## 5 RSVTM Algorithm  
RSVTM Algorithm  
_Step 1 (Initialization)_: Each visitor announces his/her visiting time constraints, i.e., \(t_{i}^{Min}\), \(t_{i}^{Max}\). At the beginning of the first time slot, i.e., \(\tau=0\), set the initial recommendation selection probability vector \(\mathbf{Pr_{i}}(\tau=0)\) as \(\Pr_{i,x}(\tau=0)=\frac{1}{3}\), \(\forall i\in\mathbb{N}\), \(x=A,B,C\). Afterwards, each museum visitor chooses a type of recommendation according to his/her recommendation selection probability vector \(\mathbf{Pr_{i}}(\tau=0)\).  
_Step 2 (Recommendation Selection)_: At every time slot \(\tau>0\), each museum visitor chooses a recommendation according to his/her recommendation selection probability vector \(\mathbf{Pr_{i}}(\tau)\) provided in relations (4a) and (4b).  
_Step 3 (Visiting Time Management)_: Given that all museum visitors have chosen a type of recommendation, then:  
_Step 3a_: Set _ite = 0_, where _ite_ denotes the iteration of the visiting time management part of the algorithm. The information of the overall visiting time is announced through the application to the museum visitors and each museum visitor determines the term \(\sum\limits_{\genfrac{}{}{0.0 pt}{}{j\in\mathbb{N}}{j\
eq i}}t_{j}\) by subtracting his/her own visiting time.  
_Step 3b_: Each museum visitor determines his optimal visiting time \(t_{i}^{*}\) in accordance with Equation (10).  
_Step 3c_: If \(\left|t_{i}^{(ite+1)}-t_{i}^{(ite)}\right|\leq\varepsilon\) (\(\varepsilon\): small positive constant), the visiting time values have converged and stop. Otherwise, return to step 3a.  
_Step 4 (Recommendation Selection)_: Given the optimal visiting time, the museum can measure the penetration of each type of recommendation, which is the reward probability \(p_{x}(\tau)=\frac{\sum\limits_{\genfrac{}{}{0.0 pt}{}{j=1}{k_{x}}}^{N}Q_{x}t_{i }^{\theta}(rt_{i})}{\sum\limits_{\genfrac{}{}{0.0 pt}{}{j=1}{k_{x}}}^{N}Q_{x}t_{i }^{\theta}(rt_{i})}\) and provides this information to museum visitors.  
_Step 5 (Recommendation Selection)_: Each museum visitor updates his/her recommendation selection probability vector via the following rule where \(0<b<1\) is a step size parameter:  
\[\Pr_{i,x}(\tau+1)=\Pr_{i,x}(\tau)-b\cdot p_{x}(t)\cdot\Pr_{i,x}(\tau),x^{(\tau+1 )}\
eq x^{(\tau)}\]\[\Pr_{i,x}(\tau+1)=\Pr_{i,x}(\tau)+b\cdot p_{x}(\tau)\cdot(1-\Pr_{i,x}(\tau)),\ x^{( \tau+1)}=x^{(\tau)}\]  
_Step 6 (Recommendation Selection)_: If one of the recommendation selection probabilities, i.e., \(\Pr_{i,x}(\tau)\), \(x=A,B,C\), converges, i.e., \(\Pr_{i,x}(\tau)=0.999\), stop. The visitor \(i\) selects the recommendation \(R_{x}\). Otherwise, return to step 2.  
It should be clarified that the RSVTM algorithm can be characterized as a low complexity algorithm due to its distributed nature and the simplicity of the calculations (i.e., closed-form expressions) that performs. In addition, as it will be shown in detail in the section of numerical results (i.e., Section 6), the recommendation selection probabilities converge fast in terms of necessary timeslots \(\tau\), i.e., there exists a probability \(\Pr_{i,j}(t)\), which is larger than a value approaching one (e.g., 0.999). Lastly, it should be highlighted that the details of the RSVTM algorithm are found from the museum visitor and the algorithm runs at the downloaded mobile application. The only information provided by the museum visitor is his/her visiting time constraints, i.e., \(t_{i}^{Min}\), \(t_{i}^{Max}\). The application after running the RSVTM algorithm will tell the museum visitor which type of recommendation to select and his/her optimal visiting time to optimize his/her perceived QoE.