# Cross-Layer Optimization for Heterogeneous MU-MIMO/OFDMA Networks
## 4 Optimization Framework
### Decision Variables  
#### 4.1.1 Data Rate  
We summarize the decision variables of the proposed model in Table 1. Let us begin with the model for the rate of flow. In our framework, the rate of flow \(f\), denoted as \(r_{f}\), is defined as the achievable end-to-end data rate of flow \(f\), and similarly, \(r_{lf}\) denotes the rate of flow \(f\) on link \(l\). Then, from the property of flow conservation [3; 6], we have the following constraints:  
\[\sum_{l\in L_{i}^{\text{out}}}r_{lf}=r_{f}, \forall f,i=f^{\text{src}} \tag{1}\] \[\sum_{l\in L_{i}^{\text{out}}}r_{lf}=\sum_{l\in L_{i}^{\text{in}} }r_{lf}, \forall f,i\
eq f^{\text{src}},f^{\text{dst}}\] (2) \[\sum_{l\in L_{i}^{\text{in}}}r_{lf}=r_{f}, \forall f,i=f^{\text{dst}} \tag{3}\]  
where \(L_{i}^{\text{out}}\) and \(L_{i}^{\text{in}}\) are the sets of outgoing links and incoming links for node \(i\), respectively. Please note that constraint (3) is automatically met if constraints (1) and (2) are satisfied [3; 6]. \begin{table}
\begin{tabular}{c c} \hline \hline
**Variable** & **Description** \\ \hline \(r_{f}\) & rate of flow \(f\) \\ \(r_{lf}\) & rate of flow \(f\) on link \(l\) \\ \(c_{lf}\) & capacity of flow \(f\) on link \(l\) \\ \(2r_{lf}\) & indicates whether \(s\) spatial streams are allocated to flow \(f\) on link \(l\) at time slot \(t\) \\ \(z_{lf}\) & indicates whether subchannel \(k\) is allocated to flow \(f\) on link \(l\) at time slot \(t\) \\ \(z_{lf}\) & indicates whether subchannel \(k\) is allocated to flow \(f\) on link \(l\) at time slot \(t\) \\ \(m_{lf}\), \(\sigma_{lf}\) & indicate whether flow \(f\) on link \(l\) at time slot \(t\) is activated in MU-MIMO/OFDMA mode \\ \(m_{sr}^{\text{src}}\), \(m_{sr}^{\text{src}}\) & indicate whether node \(i\) is in MU-MIMO transmission/reception mode at time slot \(t\) \\ \(\sigma_{a}^{\text{in}}\), \(\sigma_{b}^{\text{in}}\) & indicate whether node \(i\) is in OFDMA transmission/reception mode at time slot \(t\) \\ \(m_{sr}^{\text{src}}\), \(m_{sr}^{\text{src}}\) & indicate whether node \(i\) is in MU-MIMO transmission/reception mode in subchannel \(k\) at time slot \(t\) \\ \(\sigma_{b}^{\text{out}}\), \(\sigma_{b}^{\text{out}}\) & indicate whether node \(i\) is in OFDMA transmission/reception mode in subchannel \(k\) at time slot \(t\) \\ \(s_{bkr}^{\text{src}}\), \(s_{bkr}^{\text{src}}\) & indicate the number of outgoing/incoming spatial streams of node \(i\) in channel \(k\) at time slot \(t\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Decision Variables. Figure 1: Comparison of OFDMA, MU-MIMO, and joint OFDMA and MU-MIMO operation, where \(f\), \(s\) and \(b\) represent the index of the flow, the number of spatial streams and the bandwidth, respectively. Since before reaching the destination, the data in flow \(f\) may pass through multiple links with different capacities, we need to consider the capacity of each individual link. As mentioned earlier, the capacity is related to the number of spatial streams and the bandwidth, and these may vary for each time slot depending on the scheduling result. To take these aspects into consideration, we define two variables, \(z_{llfs}\) and \(c_{llfs}\): for flow \(f\) on link \(l\) at time slot \(t\), the binary variable \(z_{llfs}\) indicates whether or not the number of allocated spatial streams is \(s\), and \(c_{llfs}\) is a continuous variable that denotes the corresponding capacity. Then, because the value of \(r_{ll}\) cannot exceed the average capacity of link \(l\) for flow \(f\) over the whole time slot, we can impose the following constraint:  
\[r_{lf}\leq\frac{1}{T}\sum_{t,s}c_{llfs},\hskip 28.452756 pt\forall l,f. \tag{4}\]  
In addition, based on the definition of \(c_{llfs}\), we have:  
\[c_{llfs}=b_{llf}\cdot z_{llfs},\hskip 28.452756 pt\forall l,t,f,s \tag{5}\]  
where \(b_{llf}\) is the bandwidth allocated to flow \(f\) on link \(l\) at time slot \(t\), which is computed as the total number of assigned subchannels.