# QoS-Guaranteed Radio Resource Management in LTE-A Co-Channel Networks with Dual Connectivity
## 2 System Model
### Optimal Resource Allocation  
In LTE-A, the basic schedule unit is a resource block (RB). One RB consists of 12 subcarriers and a time domain, and the base station may calculate the data transfer rate of an RB when it is allocated to a specific UE. Due to the difference in channel quality, the signal status of different UE is different, so the data transfer rate will be affected by the modulation and coding rate (see Table 1). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**CQI Index** & **Modulation** & **Code Rate \(\times\) 1024** & **Efficiency** \\ \hline
0 & & out of range & \\
1 & QPSK & 78 & 0.1523 \\
2 & QPSK & 120 & 0.2344 \\
3 & QPSK & 193 & 0.3770 \\
4 & QPSK & 308 & 0.6016 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Channel Quality Indicator (CQI) table. Figure 2: Dual connectivity conditions. The impact of the resource scheduling on the data transfer rate is very significant. Since all UEs share bandwidth of a base station, to optimize the system throughput it is intuitive to assign an RB to the UE with the best channel quality. That is, we formulate a throughput optimization problem as follows. \[\max_{X_{u,k}}\sum_{u\in\textbf{U}}\sum_{k=1}^{K}TP_{u,k}\,*\,X_{u,k} \tag{1}\]  
where \(u\) is a UE belongs to the UE set \(\textbf{U}=\{u_{1},\,\ldots,\,u_{1}\}\), \(k\) is a RB of the RB set \(K=\{1,2,\,\ldots,\,\text{K}\}\), \(TP_{u,k}\) is the throughput of RB \(k\) when it is allocated to the UE \(u\), and \(X_{u,k}\) represents whether RB \(k\) is assigned to UE \(u\). \(TP_{u,k}\) can be calculated by 12 (subcarriers) * 7 (symbols) * \(CQI_{efficiency}\). There is a positive correlation between \(TP_{u,k}\) and \(CQI_{efficiency}\); however, \(CQI_{efficiency}\) is not a linear function of \(CQI_{index}\), which makes the objective function nonlinear. Therefore, we replace \(CQI_{efficiency}\) with \(CQI_{index}\) and make the optimization problem (1) to a 0-1 integer programming problem as follows:  
\[\max_{X_{u,k}}\sum_{u\in\textbf{U}}\sum_{k=1}^{K}CQI_{u,k}\,*\,X_{u,k}. \tag{2}\]  
Subject to:  
\[\sum_{u\in\textbf{U}}X_{u,k}\,\leq 1\forall k \tag{3}\]  
\[X_{u,k}\,\in\,\{\,0,\,1\}\forall u,k \tag{4}\]  
where Equation (3) constrains each RB can only be assigned to a UE, and Equation (4) states that \(X_{u,k}\) is an indication variable. Two common approaches of solving 0-1 linear programming problems are \"implicit enumeration\" and \"branch and bound\". Unfortunately, the computational complexity of these two methods is exponential time. In order to reduce the computational complexity, we can first relax the 0-1 integer programming to linear programming. The relaxed linear programming can then be solved by many existing solutions, such as \"simplex method\", \"ellipsoid method\", and \"interior point method\". For example, in MATLAB, the linprog library is provided which can obtain the optimal solution by the \"internal point method\" according to the scale of the optimization problem. After obtaining the optimal solution from linear programming, two methods can be used to transfer the solution back to the 0-1 integer programming solution. * **Max method:** For RB \(k\), the sum of all variables is at most 1 under constraint (3). The max method sorts these variables in descending order, assigns 1 to the variable with the largest value (i.e., allocate the RB to the UE) and 0 to the rest variables. * **Random method:** For RB \(k\), let \(N=\sum X_{i,k}\). The random method allocates RB \(k\) to UE \(i\) with probability \(p_{i}\) where \(p_{i}=\frac{X_{i,k}}{N}\) (i.e., set \(X_{i,k}=\)1 and \(X_{u,k}=0\,\forall u\
eq i\)).