## 3 Transmission Protocol Configuration and Resource Allocation Scheme
### Traffic Flow Resource Allocation Algorithm
#### 3.2.2 Maximization of Cell Utility  
The problem of maximizing the utility of a cell (Problem 1) can be expressed as  
\[\begin{array}{l}\text{Maximize}\sum\limits_{i\in M_{j}^{d}}U_{i}^{d}\Big{(} \alpha_{i}^{n,d}c_{i}^{n,d}+\alpha_{i}^{l,d}c_{i}^{l,d}\Big{)}+\sum\limits_{k \in M_{j}^{d}}U_{k}^{u}\Big{(}\alpha_{k}^{n,u}c_{k}^{n,u}+  
and the cell utility maximization problem should be solved. The optimal solutions of the \(N+1\) cell utility maximization problem constitute the optimal solution of the network utility maximization problem. This paper focuses on the solution of the cell utility maximization problem. For the nonconvex optimization problem, the KKT condition is a necessary condition for its optimal solution. The optimal solution of problem P1 must be in the solution set satisfying its KKT condition. In this paper, the KKT condition is used to reduce the feasible solution of problem P1 and search for the solution of the objective function of the maximization problem. Construct the Lagrange function of problem P1 as follows:  
\[ \begin{array}{l}L\big{(}S_{j},\lambda,\beta,\gamma,\
u\big{)}=-\sum\limits_ {i\in\Gamma_{j}^{d}}U_{i}^{d}\Big{(}R_{i}^{d}\Big{)}-\sum\limits_{k\in\Gamma_{ j}^{d}}U_{k}^{u}\big{(}R_{k}^{u}\big{)}+\lambda\left(\sum\limits_{k\in\Gamma_{ i}^{n,d}}-\[PERSON]^{n,u}\right)\\ +\beta\left(\sum\limits_{k\in\Gamma_{k}^{u}}\alpha_{k}^{n,u}-\[PERSON]^{n,u}\right) +\gamma\left(\sum\limits_{i\in\Gamma_{j}^{d}}\alpha_{i}^{l,d}+\sum\limits_{i \in\Gamma_{i}^{n}}\alpha_{i}^{n,u}-\Phi^{l}\right)\\ \sum\limits_{i\in\Gamma_{j}^{d}}\Big{(}\alpha_{i}^{l,d}v_{i}^{l,d}+\alpha_{i}^ {n,d}v_{i}^{n,d}\Big{)}+\sum\limits_{k\in\Gamma_{i}^{n}}\Big{(}\alpha_{k}^{l, d}v_{k}^{n,u}+\alpha_{k}^{n,u}v_{k}^{n,u}\Big{)}\end{array}  \tag{23}\]  
where, \(\lambda\geq 0\)  
service, its downlink throughput \(R_{i}^{d}=c_{i}^{n,d}\alpha_{i}^{n,d}+c_{i}^{l,d}\alpha_{i}^{l,d}>0\). Therefore, at least one of \(\alpha_{i}^{n,d}\) and \(\alpha_{i}^{l,d}\) is greater than 0, which means \(v_{i}^{l,d}\) and \(v_{i}^{n,d}\) cannot be greater than 0 at the same time. When \(\frac{c_{i}^{n,d}}{c_{i}^{l,d}}>\frac{\lambda}{\gamma}\), \(v_{i}^{n,d}>0\), \(v_{i}^{l,d}=0\), that is, \(\alpha_{i}^{n,d}\geq 0\), \(\alpha_{i}^{l,d}=0\). When \(\frac{c_{i}^{n,d}}{c_{i}^{l,d}}<\frac{\lambda}{\gamma}\), \(v_{i}^{n,d}=0\), \(v_{i}^{l,d}>0\), that is, \(s_{i}^{n,d}=0\), \(s_{i}^{l,d}>0\). According to this lemma, for the solution satisfying the KKT condition, if the frequency spectrum efficiency of user's down scheduling on the normal frame and LP-ABS of the base station meets \(c_{i}^{n,d}>(\lambda/\gamma)\), the user should prioritize scheduling on the normal frame. Otherwise, the user should give priority to scheduling on the LP-ABS. Similarly, the lemma related to user uplink scheduling can be obtained as follows. Lemma 2: _For any solution satisfying the above KKT conditions, there are specific Lagrange multipliers \(\beta\) and \(\gamma\), such that the user \(i\) obtains the uplink transmission service when \(\frac{c_{i}^{n,d}}{c_{i}^{l,d}}>\frac{\beta}{\gamma}\), \(\alpha_{i}^{n,u}>0\), \(\alpha_{i}^{l,u}=0\). when \(\frac{c_{i}^{n,u}}{c_{i}^{l,u}}<\frac{\beta}{\gamma}\), \(\alpha_{i}^{n,u}=0\), \(\alpha_{i}^{l,u}=0\)._  
According to the preceding two lemmas, perform the following operations on downstream and uplink traffic flows. Downlink traffic flows are sorted in descending order by the ratio of the spectral efficiency of the normal subframe to LP-ABS. \(d_{a}\) indicates the sorted traffic flow \(a\). Figure 4 shows the sorted traffic flow. Given set \(a\in\left\{1,2,\ldots,\left|\Gamma_{j}^{d}\right|-1\right\}\), two business flow sets \(\Gamma_{j,1}\) and \(\Gamma_{j,2}\) are constructed. Uplink traffic flows are also arranged in descending order by spectral efficiency ratio, as shown in Figure 5.  
According to [PERSON] 1 and 2, any solution satisfying the KKT condition of the cell utility maximization problem has the following properties.