# Utility-Based Wireless Routing Algorithm for Massive MIMO Heterogeneous Networks
## 3 Utility-Based Wireless Backhaul Routing
### System Model and Notations  
The system is modeled as follows. We denote with \(\mathbf{BS}=\{BS_{1},BS_{2},BS_{3},\ldots,BS_{I}\}\), the set of BSs, where \(I\) is the number of BSs. Each BS has a different number of UE that stays within its coverage and receives its service using conventional radio interface, such as 4G. These user devices mainly require a large amount of downlink traffic for applications such as web-browsing, real-time/non-real-time video playing, and location-based information services. \(BS_{1}\) represents the MBS that connects to the core network through a wired link and provides other BSs with internet access through wireless massive MIMO antennas. Without losing generality, each BS is indexed in the increasing order of its distance to \(BS_{1}\). In other words, for all \(i>j,\,BS_{i}\) is more distant to \(BS_{1}\) than \(BS_{j}\). Next, \(N_{i}\) is the number of antennas of each \(BS_{i}\). In order to connect each BS to the backhaul, a tree-shaped topology needs to be established. This topology is described as follows: for each \(BS_{i}\), let \(\hat{i}\) be the index number of its parent node and \(\hat{i}\) represent the set of its child nodes. To avoid inefficient links, the BS's children can only be more distant nodes, that is, for all \(j\in\hat{i},\,j>i\). Next, to uniquely identify each link, each link is indexed with its child-side node. Therefore, link \(l_{i}\) indicates the link between \(BS_{i}\) and \(BS_{i}\) and \(t_{i}\) is the antennas on the child-side node (i.e., uplink), while \(s_{i}\) refers to the present-side antennas (i.e., downlink). Based on the tree-shaped topology, an allocation \(A_{i}=\{t_{i},s_{j}\text{ for all }j\in\hat{i}\}\) indicates each \(l_{i}\)'s antenna allocation, while \(r_{i}^{\Sigma}\) denotes the total uplink data rate of each \(BS_{i}\), which is shared by \(BS_{i}\) and all the nodes belonging to its subtree. Next, the communication channels and data rates are modeled as follows. In general, the channel model of MIMO is represented by a \(M\times K\) matrix, where \(M\) and \(K\) mean the number of transmitting and receiving antennas, respectively. It describes the performance of each subchannel. Nonetheless, considering the complexity of the problem, it is assumed that the antenna gains of the same BS are approximately equal. This approximation is reasonable when the number of antennas is large. Based on [11], Equation (1), the channel model is set as follows:  
\[l_{i}=\min(s_{i},t_{i})\cdot\frac{B}{N_{i}}\cdot\log_{2}(1+\text{SINR}), \tag{1}\]  
where \(B\) is bandwidth. As the focus of this paper is on utility and backhaul routing, the channel quality is set to only Gaussian white noise. Basing on the above assumption, SINR (Signal to Interference plus Noise Ratio) is set as follows:  
\[\frac{\frac{P}{N_{i}}\cdot\alpha}{\sigma}, \tag{2}\]  
where \(P\), \(\alpha\) and \(\sigma\) are the emitting power (W), path loss coefficient, and Gaussian white noise, respectively. Referring to a path loss coefficient model in [26], \(\alpha=10^{-(14+3.67\log d_{i,j})}\), where \(d_{i,j}\) is the distance between \(BS_{i}\) and \(BS_{j}\) in kilometers. Based on [21], Equation (5), each BS's utility model is set as follows:  
\[U_{i}(r)=f_{i}\cdot\frac{\log(r+1)}{\log(10)}, \tag{3}\]  
where \(f_{i}\) is the loading factor presenting the number of \(BS_{i}\)'s associate UE, and \(r\) is the data rate. Figure 1 illustrates the notations, where a wireless backhaul scenario with an MBS (\(BS_{1}\)) and 6 BSs (\(BS_{2}-BS_{7}\)) is considered. Taking the red topology as an example, the link between \(BS_{1}\) and \(BS_{2}\) is \(l_{2}\). \(\hat{\mathbf{2}}\) presents \(\{BS_{1}\}\) and \(\hat{\mathbf{2}}=BS_{4}\), \(A_{2}=\{t_{2},s_{4}\}\). In a similar way, \(l_{6}\) refers to the link between \(BS_{4}\) and \(BS_{6}\), while the link between \(BS_{4}\) and \(BS_{7}\) is \(l_{7}\); \(\hat{\mathbf{4}}\) presents \(BS_{2}\) and \(\hat{\mathbf{4}}=\{BS_{6},BS_{7}\}\), \(A_{4}=\{t_{4},s_{6},s_{7}\}\).