(iv) of Fig. 1(c):

[(i)] The laminar state: All cars drive at high speed. The available space is shared evenly among the cars. The traffic is homogeneous.

[(ii)] The coexistence state: The slow cars are all together in one big jam. On the rest of the road, the cars drive at high speed. In consequence, the traffic is very inhomogeneous.

[(iii)] The jammed state: The density is so high that no single car can drive fast. As in (i), the traffic is homogeneous.

[(iv)] The single phase at high : Many small jams are distributed over the whole system. There is neither a larger area of free flow, nor a major jam. The traffic is homogeneous.

Note that ``homogeneous'' here means ``homogeneous on large scales''. This means that there is a spatial measurement length $\ell$ above which all density measurements return the same value. If a system goes from a 2-phase to a 1-phase model, then even in the regime which technically has only one phase, structure formation on small scales is still possible. Fig. 1(c) bottom right is indeed an example for this. With larger distance from the 2-phase model, i.e., the scale of these structures becomes smaller and smaller, which means that the system is homogeneous already on smaller scales. This statement can be quantified, for example via the gap distribution, i.e.distribution of the distances between jams (30).

Defining a jam

In order to make progress, one needs to define where a jam starts and where it ends. Our definition of homogeneity (see later) will not depend on this, however. A jam is a sequence of adjacent cars driving with speed less or equal . The cars between two neighbouring jams are in laminar flow.

This definition is very simple, but will not always correspond to our natural understanding of the word jam. Thus, whether a car is jammed or not according to this definition is just a starting point and not the final answer.

Initial Condition And Relaxation

For many parameters of the Krauss model, there is a unique equilibrium state, which the system will attain after a finite time t_relax, no matter how it was started. Deciding when the equilibrium is reached is not trivial.

Let be the state of the road at time $t$. To find the equilibrium value of some property, , we use the following idea: For small $t$, will depend on the initial condition. With increasing time, converges towards the equilibrium value. Assume the convergence is from above. Now we need another initial condition that approaches the equilibrium value from below. Once these two sequences are close enough together, an estimate for the equilibrium value is found. Unfortunately, it cannot be guaranteed that the value thus obtained really is the equilibrium value. We use the following two initial conditions:

Laminar start: The cars are positioned equidistant over the road with speed zero.

Jammed start: All cars are cramped together in a big jam without any gap. Their speed is zero.

An example of this method is shown in Fig. 2(a). For the number of jams was used. Since both initial conditions start with , the criterion of Sec. 3.3 finds one large jam at . After this, the following happens:

Laminar start: Vehicles accelerate, but because of interaction, many small jams form, and the number of jams increases rapidly. These jams then slowly coagulate, which slowly reduces the number jams in the system, until the equilibrium value is reached.

Jammed start: Vehicles accelerate out of the jam, but no or very few jams form in that outflow. Only very slowly does the number of jams increase, either via new jams in the outflow, or because of a ``breaking apart'' of the initial jam.

In Fig. 2(a), one sees that for both initial conditions the system eventually reaches the same number of jams. With , the system in equilibrium has, in the average, about 1.8 jams. In contrast, with , the system converges to an average of more than 20 jams.