The mathematical approaches for modeling dynamic traffic can roughly be divided into two categories: discrete packet routing models and continuous flow over time models. Despite very vital research activities on models in both categories, the connection between these approaches was poorly understood so far. In this work we build this connection by specifying a (competitive) packet routing model, which is discrete in terms of flow and time, and by proving its convergence to the intensively studied model of flows over time with deterministic queuing. More precisely, we prove that the limit of the convergence process, when decreasing the packet size and time step length in the packet routing model, constitutes a flow over time with multiple commodities. In addition, we show that the convergence result implies the existence of approximate equilibria in the competitive version of the packet routing model. This is of significant interest as exact pure Nash equilibria, similar to almost all other competitive models, cannot be guaranteed in the multi-commodity setting. Moreover, the introduced packet routing model with deterministic queuing is very application-oriented as it is based on the network loading module of the agent-based transport simulation MATSim. As the present work is the first mathematical formalization of this simulation, it provides a theoretical foundation and an environment for provable mathematical statements for MATSim.
翻译:建模动态交通模式的数学方法大致可以分为两类:离散包路由模型和随时间变化的连续流动模型。尽管对两种类型的模型都进行了非常重要的研究活动,但迄今为止对这些方法之间的联系了解甚少。在这项工作中,我们通过具体说明一个(竞争性)包路由模型来建立这一联系,该模型在流量和时间方面是独立的,并且证明它与经过密集研究的流流模式的趋同,同时有确定性排队。更确切地说,我们证明,当减少包路由模型的包件大小和时间长度时序流时,趋同过程的局限性是随时间流而流的。此外,我们表明,这种趋同结果意味着在包路由模型的竞争版本中存在近乎的平衡。这具有重大意义,因为与几乎所有其他竞争性模型相似的纯净Nash equiliconlibricrial,在多商品环境下是无法保证的。此外,采用确定性排入式汇模式的引入的包路由模型模型的极限是非常注重应用的,因为它以多种商品路由多种商品组成的网络装载模式为基础。此外,这个基于模型的数学模型的模型的模型的模型基础是用于目前的数学模拟环境。