Step dynamics on the vicinal surface at low temperature is studied by a restricted solid-on-solid model with the inter-ledge attraction of the point type (p-RSOS model). The p-RSOS model is known to show the thermal step bunching due to the singular property of the surface free energy, though the model contains no long range interaction such as elastic interactions among steps. We demonstrate the step-zipping process in the step bunching at low temperature by use of Monte Carlo simulations without special surface kinetics. (c) 2010 Elsevier B.V. All rights reserved.