On the Existence of Solutions to the Dynamic User Equilibrium Problem

Zhu, Daoli; Marcotte, Patrice

doi:10.1287/trsc.34.4.402.12322

Cited by 87 publications

(113 citation statements)

References 13 publications

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“…For each p ∈ P, we define its path departure rate h p (·) as a function of departure time t. Then, we let h(·) = � h p (·) : p ∈ P � be the vector of path departure rates. The following constraints on the departure rates are commonly employed for route choice DUE problems (Smith and Wisten, 1995;Zhu and Marcotte, 2000):…”

Section: Delay Operator As An Infinite-dimensional Mappingmentioning

confidence: 99%

“…For example, the existence of dynamic user equilibrium (DUE), which is the most widely studied form of DTA problems, depends on the continuity of the delay operators (Han et al, 2013c;Smith and Wisten, 1995;Zhu and Marcotte, 2000), while the uniqueness of DUE is guaranteed by the monotonicity of the delay operator (Mounce and Smith, 2007). Moreover, all computational procedures for DUE problems rely on certain versions of continuity and monotonicity to converge (Friesz et al, 2011;Han et al, 2015;Long et al, 2013;Mounce, 2006;Szeto and Lo, 2004).…”

Section: Introductionmentioning

confidence: 99%

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Statistical metamodeling of dynamic network loading

Song

Han

Wang

2018

Transportation Research Part B: Methodological

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Dynamic traffic assignment models rely on a network performance module known as dynamic network loading (DNL), which expresses the dynamics of flow propagation, flow conservation, and travel delay at a network level. The DNL defines the so-called network delay operator, which maps a set of path departure rates to a set of path travel times. It is widely known that the delay operator is not available in closed form, and has undesirable properties that severely complicate DTA analysis and computation, such as discontinuity, non-differentiability, non-monotonicity, and computational inefficiency. This paper proposes a fresh take on this important and difficult problem, by providing a class of surrogate DNL models based on a statistical learning method known as Kriging. We present a metamodeling framework that systematically approximates DNL models and is flexible in the sense of allowing the modeler to make trade-offs among model granularity, complexity, and accuracy. It is shown that such surrogate DNL models yield highly accurate approximations (with errors below 8%) and superior computational efficiency (9 to 455 times faster than conventional DNL procedures). Moreover, these approximate DNL models admit closed-form and analytical delay operators, which are Lipschitz continuous and infinitely differentiable, while possessing closed-form Jacobians. The implications of these desirable properties for DTA research and model applications are discussed in depth.

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Section: Delay Operator As An Infinite-dimensional Mappingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Statistical metamodeling of dynamic network loading

Song

Han

Wang

2018

Transportation Research Part B: Methodological

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“…Clearly (d * , q * ) is feasible for (D). Also (10) gives x ∈ F 0 , while (13) implies that if x e > 0 then 1 = ρ e (1, x e ). This implies that x e ≤ ν e , so x is feasible for (P).…”

Section: Steady Statesmentioning

confidence: 99%

“…Although dynamic equilibria have been around for almost fifty years (see, e.g., [2][3][4]6,7,[9][10][11][12]), their existence has only been proved recently by Zhu and Marcotte [13] though in a somewhat different setting, and by Meunier and Wagner [8] who gave the first existence result for a model that covers the case of fluid queuing networks. These proofs, however, rely heavily on functional analysis techniques and provide little intuition on the combinatorial structure of dynamic equilibria, their characterization, or feasible approaches to compute them.…”

Section: Introductionmentioning

confidence: 99%

Long Term Behavior of Dynamic Equilibria in Fluid Queuing Networks

Cominetti

Correa

Olver

2017

Integer Programming and Combinatorial Optimization

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Abstract.A fluid queuing network constitutes one of the simplest models in which to study flow dynamics over a network. In this model we have a single source-sink pair and each link has a per-time-unit capacity and a transit time. A dynamic equilibrium (or equilibrium flow over time) is a flow pattern over time such that no flow particle has incentives to unilaterally change its path. Although the model has been around for almost fifty years, only recently results regarding existence and characterization of equilibria have been obtained. In particular the long term behavior remains poorly understood. Our main result in this paper is to show that, under a natural (and obviously necessary) condition on the queuing capacity, a dynamic equilibrium reaches a steady state (after which queue lengths remain constant) in finite time. Previously, it was not even known that queue lengths would remain bounded. The proof is based on the analysis of a rather non-obvious potential function that turns out to be monotone along the evolution of the equilibrium. Furthermore, we show that the steady state is characterized as an optimal solution of a certain linear program. When this program has a unique solution, which occurs generically, the long term behavior is completely predictable. On the contrary, if the linear program has multiple solutions the steady state is more difficult to identify as it depends on the whole temporal evolution of the equilibrium.

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“…The continuousness of route cost functions has also been proved by Huang and Lam (2002) by discretising time (their study also considers departure-time-choice behaviour). For the whole-link model, a proof of existence has been shown by Zhu and Marcotte (2000), in which departure times of all vehicles are fixed. For the existence of a DUE solution with the whole-link model and simultaneous departure-time-androute-choice, Zhong et al (2011) provided a proof of existence by expanding the proof of Zhu and Marcotte (2000).…”

Section: Existence Of Due Solutionmentioning

confidence: 99%

Properties of dynamic user equilibrium solution: existence, uniqueness, stability, and robust solution methodology

Iryo

2013

Transportmetrica B: Transport Dynamics

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This paper reviews studies that deal with properties of dynamic user equilibrium (DUE) solutions to understand the extent to which they have been resolved in past studies and indicate theoretical problems that remain to be addressed in future research. DUE assignment is an equilibrium-based methodology for dynamic traffic assignment. In conventional equilibriumbased static traffic assignment, the four desirable properties of solutions -existence, uniqueness, stability of solutions, and the existence of a robust solution methodology -are all guaranteed. These properties must be ensured for planners to obtain a reliable solution for equilibriumbased assignment. Meanwhile, for DUE assignment, although some of these properties have been proved, many issues regarding these properties remain unresolved. This paper reviews studies that deal with these properties to understand how far they have been investigated and discusses what is required from future research to improve the robustness of DUE assignment.

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On the Existence of Solutions to the Dynamic User Equilibrium Problem

Cited by 87 publications

References 13 publications

Statistical metamodeling of dynamic network loading

Statistical metamodeling of dynamic network loading

Long Term Behavior of Dynamic Equilibria in Fluid Queuing Networks

Properties of dynamic user equilibrium solution: existence, uniqueness, stability, and robust solution methodology

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