INTRODUCTION

Both from an operations and from a design perspective, the capacity of a road is an important quantity. If demand exceeds capacity, queues will form, which represent a cost to the driver and thus to the economic system. In addition, such queues may impact other parts of the system, for example by spilling back into links used by drivers who are on a path that is not overloaded.

This paper discusses freeway capacity. The question concerns the maximum flows that freeways can reach, and if the maximum flows sometimes observed ($>$ 2500 vehicles per hour and lane) are sustainable flows or short-term fluctuations. Let us assume that there is traffic with a fairly high density $\rho$ on a freeway, but vehicles are still able to drive at some fast velocity $v$. Throughput is $q = \rho \, v$. The question is what will happen if density is further increased: Can $q$ further increase because $\rho$ increases more than $v$ decreases? Will $q$ gradually decrease because $\rho$ increases but $v$ decreases faster? Or is there a possibility that traffic will break down, leading to stop-and-go traffic?

More technically, the question is if there is, for each density $\rho$, a velocity $V(\rho)$ and corresponding throughput $Q(\rho) = \rho \, V(\rho)$ at which traffic flow is smooth and homogeneous. Or is there a density range where that homogeneous traffic flow is unstable, and traffic has a tendency to reorganize into a stop-and-go pattern, with possibly lower throughput?

There is in fact a long history of publications about breakdown behavior in freeway traffic, sometimes called ``reverse lambda shape of the fundamental diagram'' (2,1), ``hysteresis'' (3), ``capacity drop'' (4), ``catastrophe theory'' (5), and the like. From the modeling side, there has since long been discussions about an analogy to a gas-liquid transition (7,6), and recent work has established traffic models which display deterministic versions of a liquid-gas-like transition (9,8).

On the other hand, measurements by Cassidy (10) indicate that there can be stable homogeneous flow at all densities. Muñoz and Daganzo (11) point out correctly that many of the ``inverse lambda'' observations could also be explained by geometrical constraints, in the following way. A bottleneck downstream of a measurement location can cause the following temporal sequence of measurements:

1. The system starts with low flow at low densities.

2. Both flow and density keep increasing, along the ``free flow'' branch of the fundamental diagram.

3. This flow can be larger than what can flow through the bottleneck. Then, a queue starts forming at the bottleneck, but that does not immediately influence the measurement.

4. Eventually, the queue will have spilled back to the measurement location. At that point in time, data points will move to a much higher density, while the flow value will drop to the bottleneck capacity.

   It can take up to 20 minutes for the transition zone (transition from free flow to queue) to traverse a fixed detector location, leading to fundamental diagram data points that lie between the free flow and the queue state (11).

This mechanism generates data that looks similar to data shown in support of the breakdown hypothesis. Unfortunately, many of the published data sets do not provide enough information about the geometrical layout and the full spatio-temporal picture of the dynamics in order to resolve this question.

This question is not just academic. The correct use of technical devices such as ramp metering or adaptive speed limits (12) depends on the answer. For example, let us assume that the homogeneous solution is unstable in a certain density range, and that the alternative stop-and-go solution has a lower throughput than homogeneous traffic at the same density. In this case, the task of ramp metering might be to keep the density away from the unstable density range. If density approaches this value, on-ramp traffic should be reduced.

If, on the other hand, the homogeneous solution is stable everywhere, then ramp metering shifts capacity from the on-ramp to the through lanes, and it avoids slowdown on the freeway and its emission consequences. There would however be no net capacity effect, in the sense that -in the absence of additional obstructions- throughput downstream from the metered ramp would be the same no matter if ramp metering was switched on or not.

If, in addition, breakdown is probabilistic, that is, the homogeneous solution can survive for certain amounts of time, then the question becomes which risk of breakdown one would be willing to accept. Accepting higher flow rates in the ramp metering algorithm might increase average throughput, but it might also increase the probability of breakdown.

There is even discussion to include aspects of stochastic transitions into the Highway Capacity Manual (13). This could for example mean that, for certain flow levels, one would include a curve describing the probability that traffic flow has not broken down as a function of time. From such a curve, one could for example look up the maximum density and flow levels if one accepts a, say, 1% probability of breakdown.

Before continuing, let us make this more precise. Let us assume there is a density range where the homogeneous solution is unstable. The way this could (in principle) be tested is to have homogeneous traffic operating at a certain density, and to introduce a strong disturbance, say by stopping one car for several seconds. If the introduced disturbance heals out over time, then homogeneous traffic at that density is stable; if the disturbance grows over time, then the homogeneous solution is unstable at this density. The unstable solution needs two ingredients:

• Outflow from the jam is less than the maximally possible homogeneous flow.

• The jam, once it is there, remains compact; in other words, density inside the jam is a fixed quantity that should essentially be the same from one location to the next.

There are at least three references (Figs. 2 and 3 in (4); Fig. 3 in (14); Fig. 4 in (15)) where the data points to the existence of a stable jam, embedded both upstream and downstream in free traffic, and where the outflow from the jam is lower than the inflow. In the 2nd and the 3rd of these references, one can in addition see that the jam is growing in width, as it should in such a situation, while remaining compact. In the 1st of these references, the data to decide this question is not sufficient.

Given this state of affairs, it makes sense to look at modeling. The task is to understand which model solutions are possible at all. This understanding will lead to the predictions of additional features that will go along with one mechanism or the other, and it might be possible to measure them, and so the issue will hopefully be eventually resolved. Until then, however, there is no agreement on the issue of breakdown in freeway traffic, and in consequence all engineering relying on one or the other assumption may not work as intended.

The starting point for our work are single-lane car following models. These models are typically either of the type $v(t+\tau_v) = f(g,\Delta v, ...)$ or of the type $a(t + \tau_a) = h(g,\Delta v, ...)$, where $v(t)$ is the velocity of a car at time $t$, $\Delta v$ is the velocity difference to the car ahead, and $a$ is the acceleration. $g$ is the gap to the car ahead, where $g = \Delta x - \cellLen$, with $\Delta x$ the front-buffer-to-front-buffer distance, and $\cellLen$ is the space the car occupies in a jam. These models can for example be found in (16)

$$v(t\!+\!\tau_v) = V(g(t)) \, \hbox{\\ with\\ } V(g) = v_f - v_f \, \exp( - \lambda g / v_f ) \,$$ (1)

in (17)

$$a(t) = \alpha \cdot \Big( V(g(t)) - v(t) \Big) \hbox{\ with\ } V(g) = v_f \cdot (\tanh(g+\ell) - \tanh(\ell))$$ (2)

or in (18)

Additional parameters here are (the free speed), , , , and .

When these models are implemented on a computer, they need to be discretized in time, and one has to concern oneself with the size of the integration time step, . A typical discretization is

(4)

(5)

and

(6)

These discretizations are meant to approach the original coupled differential equations for , and there is a whole body of literature available for this (see, e.g., (19), and references therein). Once time delays (via ) are introduced into such equations, numerical treatment becomes more difficult, because the dynamical history between $t$ and needs to be memorized in increments of .

In this situation, it makes sense to look for computational models which are not based on the limit , but which generate useful results also for relatively large time steps of, say, one second. The model that we will use in this paper has been introduced by Krauss (20); it is a variant of a model used by Gipps (21). The Krauss model has been shown to be free of collisions, i.e. never occurs (22,20).

In addition to being crash free at large time steps, the Krauss model is also stochastic. The important parameter for our study is a noise amplitude , which we will vary from 0.5 to 2. For or the model leaves the range of where it is plausible for traffic.

Our main results are the following:

For medium , there are three states of traffic, which we will call laminar, coexistence, and jammed. The state depends on the density. Laminar, occurring at low density, means that nearly all vehicles have large spacing, and are driving at or near free speeds. Occurrence of some mini-jams is possible, but these mini-jams are not sustained and far apart. Jammed, occurring at high density, means that nearly all vehicles have small spacing, and are driving at low speeds or are stopped. Coexistence, occurring at intermediate density, means that the system is a mix of laminar and jammed traffic. In the coexistence state, traffic is strongly inhomogeneous.

It is important to note that there are three states (laminar, jammed, coexistence) but only two phases (laminar, jammed). The phases refer to homogeneous sections of the system; the state refers to the system as a whole.

For large , there is only one phase of traffic and therefore only one state. When going from low to high density, cars move closer and closer together, but traffic remains homogeneous at all times.

At some in between, there is a transition from the 2-phase to the 1-phase regime.

In the Krauss model, changes of also change the average acceleration. This is an unfortunate coincidence, and we believe that our general results regarding the number of phases are not related to this effect.

Deterministic models, formulated either as car following models or as fluid-dynamical models, can display 1-phase or 2-phase behavior. They can however not display stochastic transitions between the phases.

The results are important for model building as well as for understanding field measurements. In a 2-phase model, theory predicts that there can be a hysteretic transition from the laminar to the coexistence state without a change in density. This means that, at a given density, traffic can operate in the laminar flow state for long times, until it will eventually ``break down'' and switch to the coexistence state. In a 1-phase model, this is impossible, and there is only one phase for any given density.

A direct consequence of this is that, if traffic follows a 1-phase model, any initial jam will ``smear out'' and thus eventually go away, even with unchanged traffic conditions. Conversely, in a 2-phase model with density in the coexistence range, jams have a typical density and a typical shape of their upstream and downstream front. These shapes are stable under disturbances, that is, the system will restore these densities and shapes after disturbances.

This paper starts with Sec. 2 which describes the general idea of a gas-liquid transition. Sec. 3 describes the general simulation setup including the car following model that is used, discusses space-time plots of the resulting dynamics, and investigates transients vs.steady state. Sec. 4 then establishes how a coexistence state can be numerically detected for a given model. Sec. 5 reports similar results for cellular automata (CA) models. Sec. 6 discusses how these results relate to deterministic models; the paper is concluded by a discussion and a summary.