# A Comparative Analysis of Wi-Fi Offloading and Cooperation in Small-Cell Network
## 4 Comparison between Cooperation and Wi-Fi Offloading in Small-Cell Network
### Cooperation in Small-Cell BSs  
In this subsection, we will formulate a problem of sum-rate maximization in cooperative small cellular network and propose a heuristic to solve it by associating a user to multiple SBSs on RBs and by allocating power. For cooperative SC network, we formulate the problem as a sum-rate maximization (utility maximization) problem. The constraints of QoS, maximum association limit and maximum power of SBS are ensured. The problem is formulated for a time frame \(T\) to be solved by a central controller (MBS in this case). It aims to associate multiple SBSs to a user, allocates RBs and assigns power. We use a single variable for SBS association and RB allocation \(x_{i,j}^{k}\) and \(P_{i,j}^{k}\) is the power allocation variable. The problem formulated is as follows:  
\[\max_{\mathbf{x}\mathbf{P}} \sum_{j\in I}\sum_{i\in I}\sum_{k\in K}x_{ij}^{k}R_{ij}^{k}\] (13) subject to \[\sum_{i\in I}\sum_{k\in K}x_{ij}^{k}R_{ij}^{k}\geq R_{j}^{th} \quad\forall j\in I, \tag{14a}\] \[\sum_{i\in I}x_{ij}^{k}\leq M_{j}\quad\forall j\in J,\forall k\in K\] (14b) \[\sum_{j\in I}\sum_{k\in K}x_{ij}^{k}\leq N_{i}\quad\forall i\in I,\] (14c) \[\sum_{j\in I}x_{ij}^{k}\leq 1\quad\forall k\in K,\forall i\in I,\] (14d) \[\sum_{j\in I}\sum_{k\in K}x_{ij}^{k}P_{ij}^{k}\leq P_{max}\quad \forall i\in I. \tag{14e}\]  
As already discussed, our objective function as given in (13) is to maximize achievable data rate over all SC UEs. Here multiple SBSs serve a UE. Constraint (14a) is the QoS constraint with \(R_{j}^{th}\) as the minimum data rate requirement of each user \(j\). It ensures that the achievable data rate of each user is above a threshold \(R_{j}^{th}\). Constraint (14b) is the cooperative SBS constraint. This constraint dictates the number of SBS that all UEs can be associated with and is less than equal to \(M_{j}\). In case the value of \(M_{j}\) is 1, cooperative communication turns into a non-cooperative scenario i.e., each user can only associate itself to a single SBS. Number of UEs that can be served by each SBS is ensured by constraint (14c). Similarly, constraint (14d) ensures that a RB cannot be reused by a SBS. Lastly, constraint (14e) ensures that the maximum power constraint of each SBS is not violated. Our model of cooperative communication is a mixed integer non-linear programming (MINLP). As illustrated in (13), our objective function is joint resource allocation problem with controlling variables \(P_{ij}^{k}\) and \(x_{ij}^{k}\). \(P_{ij}^{k}\) is a continuous variable whereas \(x_{ij}^{k}\) is a binary variable. Our objective function is a non-convex function jointly as well as individually with respect to each variable [33] and hence it is an NP hard problem and is hard to solvein polynomial time. Owing to the complexity of the problem, we propose a heuristic algorithm to solve the joint problem in polynomial time. Sequential Cooperative Rate Enhancement (SCRE) Algorithm for Small-Cell Network  
We propose a SCRE algorithm to solve the joint problem of SBS association, RB allocation and power allocation. SCRE is a centralized algorithm and is run on frame-by-frame basis. The MBS solves the problem for a time frame \(T\) at the start of the time frame. It assumes to be aware of the quasi-static channel state information for each SBS \(i\) to UE \(j\) and central node is aware of all the thresholds applied in the network. The problem is a joint problem of RB allocation, user-to-multiple SBS association and power allocation. These three allocation variables are solved in sequence to maximize the sum rate of the network. Preference is given to the resources of the first SBS that a user is associated with. In case the user's data rate is not satisfied, the association of the edge UEs (decided based on the channel state conditions) increases whereas the power of the near users is decreased. We assume that the channel state information is known at the MBS [34]. Algorithm 1 shows the steps of the centralized SCRE algorithm.