Accurate and reliable traffic state identification is crucial to developing responsive and proactive traffic management applications. In this study, the problem of vehicle discharge state identification at signalized intersections is investigated, which focuses on the vehicle discharge process during the green interval. Instead of using detailed vehicle trajectory data and treating the observations of vehicles independently, this study formulates the vehicle discharge process in a Hidden Markov Model (HMM) framework using sequential observations of vehicle travel time and discharge headway as inputs. Three vehicle discharge states (i.e., overflow, single stop, and free arrival) are encoded as latent states, and a restricted left-to-right state transition matrix is imposed to respect the nature of the vehicle discharge process in the real world. The standard HMM is further extended to incorporate two informative covariates to parameterize the probabilities of the initial states and state transitions. The proposed models have been validated on the Next Generation Simulation (NGSIM) dataset. Compared to a benchmark model, the proposed models show their strength in correctly inferring the vehicle discharge state and are more reliable to use in presence of random missing observations. The effectiveness of covariate incorporation is also investigated, and several extended applications of the proposed models are discussed.