# Millimetre-Wave Backhaul for 5G Networks: Challenges and Solutions
## 5 Framework Design for 5G Mm-Wave Backhaul
### Scheduling
#### 5.3.1 Optimization Model  
Similar to routing, STDMA scheduling can be modelled by an integer optimization programming problem. Minimizing transmission time: The whole scheduling is divided into several consecutive and non-overlapping time periods, defined as phases. Assume that there are \(T\) phases, and multiple links can transmit simultaneously in each phase. The maximum transmission time of the \(t\)-th phase is defined by its phase time, denoted as \(\delta_{t}\), where \(1\leq t\leq T\). The goal of scheduling is to accomplish all of the transmissions with the minimum time; namely, to achieve:  
\[\min\sum_{t}\delta_{t} \tag{3}\]  
System constraints: Before scheduling, every route should be separated into several single-stream transmissions from the source to the destination, each of which is called a new flow. All new flows are renumbered starting from one. A binary variable \(h^{t}_{m,n}\) is defined to indicate that the \(n\)-th hop of new flow \(m\) is activated to send data in the \(t\)-th phase when it is assigned a value of one, and zero otherwise. The constraints of scheduling are shown as follows. \[\sum_{t}h^{t}_{m,n}=1\quad\forall\ m,\ n \tag{4a}\] \[\sum_{m}\sum_{n}h^{t}_{m,n}\leq N_{RF}\ \left\{ \begin{array}{l} \text{if}\ \{m,n\}\in\mathcal{T}(b_{i})\ (b_{i}\in\mathcal{B}),\ \forall\ t\\ \text{if}\ \{m,n\}\in\mathcal{R}(b_{i})\ (b_{i}\in\mathcal{B}),\ \forall\ t \end{array} \right.\] (4b) \[h^{t}_{m_{1},n_{1}}+h^{t}_{m_{2},n_{2}}\leq 1\quad\forall\ t,\ \forall\ \{m_{1},n_{1},\,m_{2},\,n_{2}\}\in\mathcal{I}\] (4c) \[\sum_{t}h^{t}_{m,n}\delta_{t}\geq d_{m}/r_{m,n}\quad\forall\ m,\ n\] (4d) \[\sum_{t=1}^{t^{*}}h^{t}_{m,\tilde{n}}\geq\sum_{t=1}^{t^{*}}h^{t}_{m, \tilde{n}}\quad\forall\ m,\tilde{n}<\tilde{n},\ \forall\ t^{*}\in[1,T] \tag{4e}\]
* Device and channel constraint: In order to reduce the overhead brought by beam training and switching, we assume that each hop of a new flow would only be transmitted once, as shown in Constraint (4a). In each phase, both the scheduled numbers of incoming links to an SBS and outgoing links from an SBS should not exceed the number of RF chains, while wired links are excluded from such constraints. This constraint could be written as (4b), where \(\mathcal{T}(b_{i})\) and \(\mathcal{R}(b_{i})\) stand for wireless links with SBS \(b_{i}\) as the transmitter and the receiver, respectively. Constraint (4c) shows that if there exists severe interferences between two specific links, e.g., the links corresponding to the \(n_{1}\)-th hop of the \(m_{1}\)-th new flow and the \(n_{2}\)-th hop of the \(m_{2}\)-th new flow, both of the links should not be activated at the same time. Set \(\mathcal{I}\) refers to interference set. If the SINR value of at least one link is below a certain threshold when two links transmit simultaneously, these two links become an element in set \(\mathcal{I}\). * Traffic and time constraint: Constraint (4d) shows that the time demand for the \(n\)-th hop of the \(m\)-th new flow, calculated from the data demands (\(d_{m}\)) and transmission rate (\(r_{m,n}\)) achieved by routing, should be no larger than its scheduled transmission time. For a specific new flow, the \(\widetilde{n}\)-th hop could not be scheduled ahead of the \(\widehat{n}\)-th hop, for \(\widehat{n}<\widetilde{n}\), due to the relaying sequencing, shown in Constraint (4e). The full duplex techniques may enable a group of consecutive hops of a new flow to be transmitted in the same phase, which is exempted from half-duplex-related constraints in conventional scheduling problems. The above optimization problem is a mixed integer nonlinear programming (MINLP) problem, which, however, can be converted into an MILP problem through the reformulation-linearization technique (RLT) [24]. After conducting the optimized routing scheme, Figure 4b illustrates the STDMA scheduling scheme in four independent scenarios based on the path selection result in Figure 4a. There are altogether eight new flows, with each containing two to three hops.