  This study investigates the theoretical properties of a departure time choice
problem considering commuters' heterogeneity with respect to the value of
schedule delay in corridor networks. Specifically, we develop an analytical
method to solve the dynamic system optimal (DSO) and dynamic user equilibrium
(DUE) problems. To derive the DSO solution, we first demonstrate the
bottleneck-based decomposition property, i.e., the DSO problem can be
decomposed into multiple single bottleneck problems. Subsequently, we obtain
the analytical solution by applying the theory of optimal transport to each
decomposed problem and derive optimal congestion prices to achieve the DSO
state. To derive the DUE solution, we prove the queue replacement principle
(QRP) that the time-varying optimal congestion prices are equal to the queueing
delay in the DUE state at every bottleneck. This principle enables us to derive
a closed-form DUE solution based on the DSO solution. Moreover, as an
application of the QRP, we prove that the equilibrium solution under various
policies (e.g., on-ramp metering, on-ramp pricing, and its partial
implementation) can be obtained analytically. Finally, we compare these
equilibria with the DSO state.

Optimization and Control