## 5 Numerical Results
It has been performed as follows:  
* the look-back parameter \(L\) has been choice by studying the partial autocorrelation function (PACF) [36] of the processing capacities of the training set; it has been chosen the first value in which the PACF has a negligible value (\(10^{-2}\));
* the Kernel filter size \(S_{F}\) has been chosen equal to the number of VNFIs executing in each NFVI-PoP;
* the remaining hyperparametrs (the number \(N_{nr}\) of neurons of the LSTM layer, the number \(N_{F}\) of Kernel filters and the batch size \(N_{sz}\)) have been chosen by performing a sensitivity analysis with the KerasTuner software [37]. KerasTuner is an easy-to-use, scalable hyperparameter optimization framework that solves the pain points of hyperparameter search. Easily configure the search space with a define-by-run syntax, then leverage one of the available search algorithms to find the best hyperparameter values for the models. We have chosen the KerasTuner option that performs the hyperparameter optimization with the Hyperband algorithm. We report in Table 3 the values of the optimized hyperparameters for each of the five NFVI-PoPs and a value \(S\) equal to 1. Finally the total number \(N_{ep}\) of epochs is determined by applying a early stopping procedure and using the validation set. The effectiveness of the APRPA algorithm is studied in Figure 5 where we report over time (expressed in multiples of \(T_{m}\)) the required processing capacity and the ones to be allocated to a Firewall VNFI in the NFVI-PoP\({}_{1}\). The allocation time \(T_{a}\) is chosen equal to 60 min that leads to a value of \(S\) equal to 6. We report four allocation curves: (i) the first one when SPRPA is applied and the loss function of the Convolutional/LSTM neural network is characterized by the conventional Root Mean Squared Error (RMSE); (ii) the other three are ones achieved by applying the APRPA solution and characterized by the Asymmetric Mean Absolute Error (AMAE) that appropriately weights the resource allocation and QoS degradation costs. The choice of the core costs and the VNFI type leads to an average allocation cost \(\bar{C}_{RA}\) equal to \(1.86*10^{-5}\,\$/\Gb\). The QoS degradation cost \(C_{QoS}\) is chosen equal to \(9.30*10^{-6}\,\$/\Gb\), \(9.30*10^{-5}\,\$/\Gb\), \(9.30*10^{-4}\,\$/\Gb\) for the three curves, respectively, that is values lower than, equal to and higher than the average allocation cost of the NFVI-PoP\({}_{1}\). From Figure 5 we can remark that: (i) when the QoS degradation cost \(C_{QoS}\) equals the average resource allocation cost we have similar values of allocated capacities for the APRPA and SPRPA \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(N_{nr}\) & \(L\) & \(N_{F}\) & \(S_{F}\) & \(N_{sz}\) \\ \hline
**NFVI-PoP\({}_{1}\)** & 33 & 40 & 8 & 10 & 24 \\ \hline
**NFVI-PoP\({}_{2}\)** & 19 & 40 & 16 & 10 & 24 \\ \hline
**NFVI-PoP\({}_{3}\)** & 38 & 40 & 32 & 10 & 24 \\ \hline
**NFVI-PoP\({}_{4}\)** & 34 & 40 & 32 & 10 & 24 \\ \hline
**NFVI-PoP\({}_{5}\)** & 39 & 40 & 16 & 8 & 24 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Values of Optimized Hyperparameters for each of the five NFVI-PoPs of Figure 4 and a value of \(S\) equal to 1.  
algorithms; (ii) when the QoS degradation cost is lower (higher) than the average allocation cost, we can observe that APRPA solution tends to allocate less (more) processing resources with respect to the SPRPA one. We report in Figure 6 the total network cost as a function of the QoS degradation cost \(C_{QoS}\) for the RMSE and AMAE solutions when the allocation time \(T_{a}\) is chosen equal to \(10\) min, \(30\) min and \(60\) min to which correspond values of the parameter \(S\) equal to \(1\), \(3\) and \(6\), respectively. The total network cost is given by the sum of the costs of the VNFIs executed in all of the NFVI-PoPs.