# A Comparative Analysis of Wi-Fi Offloading and Cooperation in Small-Cell Network
## 3 System Model
### Transmission Model for Wi-Fi Offloading in Small Cells  
Figure 1b shows the transmission model for the Wi-Fi offloading strategy in the SC network. Here the SBS and Wi-Fi AP are co-located with licensed band for cellular UEs and unlicensed band for Wi-Fi only (non-cellular) UEs. In this model, we assume that the Wi-Fi APs are either user deployed, or they might be deployed by another entity such as city council. Thus, the Wi-Fi operation does not incur any cost to the cellular operator. For convenient distinction, we refer to the Wi-Fi only UEs as Wi-Fi devices. Cellular UEs can also be offloaded to Wi-Fi (LTE-U). As shown in Figure 1b, three SC bounded with dashed lines are present in a MC. Each Wi-Fi BS has its own coverage region shown with a dotted line. UE \(j\) connects with a single SBS \(i\) using LTE downlink shown with a solid line to meet its data rate demand. In the case that it is unable to meet rate threshold then it is offloaded to LTE-U link of SBS \(i+2\) because it lies within the Wi-Fi range of that SBS. Similarly, UE \(j+1\) connects with a single SBS \(i+1\) using LTE downlink and in addition it can be offloaded to the Wi-Fi AP co-located with the SBS \(i+1\) to satisfy its data rate requirement. This Wi-Fi AP is also referred to as LTE-U SBS. In the SC network with Wi-Fi offloading, a joint problem involving user association with SBS, RB allocation, power allocation and leftover capacity is analyzed. The variables that indicate the UE's association with LTE SBS and LTE-U SBS are  
\[y_{ij}^{k}= \begin{cases}1,&\text{if a UE $j$ is associated with LTE SBS $i^{th}$ on a RB $k$},\\ 0,&\text{otherwise}\end{cases}  \tag{5}\]\[z_{ij}= \begin{cases}1,&\text{if a UE $j$ is associated with LTE-U of SBS $i$},\\ 0,&\text{otherwise}.\end{cases}  \tag{6}\]  
The power allocation variable \(P_{ij}^{k}\) can be allocated a value between 0 and \(P_{i}^{max}\). Dimensions of association matrices are \(\mathbf{Y}[M\times N\times K]\), \(\mathbf{Z}[M\times N]\) and the dimensions of the power allocation matrix is \(\mathbf{P}[M\times N\times K]\). Data rate \(R_{ij}^{U}\) that can be achieved by a UE \(j\) from a Wi-Fi AP \(i\) is one of our controlling variables. We assume that the leftover capacity of each Wi-Fi AP is known to MBS. The leftover capacity of an AP is defined as \(L_{i}=R_{AP_{i}}-\sum_{w\in W}r_{i}^{w}\) as in [32]. Here \(R_{AP_{i}}\) is the total capacity of a Wi-Fi AP \(i\) and \(r_{i}^{w}\) is the data rate achieved by a Wi-Fi device \(w\). Hence, the leftover capacity is equal to the rate that can be achieved by a UE (on LTE-U link) after all the Wi-Fi devices have satisfied their required data rates. The leftover capacity typically depends on the number of Wi-Fi devices and their average data rate requirement. The leftover capacity of each user is expressed as \(R_{j}^{U}\). As the number of Wi-Fi devices increase, the leftover capacity reduces and vice-versa. SINR of a cellular UE using licensed band can be represented as \(\gamma_{ij}^{LTE,k}\) as:  
\[\gamma_{ij}^{LTE,k}=\frac{P_{ij}^{k}G_{ij}}{\sigma^{2}+\sum_{i^{\prime}\in I,i^{\prime}\
eq i}P_{ij}^{k}G_{ij}}, \tag{7}\]  
where \(P_{ij}^{k}\) is the power transmitted from a SBS \(i\) to a UE \(j\) on a RB \(k\) and \(G_{ij}\) is the channel gain experienced by a UE \(j\) from a SBS \(i\). We use normalized rates in this model. Moreover, we also use Rayleigh fading channels in our model. The required power for a UE \(j\) is a product of power received by using a RB \(k\) of a SBS \(i\) and gain \(G_{ij}\) from a SBS \(i\) to a UE \(j\). \(\sigma^{2}\) is the noise power and \(\sum_{i^{\prime}\in I,i^{\prime}\
eq i}P_{ij}^{k}G_{ij}\) is interference power received from all the remaining SBSs in set \(I\).