## 3 Models and Preliminaries
### Quality of Experience Function and Modeling  
The concept of QoE function has been adopted to represent the perceived satisfaction of the visitors from (a) the time spent for the museum visit, (b) the selected type of recommendation, and (c) the fulfillment of their QoE prerequisites. A combined QoE function is adopted by each museum visitor, which consists of two parts: (a) the pure QoE function and (b) the congestion control function. Museum visitor pure QoE function is reflected via the ratio of the achievable QoE to the corresponding time spent for the museum visit. We assume that visitor's perceived satisfaction increases as he/she achieves high QoE without overspending time for the museum visit. Note that \(t_{i}^{Min}\leq t_{i}\leq t_{i}^{Max}\), which is mentioned in Section 3.1. It should also be highlighted that the optimal visiting time for a museum visitor is not always \(t_{i}^{Min}\) because, based on the selected recommendation and museum congestion (expressed via the cumulative visiting time of all visitors that simultaneously explore the museum), his/her perceived QoE can be limited. Thus, each museum visitor expresses his/her flexibility regarding the visiting time (via the bounds \(t_{i}^{Min}\) and \(t_{i}^{Max}\)) and considers his/her perceived QoE in order to determine the actual optimal visiting time. It should be highlighted that the museum congestion is implicitly expressed via the time the rest of the visitors spend in the museum, i.e., \(\sum\limits_{\genfrac{}{}{0.0 pt}{}{j\in\aleph}{j\
eq i}}t_{j}\) concurrently with visitor \(i,i\in\aleph\). This formulation stems from the observation that either a large number of visitors are simultaneously exploring the museum (i.e., increased crowd density) or there are some visitors spending more time observing exhibits, which contributes to museum congestion and also contributes to \"blocking\" visitors \(i,i\in\aleph\) from observing them. Specifically, the achievable pure QoE function is formulated as a sigmoidal function with respect to the relative visiting time ratio \(rt_{i^{\prime}}\), i.e., \(Q_{x}\cdot t_{i}^{\beta}(rt_{i}),\ x=A,B,C,\ y=a,b,f,g\) where the notation \(y\)shows the visiting style of the visitor, i.e., ant (a), butterfly (b), fish (f), and grasshopper (g). The visiting efficiency function \(\hat{t}_{i}^{g}(rt_{i})\) represents visitor's perceived satisfaction and depends on the museum congestion as well as on the time that he/she invests in the museum visit. For analysis purposes, a common sigmoidal function is adopted, i.e., \(\hat{t}_{i}^{g}(rt_{i})=(1-e^{-A_{g}rt_{i}})^{M_{g}}\), where \(A_{g},M_{g},y=a,b,f,g\) are positive parameters controlling the slope of the visiting efficiency function to guarantee the relative positions of the visiting efficiency functions, which is presented in Figure 2.  
If the target relative visiting time ratio \(rt_{i}^{target,y}\) is achieved, then the QoE prerequisites of visitor \(i\) are fulfilled. To realize this, the \(rt_{i}^{target,y}\) value is mapped at the inflection point of the sigmoidal function \(Q_{x}\cdot\hat{t}_{i}^{g}(rt_{i})\), \(x=A,B,C\), \(y=a,b,f,g\). The notion of the latter formulation is that, if the QoE of visitor \(i\) is lower than \(Q_{x}\cdot\hat{t}_{i}^{g}(rt_{i}^{target,y})\), then his/her perceived satisfaction decreases rapidly, which alerts him/her to the fact that, from his perspective, the museum is congested. In contrast, if the achieved QoE is greater than \(Q_{x}\cdot\hat{t}_{i}^{g}(rt_{i}^{target,y})\), then the museum visitor's satisfaction presents a slight increase due to the fact that his/her QoE expectations have already been fulfilled. In a nutshell, each visitor's combined QoE function can be expressed by using the equation below. \[ \begin{array}{c}Q_{i}(t_{i},\mathbf{t}_{-i})=\frac{Q_{x}\cdot\hat{t}_{i}^{g}(rt_ {i})}{t_{i}}-c_{x}N_{x}^{k}\mathrm{e}^{t_{i}},i\in\aleph,\;x=A,B,C,\\ y=a,b,f,g,\aleph=\aleph_{a}\cup\aleph_{b}\cup\aleph_{f}\cup\aleph_{g}\end{array}