Models and algorithms for the study of traffic with delay

Abstract: Trac is an essential matter to examine and understand since it is a paramount aspect of modern society. The objective in this bachelor thesis is to formulate models that describe the propagation of vehicles and analyse these using the concepts of stability as well as sensitivity. Moreover, we are to conclude the suitability of di erent algorithms for this purpose. Firstly, several di erent models are proposed, they do not only di er in respect to the parameters that govern the trac, but also in respect to the time delay. The rst main classication of models have drivers that react instantaneously, whilst the second have a time delay on the response. Furthermore, the di erent models have participants that observes the surrounding motors in varying manners; e.g., they may be aware of the preceding vehicle as well as the subsequent one, or just the vehicle ahead. After the analysis of the models the conclusion is drawn that trac propagation with delay is more dicult to modulate in a stable manner. This is a quite expected outcome since the delay entails that the reaction of the drivers is held up for a period of time, thus the chance to adapt ones' driving ecaciously is reduced. Furthermore, if the time delay > 0:5 s the model is practically impossible to modulate in a stable manner i.e., all the possible parameter values result in an unstable trac situation. The conclusion regarding stability is drawn from the eigenvalues given by a di erence system. Correspondingly, the eigenvalue problem of the models with delay is more complex than the equivalent problem without delay. This is due to the fact that the introduced delay causes the eigenvalue problem to become nonlinear, whilst the models with no delay give rise to a linear problem. Accordingly, the algorithms used to solve the nonlinear eigenvalue problem are more sophisticated. When dealing with calculations of the eigenvalues of large scale systems, special algorithms are particularly suitable.