Mean-field theory for the Nagel-Schreckenberg model with overtaking strategy

Based on the Nagel-Schreckenberg (NS) model with periodic boundary conditions, a modified model considered overtaking strategy (NSOS) has been proposed \cite{su2016occurrence,su2016the}. In this paper, we focus on the theoretical analysis of traffic flow for NSOS model by using mean-field method. In the special case of $v_{max}=1$ where vehicles can not overtake preceding ones, the features of stationary state can be obtained exactly. However, in the case of $v_{max}>1$ where overtaking happens, some approximative methods have to be took into account. The main results are that we find the reason why traffic flow is increased in the regime where densities exceed the maximum flow density, and the influence of traffic flow on the transition density is dominated by the braking probability $p$.