1. Introduction

2. System Model

2.4. Task Offloading Model

2.4.1. BackCom Data Transmission

During ${\epsilon }_1^t$, ${\mathrm{MD}}_1$ offloads tasks to ${\mathrm{MD}}_2$ by BackCom. Let $0\le {\beta }_1\le 1$ denote the reflection coefficient of ${\mathrm{MD}}_1$, where ${\beta }_1$ represents the fraction of the received signal that is utilized as a carrier for data transfer, and the remaining fraction $(1-{\beta }_1)$ is allocated for energy harvesting [35]. Let $g_{12}^t$ represent the channel gain between ${\mathrm{MD}}_1$ and ${\mathrm{MD}}_2$. According to Shannon’s theorem [36], the amount of tasks offloaded from ${\mathrm{MD}}_1$ to ${\mathrm{MD}}_2$ satisfies [8]

where $\zeta$ represents the performance gap reflecting real modulation [11], ${\sigma }^2$ is the noise power. The corresponding energy consumption by circuit is

$$H_{1,\mathrm{ba}}^t=P_1^{\mathrm{ba}}{\epsilon }_1^t$$

(8)

where $P_1^{\mathrm{ba}}$ is the circuit power consumption of ${\mathrm{MD}}_1$ by BackCom, which is a constant value depending on the circuit structure. Due to utilizing BackCom technology, data transmission and energy harvesting can be performed simultaneously. Thus, the energy harvested for ${\mathrm{MD}}_1$ during ${\epsilon }_1^t$ is as [8]

$$E_{1,\mathrm{ba}}^t=\mu h_1^t{P}_0{\epsilon }_1^t\left(1-{\beta }_1^t\right)$$

(9)

After receiving the tasks offloaded from ${\mathrm{MD}}_1$ , ${\mathrm{MD}}_2$ will relay a portion of these tasks to the HAP as a relay. Note that ${\mathrm{MD}}_2$ will also offload its own tasks to the HAP. At time ${\epsilon }_2^t$, the task data transmitted by ${\mathrm{MD}}_2$ through BackCom is constrained as follows

where ${\beta }_2^t$ is the reflection coefficient of the BackCom at ${\mathrm{MD}}_2$. $h_2^t$ represents the channel gain from the HAP to the ${\mathrm{MD}}_2$ at time slot t, and $g_{2a}^t$ denotes the channel gain from the ${\mathrm{MD}}_2$ to the HAP at time slot t. The corresponding energy consumption is

$$H_{2,\mathrm{ba}}^t=P_2^{\mathrm{ba}}{\epsilon }_2^t$$

(11)

where $P_2^{\mathrm{ba}}$ is the circuit power consumption of ${\mathrm{MD}}_2$ by BackCom. The energy harvested by BackCom during ${\epsilon }_2^t$ is

$$E_{2,\mathrm{ba}}^t=\mu h_2^t{P}_0{\epsilon }_2^t\left(1-{\beta }_2^t\right)$$

(12)

2.4.2. AC Data Transmission

After the BackCom offloading is completed, the AC mode offloading is initiated, which proceeds through an offloading process similar to that of BackCom. The AC mode can achieve a higher data transmission rate, but it is incapable of energy harvesting during the offloading phase. During the time fraction ${\epsilon }_3^t$, the upper limit of the amount of task from ${\mathrm{MD}}_1$ offloading to ${\mathrm{MD}}_2$ can be expressed as

where $0\le P_1^t\le P_{\max }$ represents the transmit power allocated to AC at ${\mathrm{MD}}_1$ [8], ${\sigma }^2$ is the noise power and $g_{12}$ represents the channel gain from ${\mathrm{MD}}_1$ and ${\mathrm{MD}}_2$. The corresponding AC offloading energy consumption is

$$H_{1,\mathrm{ac}}^t=\left(P_1^t+P_1^{ac}\right){\epsilon }_3^t$$

(14)

where $P_1^{ac}$ denotes the circuit power of ${\mathrm{MD}}_1$ through AC which is a constant value [8].

The upper limit of the amount task offloading from ${\mathrm{MD}}_2$ to HAP by AC during ${\epsilon }_4^t$ is given by

where $0\le P_2^t\le P_{\max }$ denotes the transmit power of the ${\mathrm{MD}}_2$ through AC. Let $P_2^{ac}$ denote the circuit power of ${\mathrm{MD}}_2$ through AC. The energy consumed for task offloading AC at ${\mathrm{MD}}_2$ in slot t is

$$H_{2,\mathrm{ac}}^t=\left(P_2^t+P_2^{ac}\right){\epsilon }_4^t$$

(16)

3. Problem Formulation

4. Algorithm Design

4.1. Lyapunov Optimization Formulation

To simplify the battery energy queue at ${\mathrm{MD}}_1$ and ${\mathrm{MD}}_2$, we introduce two virtual queues for battery level: ${\widehat{B}}_i\left(t\right)=B_i\left(t\right)-B_{\max },i\in \{1,2\}$. In order to optimize the task queue and the energy queue simultaneously, we define a combined queue vector $\Theta \left(t\right)\triangleq \left[Q_i\left(t\right),{\widehat{B}}_i\left(t\right)\right],i\in \{1,2\}$. Following Lyapunov optimization framework, we obtain the quadratic Lyapunov function and the Lyapunov drift [37] as

$$L\left(\Theta \left(t\right)\right)\triangleq {\displaystyle \frac{1}{2}}\sum _{i\in \{1,2\}}\left(Q_i^2\left(t\right)+{\widehat{B}}_i^2\left(t\right)\right)$$

(24)

and

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

(25)

According to the Lyapunov optimization theory [37], the one slot drift-plus-penalty expression can be expressed as

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

(26)

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 causes the algorithm to focus more on network EE, which may also result in a larger backlog of the task queue. We derive an upper bound of ${\Delta }_V\left(\Theta \left(t\right)\right)$ as Lemma 1.

Lemma 1. 

For any controlstrategy $\left\{\epsilon ,\mathbf{P},\beta ,\mathbf{d}\right\}$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:

$$\begin{array}{cc}\hfill {\Delta }_V\left(\Theta \left(t\right)\right)& \le B-V\mathbb{E}\left\{D_{\mathrm{tot}}\left(t\right)-u\left(t\right)E_{\mathrm{tot}}\left(t\right)\mid \Theta \left(t\right)\right\}\hfill \\ \hfill & +\sum _{i\in \{1,2\}}\left\{Q_i\left(t\right)\mathbb{E}\left\{A_i\left(t\right)-d_{i,\mathrm{loc}}^t-d_{i,\mathrm{off}}^t\mid \Theta \left(t\right)\right\}+{\widehat{B}}_i\left(t\right)\mathbb{E}\left\{E_i^t-H_{i,\mathrm{loc}}^t-H_{i,\mathrm{off}}^t\mid \Theta \left(t\right)\right\}\right\}\hfill \end{array}$$

(27)

where B is a constant value, which satisfies $\forall t$:

$$\begin{array}{cc}\hfill B& \ge {\displaystyle \frac{1}{2}}\sum _{i\in \{1,2\}}\left\{\mathbb{E}\left\{{\left(d_{i,\mathrm{loc}}^t+d_{i,\mathrm{off}}^t\right)}^2+A_i^2\left(t\right)\mid \Theta \left(t\right)\right\}+\mathbb{E}\left\{{E_i^t}^2+{\left(H_{i,\mathrm{loc}}^t+H_{i,\mathrm{off}}^t\right)}^2\mid \Theta \left(t\right)\right\}\right\}\hfill \end{array}$$

(28)

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 the definition of task queues and battery energy queues Eq. (3)-(4), we have

$$\begin{array}{cc}\hfill \Delta Q_i\left(t\right)& ={\displaystyle \frac{1}{2}}\left(Q_i{(t+1)}^2-Q_i{\left(t\right)}^2\right)\hfill \\ & \le {\displaystyle \frac{{\left(d_{i,\mathrm{loc}}^t+d_{i,\mathrm{off}}^t\right)}^2+A_i^2\left(t\right)}{2}}+Q_i\left(t\right)\left(A_i\left(t\right)-d_{i,\mathrm{loc}}^t-d_{i,\mathrm{off}}^t\right),i\in \{1,2\}\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill \Delta {\widehat{B}}_i\left(t\right)& ={\displaystyle \frac{1}{2}}\left({\widehat{B}}_i{(t+1)}^2-{\widehat{B}}_i{\left(t\right)}^2\right)\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}\left[{E_i^t}^2+{\left(H_{i,\mathrm{loc}}^t+H_{i,\mathrm{off}}^t\right)}^2\right]+{\widehat{B}}_i\left(t\right)\left(E_i^t-H_{i,\mathrm{loc}}^t-H_{i,\mathrm{off}}^t\right),i\in \{1,2\}\hfill \end{array}$$

(30)

Combining the above inequalities (29) and (30), we obtain the upper bound of the Lyapunov drift-plus-penalty.    □

According to the drift-plus-penalty technique in Lyapunov optimization theory, we seek to greedily minimize the upper bound of ${\Delta }_V\left(\Theta \left(t\right)\right)$ at each time slot t. Notably the value of $A\left(t\right)$, $Q_i\left(t\right)$ and ${\widehat{B}}_i\left(t\right)$ can be observed at the beginning of each time slot t. By eliminating the constant term in the RHS (right hand side) of (27), the problem P1 can be transformed into a one-time slot problem as

$$\begin{array}{cc}\hfill P2.1:\underset{\epsilon ,\mathbf{P},\beta ,\mathbf{d}}{min}& -\left(Q_1\left(t\right)+{\omega }_{1}V\right)\left(d_{1,\mathrm{loc}}^t+d_{1,\mathrm{off}}^t\right)-\left(Q_2\left(t\right)+{\omega }_{2}V\right)\left(d_{2,\mathrm{loc}}^t+d_{2,\mathrm{off}}^t\right)\hfill \\ \hfill & +\left(Vu\left(t\right)-{\widehat{B}}_1\left(t\right)\right)\left(H_{1,\mathrm{loc}}^t+H_{1,\mathrm{off}}^t\right)+\left(Vu\left(t\right)-{\widehat{B}}_2\left(t\right)\right)\left(H_{2,\mathrm{loc}}^t+H_{2,\mathrm{off}}^t\right)\hfill \end{array}$$

(31a)

$$\begin{array}{cc}\hfill & +{\widehat{B}}_1\left(t\right)E_1^t+{\widehat{B}}_2\left(t\right)E_2^t\hfill \end{array}$$

(31b)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& \left(20b\right)-\left(20c\right),\left(20e\right)-\left(20i\right)\hfill \end{array}$$

Due to the coupling between the parameters $\epsilon$, $\mathbf{P}$ and $\beta$, the proposed problem P2.1 is still a non-convex problem. To decouple these parameters, we introduce auxiliary variables ${\phi }_1^t=P_1^t{\epsilon }_3^t$, ${\phi }_2^t=P_2^t{\epsilon }_4^t$, ${\phi }_3^t={\epsilon }_1^t{\beta }_1$, ${\phi }_4^t={\epsilon }_2^t{\beta }_2$. Then the problem P2.1 can be simplified as:

$$\begin{array}{cc}\hfill P2.2:\underset{\epsilon ,\phi ,\mathbf{d}}{min}& -\sum _{i\in \{1,2\}}{C}_i\left(d_{i,\mathrm{loc}}^t+d_{i,\mathrm{off}}^t\right)\hfill \\ \hfill & +C_3{H}_{1,\mathrm{loc}}^t+\left(C_3{P}_1^{\mathrm{ba}}+C_5+C_6\right){\epsilon }_1^t+C_3{\phi }_1^t\hfill \\ \hfill & +C_4{H}_{2,\mathrm{loc}}^t+\left(C_4{P}_2^{\mathrm{ba}}+C_5+C_6\right){\epsilon }_2^t+C_4{\phi }_2^t\hfill \end{array}$$

(32a)

$$\begin{array}{cc}\hfill & +\left(C_5+C_6\right){\epsilon }_0^t-C_5{\phi }_3^t-C_6{\phi }_4^t+C_7{\epsilon }_3^t+C_8{\epsilon }_4^t\hfill \end{array}$$

(32b)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& {\epsilon }_0^t+{\epsilon }_1^t+{\epsilon }_2^t+{\epsilon }_3^t+{\epsilon }_4^t\le T\hfill \end{array}$$

$$\begin{array}{cc}\hfill & B_{\min }\le B_i\left(t\right)-H_{i,\mathrm{loc}}^t-H_{i,\mathrm{off}}^t+E_i^t\le B_{\max },i\in \{1,2\}\hfill \end{array}$$

(32c)

$$\begin{array}{cc}\hfill & d_{i,\mathrm{loc}}^t+d_{i,\mathrm{off}}^t\le Q_i\left(t\right),i\in \{1,2\}\hfill \end{array}$$

(32d)

$$\begin{array}{cc}\hfill & d_{1,\mathrm{ba}}^t+d_{1,\mathrm{ac}}^t\le d_{2,\mathrm{ba}}^t+d_{2,\mathrm{ac}}^t\hfill \end{array}$$

(32e)

$$\begin{array}{cc}\hfill & d_{1,\mathrm{ba}}^t\le {\epsilon }_1^{t}W{log}_2\left(1+{\displaystyle \frac{\zeta P_0{h}_1^t{g}_{12}^t{\phi }_3^t}{{\epsilon }_1^t{\sigma }^2}}\right)\hfill \end{array}$$

(32f)

$$\begin{array}{cc}\hfill & d_{2,\mathrm{ba}}^t\le {\epsilon }_2^{t}W{log}_2\left(1+{\displaystyle \frac{\zeta P_0{h}_2^t{g}_{2a}^t{\phi }_4^t}{{\epsilon }_2^t{\sigma }^2}}\right)\hfill \end{array}$$

(32g)

$$\begin{array}{cc}\hfill & d_{1,\mathrm{ac}}^t\le {\epsilon }_3^{t}W{log}_2\left(1+{\displaystyle \frac{g_{12}^t{\phi }_1^t}{{\epsilon }_3^t{\sigma }^2}}\right)\hfill \end{array}$$

(32h)

$$\begin{array}{cc}\hfill & d_{2,\mathrm{ac}}^t\le {\epsilon }_4^{t}W{log}_2\left(1+{\displaystyle \frac{g_{2a}^t{\phi }_2^t}{{\epsilon }_4^t{\sigma }^2}}\right)\hfill \end{array}$$

(32i)

$$\begin{array}{cc}\hfill & {\epsilon }_0^t,{\epsilon }_1^t,{\epsilon }_2^t,{\epsilon }_3^t,{\epsilon }_4^t\ge 0,\hfill \end{array}$$

(32j)

$$\begin{array}{cc}\hfill & 0\le {\phi }_1^t,{\phi }_2^t\le P_{\max }T\hfill \end{array}$$

(32k)

where $C_1=Q_1\left(t\right)+{\omega }_{1}V$, $C_2=Q_2\left(t\right)+{\omega }_{2}V$, $C_3=Vu\left(t\right)-{\widehat{B}}_1\left(t\right)$, $C_4=Vu\left(t\right)-{\widehat{B}}_2\left(t\right)$, $C_5=\mu h_1^t{P}_0{\widehat{B}}_1\left(t\right)$, $C_6=\mu h_2^t{P}_0{\widehat{B}}_2\left(t\right)$, $C_7=C_3{P}_1^{\mathrm{ac}}$, $C_8=C_4{P}_2^{\mathrm{ac}}$.

Lemma 2. 

P2.2 is a convex optimization problem, which can be efficiently solved by convex optimization tools, such as CVX [40].

Proof. 

In problem (P2.2), the objective function (32a) is linear with respect to all variables. Constraints (32b)-(32e) are all linear inequality constraints. What’s more, for constraint (32f), ${log}_2\left(1+{\displaystyle \frac{\zeta P_0{h}_1^t{g}_{12}^t{\phi }_4^t}{{\epsilon }_1^t{\sigma }^2}}\right)$ is the perspective of ${log}_2\left(1+{\displaystyle \frac{\zeta P_0{h}_1^t{g}_{12}^t{\beta }_1}{{\sigma }^2}}\right)$ which is concave with respect to ${\phi }_4$. Since the perspective operation preserves convexity [41], ${\epsilon }_1^t{log}_2\left(1+{\displaystyle \frac{\zeta P_0{h}_1^t{g}_{12}^t{\phi }_4^t}{{\epsilon }_1^t{\sigma }^2}}\right)$ is concave with ${\phi }_4$ and ${\epsilon }_1^t$ and (32f) is a convex constraint. For the same reason, (32g)-(32i) are all convex constraints. Thus, P2.2 is a convex problem.    □

According to Lemma 2, at each time slot, we only need to solve a convex problem P2.2, 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 Backscatter-Assisted WPMEC Algorithm (DOBAM), is summarized as Algorithm 1.

5. Simulation Results

6. Conclusions

References

1. Zhao, R.; Zhu, F.; Tang, M.; He, L. Profit maximization in cache-aided intelligent computing networks. Physical Communication 2023, 58, 102065. [Google Scholar] [CrossRef]
2. 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]
3. Wang, F.; Xu, J.; Cui, S. Optimal energy allocation and task offloading policy for wireless powered mobile edge computing systems. IEEE Transactions on Wireless Communications 2020, 19, 2443–2459. [Google Scholar] [CrossRef]
4. Wei, Z.; Zhang, B.; Lin, S.; Wang, C. A self-oscillation WPT system with high misalignment tolerance. IEEE Transactions on Power Electronics 2023. [Google Scholar] [CrossRef]
5. 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, 10, 12156–12168. [Google Scholar] [CrossRef]
6. Ju, H.; Zhang, R. Throughput maximization in wireless powered communication networks. IEEE Transactions on Wireless Communications 2013, 13, 418–428. [Google Scholar] [CrossRef]
7. He, H.; Zhou, C.; Huang, F.; Shen, H.; Li, S. Energy-Efficient Task Offloading in Wireless-Powered MEC: A Dynamic and Cooperative Approach. Mathematics 2024, 12. [Google Scholar] [CrossRef]
8. 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, 23, 1878–1887. [Google Scholar] [CrossRef]
9. 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]
10. Hoang, D.T.; Niyato, D.; Wang, P.; Kim, D.I.; Han, Z. Ambient backscatter: A new approach to improve network performance for RF-powered cognitive radio networks. IEEE Transactions on Communications 2017, 65, 3659–3674. [Google Scholar] [CrossRef]
11. Shi, L.; Ye, Y.; Chu, X.; Sun, S.; Lu, G. Energy-efficient resource allocation for backscatter-assisted wireless powered MEC. IEEE Transactions on Vehicular Technology 2023, 72, 9591–9596. [Google Scholar] [CrossRef]
12. Ye, Y.; Shi, L.; Chu, X.; Lu, G. Throughput fairness guarantee in wireless powered backscatter communications with HTT. IEEE Wireless Communications Letters 2020, 10, 449–453. [Google Scholar] [CrossRef]
13. Ye, Y.; Shi, L.; Chu, X.; Hu, R.Q.; Lu, G. Resource allocation in backscatter-assisted wireless powered MEC networks with limited MEC computation capacity. IEEE Transactions on Wireless Communications 2022, 21, 10678–10694. [Google Scholar] [CrossRef]
14. Ji, L.; Guo, S. Energy-efficient cooperative resource allocation in wireless powered mobile edge computing. IEEE Internet of Things Journal 2018, 6, 4744–4754. [Google Scholar] [CrossRef]
15. Sabella, D.; Vaillant, A.; Kuure, P.; Rauschenbach, U.; Giust, F. Mobile-edge computing architecture: The role of MEC in the Internet of Things. IEEE Consumer Electronics Magazine 2016, 5, 84–91. [Google Scholar] [CrossRef]
16. Huang, M.; Liu, W.; Wang, T.; Liu, A.; Zhang, S. A cloud–MEC collaborative task offloading scheme with service orchestration. IEEE Internet of Things Journal 2019, 7, 5792–5805. [Google Scholar] [CrossRef]
17. Sun, G.; Wang, Z.; Su, H.; Yu, H.; Lei, B.; Guizani, M. Profit Maximization of Independent Task Offloading in MEC-Enabled 5G Internet of Vehicles. IEEE Transactions on Intelligent Transportation Systems, 2024; 1–13. [Google Scholar] [CrossRef]
18. 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]
19. Zhang, S.; Bao, S.; Chi, K.; Yu, K.; Mumtaz, S. DRL-based computation rate maximization for wireless powered multi-AP edge computing. IEEE Transactions on Communications 2023. [Google Scholar] [CrossRef]
20. Wu, X.; Yan, X.; Yuan, S.; Li, C. Deep Reinforcement Learning-Based Adaptive Offloading Algorithm for Wireless Power Transfer-Aided Mobile Edge Computing. 2024 IEEE Wireless Communications and Networking Conference (WCNC). IEEE, 2024, pp. 1–6.
21. 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.
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). IEEE, 2019, pp. 1–7.
23. Iwaki, A.; Sanada, K.; Hatano, H.; Mori, K. Performance Analysis of Wireless Powered Communication Networks Based on Harvest-then-Access Scheduling Considering Double-near far Problem. IEICE Technical Report; IEICE Tech. Rep. 2022, 122, 42–47. [Google Scholar]
24. Zhang, W.; Zhang, G.; Mao, S. Deep-Reinforcement-Learning-Based Joint Caching and Resources Allocation for Cooperative MEC. IEEE Internet of Things Journal 2024, 11, 12203–12215. [Google Scholar] [CrossRef]
25. Sun, Y.; Xu, J.; Cui, S. User association and resource allocation for MEC-enabled IoT networks. IEEE Transactions on Wireless Communications 2022, 21, 8051–8062. [Google Scholar] [CrossRef]
26. Wu, T.; He, H. An Efficient Energy Efficiency Maximization Algorithm for Backscatter-Assisted WP-MEC Network with Relay. Proceedings of the 2024 16th International Conference on Machine Learning and Computing; ICMLC ’24; Association for Computing Machinery: New York, NY, USA, 2024; pp. 720–727. [Google Scholar] [CrossRef]
27. Ye, Y.; Shi, L.; Chu, X.; Li, D.; Lu, G. Delay minimization in wireless powered mobile edge computing with hybrid BackCom and AT. IEEE Wireless Communications Letters 2021, 10, 1532–1536. [Google Scholar] [CrossRef]
28. Correia, R.; Carvalho, N.B.; Kawasaki, S. Continuously power delivering for passive backscatter wireless sensor networks. IEEE transactions on microwave theory and techniques 2016, 64, 3723–3731. [Google Scholar] [CrossRef]
29. Han, K.; Huang, K. Wirelessly powered backscatter communication networks: Modeling, coverage, and capacity. IEEE Transactions on Wireless Communications 2017, 16, 2548–2561. [Google Scholar] [CrossRef]
30. Lyu, B.; Guo, H.; Yang, Z.; Gui, G. Throughput maximization for hybrid backscatter assisted cognitive wireless powered radio networks. IEEE Internet of Things Journal 2018, 5, 2015–2024. [Google Scholar] [CrossRef]
31. Shi, L.; Ye, Y.; Chu, X.; Lu, G. Computation bits maximization in a backscatter assisted wirelessly powered MEC network. IEEE Communications Letters 2020, 25, 528–532. [Google Scholar] [CrossRef]
32. Lin, D.; Wang, K.; Wang, T.; Ding, Z. Uplink Data Rate Maximization in Multi-Cell BackCom NOMA Systems. IEEE Open Journal of the Communications Society 2024, 5, 526–539. [Google Scholar] [CrossRef]
33. Fu, Z.; Shi, L.; Ye, Y.; Zhang, Y.; Zheng, G. Computation EE Fairness for a UAV-Enabled Wireless Powered MEC Network With Hybrid Passive and Active Transmissions. IEEE Internet of Things Journal 2024. [Google Scholar] [CrossRef]
34. Bi, S.; Zhang, Y.J. Computation rate maximization for wireless powered mobile-edge computing with binary computation offloading. IEEE Transactions on Wireless Communications 2018, 17, 4177–4190. [Google Scholar] [CrossRef]
35. Xu, Y.; Gu, B.; Hu, R.Q.; Li, D.; Zhang, H. Joint computation offloading and radio resource allocation in MEC-based wireless-powered backscatter communication networks. IEEE Transactions on Vehicular Technology 2021, 70, 6200–6205. [Google Scholar] [CrossRef]
36. Verdu, S. Fifty years of Shannon theory. IEEE Transactions on information theory 1998, 44, 2057–2078. [Google Scholar] [CrossRef]
37. Neely, M. Stochastic network optimization with application to communication and queueing systems; Springer Nature, 2022.
38. 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]
39. Dinkelbach, W. On nonlinear fractional programming. Management science 1967, 13, 492–498. [Google Scholar] [CrossRef]
40. Diamond, S.; Boyd, S. CVXPY: A Python-embedded modeling language for convex optimization. Journal of Machine Learning Research 2016, 17, 1–5. [Google Scholar]
41. 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]
42. 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]
43. He, H.; Zhou, C.; Huang, F.; Shen, H.; Li, S. Energy-Efficient Task Offloading in Wireless-Powered MEC: A Dynamic and Cooperative Approach 2024.
44. Shi, L.; Ye, Y.; Chu, X.; Lu, G. Computation Bits Maximization in a Backscatter Assisted Wirelessly Powered MEC Network. IEEE Communications Letters 2021, 25, 528–532. [Google Scholar] [CrossRef]
45. Xu, Y.; Gui, G. Optimal Resource Allocation for Wireless Powered Multi-Carrier Backscatter Communication Networks. IEEE Wireless Communications Letters 2020, 9, 1191–1195. [Google Scholar] [CrossRef]