Numerous car-following models have been developed since the 1950s. However, there still exist many traffic phenomena that cannot be demonstrated using the existing models. Therefore, this research proposed a new car-following model, the Asymmetric car-following (ACF) model based on the understanding of driver's asymmetric behavior, which can explain complex traffic phenomena. We established the asymmetric car-following (ACF) rule under the vehicle's safety constraints using eight parameters that indicate the driver's characteristics and vehicle's performances. To evaluate the ACF model, we performed the simulation for car-following pairs and conducted a comparison analysis with the existing models: Newell, Gipps, GM, and IDM. As a result, the proposed ACF model showed good fitness with the empirical trajectory and the apparent asymmetric behavior compared to others. For further investigation in the congested traffic stream, we simulated a group of vehicles by adding an error term to represent the driver's unexpected behavior. The simulation showed growth, propagation, and dissipation of the stop-and-go traffic. These results proved that the ACF model has the strength to elaborate on various traffic phenomena, such as traffic hysteresis and stop-and-go traffic.