Solving the user equilibrium departure time problem at an off-ramp with incentive compatible cost functions

We consider the equilibrium departure time problem for a set of vehicles that travel through a network with capacity restrictions and need to reach a destination at a fixed time. The vehicles incur a penalty for both any queuing delays and arriving at the destination early or late. In particular, we consider the case of a congested off-ramp, which is a common occurrence next to commercial hubs during the morning commute, and has the added negative effect of reducing the capacity on the freeway for through traffic. We study the use of incentives and tolls to manipulate the equilibrium departure times of the exiting vehicles and thereby mitigate the impact on through traffic. Our main result is to show the existence and uniqueness properties of the departure time equilibrium for a general class of delay and arrival time cost functions, which allows for discontinuities in the arrival cost function. This enables the use of step incentives or tolls, which are the mostly common strategies used in practice. Our results also apply to the Vickrey single bottleneck equilibrium, which is a special case of our network.