## 6 Numerical Results
### Analysis of Optimal Cache Deployments  
Utilizing the analytical model, in this section we investigate how the optimal distribution of the available storage capacity changes while varying:  
* The location of the MCEN cache, i.e., hop-distance of users from MCEN, \(h_{metro}\);
* The content catalog popularity skew parameter \(\alpha\);
* The total allowed storage capacity \(T\). 3.1 The Effects of the Location of Metro Edge Cache (\(h_{metro}\)) and the Popularity Skew Parameter (\(\alpha\))  
We now focus on the impact of the location of the MCEN, represented by \(h_{metro}\) and the skew parameter \(\alpha\). In Figure 5 we show the \(RO_{avg}/req\) utilized with \(n_{access}=8\), 16 or 32, while changing hop-distance of the MCEN cache \(h_{metro}\) in the topology and using four different values of \(\alpha\). In this evaluation we fix \(T\) at 160,000 GB. First, we notice that, for all values of \(\alpha\) (except for \(\alpha=1.1\)), when the MCEN cache is near to end-users (\(h_{metro}\leq 4.5\)), the minimum \(RO_{avg}/req\) is achieved when all the 32 AMEN caches are utilized. However, when the MCEN cache is relatively farther (\(h_{metro}\geq 5.5\)), the cache deployment which yields the minimum \(RO_{avg}/req\) is when only eight AMEN caches are utilized. This demonstrates that utilizing all caches locations in access is not always the optimal cache deployment. For the case where \(\alpha=1.1\), the minimal \(RO_{avg}/req\) is achieved when utilizing 32 AMEN caches. This is because for such a value of \(\alpha\), a very small number of items become extremely popular while moderately-popular items lose their popularity, making it more useful to store the popular items in nearest cache locations and to avoid storing less popular content. On the contrary, for lower values of \(\alpha\), the curve representing the popularity distribution is more skewed (see Figure 2), meaning that a significant number of VoD items have similar popularity, and therefore, it becomes more advantageous to pull more of these VoD items from the server into the MCEN cache. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{3}{c}{Case Study \#1} \\ \hline Cache Deployment & DC & MCEN Cache & AMENs Caches & RO \\ \hline Only MCEN & 0 & 100\% & 0 & 27 \\ \hline \(k-optimized\) & 9\% & 46\% & 45\% & 19 \\ \hline Only AMENs & 42\% & 0 & 58\% & 24.43 \\ \hline \multicolumn{5}{c}{Case Study \#2} \\ \hline Cache Deployment & DC & MCEN Cache & AMENs Caches & RO \\ \hline Only MCEN & 0 & 100\% & 0 & 27 \\ \hline \(k-optimized\) & 4\% & 27\% & 69\% & 16.21 \\ \hline Only AMENs & 27\% & 0 & 73\% & 19.32 \\ \hline \multicolumn{5}{c}{Case Study \#3} \\ \hline Cache Deployment & DC & MCEN Cache & AMENs Caches & RO \\ \hline Only MCEN & 0 & 100\% & 0 & 27 \\ \hline \(k-optimized\) & 5\% & 22\% & 73\% & 12.57 \\ \hline Only AMENs & 22\% & 0 & 78\% & 15.06 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Percentage of requests served from the video server, the metro-core edge node (MCEN) cache and the AMEN caches, and the average resource occupation (RO) for each cache deployment in the three case studies. We now focus on the size of AMEN caches obtained for the optimal cache deployment for different values of \(h_{metro}\) and \(\alpha\). Figure 6 shows the storage capacity in each AMEN cache returned by our formula for \(n_{access}=8\), 16 and 32, and for \(h_{metro}=3.5\), 5.5 and 7.5. When a high number of AMEN caches is utilized (\(n_{access}=32\)), the optimal solution consists of deploying relatively small capacity caches, while when a low number of caches is utilized (\(n_{access}=8\)), larger caches are preferred.