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Continuous-time point-queue models in dynamic network loading

机译：动态网络负载中的连续时间点队列模型

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摘要

Extending a recent work (Pang et al., in press) pertaining to a simple single-bottleneck model, this paper is the first of a two-part research wherein we undertake a mathemati cally rigorous study of the continuous-time dynamic user equilibrium (DUE) problem using the recently introduced mathematical paradigm of differential complementarity systems (DCSs) (Pang and Stewart, 2008). The first step in this comprehensive research is to gain a thorough understanding of some continuous-time point-queue models, which will be used as the building block of a computationally tractable model for the continuous-time DUE problem that we will study in detail in the accompanying paper (Ban et al., in press). Starting with the original point-queue model introduced by Vickrey (1969), we summarize some desirable properties that a continuous-time point-queue model should possess, and show that one of these properties-the nonnegativity of the queue lengths-is violated by Vickrey's original model. As a remedy to this drawback of Vickrey's model and with the goal of extending it to a continuous-time setting, we introduce two continuous-time point-queue models and show that they satisfy the properties we propose. Discretizations of the continuous-time models are discussed and construction of numerical trajectories is presented; convergence of such trajectories as the time step approaches zero is estab lished; regularity of a solution to the continuous-time problem is clarified, and numerical results are presented.
机译：扩展了有关简单的单瓶颈模型的最新工作（Pang等人，在印刷中），该论文是两部分研究的第一部分，其中我们对连续时间动态用户均衡进行了严格的数学研究（ DUE）问题使用最近引入的差分互补系统（DCS）的数学范式（Pang and Stewart，2008）。这项全面研究的第一步是全面了解一些连续时间点队列模型，这些模型将用作连续时间DUE问题的可计算易处理模型的基础，我们将在下文中进行详细研究。随附的论文（Ban等人，印刷中）。从Vickrey（1969）引入的原始点队列模型开始，我们总结了连续时间点队列模型应具有的一些理想属性，并表明这些属性之一（队列长度的非负性）受到了侵犯。 Vickrey的原始模型。为了弥补Vickrey模型的这一缺点，并以将其扩展到连续时间设置为目标，我们引入了两个连续时间点队列模型，并证明它们满足我们提出的特性。讨论了连续时间模型的离散化，并提出了数字轨迹的构造；建立时间步长接近零时这种轨迹的收敛；阐明了连续时间问题的解的规律性，并给出了数值结果。

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