This paper concerns the problem of vehicular traffic flows simulation. We present a continuous car-following model with explicit reaction-time delay. This model is based on the Intelligent Driver Models and eliminates some drawbacks like unrealistic distance between vehicles in a homogeneous flow and instant reaction of a driver. The model is defined by a couple of delayed differential equations. Introducing reaction time explicitly one can simulate traffic flow instabilities such as kinematic waves with a greater degree of realism. We calculate conditions for that the linear stability occurs on a ring and in the finite platoon of cars. The parameters of the model are calibrated according to NGSIM trajectory data. Another method described in this paper allows to estimate parameters according to technical characteristics of the specific vehicle type. Simulating vehicular traffic for large cities with microscopic approach one needs to solve tremendous systems of differential equations. The speed of changes in the size of the components of such systems usually lies in a wide range, as the dynamics and behaviour of the vehicles can strongly differ. We introduce a multirate numerical scheme with a self-adjusting time stepping strategy. The step size for each component is determined by its own local temporal variation. The stability analysis for the developed scheme is performed and the stability conditions are obtained. The use of multiple time steps allows parallel computing.