# Proactive Service Caching in a MEC System by Using Spatio-Temporal Correlation among MEC Servers
## 5 Performance Evaluation
### Prediction Accuracy  
We denote the error involved in predicting the popularity of each service \(k\) in a MECS \(m\) during a time slot \(t\) as \(\delta_{m,k}(t)=\
u_{m,k}(t)-\delta_{m,k}(t)\) and compare the distribution of \(\delta_{m,k}(t)\) for all \(k\in\mathcal{K}\) in different MECSs in Figure 5. In this figure, we illustrate the results for four different cases based on the applied methods and MECSs. In each subfigure, the x-axis is the service index \(k\in[0,K-1]\) and the y-axis is the prediction error. Thus, for each service \(k\), each subfigure shows the distribution of \(\delta_{m,k}(t)\) in a MECS \(m\) as a box plot. We observe in the figure that regardless of the true popularity of a service, the 75 th percentile of \(\delta_{m,k}\) obtained by our method is smaller than that resulting from the LSTM method. For example, in MECS 4, the 75 th percentile of \(\delta_{m,k}\) is 0.054 when our method is used, while it is 0.11 when the LSTM method is used. In addition, when the LSTM method is used, there are services whose median \(\delta_{m,k}\)s are much higher than those obtained by our method. For example, the \(\delta_{m,k}\) of the service \(k=33\) in MECS 4 obtained by our method is 0.080, while it is 0.90 when the LSTM method is used. We also observe that the interquartile range (IQR), which is the difference between the 75 th percentile and the 25 th percentile, is smaller when our method is applied compared to when the LSTM method is used. For example, in MECS 4, the average IQR acquired by our method is 0.034, while it is 0.046 when the LSTM method is applied. The results are attributed to the manner that each method predicts the service popularity. When the LSTM method is used, each MECS predicts the popularity of each service within its service range, regardless of the service popularity distributions in the \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline M & 9 \\ K & 64 \\ \(\eta_{m}\) & \(8,16\) \\ \(w_{a}\) & 12 \\ Service popularity dataset & MovieLens [42] \\ User mobility dataset & Divvy trip dataset [43] \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Parameters used for the simulation study. other MECSs. On the contrary, our method predicts the popularity of each service in each MECS at the same time by comprehensively considering the service popularity distributions all over the MECSs. Therefore, our method reduces the amount of error in predicting the popularity of each service in each MECS. We obtain the same results in all the other MECSs. Figure 5: _Cont._  
To investigate the influence of the service popularity prediction error on the service caching decision, we inspect the difference between the true \(S_{m}(t)\) and the predicted \(S_{m}(t)\) in each MECS \(m\). We note that the true \(S_{m}(t)\) is composed of the top \(\eta\) popular services during a time slot \(t\), which can be known at the end of a time slot \(t\). On the contrary, the predicted \(S_{m}(t)\) comprises the top \(\eta\) popular services whose popularities are estimated by a predictor at the beginning of a time slot \(t\). We denote the true \(S_{m}(t)\) as \(S_{m}^{T}(t)\) and the predicted \(S_{m}(t)\) as \(S_{m}^{P}(t)\). To quantify the similarity between \(S_{m}^{T}(t)\) and \(S_{m}^{P}(t)\), we calculate \(a_{m}(t)=|S_{m}^{T}(t)\cap S_{m}^{P}(t)|/\eta\) and show the results in Figure 6 with different \(\eta\)s. In the figure, we draw the ranges of the y-axis to be the same on purpose to make comparison easier. This figure has four subfigures. In each subfigure, the x-axis is a MECS index \(m\) and the y-axis is \(a_{m}\). Each subfigure shows the distribution of \(a_{m}(t)\) in each MECS as a box plot. Figure 5: Distribution of the prediction error for each service \(k\in\mathcal{K}\) in different MECSs. MECS 4 is located at the center of the grid, and MECS 0 is located at the bottom left corner of the grid. Each subfigure shows the distribution of \(\delta_{m,k}(t)\) in each MECS as a box plot.