## 3 Transmission Protocol Configuration and Resource Allocation Scheme
### Traffic Flow Resource Allocation Algorithm
#### 3.2.2 Maximization of Cell Utility
 \tag{31}\]  
This contradicts that \(\alpha^{n,d}\) is the optimal solution of problem P2. Therefore, at most, one soft QoS traffic flow satisfies \(du_{i}\left(R_{i}^{d}\right)/dR_{i}^{d}>0\). The traffic flows in the set \(\Gamma_{j,1}^{Q}\) have been sorted in descending order of spectral efficiency \(c_{i}^{n,d}\). Suppose that for the sorted traffic flows \(h\) and \(i\), \(h<i\leq\left|\Gamma_{j,1}^{Q}\right|\), \(\alpha_{i}^{n,d}=\alpha^{\prime}>0\), \(\alpha_{h}^{n,d}=0\). Consider another resource allocation scheme \(\widehat{\alpha}^{-n,d}\), where \(\alpha_{i}^{n,d}=\alpha^{\prime}\), \(\alpha_{i}^{n,d}=0\), and the resource allocation results of other traffic flows remain unchanged. It can be found that the resource allocation scheme \(\widehat{\alpha}^{-n,d}\) can further improve the objective function value of problem P2 compared with \(\alpha^{n,d}\). This is contrary to the fact that \(\alpha^{n,d}\) is the optimal solution of problem P2. Therefore, the traffic flow satisfying \(du_{i}\left(R_{i}^{d}\right)/dR_{i}^{d}\) = 0 must be the last soft QoS traffic flow in the sorted \(\Gamma_{j,1}^{Q}\) to obtain transmission resources. For a soft QoS traffic flow \(i\), \(du_{i}\left(R_{i}^{d}\right)/dR_{i}^{d}>0\) is equivalent to \(R_{i}^{d}<R_{thr}\) Lemma 3 shows that in the optimal solution to problem P2, at most, one soft QoS traffic flow satisfies \(R_{i}^{d}<R_{thr}\), and must be the last soft QoS traffic flow to obtain resources in the set \(M_{j,1}^{Q}\). Therefore, the optimal solution of problem P2 can be solved by assuming that the first m business flows in the set \(M_{j,1}^{Q}\) can be allocated to resources, and the corresponding \(\lambda\) value can be calculated. At the same time, m is traversed in the set \(\left\{1,2,\ldots,\left|\Gamma_{j,1}^{Q}\right|\right\}\) to find the optimal solution to the problem. Assuming that m soft QoS traffic streams can obtain services from base station \(j\), there are the following two types of situations:  
Scenario 1 indicates that thereis no business flow satisfying \(R_{i}^{d}<R_{thr}\), that is, \(R_{m}^{d}\geq R_{thr}\). Based on Lemma 3, the following equation \(\lambda\) can be obtained:  
\[\sum_{i=1}^{m}\left[\hat{u_{i}}\left(\lambda/c_{i}^{n,d}\right)^{-1}+R_{thr} \right]\left(c_{i}^{n,d}+\sum_{k\in\Gamma_{j,1}^{Q}}u_{k}^{B}\left(\lambda/c_{ k}^{n,d}\right)^{-1}/c_{k}^{n,d}=\Phi^{n,d} \tag{32}\]  
Let \(\lambda_{1}\) be the solution of the equation, then the optimal solution of problem P2 is:  
\[\alpha_{i}^{n,d}=\left\{ \begin{array}{c}\left[\hat{u_{i}}\left(\lambda_{1}/ c_{i}^{n,d}\right)^{-1}+R_{thr}\right]/c_{i}^{n,d},i=1,2,\ldots,m\\ 0,i=m+1,\ldots,\left|\Gamma_{j,1}^{Q}\right|\\ u_{k}^{B}\left(\lambda_{1}/c_{k}^{n,d}\right)^{-1}/c_{k}^{n,d},i=\left|\Gamma _{j,1}^{Q}\right|+1,\ldots,\left|\Gamma_{j,1}\right|\end{array} \right. \tag{33}\]  
Scenario 2 indicates that there is a business flow that satisfies \(R_{i}^{d}<R_{thr}\), that is, \(R_{thr}^{d}<R_{thr}\). Accordingly, the equation about \(\lambda\) can be obtained as  
\[\sum_{i=1}^{n-1}\left[\hat{u_{i}}\left(\lambda/c_{i}^{n,d}\right)^{-1}+R_{thr }\right]/c_{i}^{n,d}+u_{i}^{Q}\left(\lambda/c_{m}^{n,d}\right)^{-1}/c_{m}^{n, d}+\sum_{k\in\Gamma_{j,1}^  
in the sub-problem-solving process include traffic flow set, spectral efficiency, and total available resources. Using the results of sub-problems 1, 2 and 3, we can obtain the solution of problem P1 under fixed a and b. Finally, traverse a and b in sets \(\left\{1,2,\ldots,\left|\Gamma_{j}^{d}\right|-1\right\}\) and \(\left\{1,2,\ldots,\left|\Gamma_{j}^{u}\right|-1\right\}\) respectively. The optimal solution of \(N+1\) cell utility maximization problem constitutes the optimal solution of the network utility maximization problem.