## 3 Personalized Waiting Time  
This section provides the formulation of the proposed personalized waiting time. The notation used in the rest of the paper is summarized in Table 1.  
As mentioned above, we argue that the personalized waiting time, taking the approaching time of the tourist into account, would improve the tourist's perception of waiting when he/she arrives at the attraction. Specifically, the calculation of the personalized waiting time considers not only the queue length, the capacity, and the duration of an attraction, but also the moving time of the tourist  
Figure 1: Basic concept of the proposed personalized waiting time. from his/her starting position to the attraction. Given \(n_{i}^{Q}\), \(n_{i}^{Op}\), and \(t_{i}^{Op}\), the general waiting time of \(A_{i}\),\((t_{i}^{GW})\) can be obtained from Equation (1). \[t_{i}^{GW}=\left[\frac{n_{i}^{Q}}{n_{i}^{OP}}\right]\times t_{i}^{Op} \tag{1}\]  
Definition 1 (Requesting Time): The requesting time is defined as the time when the central subsystem receives the personalized dynamic scheduling request from the mobile app subsystem. Observed at the requesting time, this method considers the starting times of the current operation session and the next operation session. Specifically, the latter is defined as the immediate following session of the current operation session. Thus, we have \(T_{i}^{Next}=T_{i}^{Curr}+t_{i}^{Op}\)  
To calculate the personalized waiting time, we have to determine the ideal session for the tourist. The ideal session of \(A_{i}\) is defined as the session which starts later than \(T_{Req}\) and can allow one visitor getting into \(A_{i}\) immediately after the \(\left[\frac{n_{i}^{Q}}{n_{i}^{Op}}\right]\) th batch of visitors (in the queue of length \(n_{i}^{Q}\geq 0\) observed at \(T^{Req}\)) is served. In other words, if a tourist who activates the personalized dynamic scheduling function arrives before and right at the start of the ideal session, he/she can get into \(A_{i}\) at the ideal session along with the last \(n_{i}^{Q}-\left[\frac{n_{i}^{Op}}{n_{i}^{Op}}\right]\times n_{i}^{Op}\) visitors in the queue if \(n_{i}^{Q}\geq 0\) observed at \(T^{Req}\). Definition 2 (Ideal Session Time): The ideal session time is defined as the starting time of the ideal session. Theorem 1: \(T_{i}^{Ideal}=T_{i}^{Next}+\left[\frac{n_{i}^{Op}}{n_{i}^{Op}}\right]\times t _{i}^{Op}\)_._  
Proof: If all visitors in the queue plus one more visitor can be served in one single session (i.e., \(n_{i}^{Q}+1\leq n_{i}^{Op}\)), then the ideal session time is actually the starting time of the next operation session (i.e., \(T_{i}^{Ideal}=T_{i}^{Next}\)). On the other hand, if the visitors in the queue plus one more visitor cannot all be served in one single session (i.e.,\(n_{i}^{Q}+1>n_{i}^{Op}\)), then ideal session time will be the starting time of a session later than the next session observed at \(T^{Req}\) and determined according to the queue length, capacity, and duration of \(A_{i}\). As a result, we obtain \(T_{i}^{Ideal}=T_{i}^{Next}+\left[\frac{n_{i}^{Op}}{n_{i}^{Op}}\right]\times t _{i}^{Op}\). \(\square\)  
Definition 3 (Moving Time): The moving time is defined as the duration of the tourist moving from his/her location where he/she activates the personalized dynamic scheduling function to the location of an attraction. Definition 4 (Arrival Time): The arrival time is defined as the time when the tourist arrives at an attraction. Given \(T^{Req}\), we derive  
\[T_{i}^{Arr}=T^{Req}+t_{i}^{Move} \tag{2}\]  
Definition 5 (Recommendation Session Time): The personalized waiting time is defined as the actual waiting time of the tourist when he/she arrives at an attraction. Theorem 2: If \(T_{i}^{Arr}\leq T_{i}^{Ideal}\), then \(t_{i}^{Wait}=\left(T_{i}^{Curr}-T^{Req}\right)+\left(1+\left[\frac{n_{i}^{ Op}}{n_{i}^{Op}}\right]\right)\times t_{i}^{Op}-t_{i}^{Move}\); otherwise, \(t_{i}^{Wait}=\left(T_{i}^{Curr}-T^{Req}\right)+\left[\frac{T^{Req}-T_{i}^{Curr }+t_{i}^{Move}}{t_{i}^{Op}}\right]\times t_{i}^{Op}-t_{i}^{Move}\). **Proof.