1. Introduction

2. System Model

3. Algorithm Design

3.1. Lyapunov Optimization Formulation

To simply the battery energy queue at MD and helper, we introduce two virtual queues ${\widehat{B}}_m\left(t\right)=B_m\left(t\right)-b_m^{\max }$ and ${\widehat{B}}_h\left(t\right)=B_h\left(t\right)-b_h^{\max }$. Following the Lyapunov optimization framework, we define a combined queue vector $\Theta \left(t\right)\triangleq \left[\mathit{Q}\left(t\right),{\widehat{B}}_m\left(t\right),{\widehat{B}}_h\left(t\right)\right]$ and the quadratic Lyapunov function as:

$$L\left(\Theta \left(t\right)\right)\triangleq \frac{1}{2}\left(Q^2\left(t\right)+{\widehat{B}}_m^2\left(t\right)+{\widehat{B}}_h^2\left(t\right)\right)$$

(22)

Then, we obtain the one-period conditional Lyapunov drift [34] as follows:

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

(23)

Note that $L\left(\Theta \left(t\right)\right)$ represents the congestion of all queues $\mathit{Q}\left(t\right)$,${\widehat{B}}_m\left(t\right)$ and ${\widehat{B}}_h\left(t\right)$. According to the Lyapunov optimization theory [34], we derive the one slot drift-plus-penalty expression as

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

(24)

where control parameter V is a positive value, used to balance the trade-off between network EE and network stability. Actually, V acts as a weighting factor of the cost optimality in the drift-plus-penalty expression. Increasing the value of V implies algorithm pay more attention to the network EE , that also means resulting in a larger backlog of task queue Q. We derive an upper bound of ${\Delta }_V\left(\Theta \left(t\right)\right)$ as Lemma 1.

Lemma 1. 

For any controlstrategy $\left\{\overrightarrow{\tau }\left(t\right),\overrightarrow{P}\left(t\right)\right\}$at each time slot t, the one slot Lyapunov drift-plus-penalty${\Delta }_V\left(\Theta \left(t\right)\right)$ is bounded by the following inequality:

$$\begin{array}{cc}\hfill {\Delta }_V\left(\Theta \left(t\right)\right)& \le B-V\mathbb{E}\left\{D_{\mathrm{tot}}\left(t\right)-u\left(t\right)E_{\mathrm{tot}}\left(t\right)\mid \Theta \left(t\right)\right\}+Q\left(t\right)\mathbb{E}\left\{A\left(t\right)-d_m\left(t\right)\mid \Theta \left(t\right)\right\}\hfill \\ \hfill & +{\widehat{B}}_m\left(t\right)\mathbb{E}\left\{e_m^{\mathrm{eh}}\left(t\right)-e_m\left(t\right)\mid \Theta \left(t\right)\right\}+{\widehat{B}}_h\left(t\right)\mathbb{E}\left\{e_h^{\mathrm{eh}}\left(t\right)-e_h\left(t\right)\mid \Theta \left(t\right)\right\}\hfill \end{array}$$

(25)

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

$$\begin{array}{cc}\hfill B& \ge \frac{1}{2}\mathbb{E}\left\{d_m{\left(t\right)}^2+A^2\left(t\right)\mid \Theta \left(t\right)\right\}+\frac{1}{2}\mathbb{E}\left\{e_m^{\mathrm{eh}}{\left(t\right)}^2+e_m{\left(t\right)}^2\mid \Theta \left(t\right)\right\}\hfill \\ \hfill & +\frac{1}{2}\mathbb{E}\left\{e_h^{\mathrm{eh}}{\left(t\right)}^2+e_h{\left(t\right)}^2\mid \Theta \left(t\right)\right\}\hfill \end{array}$$

(26)

Proof. 

By using the inequality ${\left(max[a-b,0]+c\right)}^2\le a^2+b^2+c^2+2a\left(c-b\right)$ , $\forall a,b,c\ge 0$ and combining Eq. (3), we have

$$\begin{array}{cc}\hfill \Delta Q\left(t\right)& =\frac{1}{2}\left(Q{(t+1)}^2-Q{\left(t\right)}^2\right)\le \frac{d_m{\left(t\right)}^2+A^2\left(t\right)}{2}+Q\left(t\right)\left(A\left(t\right)-d_m\left(t\right)\right)\hfill \end{array}$$

(27)

Based on the definition of battery energy queue ${\widehat{B}}_m\left(t\right)$ and ${\widehat{B}}_h\left(t\right)$, we have

$$\begin{array}{cc}\hfill \Delta {\widehat{B}}_m\left(t\right)& =\frac{1}{2}\left({\widehat{B}}_m{(t+1)}^2-{\widehat{B}}_m{\left(t\right)}^2\right)\le \frac{1}{2}\left[e_m^{\mathrm{eh}}{\left(t\right)}^2+e_m{\left(t\right)}^2\right]+{\widehat{B}}_m\left(t\right)\left(e_m^{\mathrm{eh}}\left(t\right)-e_m\left(t\right)\right)\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill \Delta {\widehat{B}}_h\left(t\right)& =\frac{1}{2}\left({\widehat{B}}_h{(t+1)}^2-{\widehat{B}}_h{\left(t\right)}^2\right)\le \frac{1}{2}\left[e_h^{\mathrm{eh}}{\left(t\right)}^2+e_h{\left(t\right)}^2\right]+{\widehat{B}}_h\left(t\right)\left(e_h^{\mathrm{eh}}\left(t\right)-e_h\left(t\right)\right)\hfill \end{array}$$

(29)

Combining the above inequalities (27), (28) and (29), we obtain the upper bound of the Lyapunov drift-plus-penalty.    □

According to the drift-plus-penalty technique in Lyapunov optimization theory [34], we seek to greedily minimize the upper bound of ${\Delta }_V\left(\Theta \left(t\right)\right)$ at each time slot t, then we can obtain a close-to-optimal solution of problem P2. Therefore, we transform problem P2 to a minimization problem of the RHS (right hand side) of (25). Note that we can observe the value of $A\left(t\right)$, $Q\left(t\right)$, ${\widehat{B}}_m\left(t\right)$ and ${\widehat{B}}_h\left(t\right)$ at the beginning of each slot t, so we can solve the optimizing problem at each slot. Then the one time slot problem can be represented as

$$\begin{array}{cc}\hfill P2.1:\underset{\overrightarrow{\tau }\left(t\right),\overrightarrow{P}\left(t\right)}{min}& Q\left(t\right)\left[A\left(t\right)-d_m^{\mathrm{loc}}+d_m^{\mathrm{off}}\left(t\right)\right]+{\widehat{B}}_m\left(t\right)\left[e_m^{\mathrm{eh}}\left(t\right)-e_m\left(t\right)\right]\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +{\widehat{B}}_h\left(t\right)\left[e_h^{\mathrm{eh}}\left(t\right)-e_h\left(t\right)\right]-V\left[D_{\mathrm{tot}}\left(t\right)-u\left(t\right)E_{\mathrm{tot}}\left(t\right)\right]\hfill \end{array}$$

(30a)

$$\begin{array}{cc}\hfill s.t.& \left(18b\right)-\left(18d\right),\left(18f\right)-\left(18i\right)\hfill \end{array}$$

(30b)

The proposed problem P2.1 is a non-convex problem and can not be easily solved by classic convex optimization methods. To address this issue, we first introduce auxiliary variables ${\phi }_1=p_m^t{\tau }_1^t$ and ${\phi }_2=p_h^t{\tau }_2^t$. Then the problem P2.1 can be simplified as:

$$\begin{array}{cc}\hfill P2.2:\underset{\overrightarrow{\tau }\left(t\right),\overrightarrow{\phi }\left(t\right)}{min}& c_{1}W{\tau }_1^{t}log\left(1+{\displaystyle \frac{{\phi }_1}{{\tau }_1^t}}{s}_1\right)+c_2{\tau }_0^t+c_3{\phi }_1+c_4{\phi }_2+k{t}_2\hfill \end{array}$$

(31a)

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

(31b)

$$\begin{array}{cc}\hfill & a_1-{\phi }_1+{\tau }_0^t{h}_1\ge 0\hfill \end{array}$$

(31c)

$$\begin{array}{cc}\hfill & a_2-k{\tau }_2^t-{\phi }_2+{\tau }_0^t{h}_2\ge 0\hfill \end{array}$$

(31d)

$$\begin{array}{cc}\hfill & {\tau }_1^{t}Wlog\left(1+{\displaystyle \frac{{\phi }_1}{{\tau }_1^t}}{s}_1\right)\le Q_t-a_3\hfill \end{array}$$

(31e)

$$\begin{array}{cc}\hfill & a_4{\tau }_2^t+{\tau }_2^{t}Wlog\left(1+{\displaystyle \frac{{\phi }_2}{{\tau }_2^t}}{s}_2\right)\ge {\tau }_1^{t}Wlog\left(1+{\displaystyle \frac{{\phi }_1}{{\tau }_1^t}}{s}_1\right)\hfill \end{array}$$

(31f)

$$\begin{array}{cc}\hfill & {\tau }_0^t,{\tau }_1^t,{\tau }_2^t\ge 0\hfill \end{array}$$

(31g)

$$\begin{array}{cc}\hfill & 0\le {\phi }_1\le \tau P_m^{\max },0\le {\phi }_2\le \tau P_h^{\max }\hfill \end{array}$$

(31h)

where $a_1={\widehat{B}}_m\left(t\right)-\kappa {\left(f_m\right)}^3\tau$, $a_2={\widehat{B}}_h\left(t\right)-\kappa {\left(f_h\right)}^3\tau$, $a_3={\displaystyle \frac{f_m\tau }{{\varphi }_m}}$, $a_4$, $c_1=-\left[Q\left(t\right)+V\right]$,$c_2={\widehat{B}}_m\left(t\right)\mu h_m^t{\tau }_0^t{P}_0+{\widehat{B}}_h\left(t\right)\mu h_h^t{\tau }_0^t{P}_0$, $c_3=Vu-{\widehat{B}}_m\left(t\right)$, $c_4=Vu\left(t\right)-{\widehat{B}}_h\left(t\right)$, $k=c_3\kappa {\left(f_m\right)}^3$, $h_1=\mu h_m^t{P}_0$, $h_2=\mu h_h^t{P}_0$, $s_1=\frac{g_m^t}{{\sigma }^2}$, $s_2=\frac{g_h^t}{{\sigma }^2}$. The problem P2.2 is still a non-convex problem due to the non-convex constraint (31f). In (31f), both sides of the equation is concave that do not satisfy the the conditions for convex constraints. We introduce auxiliary variable $\psi$ to replace the concave function ${\tau }_{1}Wlog\left(1+{\displaystyle \frac{{\phi }_1}{{\tau }_1^t}}{s}_1\right)$. $\psi$ satisfies

$$\psi \le {\tau }_{1}Wlog\left(1+{\displaystyle \frac{{\phi }_1}{{\tau }_1^t}}{s}_1\right)$$

(32)

Bring in $\psi$ and constraint (32), the problem P2.2 can be transformed into

$$\begin{array}{cc}\hfill P3:\underset{\overrightarrow{\tau }\left(t\right),\overrightarrow{\phi }\left(t\right)}{min}& c_1\psi +c_2{\tau }_0^t+c_3{\phi }_1+c_4{\phi }_2+k{t}_2\hfill \end{array}$$

(33a)

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

(33b)

$$\begin{array}{cc}\hfill & a_1-{\phi }_1+{\tau }_0^t\mu h_m^t{P}_0\ge 0\hfill \end{array}$$

(33c)

$$\begin{array}{cc}\hfill & a_2-k{\tau }_2^t-{\phi }_2+{\tau }_0^t\mu h_h^t{P}_0\ge 0\hfill \end{array}$$

(33d)

$$\begin{array}{cc}\hfill & a_4{\tau }_2^t+{\tau }_2^{t}Wlog\left(1+{\displaystyle \frac{{\phi }_2}{{\tau }_2^t}}{s}_2\right)\ge \psi \hfill \end{array}$$

(33e)

$$\begin{array}{cc}\hfill & \psi \le {\tau }_{1}Wlog\left(1+{\displaystyle \frac{{\phi }_1}{{\tau }_1^t}}{s}_1\right)\hfill \end{array}$$

(33f)

$$\begin{array}{cc}\hfill & \psi \le Q_t-a_3\hfill \end{array}$$

(33g)

$$\begin{array}{cc}\hfill & {\tau }_0^t,{\tau }_1^t,{\tau }_2^t\ge 0\hfill \end{array}$$

(33h)

$$\begin{array}{cc}\hfill & 0\le {\phi }_1\le \tau P_m^{\max },0\le {\phi }_2\le \tau P_h^{\max },\hfill \end{array}$$

(33i)

Lemma 2. 

$\psi ={\tau }_{1}Wlog\left(1+{\displaystyle \frac{{\phi }_1}{{\tau }_1^t}}{s}_1\right)$ when the problem P3 reaches the optimal solution, which is consistent with P2. P3 is a convex optimization problem, which can be efficiently solved by convex optimization tools, such as CVX[35].

Proof. 

step1. Assuming that $\psi <{\tau }_{1}Wlog\left(1+{\displaystyle \frac{{\phi }_1}{{\tau }_1^t}}{s}_1\right)$, we can reduce the objective function by simply increasing $\psi$ to ${\tau }_{1}Wlog\left(1+{\displaystyle \frac{{\phi }_1}{{\tau }_1^t}}{s}_1\right)$. Thus, a better solution is found.

step2. In problem P3, the objective function (33a) is linear with respect to all variables. Constraints (33b)-(33d) and (33g) are all linear inequality constraints. What’s more, for constraint (33e), ${\tau }_2^{t}Wlog\left(1+{\displaystyle \frac{{\phi }_2}{{\tau }_2^t}}{s}_2\right)$ is the perspective of $Wlog(1+{\phi }_2{s}_2)$, which is a concave function of ${\phi }_2$. Since the perspective operation preserves convexity [36], ${\tau }_2^{t}Wlog\left(1+{\displaystyle \frac{{\phi }_2}{{\tau }_2^t}}{s}_2\right)$ is concave with respect to ${\phi }_2$ and ${\tau }_2^t$. It is obvious that $a_4{\tau }_2^t$ and $\psi$ are linear functions. Thus, the (33e) is a convex constraint. For the same reason, ${\tau }_{1}Wlog\left(1+{\displaystyle \frac{{\phi }_1}{{\tau }_1^t}}{s}_1\right)$ in (33f) is concave with respect to ${\phi }_1$ and ${\tau }_1^t$ so that (33f) is a convex constraint. Thus P3 is proved to be convex.    □

According to Lemma 2, at each time slot, we only need to solve a convex problem, P3, which contains a small number of variables. By doing so, we can achieve optimal long-term average EE, even without knowledge of future system information. Our proposed algorithm, the Dynamic Offloading for User Cooperation Algorithm (DOUCA), is summarized as Algorithm 1.

Algorithm 1: Dynamic Offloading for User Cooperation Algorithm (DOUCA)

Input: the task arrical queue $A\left(t\right)$; the channel gain $\{h_m^t$, $h_h^t$, $g_m^t$, $g_h^t\}$. Output: $\overrightarrow{\tau }\left(t\right),\overrightarrow{\phi }\left(t\right)$

4. Simulation Results

5. Conclusions

References

1. Wang, X.; Li, J.; Ning, Z.; Song, Q.; Guo, L.; Guo, S.; Obaidat, M.S. Wireless powered mobile edge computing networks: A survey. ACM Computing Surveys 2023, 55, 1–37. [Google Scholar] [CrossRef]
2. Wu, T.; He, H.; Shen, H.; Tian, H. Energy-Efficiency Maximization for Relay-Aided Wireless-Powered Mobile Edge Computing. IEEE Internet of Things Journal 2024, 11, 18534–18548. [Google Scholar] [CrossRef]
3. Mi, X.; He, H.; Shen, H. A Multi-Agent RL Algorithm for Dynamic Task Offloading in D2D-MEC Network with Energy Harvesting. Sensors 2024, 24. [Google Scholar] [CrossRef] [PubMed]
4. He, Y.; Wu, X.; He, Z.; Guizani, M. Energy efficiency maximization of backscatter-assisted wireless-powered MEC with user cooperation. IEEE Transactions on Mobile Computing 2023. [Google Scholar] [CrossRef]
5. Guo, H.; Liu, J.; Zhang, J.; Sun, W.; Kato, N. Mobile-edge computation offloading for ultradense IoT networks. IEEE Internet of Things Journal 2018, 5, 4977–4988. [Google Scholar] [CrossRef]
6. Hu, Y.C.; Patel, M.; Sabella, D.; Sprecher, N.; Young, V. Mobile edge computing—A key technology towards 5G. ETSI white paper 2015, 11, 1–16. [Google Scholar]
7. Shi, L.; Ye, Y.; Chu, X.; Lu, G. Computation energy efficiency maximization for a NOMA-based WPT-MEC network. IEEE Internet of Things Journal 2021, 8, 10731–10744. [Google Scholar] [CrossRef]
8. Li, G.; Zeng, M.; Mishra, D.; Hao, L.; Ma, Z.; Dobre, O.A. Latency minimization for IRS-aided NOMA MEC systems with WPT-enabled IoT devices. IEEE Internet of Things Journal 2023. [Google Scholar] [CrossRef]
9. Mao, S.; Wu, J.; Liu, L.; Lan, D.; Taherkordi, A. Energy-efficient cooperative communication and computation for wireless powered mobile-edge computing. IEEE Systems Journal 2020, 16, 287–298. [Google Scholar] [CrossRef]
10. Mao, S.; Leng, S.; Maharjan, S.; Zhang, Y. Energy Efficiency and Delay Tradeoff for Wireless Powered Mobile-Edge Computing Systems With Multi-Access Schemes. IEEE Transactions on Wireless Communications 2019, PP, 1–1. [Google Scholar] [CrossRef]
11. Zhou, F.; Hu, R.Q. Computation efficiency maximization in wireless-powered mobile edge computing networks. IEEE Transactions on Wireless Communications 2020, 19, 3170–3184. [Google Scholar] [CrossRef]
12. Sun, M.; Xu, X.; Huang, Y.; Wu, Q.; Tao, X.; Zhang, P. Resource management for computation offloading in D2D-aided wireless powered mobile-edge computing networks. IEEE Internet of Things Journal 2020, 8, 8005–8020. [Google Scholar] [CrossRef]
13. Hu, X.; Wong, K.K.; Yang, K. Wireless powered cooperation-assisted mobile edge computing. IEEE Transactions on Wireless Communications 2018, 17, 2375–2388. [Google Scholar] [CrossRef]
14. He, B.; Bi, S.; Xing, H.; Lin, X. Collaborative Computation Offloading in Wireless Powered Mobile-Edge Computing Systems. 2019 IEEE Globecom Workshops (GC Wkshps), 2019, pp. 1–7. [CrossRef]
15. Hu, H.; Song, W.; Wang, Q.; Hu, R.Q.; Zhu, H. Energy Efficiency and Delay Tradeoff in an MEC-Enabled Mobile IoT Network. IEEE Internet of Things Journal 2022, 9, 15942–15956. [Google Scholar] [CrossRef]
16. Boyd, S.P.; Vandenberghe, L. Convex optimization; Cambridge university press, 2004.
17. Neely, M. Stochastic network optimization with application to communication and queueing systems; Springer Nature, 2022.
18. Zhou, F.; Hu, R.Q. Computation Efficiency Maximization in Wireless-Powered Mobile Edge Computing Networks. IEEE Transactions on Wireless Communications 2020, 19, 3170–3184. [Google Scholar] [CrossRef]
19. MEC, N.A.C.T.S. Energy-Efficient Optimization of Multi-User NOMA-Assisted Cooperative THz-SIMO MEC Systems 2023.
20. Ernest, T.Z.H.; Madhukumar, A. Computation Offloading in MEC-Enabled IoV Networks: Average Energy Efficiency Analysis and Learning-Based Maximization. IEEE Transactions on Mobile Computing 2023. [Google Scholar] [CrossRef]
21. Park, J.; Lim, Y. Bio-inspired sleep control for improving the energy efficiency of a mec system. Applied Sciences 2023, 13, 2620. [Google Scholar] [CrossRef]
22. He, B.; Bi, S.; Xing, H.; Lin, X. Collaborative Computation Offloading in Wireless Powered Mobile-Edge Computing Systems. 2019 IEEE Globecom Workshops (GC Wkshps), 2019.
23. Su, B.; Ni, Q.; Yu, W.; Pervaiz, H. Optimizing computation efficiency for NOMA-assisted mobile edge computing with user cooperation. IEEE Transactions on Green Communications and Networking 2021, 5, 858–867. [Google Scholar] [CrossRef]
24. Wang, R.; Chen, J.; He, B.; Lv, L.; Zhou, Y.; Yang, L. Energy consumption minimization for wireless powered NOMA-MEC with user cooperation. 2021 13th International Conference on Wireless Communications and Signal Processing (WCSP). IEEE, 2021, pp. 1–5.
25. Wu, X.; He, Y.; Saleem, A. Computation rate maximization in multi-user cooperation-assisted wireless-powered mobile edge computing with OFDMA. China Communications 2023, 20, 218–229. [Google Scholar] [CrossRef]
26. Zeng, Y.; Chen, S.; Cui, Y.; Du, J. Efficient Trajectory Planning and Dynamic Resource Allocation for UAV-Enabled MEC System. IEEE Communications Letters 2024. [Google Scholar] [CrossRef]
27. Luo, Y.; Lin, S.; Hong, X.; Shi, J. DRL-Assisted Resource Allocation for Non-Completely Overlapping NOMA Based Dynamic MEC Systems. IEEE Internet of Things Journal 2024. [Google Scholar]
28. Liu, L.; Yuan, X.; Chen, D.; Zhang, N.; Sun, H.; Taherkordi, A. Multi-User Dynamic Computation Offloading and Resource Allocation in 5G MEC Heterogeneous Networks With Static and Dynamic Subchannels. IEEE Transactions on Vehicular Technology 2023. [Google Scholar] [CrossRef]
29. Sun, W.; Zhao, Y.; Ma, W.; Guo, B.; Xu, L.; Duong, T.Q. Accelerating convergence of federated learning in mec with dynamic community. IEEE Transactions on Mobile Computing 2023. [Google Scholar] [CrossRef]
30. Dinkelbach, W. On nonlinear fractional programming. Management science 1967, 13, 492–498. [Google Scholar] [CrossRef]
31. Yu, Y.; Bu, X.; Yang, K.; Wu, Z.; Han, Z. Green large-scale fog computing resource allocation using joint benders decomposition, Dinkelbach algorithm, ADMM, and branch-and-bound. IEEE Internet of Things Journal 2018, 6, 4106–4117. [Google Scholar] [CrossRef]
32. Zappone, A.; Jorswieck, E.; others. Energy efficiency in wireless networks via fractional programming theory. Foundations and Trends® in Communications and Information Theory 2015, 11, 185–396. [Google Scholar] [CrossRef]
33. Mao, S.; Leng, S.; Maharjan, S.; Zhang, Y. Energy efficiency and delay tradeoff for wireless powered mobile-edge computing systems with multi-access schemes. IEEE Transactions on Wireless Communications 2019, 19, 1855–1867. [Google Scholar] [CrossRef]
34. Neely, M.J. Stochastic network optimization with non-convex utilities and costs. 2010 Information Theory and Applications Workshop (ITA). IEEE, 2010, pp. 1–10.
35. Diamond, S.; Boyd, S. CVXPY: A Python-embedded modeling language for convex optimization. Journal of Machine Learning Research 2016, 17, 1–5. [Google Scholar]
36. Zawawi, Z.B.; Huang, Y.; Clerckx, B. Multiuser wirelessly powered backscatter communications: Nonlinearity, waveform design, and SINR-energy tradeoff. IEEE Transactions on Wireless Communications 2018, 18, 241–253. [Google Scholar] [CrossRef]
37. Bi, S.; Huang, L.; Wang, H.; Zhang, Y.J.A. Lyapunov-guided deep reinforcement learning for stable online computation offloading in mobile-edge computing networks. IEEE Transactions on Wireless Communications 2021, 20, 7519–7537. [Google Scholar] [CrossRef]