1. Introduction

2. System Model

3. Algorithm Design

3.1. Lyapunov Optimization

To address the average power constraints, we introduce a virtual energy queue $\mathrm{Y}\left(t+1\right)={\left[\mathrm{Y}\left(\mathrm{t}\right)-\gamma +P_0^t{\tau }_0^t\right]}^{+}$, where ${\left[x\right]}^{+}\triangleq max\left\{0,x\right\}$[44]. Here $\mathrm{Y}\left(\mathrm{t}\right)$ can be seen as a queue with random "energy arrivals" $P_0^t{\tau }_0^t$ and a fixed "service rate" $\gamma$. Thus, we derive the following Lemma 1.

Lemma 1.

The long-term average power constraints will be met if the virtual queue $\mathrm{Y}\left(\mathrm{t}\right)$ satisfies average rate stability

$$\underset{K\to \infty }{\lim }{\displaystyle \frac{1}{K}}\mathbb{E}\left\{\mathrm{Y}\left(\mathrm{t}\right)\right\}=0$$

Proof. 

According to the above formula $\mathrm{Y}\left(\mathrm{t}+1\right)={[\mathrm{Y}\left(\mathrm{t}\right)-\gamma +P_0^t{\tau }_0^t]}^{+}$ , we can conclude that

$$\mathrm{Y}\left(\mathrm{t}+1\right)\ge \mathrm{Y}\left(\mathrm{t}\right)-\gamma +P_0^t{\tau }_0^t$$

(23)

we can expand all the terms of $t\in \{0,1,2,...,K-1\}$, sum them up, and then take the average, yielding

$${\displaystyle \frac{\mathrm{Y}\left(K\right)-\mathrm{Y}\left(0\right)}{K}}\ge {\displaystyle \frac{1}{K}}\stackrel{K-1}{{\displaystyle \sum }_{t=0}}{P}_0^t{\tau }_0^t-\gamma$$

(24)

Next, we simultaneously take the expectation on both sides and set K, obtaining:

$$\underset{K\to \infty }{\lim \sup }{\displaystyle \frac{\mathbb{E}\left\{\mathrm{Y}\left(\mathrm{t}\right)\right\}}{K}}\ge \underset{K\to \infty }{\lim \sup }{\displaystyle \frac{1}{K}}\stackrel{K-1}{{\displaystyle \sum }_{t=0}}\mathbb{E}\left\{P_0^t{\tau }_0^t\right\}-\gamma$$

(25)

By our assumption$\underset{K\to \infty }{\lim }{\displaystyle \frac{1}{K}}\mathbb{E}\left\{\mathrm{Y}\left(\mathrm{t}\right)\right\}=0$, it follows that $\underset{K\to \infty }{\lim \sup }{\displaystyle \frac{1}{K}}{\displaystyle {\displaystyle \sum }_{t=0}^{K-1}}\mathbb{E}\left\{P_0^t{\tau }_0^t\right\}-\gamma \le 0$, i.e., $\overline{P_0^t{\tau }_0^t}\le \gamma$ , the lemma is proven.    □

By defining a network queue vector $\mathsf{\Theta }\left(t\right)=\left\{Q_n\left(t\right),Q_f\left(t\right),\mathrm{Y}\left(\mathrm{t}\right)\right\}$, which encompasses the queue lengths for the NU, FU, and the virtual queue, respectively, we can obtain the associated Lyapunov function $L\left(\mathsf{\Theta }\left(t\right)\right)$ and the Lyapunov drift $\Delta \left(\mathsf{\Theta }\left(t\right)\right)$ as

$$L\left(\mathsf{\Theta }\left(t\right)\right)\triangleq {\displaystyle \frac{1}{2}}\left(Q_n{\left(t\right)}^2+Q_f{\left(t\right)}^2+\mathrm{Y}{\left(\mathrm{t}\right)}^2\right)$$

(26)

$$\Delta \left(\mathsf{\Theta }\left(t\right)\right)\triangleq \mathbb{E}\left\{\mathit{L}\left(\mathit{t}+\mathit{1}\right)-\mathit{L}\left(\mathit{t}\right)\mid \mathsf{\Theta }\left(\mathit{t}\right)\right\}$$

(27)

Employing the Lyapunov optimization theory, we drive the drift-plus-penalty as

$${\Delta }_V\left(\mathsf{\Theta }\left(t\right)\right)=\Delta \left(\mathsf{\Theta }\left(t\right)\right)-V\mathbb{E}\left\{D_{\mathrm{tot}}\left(t\right)\mid \mathsf{\Theta }\left(t\right)\right\}$$

(28)

Here,$V>0$ signifies the penalty’s importance weight. The Lyapunov optimization method aims to minimize the upper bound of the drift plus penalty, thereby maximizing the objective function while ensuring queue stability. Optimizing the objective function across each time slot leads to long-term optimality. It’s important to note that a higher value of V prioritizes objective maximization, whereas a lower value emphasizes queue stability. To obtain the upper bound of ${\Delta }_V\left(\mathsf{\Theta }\left(t\right)\right)$, we derive the following Lemma 2.

Lemma 2.

At each time slot t, for any control strategy $\left\{{\tau }^t,{\mathit{P}}^t\right\}$, the one-slot Lyapunov drift-plus-penalty${\Delta }_V\left(\mathsf{\Theta }\left(t\right)\right)$is bounded as per the following inequality

$$\begin{array}{cc}\hfill {\Delta }_V\left(\mathsf{\Theta }\left(t\right)\right)& \le B-V\mathbb{E}\left\{D_{\mathrm{tot}}\left(t\right)\mid \mathsf{\Theta }\left(t\right)\right\}+Q_f\left(t\right)\mathbb{E}\left\{A_f\left(t\right)-d_f^{\mathrm{loc}}\left(t\right)-d_{f,n}^{\mathrm{off}}\left(t\right)\mid \mathsf{\Theta }\left(t\right)\right\}\hfill \\ \hfill & +Q_n\left(t\right)\mathbb{E}\left\{A_n\left(t\right)-d_n^{\mathrm{loc}}\left(t\right)-d_{n,a2}^{\mathrm{off}}\left(t\right)\mid \mathsf{\Theta }\left(t\right)\right\}+\mathrm{Y}\left(\mathrm{t}\right)\mathbb{E}\left\{P_0^t{\tau }_0^t-\gamma \mid \mathsf{\Theta }\left(t\right)\right\}\hfill \end{array}$$

(29)

where B is a constant that satisfies the following $\forall t$

$$\begin{array}{cc}\hfill B& \ge {\displaystyle \frac{1}{2}}\mathbb{E}\left\{{\left(d_f^{\mathrm{loc}}\left(t\right)+d_{f,n}^{\mathrm{off}}\left(t\right)\right)}^2+A_f{\left(t\right)}^2\mid \mathsf{\Theta }\left(t\right)\right\}\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\mathbb{E}\left\{{\left(d_n^{\mathrm{loc}}\left(t\right)+d_{n,a2}^{\mathrm{off}}\left(t\right)\right)}^2+A_n{\left(t\right)}^2\mid \mathsf{\Theta }\left(t\right)\right\}\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\mathbb{E}\left\{{\gamma }^2+{\left(P_0^t{\tau }_0^t\right)}^2\mid \mathsf{\Theta }\left(t\right)\right\}\hfill \end{array}$$

(30)

Proof. 

For all $\forall a,b,c\ge 0$, we have the inequality ${\left(max[a-b,0]+c\right)}^2\le a^2+b^2+c^2+2a\left(c-b\right)$. By using the inequality, we have

$$\begin{array}{cc}\hfill \Delta Q_f\left(t\right)& ={\displaystyle \frac{1}{2}}\left(Q_f{\left(t+1\right)}^2-Q_f{\left(t\right)}^2\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le {\displaystyle \frac{{\left(d_f^{\mathrm{loc}}\left(t\right)+d_{f,n}^{\mathrm{off}}\left(t\right)\right)}^2+A_f{\left(t\right)}^2}{2}}+Q_f\left(t\right)\left(A_f\left(t\right)-d_f^{\mathrm{loc}}\left(t\right)-d_{f,n}^{\mathrm{off}}\left(t\right)\right)\hfill \\ \hfill \Delta Q_n\left(t\right)& ={\displaystyle \frac{1}{2}}\left(Q_n{\left(t+1\right)}^2-Q_n{\left(t\right)}^2\right)\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill & \le {\displaystyle \frac{{\left(d_n^{\mathrm{loc}}\left(t\right)+d_{n,a2}^{\mathrm{off}}\left(t\right)\right)}^2+A_n{\left(t\right)}^2}{2}}+Q_n\left(t\right)\left(A_n\left(t\right)-d_n^{\mathrm{loc}}\left(t\right)-d_{n,a2}^{\mathrm{off}}\left(t\right)\right)\hfill \\ \hfill \Delta \mathrm{Y}\left(\mathrm{t}\right)& ={\displaystyle \frac{1}{2}}\left(\mathrm{Y}{\left(\mathrm{t}+1\right)}^2-\mathrm{Y}{\left(\mathrm{t}\right)}^2\right)\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill & \le {\displaystyle \frac{{\left(P_0^t{\tau }_0^t\right)}^2+{\gamma }^2}{2}}+Y\left(t\right)\left(P_0^t{\tau }_0^t-\gamma \right)\hfill \end{array}$$

(33)

Upon combining inequalities (31), (32), and (33), the resulting expression yields the upper bound of the Lyapunov drift-plus-penalty.    □

By applying the drift-plus-penalty minimization technique, and eliminating the constant term observable at the start of time slot t in (29), we can obtain the optimal solution to problem (P1) by solving the following problem in each individual time slot.

$$\begin{array}{cc}\hfill P2:\underset{{\tau }^t,{\mathit{P}}^t}{min}& \mathrm{Y}\left(\mathrm{t}\right)P_0^t{\tau }_0^t-\left[Q_f\left(t\right)+{\omega }_{1}V\right]\left\{{\tau }_1^{t}W{log}_2\left(1+{\displaystyle \frac{P_1^t{g}_f^t}{{\sigma }^2}}\right)+d_f^{\mathrm{loc}}\left(t\right)\right\}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & -\left[Q_n\left(t\right)+{\omega }_{2}V\right]\left\{{\tau }_3^{t}W{log}_2\left(1+{\displaystyle \frac{P_3^t{g}_n^t}{{\sigma }^2}}\right)+d_n^{\mathrm{loc}}\left(t\right)\right\}\hfill \end{array}$$

(34a)

$$\begin{array}{cc}\hfill s.t.& \left(22b\right)-\left(22d\right),\left(22g\right)-\left(22k\right)\hfill \end{array}$$

(34b)

Due to the the non-convex constraints (34b), (P2) remains a non-convex problem. We introduce auxiliary variables ${\phi }_0=p_0^t{\tau }_0^t,{\phi }_1=p_1^t{\tau }_1^t,{\phi }_2=p_2^t{\tau }_2^t,{\phi }_3=p_3^t{\tau }_3^t$ here. According to equation (22j), we have $0\le {\phi }_0\le {\tau }_0{P}_a^{\max },0\le {\phi }_1\le {\tau }_1{P}_f^{\max },0\le {\phi }_2\le {\tau }_2{P}_n^{\max },0\le {\phi }_3\le {\tau }_3{P}_n^{\max }$. We denote $\phi =\left[{\phi }_0,{\phi }_1,{\phi }_2,{\phi }_3\right]$. To simplify the mathematical expressions, we omit t here. So (P2) can be equivalently substituted as

$$\begin{array}{cc}\hfill P3:\underset{\tau ,\phi }{max}& \mathrm{Y}{\phi }_0+C_1\left\{{\tau }_{1}W{log}_2\left(1+{\displaystyle \frac{{\phi }_{1}A}{{\tau }_1}}\right)+d_f^{\mathrm{loc}}\right\}+C_2\left\{{\tau }_{3}W{log}_2\left(1+{\displaystyle \frac{{\phi }_{3}B}{{\tau }_3}}\right)+d_n^{\mathrm{loc}}\right\}\hfill \end{array}$$

(35a)

$$\begin{array}{cc}\hfill s.t.& {\tau }_0+{\tau }_1+{\tau }_2+{\tau }_3\le \tau ,\hfill \end{array}$$

(35b)

$$\begin{array}{cc}\hfill & {\phi }_1+k{\left(f_f\right)}^{3}T\le {\alpha }_1{\phi }_0,\hfill \end{array}$$

(35c)

$$\begin{array}{cc}\hfill & {\phi }_2+{\phi }_3+k{\left(f_n\right)}^{3}T\le {\alpha }_2{\phi }_0,\hfill \end{array}$$

(35d)

$$\begin{array}{cc}\hfill & {\tau }_{1}W{log}_2\left(1+{\displaystyle \frac{{\phi }_{1}A}{{\tau }_1}}\right)\le {\tau }_{2}W{log}_2\left(1+{\displaystyle \frac{{\phi }_{2}B}{{\tau }_2}}\right),\hfill \end{array}$$

(35e)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{f_{f}T}{{\varphi }_f}}+{\tau }_{1}W{log}_2\left(1+{\displaystyle \frac{{\phi }_{1}A}{{\tau }_1}}\right)\le Q_f,\hfill \end{array}$$

(35f)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{f_{n}T}{{\varphi }_n}}+{\tau }_{3}W{log}_2\left(1+{\displaystyle \frac{{\phi }_{3}B}{{\tau }_3}}\right)\le Q_n,,\hfill \\ \hfill & 0\le {\phi }_0\le {\tau }_0{P}_a^{\max },0\le {\phi }_1\le {\tau }_1{P}_f^{\max },\hfill \end{array}$$

(35g)

$$\begin{array}{cc}\hfill & 0\le {\phi }_2\le {\tau }_2{P}_n^{\max },0\le {\phi }_3\le {\tau }_3{P}_n^{\max },\hfill \end{array}$$

(35h)

$$\left(22j\right)$$

(35i)

The coefficients are defined as $C_1=-\left[Q_f+{\omega }_{1}V\right]$ and $C_2=-\left[Q_n+{\omega }_{2}V\right]$, where $A={\displaystyle \frac{g_f}{{\sigma }^2}}$, and $B={\displaystyle \frac{g_n}{{\sigma }^2}}$. Owing to the non-convex constraint (35e), problem (P3) remains non-convex. To address this, we introduce auxiliary variables ${\psi }_1$ and ${\psi }_2$, defined such that ${\psi }_1\le {\tau }_{1}W{log}_2\left(1+{\displaystyle \frac{{\phi }_{1}A}{{\tau }_1}}\right),{\psi }_2\le {\tau }_{3}W{log}_2\left(1+{\displaystyle \frac{{\phi }_{3}B}{{\tau }_3}}\right)$. By making these definitions, problem (P3) can be reformulated in terms of these auxiliary variables as follows:

$$\begin{array}{cc}\hfill P4:\underset{\tau ,\phi }{max}& \mathrm{Y}{\phi }_0+C_1\left\{{\tau }_{1}W{log}_2\left(1+{\displaystyle \frac{{\phi }_{1}A}{{\tau }_1}}\right)+d_f^{\mathrm{loc}}\right\}+C_2\left\{{\tau }_{3}W{log}_2\left(1+{\displaystyle \frac{{\phi }_{3}B}{{\tau }_3}}\right)+d_n^{\mathrm{loc}}\right\}\hfill \end{array}$$

(36a)

$$\begin{array}{cc}\hfill s.t.& {\tau }_0+{\tau }_1+{\tau }_2+{\tau }_3\le \tau ,\hfill \end{array}$$

(36b)

$$\begin{array}{cc}\hfill & {\phi }_1+k{\left(f_f\right)}^{3}T\le {\alpha }_1{\phi }_0,\hfill \end{array}$$

(36c)

$$\begin{array}{cc}\hfill & {\phi }_2+{\phi }_3+k{\left(f_n\right)}^{3}T\le {\alpha }_2{\phi }_0,\hfill \end{array}$$

(36d)

$$\begin{array}{cc}\hfill & {\psi }_1\le {\tau }_{2}W{log}_2\left(1+{\displaystyle \frac{{\phi }_{2}B}{{\tau }_2}}\right),\hfill \end{array}$$

(36e)

$$\begin{array}{cc}\hfill & {\psi }_1\le {\tau }_{1}W{log}_2\left(1+{\displaystyle \frac{{\phi }_{1}A}{{\tau }_1}}\right),\hfill \end{array}$$

(36f)

$$\begin{array}{cc}\hfill & {\psi }_1\le Q_f-{\displaystyle \frac{f_{f}T}{{\varphi }_f}},\hfill \end{array}$$

(36g)

$$\begin{array}{cc}\hfill & {\psi }_2\le {\tau }_{3}W{log}_2\left(1+{\displaystyle \frac{{\phi }_{3}B}{{\tau }_3}}\right),\hfill \end{array}$$

(36h)

$$\begin{array}{cc}\hfill & {\psi }_2\le Q_n-{\displaystyle \frac{f_{n}T}{{\varphi }_n}},,\hfill \\ \hfill & 0\le {\phi }_0\le {\tau }_0{P}_a^{\max },0\le {\phi }_1\le {\tau }_1{P}_f^{\max },\hfill \end{array}$$

(36i)

$$\begin{array}{cc}\hfill & 0\le {\phi }_2\le {\tau }_2{P}_n^{\max },0\le {\phi }_3\le {\tau }_3{P}_n^{\max },\hfill \end{array}$$

(36j)

$$\left(22j\right),$$

(36k)

${\psi }_1={\tau }_{1}W{log}_2\left(1+{\displaystyle \frac{{\phi }_{1}A}{{\tau }_1}}\right),{\psi }_2={\tau }_{3}W{log}_2\left(1+{\displaystyle \frac{{\phi }_{3}B}{{\tau }_3}}\right)$ , when the problem (P4) reaches the optimal solution, it aligns with the original problem (P3). Specifically, for constraint (36e), ${\tau }_{2}W{log}_2\left(1+{\displaystyle \frac{{\phi }_{2}B}{{\tau }_2}}\right)$ is a concave function of ${\tau }_2$ since the perspective operation applied to $W{log}_2\left(1+{\phi }_{2}B\right)$ preserves convexity [45] . Thus, constraint (36e) is convex, making problem (P4) a convex optimization problem. To further analyze the solution, we will employ convex optimization tools, such as CVX [46].

We introduce an efficient dynamic task offloading algorithm with user assistance to tackle Problem 4. Additionally, we apply the Lagrange method to gain meaningful insights into the optimal solution’s properties.

Theorem 1.

Given non-negative Lagrange multipliers ${\lambda }_i$, $i=1,2,\dots ,12$, the optimal power allocation $\mathit{P}=\left[P_0,P_1,P_2,P_3\right]$ must fulfill certain conditions

$$\begin{array}{cc}\hfill & P_0=\left\{\begin{array}{cc}0,\hfill & \mathit{if}{\tau }_0=0\hfill \\ {\left[P_a^{\max }\right]}^{+},\hfill & \mathit{others}\hfill \end{array}\right.\hfill \end{array}$$

(37a)

$$\begin{array}{cc}\hfill & P_1=\left\{\begin{array}{cc}0,\hfill & \mathit{if}{\tau }_1=0\hfill \\ {\left[{\displaystyle \frac{W\left(Q_f+V+{\lambda }_5\right)}{ln2\left({\lambda }_2+{\lambda }_{10}\right)}}-{\displaystyle \frac{{\sigma }^2}{g_f}}\right]}^{+},\hfill & \mathit{others}\hfill \end{array}\right.\hfill \end{array}$$

(37b)

$$\begin{array}{cc}\hfill & P_2=\left\{\begin{array}{cc}0,\hfill & \mathit{if}{\tau }_2=0\hfill \\ {\left[{\displaystyle \frac{{\lambda }_{4}W}{ln2\left({\lambda }_3+{\lambda }_{11}\right)}}-{\displaystyle \frac{{\sigma }^2}{g_n}}\right]}^{+},\hfill & \mathit{others}\hfill \end{array}\right.\hfill \end{array}$$

(37c)

$$\begin{array}{cc}\hfill & P_3=\left\{\begin{array}{cc}0,\hfill & \mathit{if}{\tau }_3=0\hfill \\ {\left[{\displaystyle \frac{W\left(Q_n+V+{\lambda }_7\right)}{ln2\left({\lambda }_3+{\lambda }_{12}\right)}}-{\displaystyle \frac{{\sigma }^2}{g_n}}\right]}^{+},\hfill & \mathit{others}\hfill \end{array}\right.\hfill \end{array}$$

(37d)

Proof. 

Let ${\lambda }_i\ge 0\left(i=1,2,\dots ,12\right)$ for denote the Lagrange multipliers corresponding to the constraints. The Lagrangian function for problem (P4), constructed based on these multipliers, is as follow

$$\begin{array}{cc}\hfill L\left(\tau ,\phi ,\lambda \right)& =\mathrm{Y}\left(\mathrm{t}\right){\phi }_0+C_1\left\{{\tau }_{1}W{log}_2\left(1+{\displaystyle \frac{{\phi }_{1}A}{{\tau }_1}}\right)+{\displaystyle \frac{f_{f}T}{{\varphi }_f}}\right\}+C_2\left\{{\tau }_{3}W{log}_2\left(1+{\displaystyle \frac{{\phi }_{3}B}{{\tau }_3}}\right)+{\displaystyle \frac{f_{n}T}{{\varphi }_n}}\right\}\hfill \\ \hfill & +{\lambda }_1\left[{\tau }_0^t+{\tau }_1^t+{\tau }_2^t+{\tau }_3^t-\tau \right]\hfill \\ \hfill & +{\lambda }_2\left[{\phi }_1+k{\left(f_f\right)}^{3}T-{\alpha }_1{\phi }_0\right]\hfill \\ \hfill & +{\lambda }_3\left[{\phi }_2+{\phi }_3+k{\left(f_n\right)}^{3}T-{\alpha }_2{\phi }_0\right]\hfill \\ \hfill & +{\lambda }_4\left[{\psi }_1-{\tau }_{2}W{log}_2\left(1+{\displaystyle \frac{{\phi }_{2}B}{{\tau }_2}}\right)\right]\hfill \\ \hfill & +{\lambda }_5\left[{\psi }_1-{\tau }_{1}W{log}_2\left(1+{\displaystyle \frac{{\phi }_{1}A}{{\tau }_1}}\right)\right]\hfill \\ \hfill & +{\lambda }_6\left[{\psi }_1-Q_f-{\displaystyle \frac{f_{f}T}{{\varphi }_f}}\right]\hfill \\ \hfill & +{\lambda }_7\left[{\psi }_2-{\tau }_{3}W{log}_2\left(1+{\displaystyle \frac{{\phi }_{3}B}{{\tau }_3}}\right)\right]\hfill \\ \hfill & +{\lambda }_8\left[{\psi }_2-Q_n-{\displaystyle \frac{f_{n}T}{{\varphi }_n}}\right]\hfill \\ \hfill & +{\lambda }_9\left[{\phi }_0-{\tau }_0{P}_a^{\max }\right]\hfill \\ \hfill & +{\lambda }_{10}\left[{\phi }_1-{\tau }_1{P}_f^{\max }\right]\hfill \\ \hfill & +{\lambda }_{11}\left[{\phi }_2-{\tau }_2{P}_n^{\max }\right]\hfill \\ \hfill & +{\lambda }_{12}\left[{\phi }_3-{\tau }_3{P}_n^{\max }\right]\hfill \end{array}$$

(38a)

We can use the first-order optimality conditions. Taking the derivative of the Lagrangian function yields

$$\begin{array}{cc}\hfill & {\phi }_0={\left[{\tau }_0{P}_a^{\max }\right]}^{+},\hfill \end{array}$$

(39a)

$$\begin{array}{cc}\hfill & {\phi }_1={\left[{\displaystyle \frac{{\tau }_{1}W\left(Q_f+V+{\lambda }_5\right)}{ln2\left({\lambda }_2+{\lambda }_{10}\right)}}-{\displaystyle \frac{{\tau }_1{\sigma }^2}{g_f}}\right]}^{+},\hfill \end{array}$$

(39b)

$$\begin{array}{cc}\hfill & {\phi }_2={\left[{\displaystyle \frac{{\tau }_2{\lambda }_{4}W}{ln2\left({\lambda }_3+{\lambda }_{11}\right)}}-{\displaystyle \frac{{\tau }_2{\sigma }^2}{g_n}}\right]}^{+},\hfill \end{array}$$

(39c)

$$\begin{array}{cc}\hfill & {\phi }_3={\left[{\displaystyle \frac{{\tau }_{3}W\left(Q_n+V+{\lambda }_7\right)}{ln2\left({\lambda }_3+{\lambda }_{12}\right)}}-{\displaystyle \frac{{\tau }_3{\sigma }^2}{g_n}}\right]}^2,\hfill \end{array}$$

(39d)

By examining the first-order derivatives, we can establish the necessary conditions for optimality. The relationship between the auxiliary variables and the original variables is leveraged to derive the theorem.    □

According to this theorem, we can infer that during the process of radio frequency energy transfer, higher power leads to better results. As the value of W increases, both FU and NU entities are more motivated to offload data, which results in a reduction of the computational tasks performed locally. Furthermore, an increase in V prompts FU and NU to offload a larger portion of their tasks. This shift towards offloading is driven by the desire to enhance the computation rate, which in turn leads the MDS to increase the volume of data offloaded to meet its objectives.

The process of solving the original MSSO problem, denoted as (P1), is encapsulated within Algorithm 1.

Algorithm 1: User-Assisted Dynamic Resource Allocation Algorithm

4. Simulation Results

5. Conclusions

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