A new mechanism for metastability of under-saturated traffic responsible for time-delayed traffic breakdown at the signal

In this paper we introduce a newmechanism of metastability of under-saturated traffic at the signal that is responsible for a time-delayed traffic breakdown revealed by Kerner (2011). In our model, we assume that the metastability of under-saturated traffic at the signal is caused by a dependence of the mean time of driver acceleration from a queue at the signal on the driver’s stopped time within the queue. With the use of Nagel–Schreckenberg model, we demonstrate that this mechanism of the metastability of city traffic can lead to the time-delayed traffic breakdown at the signal. © 2014 Elsevier B.V. All rights reserved. Vehicular flow is a many-particle system far from equilibrium and it exhibits diverse interesting non-equilibrium features. For instance, capacity drop phenomenon is widely observed, which is believe to be related to first-order phase transition from free flow to congested flow. Phantom jam is another interesting phenomenon observed in real traffic [1–7]. However, there is controversy about whether phantom jam is usually formed in a sequence free flow → synchronized flow → jam (F→S→J transitions) (as stated in Kerner’s three-phase theory [9]) or from free flow directly (F→J transition) (as stated in two-phase traffic flow theories [1–4, 8,11–15]). The mechanism of wide scattering of statistical data in congested flow in the flow rate density plane is not clear, either [8–13]. To explain the features of vehicular flow,many traffic flowmodels have been proposed, which can be roughly classified into three types. In Lighthill–Whitham–Richards model, the traffic flow is always stable [14,15]. It is believed that complex traffic flow phenomena are induced by external factors (e.g., traffic bottlenecks, lane changing behaviors) [8]. On the other hand, the two-phase traffic theory (see [1,2,11–13] and references therein) and Kerner’s three-phase traffic theory [9] assume that traffic flow is not always stable, and the complex traffic phenomena are related to traffic instability. Nevertheless, there are several heated debates of the ∗ Corresponding author at: School of Engineering Science, University of Science and Technology of China, Hefei 230026, China. Tel.: +86 551 63600127. E-mail address: [email protected] (R. Jiang). http://dx.doi.org/10.1016/j.cpc.2014.02.011 0010-4655/© 2014 Elsevier B.V. All rights reserved. two theories about whether a unique relationship between flow rate and density exists or a two-dimensional region exists in the steady state, and traffic breakdown is associated with F→S transition [9] or F→J transition [1,2,11–13]. The latter point is the most important one. This is because if the F→S transition determines traffic breakdown at highway bottlenecks as stated in Kerner’s three-phase theory [9], then the reliable application of all classical methods for traffic control and optimization in transportation networks is a very questionable one as explained in a recent review [10]. The above mentioned controversy about traffic flow mainly arises in uninterrupted flow observed on highways, freeways or expressways. In city traffic, the Biham–Middleton–Levinemodel is the first city traffic model in the community of traffic physics [16]. Although the model is extremely simple, it displays some interesting behaviors, e.g., self-organization of alternating stripes in free flow. Since its proposal, many extensive studies have been carried out based on the model (see e.g. Refs. [17–19]). Since vehicles have to stop and start frequently at the traffic lights, it is generally believed that whether to consider instabilities or not is not important. However, recently, Kerner showed that, in undersaturated traffic, spontaneous traffic breakdown can occur at the light signal after a random time delay [20]. It has been argued that this traffic breakdown is initiated by a phase transition from free flow to synchronized flow (in the three-phase traffic model) or by a phase transition from free flow to jam (in the two-phase model) occurring upstream of the queue. Kerner further studied the green-wave breakdown at traffic signal [21–23]. He proposed 1440 R. Jiang et al. / Computer Physics Communications 185 (2014) 1439–1442