# Cross-Layer Optimization for Heterogeneous MU-MIMO/OFDMA Networks
## 5 Application Examples  
### Settings  
In this section, we demonstrate the feasibility of the proposed model based on two optimization problems. We first consider a sum-rate maximization problem for which the objective is given below, and then discuss max-min fair allocation, in the following section:  
\[\max\ \sum_{f\in F}r_{f}. \tag{47}\]  
The model parameters are given in Table 2. We randomly locate 10 nodes within a space of size 400 \(\times\) 400 (unit: m\({}^{2}\)), and the numbers of antennas (i.e., \(S_{i}\)) and the available bandwidth (i.e., \(B_{i}\)) of the nodes are set randomly, as shown in Table 3. The number of flows is set to two, and the source and destination nodes of each flow are selected randomly from the nodes. The data range and interference range are set to 200 and 300 m, respectively. We use a MATLAB optimization solver on an Intel i9 machine with 16 GB RAM, and before running the solver, we compute \(Q_{lf}\) and apply it to the model to reduce the complexity. For a given network, we perform optimizations for four cases depending on the transmission mode as follows:  
1. Case 1: the nodes selectively use either MU-MIMO or OFDMA. 2. Case 2: the nodes can use both at the same time. 3. Case 3: the nodes use only MU-MIMO. 4. Case 4: the nodes use only OFDMA. Please note that Cases 3 and 4 are added as a baseline, since they could make more sense to current wireless communication systems than Case 1 and Case 2. For example, in the latest Wi-Fi standard, 802.11 ax [1], multi-user transmissions using OFDMA and MU-MIMO, corresponding to Cases 1 and 2, are theoretically available, but it might be difficult for a node to change its transmission mode without any restrictions since their actual access mechanisms are totally different [31]. In addition, for some networks with low-capability devices, it is obviously more practical to consider Case 3 and Case 4. First, we can see that multi-path routing is activated in all cases. In flow 1, node 7 is the source of flow 1 and is supposed to transmit data to nodes 5 and 8 with a unit data rate. In particular, in Case 1, these transmissions are conducted over OFDMA with a total bandwidth of 40 MHz, as shown in Table 4. From the fact that the maximum available bandwidth of node 7 is 40 MHz, as shown in Table 3, we can see that it fully uses OFDMA for these transmissions. Although all the resources allocated to flow 1 are OFDMA in Case 1, the transmissions for flow 2 are allocated to MU-MIMO resources. Links \(4\to 9\) and \(9\to 10\) are assigned two and three spatial streams, respectively. The outcome that all links for the same flow use an identical transmission method is not intentional, but is the result of chance. Recall that the proposed model allows a node to switch between transmission modes in different time slots. We assume that the reason OFDMA is preferred for flow 1 in Case 1 is because most of the nodes on the path of flow 1, i.e., nodes 5, 7 and 8, have only a single antenna. The results for Case 2 are almost the same as those for Case 1, except that more links are allocated for flow 2. This is because a more flexible resource allocation is available in Case 2, which results in an increased number of multi-user transmissions. In Case 1, multi-user transmission occurs only at node 7, whereas in Case 2, multi-user transmissions are used at nodes 3, 5 and 9. More specifically, they differ in terms of the number of spatial streams allocated to each subchannel in OFDMA transmission. For example, nodes 4 and 5 simultaneously transmit data to node 3 over two time slots, using two spatial streams and one spatial stream over different subchannels. As another example, node 9 handles three incoming streams over subchannel 8: one spatial stream from node 4, and two spatial streams from node 3. Since the number of antennas for node 9 is three, this transmission is valid. This flexible resource allocation due to the joint use of MU-MIMO and OFDMA helps  
Figure 3: Resource allocation results for sum-rate maximization.