This paper establishes that stochastic models either possess one homogeneous phase of traffic across the whole density range (1-phase behavior), or they possess two disjoint homogeneous phases, ``laminar'' and ``jammed'', which are separated by a density regime where the two phases coexist (2-phase behavior). Speculations about this have been around for a rather long time (e.g. (3,6,7)); corresponding deterministic models have been established more recently (e.g. (9,8)). However, despite much discussion (e.g. (32,35,33,31,34)) no clear picture for stochastic models was established. Only stochastic models allow to look at meta-stable states, spontaneous transitions, and fractal-like structure, all of which are important for real world traffic. Importantly, 1-phase and 2-phase behavior can be obtained from the same model by just changing one parameter.
With respect to reality, there is no general agreement if measurements show 1-phase/1-state or 2-phase/3-state traffic. As discussed in the introduction, there is some evidence for 2-phase behavior in German data (4,14,15). Measurements in Northern America (10) point towards 1-phase behavior. In addition, many of the earlier measurements that point towards 2-phase behavior can in fact be explained by 1-phase models together with geometric constraints (11). To make matters worse, newer publications claim the existence of three (e.g. (14)) or even more (e.g. (40)) phases, while other publications (e.g. (41)) claim that these different phases are just queues.
Since there is discussion of entering the notion of stochastic breakdown into the Highway Capacity Manual, and since, as discussed in the introduction, the correct operation of devices, such as ramp metering and adaptive speed limit, depends on the answer of the breakdown question, it seems critical to fully understand these issues. It also seems critical to consider stochastic models, in order to not base the notion of stochastic breakdown on deterministic models. This paper's contribution is a solid step towards understanding the consequences of stochastic traffic breakdown, if it exists. In other words, this model will allow the development of further predictions, which are impossible to make by deterministic models, and these predictions could be tested against field data. For example, a stochastic model would predict a certain wave structure inside a queue caused by a downstream bottleneck, similar to (42), although a bottleneck with fixed capacity would be better suited to test the theory.
The basic theory of phase transitions, which is behind the much of this modeling work, applies in the so-called thermodynamic limit, which refers to infinitely large systems. Since traffic systems are small when compared to thermodynamic systems, the theory needs to be modified for those smaller-scale systems. Both the theory and computer modeling provide the tools for this, but great care has to be taken to find predictions which could actually be tested in the real world with finite queue lengths and finite durations. In consequence, such comparisons are highly desirable, but outside the scope of this paper.