Queue Replacement Approach to Dynamic User Equilibrium Assignment with Route and Departure Time Choice

This study develops a hybrid analytical and numerical approach for dynamic user equilibrium (DUE) assignment with simultaneous route and departure time choice (RDTC) for homogeneous users. The core concept of the proposed approach is the generalized queue replacement principle (GQRP), which establishes an equivalence between the equilibrium queueing-delay pattern and the solution to a linear programming (LP) problem obtained by relaxing some conditions in the original DUE-RDTC problem. We first present a method for determining whether the GQRP holds. Based on the GQRP, we then develop a systematic procedure to obtain an exact DUE solution by sequentially solving two LPs: one for the equilibrium cost pattern, including queueing delays, and the other for the corresponding equilibrium flow pattern. Computational results on networks of varying scales confirm the effectiveness of the proposed method.

💡 Research Summary

This paper tackles the dynamic user equilibrium (DUE) problem in traffic networks when travelers simultaneously choose routes and departure times (RDTC). Traditional solution approaches based on variational inequalities (VI) or differential variational inequalities (DVI) suffer from several drawbacks: the cost functions are often non‑monotone, the equilibrium conditions involve nested link‑and‑route travel‑time expressions in an Eulerian (absolute‑time) framework, and fixed‑point algorithms frequently fail to converge on large‑scale networks. To overcome these issues, the authors introduce a hybrid analytical‑numerical methodology built on two key concepts.

First, they adopt a Lagrangian‑like coordinate system, which re‑parameterizes all flow variables with respect to the destination arrival time rather than the departure time. In this representation, each user’s total travel cost—comprising free‑flow travel time, queueing delay, and schedule‑delay penalty—is expressed directly as a function of the destination arrival time and the chosen path. This eliminates the cumbersome nested structure of conventional formulations and yields a mixed linear complementarity problem (LCP) that captures demand conservation, flow conservation, and point‑queue dynamics in a compact linear form.

Second, the authors generalize the queue‑replacement principle (QRP) to a Generalized Queue Replacement Principle (GQRP). The classic QRP states that, for a single bottleneck, the equilibrium queueing‑delay pattern can be replaced by a congestion‑pricing pattern (the shadow price of the capacity constraint) that eliminates queues. The GQRP extends this equivalence to networks with multiple bottlenecks and simultaneous route choice. It shows that, if the GQRP holds, the equilibrium queueing‑delay pattern is exactly the optimal Lagrange multiplier of an auxiliary linear programming (LP) problem obtained by relaxing the queueing condition in the original DUE‑RDTC formulation.

The solution procedure consists of two sequential LPs.

1. Cost‑determination step: The queueing condition is relaxed, producing a skew‑symmetric quadratic program (COST‑QP). By exploiting the skew‑symmetry, the problem decomposes into a primal LP (COST‑LP‑P) and its dual (COST‑LP‑D). Under the GQRP, the optimal solution of the dual LP yields the exact equilibrium travel‑cost pattern (including queueing delays).
2. Flow‑determination step: The candidate cost pattern from the first step is substituted back into the original DUE‑LCP. This substitution reduces the remaining equilibrium conditions to a flow‑determination LP (FLOW‑LP). If FLOW‑LP is feasible and its optimal objective value is zero, the candidate cost pattern is confirmed as the true equilibrium cost pattern, and the optimal solution of FLOW‑LP provides the exact equilibrium flow pattern.

Both LPs are linear, convex, and can be solved efficiently with standard solvers even on networks with thousands of links. The authors validate the approach on a suite of test networks ranging from small illustrative examples to realistic large‑scale networks. Results demonstrate that the proposed method consistently produces exact equilibrium cost and flow patterns, whereas state‑of‑the‑art VI/DVI algorithms often exhibit large cost gaps or fail to converge, especially when route choice is involved.

The paper’s contributions are threefold: (i) a novel DUE‑RDTC formulation in a Lagrangian‑like coordinate system, (ii) the introduction and rigorous verification of the Generalized Queue Replacement Principle, and (iii) an algorithmic framework that reduces the complex DUE problem to two tractable LPs without sacrificing exactness. From a policy perspective, the GQRP implies that congestion pricing schemes derived from the shadow prices can replicate the equilibrium queueing pattern, offering a practical pathway to achieve dynamic system optimal (DSO) conditions through price‑based interventions. The authors also discuss extensions to heterogeneous users, multi‑destination settings, and real‑time information environments, outlining promising directions for future research.