# Millimetre-Wave Backhaul for 5G Networks: Challenges and Solutions
## 5 Framework Design for 5G Mm-Wave Backhaul
### Transmission Path Selection  
Multi-hop relaying is required in mm-wave backhaul to overcome blockage between SBSs or a long distance between communicating SBSs. Typically, the routing problem can be depicted by a graph, with vertices representing SBSs and edges standing for wireless or wired links. [PERSON]'s algorithm and [PERSON]'s algorithm are two typical methods to search for the minimum cost path in a weighted graph. The selection of edge cost is rather flexible. For example, if we set the edge weights to one for all connected links and \(\infty\) for blocked ones, the solution achieves the minimum number of relaying hops. The weight can also be distance for attaining the shortest-path solution and delay for obtaining the minimum-latency solution, _etc._  
Most existing wireless routing methods for mm-wave systems assume single-stream transmission. Unlike in a conventional wireless system, the routing in the proposed framework can establish a number of routes for a single flow, _i.e._, a source may select multiple paths to relay data to the destination as a benefit of hybrid beamforming. Moreover, multiple flows can be transmitted simultaneously between two devices. Here, we introduce an optimization model for graph routing, taking physical layer techniques of hybrid beamforming and full duplex into consideration. Based on the system architecture depicted in Figure 1, we assume that there are \(N\) SBSs in the system denoted by \(\mathcal{B}=\{b_{1},b_{2},\ldots,b_{N}\}\). The number of required flows are \(K\), with \(\{s_{k},d_{k}\}\) representing the source and destination of the \(k\)-th flow, respectively, for \(1\leq k\leq K\). We can then formulate the path selection in mm-wave backhaul by the following optimization problem. Optimization objective: Network capacity is crucial for the system performance. Thus, we choose to maximize the total transmission rate as the optimization objective:  
\[\max\sum_{k}\sum_{\psi\in\mathcal{B},b\
eq s_{k}}r_{s_{k}\to b}^{k} \tag{1}\]where \(r^{k}_{s_{k}\to b}\) is the transmission rate from the source to the next SBS \(b\) of the \(k\)-th flow. The objective can also be QoS [22], by choosing the cost function arguments \(r^{k}_{s_{k}\to b}\) based on some appropriate QoS metrics. System constraints: Related to the physical layer techniques in mm-wave backhauling, the constraints of the routing problem are listed as follows, which fall into two categories. \[r^{k}_{b_{i}\to b_{j}}\left\{ \begin{array}{ll}=0,&\text{if }p^{k}_{b_{i}\to b_{j}}=0\\ >0,&\text{if }p^{k}_{b_{i}\to b_{j}}=1\end{array} \right.\quad\forall\ b_{i},b_{j}\in \mathcal{B},\ \forall\ k \tag{2a}\] \[p^{k}_{b_{i}\to b_{j}}+p^{k}_{b_{j}\to b_{j}}\leq 1\quad\forall\ b_{i},b_{j}\in \mathcal{B},\forall\ k\] (2b) \[\sum_{k}\sum_{b_{j}}p^{k}_{b_{i}\to b_{j}}\leq f(N_{RF})\quad \forall\ b_{i}\in\mathcal{B},\quad\sum_{k}\sum_{b_{i}}p^{k}_{b_{i}\to b_{j}} \leq f(N_{RF})\quad\forall\ b_{j}\in\mathcal{B}\] (2c) \[\sum_{b_{j}}p^{k}_{s_{k}\to b_{j}}>0,\ \sum_{b_{j}}p^{k}_{b_{i}\to d_{k}}>0\quad \forall\ k\] (2d) \[\sum_{b_{j}}p^{k}_{b_{i}\to s_{k}}=0,\ \sum_{b_{j}}p^{k}_{d_{k}\to b_{j}}=0\quad\forall\ k\] (2e) \[\sum_{b_{j}}r^{k}_{b_{i}\to b_{j}}=\sum_{b_{j}}r^{k}_{b_{i}\to b_{j}} \quad\forall\ b_{l}\in\mathcal{B},\ b_{l}\
eq s_{k},\ b_{l}\
eq d_{k},\ \forall\ k\] (2f) \[\sum_{k}r^{k}_{b_{i}\to b_{j}}\leq C_{b_{i}\to b_{j}}\quad \forall\ b_{i},b_{j}\in\mathcal{B} \tag{2g}\]
* Device and channel constraint: We define a binary variable \(p^{k}_{b_{i}\to b_{j}}\) to indicate that \(b_{i}\) sends data to \(b_{j}\) of the \(k\)-th flow if it equals one, and otherwise, it is zero. The relation between \(p^{k}_{b_{i}\to b_{j}}\) and transmission rate \(r^{k}_{b_{i}\to b_{j}}\) is shown as Constraint (2a). Transmission cycles should be avoided in the routing, which means that \(p^{k}_{b_{i}\to b_{j}}\) and \(p^{k}_{b_{j}\to b_{j}}\) cannot be both activated to one (Constraint (2b)).