We consider a dynamic model of traffic that has received a lot of attention in the past few years. Infinitesimally small agents aim to travel from a source to a destination as quickly as possible. Flow patterns vary over time, and congestion effects are modeled via queues, which form based on the deterministic queueing model whenever the inflow into a link exceeds its capacity. Are equilibria in this model meaningful as a prediction of traffic behavior? For this to be the case, a certain notion of stability under ongoing perturbations is needed. Real traffic consists of discrete, atomic ''packets'', rather than being a continuous flow of non-atomic agents. Users may not choose an absolutely quickest route available, if there are multiple routes with very similar travel times. We would hope that in both these situations -- a discrete packet model, with packet size going to 0, and $\epsilon$-equilibria, with $\epsilon$ going to 0 -- equilibria converge to dynamic equilibria in the flow over time model. No such convergence results were known. We show that such a convergence result does hold in single-commodity instances for both of these settings, in a unified way. More precisely, we introduce a notion of ''strict'' $\epsilon$-equilibria, and show that these must converge to the exact dynamic equilibrium in the limit as $\epsilon \to 0$. We then show that results for the two settings mentioned can be deduced from this with only moderate further technical effort.

翻译：我们考虑一种在过去几年受到广泛关注的动态交通模型。无限小的出行者试图以最快速度从起点到达终点。交通流模式随时间变化，拥堵效应通过队列建模——基于确定性排队模型，当进入路段的流量超过其容量时形成队列。该模型中的均衡能否作为交通行为的有意义预测？为此，需要某种在持续扰动下的稳定性概念。实际交通由离散的原子化“分组”构成，而非连续的非原子化流。若存在多条行程时间相近的路径，用户可能不会选择绝对最快的路径。我们期望在这两种情形——分组尺寸趋近于0的离散分组模型，以及ε趋近于0的ε-均衡——中，均衡结果都能收敛到时变流模型中的动态均衡。此前尚无此类收敛性结论。我们证明，在单商品实例中，这两种情形均能以统一方式实现收敛。更精确地说，我们引入了“严格”ε-均衡的概念，并证明当ε→0时，这类均衡必收敛到精确动态均衡。进而表明，上述两种情形下的结论可通过适度额外的技术推导得出。