# SDN-Based Routing for Backhauling in Ultra-Dense Networks
## 4 System Model  
We consider here a heterogeneous network where micro-, pico-, and femto-cells (generally referred as small cells or access points in what follows) are densely and randomly deployed in the macrocell area. The total number of cells is \(C\) (i.e., small cells plus macrocell). Each cell base station (BS) provides access to UEs connected with it, hence, the cell BS has a certain amount of traffic generated by the associated-UEs that must be routed in the backhaul network. Due to differences in cell coverage areas the amount of traffic generated by different cells can significantly vary. Following the SDN principle, we consider two types of traffic flow: _data_ and _control_. Indeed, cell coordination can be split in control plane coordination, which allows optimization of network parameters (e.g., interference management, load balancing, mobility management), and data plane coordination, which allows data exchange and resource scheduling. The two flows are routed following different principles, and control traffic has higher priority.  
Each cell BS has a backhaul connection with the CN and/or with one/more other cell BSs by means of heterogeneous links with different communication capacity. In particular, macro- and micro-cell placement usually follows operator's network planning; hence, macro- and micro-cellshave high-capacity links toward the CN and among them. Conversely, pico and femto-cells can be placed anywhere by operators and end users following a random deployment. Some small-cell BSs can have direct backhaul links toward the CN while others use multi-hop connections toward the CN, exploiting other cells as forwarding elements. The UDN scenario is represented in Figure 0(a). The UDN backhaul network can be seen from an SDN perspective as a network where the CN and the cell BSs are both ingress/egress nodes and forwarding elements. Indeed, BSs generate and receive traffic, and at the same time forward traffic of other neighbor BSs. The UDN backhaul from an SDN perspective is represented in Figure 0(b). Decisions on paths for data forwarding are taken by the SDN controller that is placed in the CN. The SDN controller implements routing policies that are indicated by the application plane, which in this case requires different treatments for control and data traffic.  
From the figure we can see that the backhaul network can be modeled with an undirected graph \(G\) where vertexes are cell BSs plus the CN and edges are the backhaul links. In particular, the set of vertexes \(\mathcal{V}\) has cardinality \(V=C+1\), and the set of edges \(\mathcal{E}\) has cardinality \(E\). We define a capacity matrix \(\mathbf{C}\) whose element \(c_{v,w}\) is the capacity of the edge connecting vertexes \(v\) and \(w\), \(e_{v,w}\). If the capacity is zero it means that the edge does not exist. Vertex \(v\) has a demand of data and control traffic to route in the backhaul network. We define two traffic matrices \(\mathbf{D}^{D}\) and \(\mathbf{D}^{C}\) that represent the data- and control-traffic demands from the sources to the destinations. In particular, the element \(d^{X}_{v,w}\), with \(X=D\) or \(C\) (\(D\) for data traffic and \(C\) for control traffic), represents the amount of data/control traffic from ingress node \(v\) to egress node \(w\). Control traffic generated by cells is directed toward the CN, hence assuming that the CN is the first vertex, we have that \(d^{C}_{v,w}=0\)\(\forall v,w\
eq 1\), while the control traffic generated by the CN can be directed to any cell.