This paper investigated the performances of a well-known car-following model with numerical simulations in describing the deceleration process induced by the motion of a leading car. A leading car with a pre-specified speed profile was used to test the above model. The results show that this model is to some extent deficient in performing the process aforementioned. Modifications of the model to overcome these deficiencies were demonstrated and a modified car-following model was proposed accordingly. Furthermore, the delay time of car motion of the new model were studied.