Earthquake Model Confirms Traffic Jams Caused by Tiredness

The rapid growth of the vehicle number during the last century caused an increased complexity in our road traffic and transportation systems. Unfortunately, in such conditions, traffic congestion becomes an everyday problem for drivers. The variety of complex non-linear phenomena present in such agglomerated traffic systems has attracted the attention of a large number of researchers (for a review please consult the recent work of Nagatani [1], or Kerner's book [2]). Accordingly, since the early '30 many empirical data on different highways have been collected [3][4][5] and traffic data collection systems have been substantially evolved [6,7]. On the theoretical side, many traffic models have been developed. The models are usually classified into four categories: microscopic models, macroscopic models, cellular automata models and non-traditional models. Detailed description and analysis of such traffic models can be found in many review articles [8][9][10][11]. One recent example of a non-traditional stochastic model could be the probabilistic traffic flow theory [12] that includes the nonlinear effects of small perturbations. It was also shown [13] that finite reaction times are clearly essential factors of driver behavior that affects the performance and stability of traffic. Despite of many existing studies in the field, the phenomenon of spontaneous traffic jam formation is far from a complete understanding.

The simplest, but already quite complex form of the traffic is the accident-free and single-lane motion of a chain of cars. The motion of the queue in this simple form of traffic is primarily governed by the leading car and the statistics of driving attitudes. From empirical observations we know that this kind of traffic can be either “free” with a continuous flow structure or congested with a stop-and-go motion of cars. The congested traffic usually appears if the leading car is moving slowly and the differences between the driving attitudes of drivers are substantial. In such situations the car row will evolve noncontinuously in avalanches of widely different sizes, and a sequence of jams of different magnitudes will appear and propagate backward through the system. This form of traffic is also known as shockwave traffic jam or ‘phantom traffic jam’. A systematic analysis of empirical traffic states of a 30 kilometer long section of the German freeway A5 near Frankfurt [14,15] reported a rich variety of congested traffic states, interpreted as a spatial coexistence of altogether six different traffic states.

“Why do traffic jams appear on highways?” -is the natural question asked by the authors of the first experimental article that investigates the spontaneous shockwave traffic jam formation [16,17]. By a planned experiment performed on a circuit they show that the emergence of a traffic jam may occur even in the absence of a bottleneck. If one looks on their experiments (i.e. their video recording posted on youtube [18]) it is immediately observable that at the initial stage, vehicles are running continuously each with the same velocity, but roughly 10 min later a first shockwave traffic jam emerges spontaneously and propagates in the system. Shortly after this experiment these emergent shockwaves have been identified as nonlinear traveling wave solutions called ‘jamiton solutions’ of the purely deterministic hyperbolic continuum traffic equations [19].

The present work is inspired and motivated by the experiments of Sugiyama et al. Based on their work, the formation of a traffic jam is understood as a collective phenomena. For a physicist, the most straightforward way to approach this phenomenon is by using the simple spring-block chain with asymmetric interactions, where block will model the cars and the springs the distance keeping tendency of drivers. Using this simple approach proved to be successful in modeling other collective phenomena such as earthquakes, fracture, fragmentation or even magnetization processes. Here, through large-scale computer simulations on simple model system the minimum conditions that are absolutely necessary to produce the self-organized congested traffic jam conditions are identified. Contrary to most of the studies in the field of highway traffic we focus on a measure characteristic for one car (block) in the row: the distribution of the rest times. Based on these distributions an order parameter will be defined and the parameter space of the model is explored and analyzed. It is shown that the transition between the free and congested traffic states is realized through a disorder-induced second order phase transition. Based on evident analogies with the model elements, it is concluded that besides the disorder level in driver’s driving attitude the tiredness of drivers plays also an important role in the formation of traffic jams.

The spring-block approach of the single-lane highway traffic was described in detail in our previous work [20]. However, for t

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