## 4 Problem Solution  
### Problem Conversion  
To avoid divided-by-zero exception, we first introduce a microscale \(\epsilon\) and then define an auxiliary variable \(\alpha_{i}(t)=(\beta_{i,o}(t)+\epsilon)^{-1}\), Therefore, \(H_{i}(t)(q_{i}(t),T_{i,h}(t),\beta_{i,o}(t))\) can be converted to \(H_{i}^{\prime}(t)(q_{i}(t),T_{i,h}(t),\alpha_{i}(t))\) as given by:  
\[\begin{array}{ll}H_{i}^{\prime}(t)(q_{i}(t),T_{i,h}(t),\alpha_{i}(t))&=( \eta_{i}  
P2 is obviously nonconvex because of the second order term in the form of \(q_{i}(t)a_{i}(t)\). We can linearize the second order term of the problem by the Reformulation Linearization Technique (RLT). Therefore, we adopt RLT to linearize the second order term contained in the objective function and corresponding constraints. Specifically, to eliminate the second order term \(q_{i}(t)\alpha_{i}(t)\), we define an auxiliary variable \(Q_{i}(t)=q_{i}(t)\alpha_{i}(t)\), \(q_{i}(t)\) is bounded as \(0\leq q_{i}(t)\leq Z_{i}(t)\) and \(\alpha_{i}(t)\) is bounded as \(\frac{1}{B+\epsilon}\leq\alpha_{i}(t)\leq\frac{1}{\epsilon}\). We can obtain the RLT bound-factor product constraints for \(Q_{i}(t)\) is:  
\[\left\{ \begin{array}{l}\{[q_{i}(t)-0]\cdot[\alpha_{i}(t)-\frac{1}{B+\epsilon }]\}_{LS^{\prime}}\\ \{[Z_{i}(t)-q_{i}(t)]\cdot[\alpha_{i}(t)-\frac{1}{B+\epsilon}]\}_{LS^{\prime}} \\ \{[q_{i}(t)-0]\cdot[\frac{1}{\epsilon}-\alpha_{i}(t)]\}_{LS},\\ \{[Z_{i}(t)-q_{i}(t)]\cdot[\frac{1}{\epsilon}-\alpha_{i}(t)]\}_{LS}\end{array} \right. \tag{21}\]  
where \(\{.\}\)\(LS\) represents a linearization step under \(Q_{i}(t)=q_{i}(t)\alpha_{i}(t)\). By substituting \(Q_{i}(t)\) into (21), we can get:  
\[\left\{ \begin{array}{l}Q_{i}(t)-\frac{1}{B+\epsilon}q_{i}(t)\geq 0\\ Z_{i}(t)\alpha_{i}(t)-\frac{1}{B+\epsilon}Z_{i}(t)-Q_{i}(t)+\frac{1}{B+ \epsilon}q_{i}(t)\geq 0\\ \frac{1}{\epsilon}q_{i}(t)-Q_{i}(t)\geq 0\\ \frac{1}{\epsilon}Z_{i}(t)-Z_{i}(t)\alpha_{i}(t)-\frac{1}{\epsilon}q_{i}(t)+Q _{i}(t)\geq 0\end{array} \right. \tag{22}\]  
Substituting \(Q_{i}(t)\) into the \(H_{i}^{\prime}(t)(q_{i}(t),T_{i,h}(t),\alpha_{i}(t))\), we can get:  
\[ \begin{array}{ll}H_{i}^{\prime\prime}(t)(q_{i}(t),T_{i,h}(t),a_{i}(t),Q_{i} (t))&=(\eta_{i}T_{i,h}(t)P_{d}(t)g_{i,d}(t)-\frac{\zeta_{i}\zeta_{i}^{\prime }(Z_{i}(t)-q_{i}(t))^{3}}{T^{2}}\\ &-(p_{i,o}(t)+p_{c,i})(\frac{Q_{i}(t)}{\log_{2}\left(1+\frac{p_{i,o}(W_{i,o}(t ))}{\sigma^{2}}\right)})s_{i}(t)\end{array}  \tag{23}\]  
Besides, by substituting \(Q_{i}(t)\) into the constraints in P2, the problem P3 can be obtained as follows:  
\[\begin{array}{l}\text{P3:}\max\limits_{q_{i},T,A,Q_{i}}\sum\limits_{t=1}^{N }\sum\limits_{i=1}^{K}H_{i}^{\prime\prime}(t)(q_{i}(t),T_{i,h}(t),a_{i}(t),Q_{ i}(t))\\ s.t.\text{~{}C11:}\frac{\zeta_{i}(\zeta)(Z_{i}(t)-q_{i}(t))^{3}}{T^{2}}+(p_{i,o} (t)+p_{c,i})(\frac{Q_{i}(t)}{\log_{2}\left(1+\frac{p_{i,o}(W_{i,o}(t))}{\sigma ^{2}}\right)})\leq\eta_{i}T_{i,h}(t)P_{d}(t)g_{i,d}(t)\\ \text{~{}C12:}T_{i,h}(t)+\frac{Q_{i}(t)}{\log_{2}\left(1+\frac{p_{i,o}(W_{i,o}(t ))}{\sigma^{2}}\right)}\leq T,\forall i\in K,\forall t\in\mathcal{T}\\ \text{~{}C1,C4,C6,C7,C10}\\ \text{~{}C13:}\left(\ref{eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq  
problem (31) under the given \((\mu,\theta,\lambda,\gamma,\kappa,\delta,\rho)\). As a result, we can obtain N*K subproblems of jointly optimizing \(q_{i}(t)\), \(\alpha_{i}(t)\) and \(Q_{i}(t)\) as follows:  
\[ \begin{split}&\min_{q,\lambda,Q}(1+\rho_{i}(t))H_{i}^{rem}(t)(q_{i}(t ),a_{i}(t),Q_{i}(t))+\mu(t)q_{i}(t)C_{i}+\theta(t)\frac{1}{\alpha_{i}(t)}+( \lambda_{i}(t)-\gamma_{i}(t))\frac{1}{B+\varepsilon}q_{i}(t)\\ &+(\delta_{i}(t)-\kappa_{i}(t))\frac{1}{\alpha}q_{i}(t)+(\delta_{ i}(t)-\gamma_{i}(t))Z_{i}(t)\alpha_{i}(t)+(\kappa_{i}(t)-\lambda_{i}(t)+\gamma_{i}(t)- \delta_{i}(t))Q_{i}(t)\\ & s.t.0\leq q_{i}(t)\leq Z_{i}(t),0<Q_{i}(t)<\ T\log_{2}\left(1+ \frac{p_{i,\rho}(t)h_{i,\rho}(t)}{\sigma^{2}}\right)\end{split}