Model of subway commuters' departure time choice with in-carriage congestion and arrival early/late penalty

Considering a subway connecting a resident area and a workplace, every morning, commuters use the subway for traveling from the residential area to the workplace. During rush hour, some commuters depart earlier or later for avoiding in-carriage congestion incurred by a large number of commuters. This paper presents a model for studying die subway commuters' departure time choice behavior and the private travel total cost. The in-carriage congestion and arrival early/late penalty are considered in the model. At equilibrium, no user starts on the latest or earliest departure time. The departure rate at equilibrium state is linear to departure time. At equilibrium, for any given number of commuters, the solutions to the latest and earliest departure time are unique. Finally, numerical results are used to verify the analytical findings and explore the relationship between departure rate and some model parameters.