## 4 Layered SDN-Based Optimization Framework
### Operator Resource Constraints  
There are two potential problems with solving (2): (1) the allocation of network resources at the level of granularity of individual eNBs may result in scalability problems; and (2) without any long-term constraints, some operators may hoard backhaul resources. In order to create multiple layers to distribute the decision-making processes, we rewrite the maximization in (2) by introducing variables that for the sake of solving (2), are slack variables. As we will see, the additional variables represent actual network decisions in the distributed and time-decomposed implementation of the centralized scheduler. \begin{table}
\begin{tabular}{l c c} \hline \hline  & & **Values** \\ \hline
**Parameter** & **Notation** & **(For Eval. in Section 5)** \\ \hline \multicolumn{3}{c}{Backhaul Netw. Architecture} \\ \hline \# of Operators (indexed \(o=1,\ldots,O\)) & \(O\) & 2 \\ \# of GWs per oper. \(o\) & \(|\mathcal{G}_{o}|\) & 3 \\ \# of eNBs per GW \(g\) & \(|\mathcal{N}_{g}|\) & 10 \\ Total Backhaul Cap. (Mbps) & \(Z\) & 20 \\ Operator Backhaul Cap. (Mbps) & \(Z_{o}\) & 10 \\ eNB-to-GW RTT (ms) & \(\tau_{\Omega}^{G}\) & 1 \\ GW to Operator RTT (ms) & \(\tau_{\Omega}^{G}\) & 100 \\ Operat. to SDN Orch. RTT (s) & \(\tau_{\Omega}^{S}\) & 1 \\ \hline \multicolumn{3}{c}{Resource Allocations} \\ \hline Cap. alloc. to Oper. \(o\) & \(x_{o}\) & \\ Vector of Oper. alloc. & \(\mathbf{x}=\{x_{1},\ldots,x_{O}\}\) & \\ Cap. alloc. to GW \(g\) & \(y_{g}\) & \\ Vector of alloc. to GWs at Op. \(o\) & \(\mathbf{y}_{o}=\{y_{g}:g\in\mathcal{G}_{o}\}\) & \\ Cap. alloc. to eNB \(n\) & \(z_{n}\) & \\ Vector of alloc. to eNBs at GW \(g\) & \(\mathbf{z}_{g}=\{z_{n}:n\in\mathcal{N}_{g}\}\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of model notations. In particular, let us denote by \(x_{o}\) the portion of the wireless service rate \(Z\) that is distributed to operator \(o\) and let \(\mathbf{x}=\{x_{1},x_{2},\ldots,x_{O}\}\) denote the vector of allocated operator service rates. Each operator \(o\), \(o=1,\ldots,O\), redistributes the resources, by giving a portion \(y_{g}\) of \(x_{o}\) to each of its GWs \(g\in\mathcal{G}_{o}\), whereby we denote \(\mathbf{y}_{o}=\{y_{g}:g\in\mathcal{G}_{o}\}\) for the vector of GW rate allocations of operator \(o\). In turn, each GW \(g\) redistributes the resources, by giving a portion \(z_{n}\) of \(y_{g}\) to each of its eNBs \(n\in\mathcal{N}_{g}\), whereby we denote \(\mathbf{z}_{g}=\{z_{n}:n\in\mathcal{N}_{g}\}\). If all these assignments could happen at the same timescale indexed by \(t\), distributing the constraints at each layer, the optimization could be solved as follows:  
\[\max_{\mathbf{x}}\sum_{o=1}^{O}\mathcal{U}_{o}^{\star}\left(x_{o};t\right)\ \ \text{s.t.}\sum_{o=1}^{O}x_{o}\leq Z \tag{3}\]  
with \(\mathcal{U}_{o}^{\star}\left(x_{o};t\right)\) being the optimal value of the subproblem:  
\[\max_{\mathbf{y}_{o}}\sum_{g\in\mathcal{G}_{o}}\mathcal{U}_{g}^{\star}\left(y_{g} ;t\right)\ \ \text{s.t.}\sum_{g\in\mathcal{G}_{o}}y_{g}\leq x_{o} \tag{4}\]  
and \(\mathcal{U}_{g}^{\star}\left(y_{g};t\right)\) being the optimal value of  
\[\max_{\mathbf{z}_{g}}\sum_{n\in\mathcal{N}_{g}}\!\!Q_{n}[t]z_{n}\ \text{s.t.}\sum_{n\in \mathcal{N}_{g}}\!\!z_{n}\leq y_{g},\ 0\leq z_{n}\leq Q_{n}[t]\ \forall n\in \mathcal{N}_{g}. \tag{5}\]  
It is important, however, to remark that the allocation of \(\mathbf{x}\) to solve (3) needs to respect an \"economic\" constraint across the operators that defines a contractual service obligation and prevents any operator from gaming the system (i.e., consistently acquiring more resources than what it paid for). This constraint on the long-run average of the decisions \(\mathbf{x}\) is:  
\[\limsup_{\tau\to\infty}\frac{1}{\tau}\sum_{t=0}^{\tau-1}x_{o}[t]\leq Z_{o}, \tag{6}\]  
where for consistency of the problem, it is necessary to have \(\sum_{o=1}^{O}Z_{o}\leq Z\).