## 2 An Overview of the Joint Recommendation Selection & Visiting Time Management Framework  
In this section, we provide an overview of the overall framework proposed in this research work while, in the following sections, its individual components will be analyzed in detail. Specifically, we formulate and address the problem of joint Recommendation Selection and Visiting Time Management (RSVTM) in a museum environment. We note that the same problem formulation can more generally be applied to any exhibition area and it is not necessarily limited to specific museum assumptions like the majority of prior literature studies. In a nutshell, the proposed problem and process formulation are as follows. Before starting their museum tour, arriving visitors select a type of recommendation among the various ones offered, which will drive their visit. The types of recommendations range from a simple map for a self-guided tour to a guided visit in their own language, which offer different levels of QoE. The recommendation selection is based on a reinforcement-learning framework through which the museum visitors act as learning automata. As time evolves and before the visitors make their final selection of the recommendation, they gain knowledge and experience. Figure 1 presents the examined RSVTM problem as a learning system. Each museum visitor/ learning automaton \(i,i\in\aleph\) (where \(\aleph\) denotes the set of visitors) at each operation time slot \(\tau\) of the reinforcement learning framework/loop has a set of actions \(a_{\tau}(\tau)\). Each action represents a different choice \(x\) of recommendation selection for his/her museum tour. Towards making their decision, museum visitors consider the museum environment and specifically the output set \(\beta(\tau)=\{\mathbf{Q}(\tau),\mathbf{t}(\tau)\}\) where \(\mathbf{Q}(\tau)\) and \(\mathbf{t}(\tau)\) are the vectors of the combined QoE and visiting time of all museum visitors that are simultaneously in the museum. The output \(\beta(\tau)=\{\mathbf{Q}(\tau),\mathbf{t}(\tau)\}\) is determined from the visiting time management framework. The solution to the visiting time management problem results in visitors' optimal visiting time. Based on the chosen actions of the visitors and the corresponding reaction of their environment, we are able to determine the reward probability \(p_{x}(\tau)\), which is associated with the action \(a_{x}(\tau)\). The reward probability represents the penetration of the \(R_{x}\)_th_ recommendation in the pool of museum visitors. The action probability vector of a museum visitor/learning automaton \(i,i\in\aleph\) is \(\mathbf{Pr}_{i}(\tau)\) where each one of its elements \(\Pr_{i,\aleph}(\tau)\) expresses the probability of selecting the \(R_{x}\)_th_ recommendation and is determined based on the model of learning automata. Given the actions of the museum visitors in terms of recommendation selection, each museum visitor \(i,i\in\mathbb{N}\) aims to determine his/her optimal visiting time \(t_{i}^{*}\) to maximize his/her perceived Quality of Experience. Therefore, this goal is formulated as a distributed maximization problem of each museum visitor combined QoE function with respect to his/her visiting time. Considering the distributed nature of the problem and the selfish behavior of the visitors as they optimize their own perceived QoE, a game theoretic approach is adopted to determine the optimal visiting time vector \(\mathbf{t}^{*}=\left\{t_{1}^{*},\ldots,t_{i}^{*},\ldots,t_{N}^{*}\right\}\) for visitors. The Nash equilibrium concept is adopted to analytically solve the non-cooperative visiting time management game. To demonstrate the existence and uniqueness of the Nash equilibrium in the non-cooperative visiting time management game, we prove that the museum visitor combined QoE function \(Q_{i}\) is quasi-concave with respect to \(t_{i}\).