Deterministic Time-Delayed Traffic Flow Models: A Survey

Research in understanding traffic flow is conducted since 1930s in mathematics, physics, and engineering fields. The main interest is to reveal the characteristics of traffic dynamics and consequently propose ways to reduce undesirable impacts of traffic flow to social and economical life. This can be achieved only if rigorous and reliable mathematical models are constructed. The first part of this work covers the classification of such models as well as empirical and software tools used to study and predict traffic flow. The second part is devoted to a critical parameter in the traffic dynamics: time delay, which is recognized in this particular area as early as 1958. Delay originates from the time needed by human drivers to become conscious, make decision, and perform control actions in traffic. Such a definition states that human beings actively control the time evolution of traffic by their time-delayed behaviors (human as a controller/plant), and thus traffic dynamics becomes inherently time delayed. This dynamical structure, in a global sense, can also be seen as an interconnection of dynamics that transfer information/energy/momentum among each other, but under the presence of communication/transportation delays. For the specific problem considered, we first discuss the source of time delay, its physical interpretations, and mathematical nature, and next present a survey on mathematical models that explicitly account for delays. We conclude with interesting research topics at the intersection of control theory and time-delay systems. In this context, an example traffic flow scenario is covered to both demonstrate this intersection and show the consequences of delay presence in traffic flow dynamics, especially from the stability point of view.