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.

中文翻译：

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异构通勤者走廊问题的队列替换原理


本研究研究了出发时间选择问题的理论特性，考虑到通勤者的异质性与走廊网络中时间表延误值的关系。具体来说，我们开发了一种分析方法来解决动态系统最优（DSO）和动态用户平衡（DUE）问题。为了推导DSO解决方案，我们首先证明基于瓶颈的分解性质，即DSO问题可以分解为多个单一瓶颈问题。随后，我们通过将最优交通理论应用于每个分解问题来获得解析解，并推导出达到 DSO 状态的最优拥堵价格。为了导出 DUE 解决方案，我们证明了队列替换原理（QRP），即时变最优拥塞价格等于每个瓶颈处 DUE 状态的排队延迟。这一原理使我们能够在 DSO 解决方案的基础上推导出封闭式 DUE 解决方案。此外，作为QRP的应用，我们证明可以通过分析获得各种政策（例如入口匝道计量、入口匝道定价及其部分实施）下的均衡解。最后，我们将这些平衡状态与 DSO 状态进行比较。