## 4 Resource Allocation Design for Delay Minimization
Furthermore, by introducing the variable \(\zeta_{k}\), the Equation (32) can be rewritten as  
\[\sigma_{h,k}^{2}(\frac{1}{\zeta_{k}}-2)\leq h_{d,k}^{H}h_{d,k}+T \mathbf{\Gamma}(\mathbf{\Xi}_{k}\mathbf{\Lambda}),\ \forall k\in\mathcal{K}, \tag{35}\] \[-\frac{1}{\log(1-\delta_{0})}\frac{\hat{\gamma}_{k}\sigma^{2}}{ \sigma_{h,k}^{2}}\leq\frac{q_{k}^{2}}{\zeta_{k}},\ \forall k\in\mathcal{K}. \tag{36}\]  
It is obvious that the Equation (32) is converted into Equation (35) by introducing variables, which is convex. However, it also brings a non-convex Equation (36). Next, the CCCP is invoked to solve this issue, which main idea is that when the original problem is difficult to optimize, the algorithm does not directly seek its local optimal solution, but looks for its approximation problem to solve iteratively [25]. For a given point \((q^{(n)},\xi^{(n)})\) at the \(n\)-th iteration, the function \(f_{k}(q_{k},\zeta_{k})=q_{k}^{2}/\zeta_{k}\) has a lower bound, that is  
\[f_{k}(q_{k},\zeta_{k})=2\beta_{k}^{(n)}q_{k}-(\beta_{k}^{(n)})^{2}\zeta_{k}, \tag{37}\]  
where \(\beta_{k}^{(n)}=q_{k}^{(n)}/\zeta_{k}^{(n)}\). Thus, Equation (36) can be rewritten as  
\[-\frac{1}{\log(1-\delta_{0})}\frac{\hat{\gamma}_{k}\sigma^{2}}{\sigma_{h,k}^{2 }}\leq 2\beta_{k}^{(n)}q_{k}-(\beta_{k}^{(n)})^{2}\zeta_{k},\;\forall k\in \mathcal{K}. \tag{38}\]  
Now, Equation (38) is convex. For the convenience of derivations, we rewrite Equation (27) as  
\[g_{1,k}(\mu_{k},f_{k},\
u_{k})+g_{2,k}(\mu_{k},f_{k},\
u_{k})+g_{3,k}(\mu_{k}) \leq T,\forall k\in\mathcal{K}, \tag{39}\]  
where  
\[g_{1,k}(\mu_{k},f_{k},\
u_{k}) =\frac{1}{4}\bigg{(}\mu_{k}+(L\frac{c_{k}D_{k}}{f_{k}}+\frac{z}{ \
u_{k}})\bigg{)}^{2}, \tag{40}\] \[g_{2,k}(\mu_{k},f_{k},\
u_{k}) =-\frac{1}{4}\bigg{(}\mu_{k}-(L\frac{c_{k}D_{k}}{f_{k}}+\frac{z}{ \
u_{k}})\bigg{)}^{2},\] (41) \[g_{3,k}(\mu_{k}) =-\eta\sum_{k\in\mathcal{K}}\mu_{k}(\mu_{k}-1). \tag{42}\]  
Note that \(g_{1,k}(\mu_{k},f_{k},\
u_{k})\) is convex, while \(g_{2,k}(\mu_{k},f_{k},\
u_{k})\), and \(g_{3,k}(\mu_{k})\) are non-convex. For a given point \((\mathbf{n}^{(n)},\mathbf{f}^{(n)},\mathbf{v}^{(n)})\) at the \(n\)-th iteration,both \(g_{2,k}(\mu_{k},f_{k},\
u_{k})\) and \(g_{3,k}(\mu_{k})\) have an upper bound, which are given as  
\[\tilde{g}_{2,k}(\mu_{k},f_{k},\
u_{k})=-\frac{1}{4}\bigg{(}\mu_{k }^{(n)}-(L\frac{c_{k}D_{k}}{f_{k}^{(n)}}+\frac{z}{\
u_{k}^{(n)}})\bigg{)}^{2} +\frac{1}{2}\mu_{k}^{(n)}\bigg{(}\frac{z\
u_{k}}{(\
u_{k}^{(n)})^{2}}+\frac{z }{\
u_{k}}-\frac{2z}{\
u_{k}^{(n)}}\bigg{)}\] \[-\frac{1}{2}\bigg{(}\mu_{k}^{(n)}-(L\frac{c_{k}D_{k}}{f_{k}^{(n)} }+\frac{z}{\
u_{k}^{(n)}})\bigg{)}\bigg{(}(\mu_{k}-\mu_{k}^{(n)})+\frac{Lc_{k} D_{k}}{(f_{k}^{(n)})^{2}}(f_{k}-f_{k}^{(n)})+\frac{z}{(\
u_{k}^{(n)})^{2}}( \
u_{k}-\
u_{k}^{(n)})\bigg{)}\] \[-\frac{1}{\log(1-\delta_{0})}\frac{\hat{\gamma}_{k}\sigma^{2}}{ \sigma_{h,k}^{2}}\leq\frac{q_{k}^{2}}{\zeta_{k}},\ \forall k\in\mathcal{K}. \tag{36}\]0  
\[-\frac{1}{\log(1-\delta_{0})}\frac{\hat{\gamma}_{k}\sigma^{2}}{ \sigma_{h,k}^{2}}\leq\frac{q_{k}^{2}}{\zeta_{k}},\ \forall k\in\mathcal{K}. \tag{36}\]1  
As such, Equation (27) can be approximated by  
\[-\frac{1}{\log(1-\delta_{0})}\frac{\hat{\gamma}_{k}\sigma^{2}}{ \sigma_{h,k}^{2}}\leq\frac{q_{k}^{2}}{\zeta_{k}},\ \forall k\in\mathcal{K}. \tag{36}\]2  
Now, Equation (45) is also convex. For a given point \((\mathbf{q}^{(n)},\mathbf{\zeta}^{(n)},\mathbf{\mu}^{(n)},\mathbf{f}^{(n)},\mathbf{v}^{(n)})\) at the \((n+1)\)-th iteration, the problem \(P4\) can be reformulated as  
\[-\frac{1}{\log(1-\delta_{0})}\frac{\hat{\gamma}_{k}\sigma^{2}}{ \sigma_{h,k}^{2}}\leq\frac{q_{k}^{2}}{\zeta_{k}},\ \forall k\in\mathcal{K}. \tag{36}\]3 (46) \[s.t. (\ref{eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq: eqnote that \(P5\) is a relaxed problem, which usually cannot lead to a rank-one solution. Thus, Gaussian randomization is invoked to construct a rank-one solution [27]. For the optimal \(\mu_{k}\) obtained by solving problem \(P5\), it is close to 0 or 1 due to the punishment item. Rounding operation is run to obtain 0 or 1.