Queue replacement principle for corridor problems with heterogeneous commuters

TL;DR

This paper develops an analytical framework for corridor traffic problems with heterogeneous commuters, introducing the queue replacement principle to derive explicit solutions for dynamic user equilibrium and system optimal states.

Contribution

It introduces the queue replacement principle and an analytical method to solve DSO and DUE problems considering commuter heterogeneity in corridor networks.

Findings

Derived closed-form DUE solutions based on DSO solutions.

Proved queue replacement principle linking congestion prices and queue delays.

Analytically characterized equilibrium under various traffic policies.

Abstract

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.