## 2 Preliminaries  
Figure 1 shows an example to illustrate the concept of the proposed personalized waiting time. Without the loss of generality, the next sessions of all attractions are assumed to start at the same time (i.e., all attractions have the same general waiting time). The current time and the next sessions of three attractions are 12:00 and 12:15, respectively. Thus, the general waiting time of three attractions is 15 min. Consider that there are two visitors who want to take Attraction A. They both find that the wait time is 15 min, and then decide to go to Attraction A right away. Assume that one of them (i.e., Visitor 1) should take 10 min to walk to it, while the other (Visitor 2) needs only 2 min to reach it. However, neither has the idea of how much time each of them needs to take to arrive at Attraction A if there is no mobility time information provided for them. They will just go to Attraction A (the same destination), with Visitor 1 finding that he/she can take the ride only after 5 min wait, while Visitor 2 realizing that she/he needs to wait for the ride for 13 min. Now, let us consider the feelings of the two visitors. In our opinion, people would like to keep moving to a certain destination rather than wait at a certain place, which is a common phenomenon that can be observed among car drivers or someone who is on a quest. Hence, we believe that Visitor 1's feeling would be better than Visitor 2's.  
The central idea of our study is the personalized waiting time, which is defined as the actual waiting time when the tourist arrives at an attraction and considered in the personalized dynamic scheduling function of the TPTS system. Recall that the waiting time considered in prior studies is the general waiting time, which is defined as the waiting time of the last tourist in the queue of length at an attraction. The general waiting time is computed based only on the queue length, the capacity, and the duration of an attraction, and totally irrelevant to any personal attribute of the tourist who is requesting any waiting-time related services. As shown in Figure 1, if considering the general waiting time only, the tourist has no preference among the three attractions and thus makes the final decision by himself/herself. The approaching times of the tourist moving towardsAttractions A-C are, respectively, 12, 5, and 8 min. Thus, the actual waiting times when the tourist arrives at Attractions A-C are 3, 10, and 7 min, respectively. If the approaching times of the tourist moving towards these attractions are different, the tourist would feel or perceive that Attraction A costs the shortest waiting time among the three attractions. Therefore, the tourist is more likely to select Attraction A to visit next. This is how the personalized waiting time, taking the approaching time of the tourist into account, would improve the tourist's perception of waiting as he/she arrives at the attraction.  
This study considers the following assumptions to calculate the proposed personalized waiting time.  
1. The inter-session time between operation sessions of the attraction is ignored.
2. The moving time of an attraction is derived as if the tourist starts to move to the attraction right after the tourist activates the personalized dynamic scheduling function.
3. The processing time at both the mobile app subsystem and the central subsystem as well as the network propagation delay between the two subsystems are ignored. In other words, the time when the tourist activates the personalized dynamic scheduling function at the mobile app subsystem and the time when the central subsystem receives the personalized dynamic scheduling request from the mobile app subsystem are considered as the same moment.