On the Mean Speed in the “Boltzmann-Like” Traffic Theory: Analytical Derivation

In a Boltzmann-like description of traffic developed by Prigogine, et al., the speed distribution has been expressed as an implicit function in terms of its own mean. In order to determine the speed distribution, it is, therefore, necessary to first determine the mean speed of the distribution. In this paper, the mean speed of the steady-state speed distribution is investigated. The mean speed is determined on the basis of the intersection of two functions; one of the functions is given by the desired speed distribution, which is a description of the goals that drivers want to attain in traffic; the other function is given by the concentration-dependent traffic parameters of the model. These parameters give a description of the restrictions that drivers exert on each other in the process of attaining their individual goals. Since the model is basically a description of the interaction between these two conflicting traffic characteristics, the effect on the mean speed due to each characteristic can hence be investigated separately through the properties of the corresponding function. In addition, all the parameters in the model, at least for the determination of the mean speed, can be effectively consolidated into a single parameter. Consequently, such a formulation of the mean speed would allow easier analysis of the assumptions and validation of the predictions.