## 5 Numerical Results
The values of normalized RMSE and AMAE of the SPRPA and APRPA algorithms for the five NFVI-PoPs are reported in Table 4 to show the prediction effectiveness in the case of \(S\) equal to \(1\), \(3\) and \(6\) and QoS degradation costs \(C_{QoS}\) equal to \(1.85\cdot 10^{-5}\)\(\$/\)Gb, \(1.85\cdot 10^{-4}\)\(\$/\)Gb, \(5.56\cdot 10^{-4}\)\(\$/\)Gb and \(9.26\cdot 10^{-4}\)\(\$/\)Gb. Figure 5: Required and allocated capacities over time to a Firewall VNFI in the NFVI-PoP\({}_{1}\). Monitoring and Allocation Times \(T_{m}\) and \(T_{a}\) are chosen equal to \(10\) min and \(60\) min, respectively. Four allocation curves are reported: The one based on RMSE and the ones based on AMAE with \(C_{QoS}=9.30\cdot 10^{-6}\)\(\$/\)Gb, \(C_{QoS}=9.30\cdot 10^{-5}\)\(\$/\)Gb and \(C_{QoS}=9.30\cdot 10^{-4}\)\(\$/\)Gb.  
We also report the two total resource allocation and QoS degradation cost components in Figures 7 and 8, respectively. We show the results in the case of interest in which the QoS degradation cost \(C_{QoS}\) is higher than the average allocation cost \(\tilde{C}_{RA}\).  
We can observe that for both SPRPA and APRPA solutions a decrease in \(T_{a}\) leads to lower total costs. As a matter of example, when \(C_{QoS}\) equals \(7.41*10^{-4}\) $/Gb and the APRPA solution is applied, the total cost equals 219 $, 255 $ and 266 $ for \(T_{a}\) equal to 10 min, 30 min and 60 min, respectively. The reason of this cost decrease is consequence of the possibility to allocate cloud resources so as to appropriately follow the required processing capacity. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **SPRPA (S = 1)** & **APRPA (S = 1)** & **SPRPA (S = 3)** & **APRPA (S = 6)** & **SPRPA (S = 6)** \\ \hline \(C_{QoS}=1.85\cdot 10^{-5}\) & \(2.29\cdot 10^{-1}\) & \(3.04\cdot 10^{-3}\) & \(3.23\cdot 10^{-1}\) & \(4.45\cdot 10^{-3}\) & \(3.67\cdot 10^{-1}\) & \(5.09\cdot 10^{-3}\) \\ \hline \(C_{QoS}=1.80\cdot 10^{-4}\) & \(2.29\cdot 10^{-1}\) & \(7.07\cdot 10^{-3}\) & \(3.23\cdot 10^{-1}\) & \(1.09\cdot 10^{-2}\) & \(3.67\cdot 10^{-1}\) & \(1.25\cdot 10^{-2}\) \\ \hline \(C_{QoS}=5.56\cdot 10^{-4}\) & \(2.29\cdot 10^{-1}\) & \(7.55\cdot 10^{-3}\) & \(3.23\cdot 10^{-1}\) & \(1.09\cdot 10^{-2}\) & \(3.67\cdot 10^{-1}\) & \(1.43\cdot 10^{-2}\) \\ \hline \(C_{QoS}=9.26\cdot 10^{-4}\) & \(2.29\cdot 10^{-1}\) & \(7.65\cdot 10^{-3}\) & \(3.23\cdot 10^{-1}\) & \(1.11\cdot 10^{-2}\) & \(3.67\cdot 10^{-1}\) & \(1.35\cdot 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Values of normalized RMSE and AMAE of the APRPA and APRPA algorithms, respectively, in the case of \(S\) equal to 1, 3 and 6 and QoS degradation costs \(C_{QoS}\) equal to \(1.85*10^{-5}\) $/Gb, \(1.85*10^{-4}\) $/Gb, \(5.56*10^{-4}\) $/Gb and \(9.26*10^{-4}\) $/Gb. Figure 6: Comparison of the SPRPA and APRPA algorithms in terms of total cost as a function of the QoS degradation cost \(C_{QoS}\) when the MI duration \(T_{m}\) equals 10 min and the allocation time \(T_{a}\) is chosen to be 10 min (\(S=1\)), 30 min (\(S=3\)) and 60 min (\(S=6\)). Figure 6 again confirms how the minimization of a loss function taking into account both the allocation and QoS degradation costs allows for better performance in term of total network cost. For instance when \(T_{a}\) and \(C_{QoS}\) are equal to 60 min and \(4.63*10^{-4}\$/\text{Gb}\), respectively, the total network cost is equal to 425 $ and 248 $, respectively, for SPRPA and APRPA. The only minimization of the RMSE may lead to worse performance because of the error sign that differently impacts on the total network cost when the allocation and QoS degradation costs are different.