## 6 Numerical Results
### Analysis of Optimal Cache Deployments
Furthermore, we see that for the case where \(h_{metro}\leq 4.5\), the cache deployment that yields the minimum \(RO_{avg}/req\) is the one with higher \(n_{access}\) (i.e., 32) and relatively small caches (between 1000 and 5000 GB), whereas when \(h_{metro}\) is distant from end-users, the preferred cache deployment is the one with eight caches of relatively high storage capacity (ranging between 3500 and 20,000 GB). This is due to the fact that, when the location of the MCEN cache is near to end-users, it becomes more advantageous to deploy more storage capacity at the MCEN cache, thereby pulling many items from the DC, whereas when the MCEN location is farther, it is preferable to deploy the storage capacity at AMEN caches while utilizing a low number of caches (\(n_{access}=8\)), thereby pulling a high number of items from the DC into the AMEN caches. In conclusion, the dimensions of the metro network impact the optimal cache deployment strategy. Furthermore, we notice that for a given value of \(h_{metro}\), the cache deployment for lower values of \(\alpha\) reveals AMEN caches of smaller size with respect to when the value of \(\alpha\) is higher. For example, for \(h_{metro}=3.5\), the sizes of the AMEN caches in the case when 8 and 32 AMEN caches are utilized for \(\alpha=0.8\) are 3000 GB and 900 GB, respectively. For \(\alpha=1\), the caches are of greater size--7000 GB and 2000 GB. Figure 5: \(RO_{avg}/req\) for cache deployments with different numbers of AMEN caches utilized in the network for different values of \(\alpha\) with respect to \(h_{metro}\), the average hop-distance between the MCEN cache and end-users. Figure 6: The amounts of storage capacity for the AMEN caches corresponding to the best cache deployment when utilizing 8, 16 or 32 caches for different values of \(\alpha\) and \(h_{metro}\). GB, respectively. This is because for higher values of \(\alpha\), the popularity distribution is characterized by a shorter head (as shown in Figure 2), meaning that popular items are more popular with respect to the case where the value of \(\alpha\) is lower, and thus it becomes more significant to deploy larger caches close to end-users so as to deliver the most popular items from a close location. This shows that the optimal cache deployment (number of caches and their storage capacity) which guarantees the minimal utilization of the network resource occupation hugely depends on the existence and the location of a centralized cache in the metro segment of the network, and on the content catalog characteristics (size and popularity distribution) of the content provider. In Table 5 we summarize the important takeaways on the deployment of caches in different network levels. 3.2 Effects of the Location of the Metro Edge Cache (\(h_{metro}\)) and the Available Storage Capacity (\(T\))  
Finally, we investigate how useful it is to increase the total amount of storage capacity \(T\) in the network. We study the variation of the \(RO_{avg}/req\) as a function of \(T\) and for different values of \(\alpha\) and \(h_{metro}\), as shown in Figure 7.  
We observe that when \(T\) increases and becomes much higher than the size of the content catalog (in this case 160,000 GB), the curve asymptotically saturates, revealing only a slight improvement in \(RO_{avg}/req\). In other words, once popular items are already stored in network caches, it becomes very costly to further improve the \(RO_{avg}/req\), as it requires allocating huge amounts of storage capacity in order to store many unpopular items. However, when the MCEN cache is not utilized (due to its location being far from end-users), increasing \(T\) leads to a significant decrease in the \(RO_{avg}/req\) per VoD request, as it allows AMEN caches to store more popular items. More specifically, for \(h_{metro}\) = 3.5 and for all values of \(\alpha\), the results show and confirm that utilizing 32 AMEN caches has the best performance (i.e., leads to the lowest \(RO_{avg}/req\)), independently of