# Cross-Layer Optimization for Heterogeneous MU-MIMO/OFDMA Networks
## 3 Network Model and Assumptions  
In this section, we present a network model and explain the assumptions made in this paper. Consider a heterogeneous wireless network with \(|N|\) nodes and \(|L|\) links, where \(N\) and \(L\) are sets of nodes and links, respectively. Each node \(i\in N\) has different communication capabilities in terms of the number of antennas (denoted as \(S_{i}\)) and the bandwidth (denoted as \(B_{i}\)). In this network, a total bandwidth of \(B^{\max}\) is available, consisting of \(K\) subchannels, each of which has \(B^{\min}\) bandwidth. We consider a total number of flows \(|F|\) in the network, where each flow \(f\in F\) is characterized by its source and destination nodes, denoted as \(f^{\rm src}\) and \(f^{\rm dst}\), respectively. When a data flow \(f\) passes through a link \(l\), it is transmitted at a certain data rate, which is determined by the transmission mode and the allocated resources. More specifically, the number of spatial streams and the allocated bandwidth determine the data rate of flow \(f\) on link \(l\). In this paper, we use a unit data rate in which one data stream with \(B^{\min}\) bandwidth (i.e., one subchannel) corresponds to one unit data rate [3; 4; 7]. Based on this value, the actual data rate is computed as the product of the number of allocated spatial streams and the bandwidth. We assume that power allocation is properly conducted if the bandwidth is allocated. We use a scheduling scheme based on time slots: within a given time slot \(t\) (\(1\leq t\leq T\)), only a subset of links can be active. Uppercase notation is used to represent the model parameters, and lowercase for variables. In addition, for simplicity, it is assumed that the notation for the sets also indicates their size; for example, \(N\) is a set of nodes, and at the same time represents the number of nodes. When a node is in communication, it can operate in one of two modes: OFDMA or MU-MIMO. Initially, we assume that the nodes cannot use both OFDMA and MU-MIMO at the same time, and in Section 4.4, we relax this restriction and show that the proposed model can be easily extended to the case of joint MU-MIMO and OFDMA optimization. If a node uses OFDMA, it uses only a single antenna, meaning that only a single spatial stream can be activated. Conversely, if it uses MU-MIMO, it can exploit multiple spatial streams, but these share the same network bandwidth, which generally requires a sufficient value for the channel bandwidth (e.g., at least 20 MHz [26; 27]). We adopt different minimum bandwidth requirements for OFDMA and MU-MIMO, denoted as \(B^{\text{min}}\) and \(B^{\text{(MIMO-min)}}\) respectively. Nodes can deal with multiple outgoing and incoming streams via OFDMA or MU-MIMO if the total bandwidth and the DoFs consumed by the outgoing/incoming streams do not exceed the given limits. In addition, we assume that streams with different transmission modes cannot be processed together. The illustrations in Figure 1 show three transmission modes: OFDMA, MU-MIMO, and joint OFDMA and MU-MIMO. In each figure, the four blocks on the left show the four subchannels, and \(T\) is set to one. For node 0 with three antennas that wants to transmit frames, Figure 1a shows an example of resource allocation when it chooses OFDMA. We can see that node 0 concurrently transmits three frames: for flow 1, via link \(0\to 1\); for flow 2, via link \(0\to 2\); and for flow 3, via link \(0\to 3\). All three OFDMA transmissions are conducted through a single spatial stream (and hence \(s=1\)), but with different bandwidths: 20 MHz for flows 1 and 2, and 40 MHz for flow 3. Assuming that \(B^{\text{min}}\) is 20, the sum-rate in this case is 4 (\(=1+1+2\)). We now turn to the case of MU-MIMO shown in Figure 1b. In this case, node 0 transmits two frames for flow 2 and flow 3 via links \(0\to 2\) and \(0\to 3\), respectively. Unlike in OFDMA, both frames are transmitted with the same bandwidth (i.e., \(b=80\)), but the numbers of allocated spatial streams are different.