The frequency responses of a power system following a generator outage are closely related to the commitment states of generators, which describe the droop and inertial characteristics and system pre-contingency power dispatch. This study presents a new model to estimate the transient and steady state frequency responses of two-area power systems at any moment following the loss of an online generator. In the proposed model, the deviations of tie-flow, area frequencies and rate of change of area frequencies as well as the extreme values of these functions and their associated time of occurrence following the outage of a generation unit are derived as modified analytical functions. A new approach is also proposed to linearise the modified extreme deviations based on pseudo-Boolean function theorem. In addition, a frequency-constrained unit commitment model with modified frequency-dependent constraints incorporating primary frequency regulation reserve is presented as a mixed-integer linear programming problem. The results indicate that the proposed method gives acceptable approximations for post-contingency responses and can ensure power system security from frequency viewpoint.