## 3 Models and Preliminaries  
### System Model  
Based on the following information, we assume a single-exhibition, e.g., museum where the optimal visiting time of each museum visitor is studied. Within the museum, we assume that \(\Re\) different types of recommendations are provided to the museum visitors. Even though several different recommendations may exist, in this paper, we have assumed three different types of recommendations. Let us denote by \(\Re=\{R_{A},R_{B},R_{C}\}\) the set of recommendations available to visitors. Each one provides a different Quality of Experience (QoE) for the visitors while it asks for a different investment time on the part of the visitor. The latter will be discussed in detail in Section 3.2. As mentioned before, we denote by \(\Re=\{1,\ldots,i,\ldots,N\}\) the set of museum visitors that consists of visitors with different visiting styles, i.e., ant, butterfly, fish, and grasshopper. Consequently, we have the corresponding subsets: \(\Re_{a},\ \Re_{b},\ \Re_{f},\ \Re_{g}\) and \(\Re=\Re_{a}\cup\ \Re_{b}\cup\ \Re_{f}\cup\ \Re_{g}\). Each museum visiting style has different characteristics regarding the time spent for the visit as well as its tolerance to the level of crowd density in the museum [2; 3]. Specifically, ant visitors follow a long path in the museum and they are inclined towards extensive walking by spending a lot of time at the museum during their visit. Moreover, the increased crowd density does not discourage them from visiting the museum because they are dedicated to their visit. However, their perceived satisfaction decreases when the crowd density increases. Butterfly visitors are also interested in visiting the whole exhibition. However, the walking path that they follow may be easily redirected if the conditions favor it and suggest such an alteration. For example, if a specific part of the museum has increased crowd density, unlike ant visitors, butterfly visitors will choose to visit it later in order to avoid long queues. Thus, butterfly visitors also invest a lot of time in their museum visit, but they are more tolerant of increased crowd density when compared to ant visitors. Fish visitors are the \"laziest\" ones since they prefer to stand in the center of an exhibition room and observe many exhibits concurrently. Consequently, fish visitors spend limited time on their visit and their satisfaction is affected dramatically by increased crowd density. Standing in the center of an exhibition room, a larger crowd easily blocks their vision. Lastly, grasshoppers have clear preferences and make plans about the specific exhibits that they will visit. Thus, they spend an average amount of time on their museum visit to observe the details of the few specific exhibits that they have selected to visit and they get annoyed with increased crowd density because it adds a delay to their pre-scheduled and well-organized museum visit. Museum visitor visiting time is denoted by \(t_{i}\) and, due to museum physical characteristics (e.g., size) and visitor physical limitations and personal preferences, it is lower and upper bounded, i.e., \(t_{i}^{Min}\leq t_{i}\leq t_{i}^{Max}\). Let us also denote by \(\mathbf{t}_{-i}\), the visiting time vector of the rest of the museum visitors that are in the museum at the same time as the visitor \(i,i\in\Re\). Each type of recommendation \(R_{x},x=A,B,C\) and \(R_{x}\in\Re\) offers a different level of QoE to the visitor that selects it. Let us denote by \(Q_{x},\ x=A,B,C\) the QoE values offered by the recommendations \(R_{A},R_{B}\), and \(R_{C}\), respectively. Without loss of generality, the QoE values offered by each recommendation are sorted as \(Q_{A}<Q_{B}<Q_{C}\) with the interpretation that a higher value of QoE reflects an enhanced level of recommendation service, which is discussed in more detail in Section 3.2. A simple but fundamental and effective performance measure that verifies that the visitors' QoE prerequisites are fulfilled can be defined as the relative visiting time ratio.