## 3 Transmission Protocol Configuration and Resource Allocation Scheme
### Traffic Flow Resource Allocation Algorithm
#### 3.2.2 Maximization of Cell Utility
\[\text{Sub-problem 1}:\text{ maximize }\sum\limits_{i\in\Gamma_{j,1}}U_{i}^{d} \Big{(}\alpha_{i}^{n,d}c_{i}^{n,d}\Big{)} \sum\limits_{i\in\Gamma_{j,1}}\alpha_{i}^{n,d}\leq\Phi^{n,d}\]1 Subject to \[\text{Sub-problem 1}:\text{ maximize }\sum\limits_{i\in\Gamma_{j,1}}U_{i}^{d} \Big{(}\alpha_{i}^{n,d}c_{i}^{n,d}\Big{)} \sum\limits_{i\in\Gamma_{j,1}}\alpha_{i}^{n,d}\leq\Phi^{n,d}\]2  
where the optimization variable is \(\alpha^{n,d}=\left\{\alpha_{i}^{n,d},i\in\Gamma_{j,1}\right\}\). \(U_{i}^{Q}\) represents the utility of soft QoS traffic flow \(i\), and \(U_{k}^{B}\) represents the utility of best-effort service traffic flow \(k\). For best-effort traffic flow \(k\), its marginal utility function can be expressed as  
\[\text{Sub-problem 1}:\text{ maximize }\sum\limits_{i\in\Gamma_{j,1}}U_{i}^{d} \Big{(}\alpha_{i}^{n,d}c_{i}^{n,d}\Big{)} \sum\limits_{i\in\Gamma_{j,1}}\alpha_{i}^{n,d}\leq\Phi^{n,d}\]3  
Equation (28) is a subtractive function of the reachable rate \(R\). For soft QoS traffic flow, marginal utility function can be expressed as  
\[U_{i}^{Q}(R)=\left\{ \begin{array}{l}(1-p_{1})q_{1}e^{q_{1}(R-R_{th})},R<R_{ th}\\ p_{1}q_{1}e^{-q_{1}(R-R_{th})},R\geq R_{th}\end{array} \right. \tag{29}\]  
When \(R<R_{th}\), \(U_{i}^{Q}(R)\) is an increasing function of the reachable rate \(R\). And when \(R>R_{th}\), \(U_{i}^{Q}(R)\) is a decreasing function of the reachable rate \(R\). For problem P2, the necessary conditions for the optimal solution are given by the following lemma. **Lemma 3**. : _For the optimal solution \(s^{n,d}\) of problem P2, there is a Lagrange multiplier \(\lambda>0\), so that._  
\[ \begin{array}{l}c_{i}^{n,d}\frac{U_{i}^{Q}}{\left(\varepsilon_{i}^{n,d} \alpha_{i}^{n,d}\right)}=c_{k}^{n,d}\frac{A_{k}^{B}}{u_{k}^{B}}{\left( \varepsilon_{k}^{n,d}\alpha_{k}^{n,d}\right)}=\lambda,\forall a_{i}^{n,d}, \alpha_{k}^{n,d}>0\\ \sum\limits_{i\in\Gamma_{j,1}}\tilde{\alpha}_{i}^{n,d}=0\text{ and }\end{array}  \tag{30}\]  
At most, there is only one traffic flow that meets \(du_{i}\left(R_{i}^{d}\right)/dR_{i}^{d}>0\). This traffic flow must be the last soft QoS traffic flow in \(\Gamma_{j,1}^{Q}\) after sorting to obtain transmission resources. It is proven that if \(\tilde{\alpha}^{n,d}\) is the optimal solution of problem P2, \(\alpha^{n,d}\) satisfies the KKT condition of problem P2, that is, Equation (30). For any best-effort service business flow \(k\), \(du_{i}\left(R_{i}^{d}\right)/dR_{i}^{d}<0\) is always true. Therefore, only soft QoS traffic flow will meet \(du_{i}\left(R_{i}^{d}\right)/dR_{i}^{d}>0\). It is assumed that at least two soft QoS traffic flows meet the above conditions, in which traffic flows \(i\) and \(h\) meet \(du_{i}\left(R_{i}^{d}\right)/dR_{i}^{d}>0\) and \(du_{h}\left(R_{h}^{d}\right)/dR_{h}^{d}>0\), respectively. Considering another resource allocation scheme \(\widetilde{\alpha}^{n,d}\), the resources allocated to traffic flows \(i\) and \(h\) are \(\alpha_{i}^{n,d}-\Delta\alpha\) and \(\tilde{\alpha}_{h}^{n,d}+\Delta\alpha\) respectively, and the resource allocation results of other traffic flows remain unchanged. Since the utility function is continuous and monotonically increasing, the difference between the objective function value of problem P2 under resource allocation \(\widetilde{\alpha}^{n,d}\) and resource allocation \(\alpha^{n,d}\) is  
\[ \begin{array}{l}\left[U_{i}^{Q}\left(c_{i}^{n,d}\left(\varepsilon_{i}^{n,d} -\Delta\alpha\right)\right)+U_{h}^{Q}\left(c_{h}^{n,d}\left(\varepsilon_{h}^{ n,d}+\Delta\alpha\right)\right)\right]-\left[U_{i}^{Q}\left(c_{i}^{n,d} \varepsilon_{i}^{n,d}\right)+U_{h}^{Q}\left(c_{h}^{n,d}\varepsilon_{h}^{n,d} \right)\right]\\ =\int\limits_{s_{i}^{n,d}}^{s_{n,d}}\varepsilon_{h}^{n,d}u_{h}^{Q}\left(c_{h} ^{n,d}\alpha\right)da-\int\limits_{s_{i}^{n,d}-\Delta\alpha}^{s_{i}^{n,d}}c_{i} ^{n,d}u_{i}^{Q}\left(c_{i}^{n,d}\alpha\right)da\\ >c_{h}^{n,d}u_{h}^{Q}\left(c_{h}^{n,d}\varepsilon_{h}^{n,d}\right)-c_{i}^{n,d}u _{i}^{Q}\left(c_{i}^{n,d}\varepsilon_{i}^{n,d}\right)=0\end{array}