This paper extends the work of Zhang et al. (2008) to investigate the daily commuting patterns allowing both late arrival in the morning and early departure in the evening in a linear city. Given the fixed parking locations of morning commuters, the Nash equilibrium principle is employed to formulate commuters’ departure time choice in the evening and the user equilibrium principle in terms of daily travel cost is adopted to formulate the departure time choice of morning commuters. A combined regime of linear location-dependent parking fee and time-varying road toll is then proposed to reduce the total social cost (TSC) for the linear city. All the possible equilibrium traffic flows are analytically derived. The analytical results show that (1) commuters, who arrive at work early in the morning, may not depart early in the evening, but their evening trip costs must be lower; (2) when the parking spaces become denser, the evening commute starts earlier and the morning commute starts later; (3) a linear location-dependent parking fee could either advance or postpone the earliest departure time in the morning; (4) three parking modes, distinguished by the order in which commuters park, occur successively as the parking fee rate increases; and (5) an appropriate parking fee can always improve the system efficiency, but no matter how the parking fee is changed, it is impossible to completely eliminate the efficiency loss in the morning. Furthermore, a parking reservation regime joint with time-varying road toll is designed to minimize the TSC for the linear city. Finally, a set of numerical examples are presented to demonstrate the model's properties and these proposed regimes’ performance.

本文扩展了张等人的工作。(2008) 调查了线性城市中允许早上迟到和晚上早退的日常通勤模式。考虑到早上上班族的停车位固定，采用纳什均衡原理制定晚上上班族的出发时间选择，采用每日出行成本的用户均衡原则制定早晨上班族的出发时间选择。然后提出了一种线性位置相关停车费和随时间变化的道路收费的组合制度，以降低线性城市的总社会成本 (TSC)。所有可能的平衡交通流量都是通过分析推导出来的。分析结果表明：（1）早上上班的上班族，晚上可能不会早早下班；但他们晚上的旅行费用必须更低；(2)停车位越密集，晚上通勤越早，早上通勤越晚；(3) 线性位置相关停车费可以提前或推迟早上最早的出发时间；(4)三种停车方式，以通勤者停车的先后顺序区分，随着停车费率的提高而先后出现；(5)适当的停车费总能提高系统效率，但不管停车费怎么变，也不可能完全消除早上的效率损失。此外，停车预订制度与随时间变化的道路收费相结合，旨在最小化线性城市的 TSC。最后，给出了一组数值例子来证明模型'