## 5 Numerical Results  
The performance of the APRPA algorithm is compared to the ones of two benchmark algorithms: (i) SPRPA in which the allocation is performed by applying a traditional approach with the objective to exactly predict the needed processing capacity and based on the minimization of the Root Mean Squared Error; (ii) ATPA [5,6] in which the resources are allocated on the basis of a traffic prediction based on both resource allocation and QoS degradation costs. The comparison is carried out for the USAnet network of Figure 4 with 24 switches and 47 links. It is equipped with five NFVI-PoPs, all provided with the same number \(N_{core}=48\) of cores. The core costs are chosen according to real prices [33] and their values are differentiated and equal to \(c_{core}^{(1)}=4.56\cdot 10^{-3}\$/\)h, \(c_{core}^{(2)}=6.40\cdot 10^{-3}\$/\)h, \(c_{core}^{(3)}=8.95\cdot 10^{-3}\$/\)h, \(c_{core}^{(4)}=1.25\cdot 10^{-2}\$/\)h and \(c_{core}^{(5)}=1.75\cdot 10^{-2}\$/\)h for the NFVI-PoP\({}_{1}\), NFVI-PoP\({}_{2}\), NFVI-PoP\({}_{3}\), NFVI-PoP\({}_{4}\) and NFVI-PoP\({}_{5}\), respectively. We assume that \(Q=4\) types of VNFIs can be instantiated. The \(i\)-th type VNFI is executing Firewall (FW), Intrusion Detection System (IDS), Network Address Translator (NAT) and Proxy SFs for \(i\)respectively. We report in Table 1 the processing capacity and the allocated cores for the various operation modes of the VNFIs. The chosen core cost leads to the values \(C_{R,A,j}^{d}\) (\(j\in[1..Q]\)); \(\vec{u}\in\tilde{\mathcal{U}}_{NP}\)) of resource allocation costs reported in Table 2 for the various VNFI types and NFVI-PoPs. The costs are evaluated according to the expression (1). Finally the network links are provided with a capacity of 100 Gbps. The SFC bandwidths are characterized by real traffic values extracted by the database [34] with average bandwidths evaluated in intervals of duration 10 min. The SFCs are offered for each tuple of nodes of the USAnet network and are composed by four SFs executed according to the following order: Firewall, IDS, NAT and PROXY. \begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline
**Number of Allocated Cores** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** \\ \hline
1-st type VNFI (Firewall) & 225 & 450 & 675 & 900 & — & — & — & — \\ \hline
2-nd type VNFI (IDS) & 75 & 150 & 225 & 300 & 375 & 450 & 525 & 600 \\ \hline
3-rd type VNFI (NAT) & 450 & 900 & — & — & — & — & — \\ \hline
4-th type VNFI (PROXY) & 150 & 300 & 450 & 600 & — & — & — & — \\ \hline \end{tabular}
\end{table} 
Table 1: Four VNFI types are considered. The maximum processing capacities \(C_{i}^{p,max}\) (\(i\in[1..Q]\)) are 900 Mbps, 600 Mbps, 900 Mbps and 600 Mbps when the number of cores allocated \(n_{i}^{c}\) (\(i\in[1..Q]\)) equals 4, 8, 2 and 4, respectively. The table reports the processing capacities \(C_{i,j}^{p\prime}\) (\(i\in[1..Q]\)), \(j\in[1..n_{i}^{c}]\) expressed in Mbps when vertical scaling techniques are applied and for the various operation modes of the VNFIs. Figure 4: USAnet network equipped with five NFVI-PoPs. The SFCs are routed by applying the heuristic proposed in [27] aiming at minimizing the cloud resource cost and respecting the processing and bandwidth capacity constraints. The application of the heuristic also allows for the VNFI placement by determining how many VNFIs to use and where to instantiate them. The monitoring procedure is performed by the MAs every \(T_{m}=10\) min. The VIM determines the processing capacities to be allocated to the VNFIs of an NFVI-PoP for a time period \(T_{a}\) 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 high number of hyperparameters makes their optimization not as simple as in [35]. Their choice has been optimized for each of the five NFVI-PoPs and for each value of \(S\).