## 3 Personalized Waiting Time
** In the proposed system, \(t_{i}^{Wait}=T_{i}^{Recomm}-T_{i}^{Arr}\) according to the definition of the recommend session time. Because \(T_{i}^{Arr}\) and \(T_{i}^{Ideal}\) directly influence \(T_{i}^{Recomm}\), we consider the following two cases to derive \(T_{i}^{Recomm}\), and further derive \(t_{i}^{Wait}\). \(\square\)  
**Case 1. **_The arrival time is earlier than or equal to the ideal session time (i.e., \(T_{i}^{Arr}\leq T_{i}^{Ideal}\))._  
According to the above discussion, we infer that the tourist (who activates the personalized dynamic scheduling request at \(T^{Req}\)) can get into \(A_{i}\) immediately after the \(\left[\frac{n_{i}^{Q}}{n_{i}^{Op}}\right]\)_th_ batch of visitors in the queue is served for all \(n_{i}^{Q}\geq 0\). Thus, \(T_{i}^{Recomm}=T_{i}^{Ideal}\). Because \(T_{i}^{Ideal}=T_{i}^{Next}+\left[\frac{n_{i}^{Q}}{n_{i}^{Op}}\right]\times t_ {i}^{Op}\), \(T_{i}^{Recomm}\) can be derived from  
\[T_{i}^{Recomm}=T_{i}^{Next}+\left[\frac{n_{i}^{Q}}{n_{i}^{Op}}\right]\times t_ {i}^{Op} \tag{3}\]  
According to Equations (2) and (3), \(t_{i}^{Wait}\) can be rewritten as \(t_{i}^{Wait}=\left(T_{i}^{Next}+\left[\frac{n_{i}^{Q}}{n_{i}^{Op}}\right] \times t_{i}^{Op}\right)-\left(T^{Req}+t_{i}^{Move}\right)\). Because \(T_{i}^{Next}=T_{i}^{Curr}+t_{i}^{Move}\), we obtain  
\[ \begin{split}& t_{i}^{Wait}=\left(T_{i}^{Curr}+t_{i}^{Op} \right)+\left[\frac{n_{i}^{Q}}{n_{i}^{Op}}\right]\times t_{i}^{Op}-\left(T^{ Req}+t_{i}^{Move}\right)\\ &=\left(T_{i}^{Curr}-T^{Req}\right)+\left(1+\left[\frac{n_{i}^{ Q}}{n_{i}^{Op}}\right]\right)\times t_{i}^{Op}-t_{i}^{Move}\end{split}  \tag{4}\]  
**Case 2. **_The tourist's arrival time is later than the ideal session time (i.e., \(T_{i}^{Arr}>T_{i}^{Ideal}\))._  
In this case, the tourist will miss the ideal session and has to wait only for a short period within the duration of \(A_{i}\) after he/she arrives, because all the visitors waiting in line observed at \(T^{Req}\) have been served before or at \(T_{i}^{Ideal}\). Therefore, we have  
\[T_{i}^{Recomm}=T_{i}^{Next}+\left[\frac{T_{i}^{Arr}-T_{i}^{Next}}{t_{i}^{Op}} \right]\times t_{i}^{Op} \tag{5}\]  
According to Equations (2) and (5), \(t_{i}^{Wait}\) can be rewritten as \(t_{i}^{Wait}=\left(T_{i}^{Next}+\left[\frac{\left(T^{Req}+t_{i}^{Move}\right) -T_{i}^{Next}}{t_{i}^{Op}}\right]\times t_{i}^{Op}\right)-\left(T^{Req}+t_{i}^{Move}\right)\). Because \(T_{i}^{Next}=T_{i}^{Curr}+t_{i}^{Move}\), we obtain  
\[ \begin{split}& t_{i}^{Wait}=\left(T_{i}^{Curr}+t_{i}^{Op} \right)+\left[\frac{\left(T^{Req}+t_{i}^{Move}\right)-\left(T_{i}^{Curr}+t_{ i}^{Op}\right)}{t_{i}^{Op}}\right]\times t_{i}^{Op}-\left(T^{Req}+t_{i}^{Move}\right)\\ &=\left(T_{i}^{Curr}-T^{Req}\right)+\left[\frac{T^{Req}-T_{i}^{ Curr}+t_{i}^{Move}}{t_{i}^{Op}}\right]\times t_{i}^{Op}-t_{i}^{Move}\end{split}  \tag{6}\]  
**4. TPTS System Design**  
Figure 2 shows the system architecture of the proposed TPTS system. The TPTS system consists of four subsystems: central subsystem, mobile app subsystem, ticket-scanning subsystem, and detecting/counting subsystem. Each subsystem includes many modules to perform the corresponding functions. The mobile app subsystem is a mobile app for visitors to offer the tourist services. The subsystem provides tourists with many interfaces to obtain the park information, personalized tour suggestion, booking record, and so on.