## 3 System Framework
### Problem Formulation
#### 3.3.1 Problem 1  
The objective function in (3) represents the total number of cache hits of the edge cache network. In (4) and (5), the decision variables (\(c_{m_{n}}^{q_{f}}\) and \(c_{m_{k}}^{q_{f}}\)) are constrained to 1 only if \(f_{\text{v}}\) is cached on the MAN server \(n\) or \(k\) respectively. The constraints in (6) and (7) set \(\psi_{m_{n}}^{\hat{q}_{f_{\text{v}}}}\) and \(\psi_{m_{k}}^{\hat{q}_{f_{\text{v}}}}\) to one when the file \(f_{\text{v}}\) is cached in a higher bitrate presentation \(\hat{q}\) which requires transcoding to the requested bitrate presentation \(q\). The constraint in (8) allows content to be fetched from only one place at a time. In (9), a cache capacity violation constraint is set for the system. An upper bound on the consumed processing power needed to transcode a video from a higher presentation to a requested one is set in (10).  
\[\text{Maximize }H_{\text{C}}=\sum_{n\in N}\sum_{q_{f_{\text{v}}}\in R_{n}}(W_{m _{n}}^{q_{f_{\text{v}}}}+W_{m_{k}}^{q_{f_{  
and cached at MAN server \(n\) should be transmitted. The condition of reaching the threshold, i.e., \(f_{WT_{p}}\geq\lambda_{v}\) gives \(d_{m_{n}}^{f_{p}}=1\), whereas unsatisfied threshold condition (\(f_{WT_{p}}<\lambda_{v}\)) gives \(d_{m_{n}}^{f_{p}}=0\). Assuming the backhaul bandwidth is divided into \(\beta\) subcarriers and each subcarrier is shared by multiple users in a time-division manner, then the transmission rate \(R\) for the \(k\)-th user in the \(n\)-th cell on the \(m\)-th subchannel is given by Equation (11), where \(\alpha_{k,n}^{m}\) denotes the time-sharing factor of user \(k\) in the MAN cell \(n\) on subcarrier \(m\), \(\chi_{k,n}^{m}\) is the channel access indicator (the decision variable). If the \(n\)-th MAN server serves the \(k\)-th user, \(\chi_{k,n}^{m}=1\), and zero otherwise.  
Minimize  
\[R_{k,n}^{m}=\sum_{m=1}^{M}\alpha_{k,n}^{m}\frac{B}{N}\log_{2}\left(1+\frac{ \chi_{k,n}^{m}\cdot SINR_{k,n}^{m}}{\alpha_{k,n}^{m}\frac{B}{N}}\right) \tag{11}\]  
Subject to  
\[f_{WT_{p}}\geq\lambda_{v},\ \forall f\in F \tag{12}\]  
\[\sum_{k\in K,\ n\in N}\chi_{k,n}^{m}=1,\ \ \forall n\in N,n\
eq k \tag{13}\]  
The second optimization objective is to maximize the efficient utilization of the backhaul resources by minimizing the content transmission rate when the view threshold is not reached. In (11), the objective function is to minimize a user's channel utility. The constraint in (12) controls the decision variable \(d_{m_{n}}^{f_{p}}\), which instantiates the final channel access indicator variable \(\chi_{k,n}^{m}\). The constraint in (13) sets all channel access indicators to one when all the view-time thresholds are met. The formulated problems are NP-complete ILP problems, hence obtaining an optimal solution in polynomial time is infeasible for the reason that in a clustered cooperative environment, knowledge of all the possible video requests is required to solve the ILP and that is unattainable. Therefore, to achieve our objective of maximizing the cache hit rate of the edge caching network, we design a distributed part of media (DPoM) caching method which follows a multiple knapsack optimization, considering the MAN servers in a cluster as knapsacks with capacities equivalent to the cache sizes of the MAN servers, the profit of caching, i.e., users served from the edge, as the item's value, and the cache size requirements for each content file as the weights of the items. Heuristically, the method randomizes the selection of cacheable contents from the entire catalog by dynamically caching the first parts of contents having maximum popularity at every content drawing.