I Introduction
Ia Background and Related Works
The emerging intelligent applications (e.g., automatic navigation, face recognition, unmanned driving, etc.) have imposed great challenges for mobile devices since most of those applications have computationintensive and latencysensitive tasks to be executed
[1], [2]. However, most mobile devices have low computing capability and finite battery capacity. Mobile edge computing (MEC) can significantly augment the computing capability of mobile devices by offloading tasks from mobile devices to the nearby MEC servers in a low latency manner [2]. There exist two operation modes in MEC networks, namely, partial computation offloading and binary computation offloading. In the partial offloading mode, computation tasks can be divided into two parts, one part is executed locally at mobile devices and the other part is offloaded to the MEC server for computing. For the binary computation offloading mode, computation tasks cannot be partitioned. The entire task is either locally executed or completely offloaded to the MEC server for computing [2], [3]. In this paper, both modes are considered.To further improve the energy efficiency, wireless powered techniques that exploit radio frequency (RF) signals as the energy sources for powering the energylimited mobile devices are considered promising and viable approaches in MEC networks since they can provide stable and controllable amount of energy and prolong the battery life of mobile devices [4], [5]. Recently, an increasing attention has been paid to the wireless powered MEC networks [5]. It was shown that user quality of experience (QoE) can be improved by integrating wireless powered techniques into MEC networks since the duration of having MEC services is extended [6]. However, due to the everincreasing greenhouse gas emission concerns and the rapid growth of the operational cost, future MEC networks will more and more focus on maximizing the computation efficiency (CE) [7], which is defined as the ratio of the total computed bits to the consumed energy. According to [7], information and communication technologies account for about 2 of the greenhouse gas and 2 to 10 of global energy consumption. In order to achieve a sustainable and green operation of MEC networks, it is crucial to design resource allocation strategies for maximizing CE of MEC networks. To the authors’ best knowledge, there have been only a few studies in this area. The related works are summarized as follows.
The related works can be classified into three categories. The first category has focused on designing energyefficient resource allocation schemes in the conventional MEC networks with orthogonal multiple access (OMA) in
[8][14] or with NOMA in [15][18]. In the second category resource allocation strategies have been designed in wireless powered MEC networks [5], [6], [19][25]. The third category has designed energyefficient resource allocation schemes in wirelesspowered networks relying on either OMA or NOMA [26][31].IA1 EnergyEfficient Resource Allocation in The Conventional MEC Networks
In the OMAbased MEC networks, efforts have been dedicated to jointly optimizing the communication and computation resources for achieving energyefficient computing. Specifically, in [8], the consumed energy and execution latency were minimized by jointly optimizing the local computing frequency and offloading power of users. The authors in [9] and [10] extended the energy consumption minimization problems into multiuser MEC networks with TDMA and orthogonal frequencydivision multiple access (OFDMA), respectively. In [9], the computation performance of each user was guaranteed by optimizing the offloading computation bits ratio and offloading time. In [10], it was shown that the users with strong channel state information (CSI) prefer to offload their computation task to the MEC server while users with weak CSI chooses to perform local computing. In order to further reduce the energy consumption, the authors in [11] studied the coexistence of MEC and cloud computing servers and proposed optimal scheduling policies. Different form the works in [8][11], multiantenna techniques were exploited to improve the offloading efficiency [12], [13]
. The energy was minimized by jointly optimizing the beamforming vector and the computation frequency. Recently, the authors in
[14] studied secure offloading in multiuser MEC networks. The works in [8][14] focused on optimizing a single objective, which cannot achieve a good tradeoff among different performance metrics. In [7], the authors studied CE maximization problem in an MEC system with TDMA, which achieves a good tradeoff between the achievable computation bits and the energy consumption.Recently, in order to increase connectivity and reduce access latency, resource allocation problems were studied in MEC networks with NOMA [15][18]. In [15], the impact of NOMA on offloading was analyzed in MEC networks. It was proved that the application of NOMA can efficiently reduce the energy consumption and offloading delay compared to OMA. The authors in [16][18] designed resource allocation strategies for minimizing the consumption energy of various MEC networks with NOMA. Specifically, the user clustering, communication and computing resources were jointly optimized in multicell MEC networks in [16] while the communication and computing resource and the trajectory of the unmanned aerial vehicle (UAV) were jointly optimized in [17]. In [18], the authors studied a multiantenna MEC network with NOMA. It was shown that the exploitation of multiantenna techniques can significantly improve the offloading efficiency.
IA2 Resource Allocation in Wireless Powered MEC Networks
Recently, the authors in [5], [6], [19][25] studied resource allocation problems in various wireless powered MEC networks. Specifically, in [19]
, the central processing unit (CPU) frequency of the user and the mode selection were jointly optimized for minimizing the consumed energy under both causal and noncausal CSI conditions. The reinforcement learning and Lyapunov optimization theory were exploited to design resource allocation strategies for minimizing the system cost of wireless powered MEC networks in
[20] and [21], respectively. Although the computation performance can be improved by integrating wireless powered techniques into MEC networks [19][21], the performance improvement is limited by the harvested energy. In order to improve the energy conservation efficiency, the authors in [5] exploited multiantenna techniques and jointly optimized the energy transmit beamformer, the CPU frequencies and the offloading bits for minimizing the total energy consumption. In [22], the authors extended the energy consumption minimization problem into a cooperationassisted wireless powered MEC network. Different from the works in [5], [19][21], the authors in [23] and [24] proposed optimal resource allocation strategies for maximizing the CE of multiuser and fullduplex wireless powered MEC networks. In contrast to the works in [5], [19][24], resource allocation strategies were designed for maximizing the computation bits of multiuser and UAVenabled wireless powered MEC systems in [6] and [25] under the binary computation offloading mode.IA3 EnergyEfficient Resource Allocation in Wireless Powered Networks
In conventional wireless powered networks where the computing process was not considered, the energy efficiency maximization problems have been well studied in [26][31]. Specifically, the authors in [26] and [27] designed resource allocation strategies for maximizing the system and usercentric energy efficiency, respectively. In order to improve the achievable energy efficiency, multipleinput multipleout (MIMO) and massive MIMO techniques have been applied in wireless powered networks in [28] and [29], respectively. Different from the works in [26][29] where TDMA was applied for serving multiple users, the authors in [30] and [31] have designed energyefficient resource allocation schemes in the wireless powered networks with NOMA. It was interesting to find that NOMA does not necessarily guarantee to achieve a better energy efficiency compared to TDMA.
IB Motivations and Contributions
Note that the resource allocation strategies proposed in the conventional MEC networks [8][18] cannot be readily applied to wireless powered MEC systems, where the resource allocation problems should consider the EH causal constraints and the relationship among the EH, offloading and computing process. Moreover, the resource allocation strategies proposed for wireless powered MEC networks in [5], [6], [19][25] are based on an ideally linear EH model. These schemes may not achieve good performance in reality as the practical EH circuits can result in a nonlinear endtoend wireless power conversion [32][34]. Furthermore, resource allocation strategies designed for maximizing the computation bits under the binary computation offloading mode [6] and [25] and designed for maximizing CE under partial offloading mode cannot guarantee to maximize the CE under the binary computation offloading mode. Additionally, although energyefficiency resource allocation strategies have been designed in the conventional wireless powered networks [26][31], only the efficiency of the computation task offloading process, i.e., communications process, was considered. They are inappropriate in wirelesspowered MEC networks for maximizing the CE since energy consumed in both the offloading and local computation processes should be considered.
To the authors’ best knowledge, this is the first work that comprehensively studies resource allocation problems for maximizing CE under both partial and binary computation offloading modes. Please note that in [1], we only studied the CE problem under the partial offloading mode with TDMA. Moreover, we did not consider the effect of the power amplifier coefficient on CE [1]. The main contributions of our work in this paper are summarized as follows.

It is the first time that the CE maximization framework is formulated in wireless powered MEC networks under both partial and binary computation offloading modes. Both TDMA and NOMA are considered for offloading transmission. With TDMA, the closedform expressions for the optimal CPU frequency and the optimal offloading power of users are derived under the partial offloading mode and the closeform expression for the optimal operational mode selection is given under the binary computation offloading mode. An iterative algorithm and an alternative optimization algorithm are proposed to solve the CE maximization under the partial offloading and binary computation offloading mode, respectively.

With NOMA, an iterative algorithm and an alternative optimization algorithm based on the successive convex approximation (SCA) method are proposed for the partial offloading mode and the binary computation offloading mode, respectively. The closedform expression for the operational mode selection on whether users choose to locally compute or to offload tasks is derived. It is shown that the selection of the operational mode depends on the tradeoff between the achievable computation bits and the energy consumption cost of users.

Simulation results show that our proposed resource allocation strategies can improve fairness among users in terms of CE compared to the benchmark scheme. Moreover, it is shown that the partial offloading mode and NOMA can achieve CE gains compared with the binary computation offloading mode and TDMA, respectively. Furthermore, the tradeoff between the achievable CE and the computation bits is firstly elucidated.
The remainder of this paper is organized as follows. Section II presents the system model. maximization problems are investigated for wireless powered MEC networks with TDMA in Section III. Section IV presents the CE maximization problems in wireless powered MEC networks with NOMA. Simulation results are presented in Section V. The paper is concluded in Section VI.
Ii System Model
A wireless powered MEC network is considered in Fig. 1, where the wireless power station provides the wireless power transfer (WPT) services for users. Similar to the works in [6], [19] and [20], in order to clarify the issues pertaining to CE and permit reaching meaningful insights into the CE maximization problem, it is assumed that all devices are equipped with a single antenna. In this paper, both partial and binary computation offloading modes are considered. Similar to [5], [6], [21][25], local computation and downlink WPT can be simultaneously executed while the downlink WPT and the uplink computation offloading cannot be simultaneously performed. Thus, a harvestthenoffload protocol is applied for downlink WPT and uplink computation offloading. Moreover, both TDMA and NOMA protocols are exploited for achieving multiuser offloading during the offloading process. All the nodes and devices are equipped with a single antenna. Similar to [19][25], all the channels have block fading thus the channel power gains are static within each time frame but may change across time frames. In order to obtain the upper bound of CE and provide theoretical support for the practical system design, It is assumed that the perfect CSI can be obtained [5], [6], [21][25].
The frame structure shown in Fig. 2 consists of four stages. The frame duration is denoted by , which is selected based on the correlated time of the channel in order to guarantee that the channel power gains are constant within one frame duration [5]. In the first stage, the wireless power station transfers energy to users. In the second stage, users offload their computation tasks to the MEC by using TDMA or NOMA protocol. In the third stage, the MEC executes the computation tasks from users. In the fourth stage, the MEC downloads the computation results to users. Similar to [19][25], the computation time and the downloading time of the MEC are neglected as the MEC has a strong computation capability compared with users and the number of the bits related to the computation results is relatively small.
Remark 1
The major application scenarios for wireless powered MEC networks include two cases. One is in the wireless sensor or wearable networks where the mobile sensors and wearable computing devices are with milliwatt power consumption while they need to perform computation tasks, such as environmental parameter or physical condition monitoring [6]. The other one is in the areas, such as wildernesses and complex terrains, where the government needs to keep monitoring the environment so that the corresponding strategies can be taken to protect the environment. In those areas, neither cable charging or battery replacement can be conveniently established nor the cost for establishing cable charging systems is affordable. The wireless powered MEC network becomes a desirable alternative [19].
Iia NonLinear Energy Harvesting Model
Let denote the duration of the WPT stage. In this paper, different from the works in [5], [6], [19][25], a practical nonlinear EH model is applied while a sensitivity property is considered. Specifically, the harvested energy is zero when the input RF power is smaller than the sensitivity threshold. Based on the work in [32], the harvested energy of the th user denoted by can be given as
(1) 
where is the transmit power of the power station; is the maximum harvested power of the th user, and ; is the sensitivity threshold; and are the parameters for controlling the steepness of the function; is the instantaneous channel power gain from the power station to the th user; and denotes the bigger value of and .
In order to better illustrate the nonlinear EH model, Fig. 3 compares the harvested power obtained with the nonlinear EH model [32] to those achieved with the linear EH model and the experimental results given in [32]. The harvested power of the linear EH model is , where is the energy conversion efficiency and selected as . The parameters in the nonlinear EH model are set as W, W, and .
IiB Partial Offloading
In this mode, the computation tasks of each user can be divided into two parts, one for local computing and one for offloading.
IiB1 Local Computation
Similar to [5], [6], [19][25], each user can perform local computation in the entire frame as each user can have the circuit architecture to separate the computing unit and the offloading unit. cycles are required for computing one bit of raw data at the user side CPU. Let denote the CPU frequency of the th user. Thus, the number of locally computed bits at the th user and the consumed energy are and , respectively. is the effective capacitance coefficient of the processor’s chip, and is dependent on the chip architecture.
IiB2 Offloading with TDMA
Let and denote the offloading time and the transmit power for offloading of the th user, respectively. Similar to the work in [6], the offloaded task of the th user consists of raw data and communication overhead, such as the encryption and packet header. Let indicate the communication overhead. According to [6], the number of bits that the th user offloads to the MEC server using TDMA is given as where is the communication bandwidth, denotes the noise power, and is the instantaneous channel power gain from the th user to the MEC server. Thus, the CE of the th user is defined as
(2a)  
(2b)  
(2c) 
where is the CE of the th user; and are the total number of computed bits at the MEC server for the th user and the consumption energy of the th user, respectively; and denote the received power for the received signal processing during the WPT stage and the constant circuit power consumption of the th user during the computation offloading process, respectively [27], and denotes the amplifier coefficient.
IiB3 Offloading with NOMA
Different from TDMA, NOMA enables all the users to simultaneously offload their offloading tasks on the same frequency band so that offloading throughput can be improved. Let denote the duration of the offloading process. Without loss of generality, the channel gains for the NOMA users have a descending order . Thus, using the simple decoding order based on the descending order of the channel power channel [15][17], the CE of the th user can be expressed as
(3a)  
(3b)  
(3c) 
where denotes the CE of the th user; and denote the total number of computed bits at MEC and the total energy consumption of the th user respectively.
IiC Binary Offloading
Under the binary offloading mode, the computation task can be either completely computed at the local device or completely offloaded to the MEC server for computing. Let and denote the set of users that choose to perform task offloading and the set of users that choose to perform local computation, respectively. Thus, and , where denotes the null set.
IiC1 Local Computation
In this case, all the harvested energy is used for local computing. Thus, the CE of the th user denoted by can be given as
(4a)  
(4b)  
(4c) 
where and are the total number of locally computed bits and energy consumption for computation of the th user respectively.
IiC2 Offloading with TDMA
With TDMA for offloading, the computation efficiency of the th user denoted by can be given as
(5a)  
(5b)  
(5c) 
where and are the number of computed bits and energy consumption for offloading of the th user respectively.
IiC3 Offloading with NOMA
With NOMA, the CE of the th user in denoted by can be given as
(6a)  
(6b)  
(6c) 
Iii CE Maximization In Wireless Powered MEC Networks: TDMA based
Iiia Partial Offloading Mode
IiiA1 Problem Formulation
When the TDMA protocol and the partial computation offloading mode are applied, the CE maximization problem is formulated under the maxmin fairness criterion as
(7a)  
(7b)  
(7c)  
(7d)  
(7e)  
(7f)  
(7g) 
is the minimum number of computed bits required by the th user and is the maximum transmission power of the wireless power station. The constraint is the minimum computed bits constraint. The constraint is the EH causal constraint that the total consumption energy cannot be larger than the harvested energy. The constraints and are the constraints on the EH time and the computation offloading time. is a nonconvex fractional optimization problem. It is challenging to solve due to the existence of coupling relationship among different optimization variables, such as the coupling between and , as well as due to the nonconvex constraints and .
IiiA2 Solution and Iterative Algorithm
In order to solve , Theorem 1 is presented as follows.
Theorem 1
In wireless powered MEC networks with TDMA under the partial computation offloading mode, the maximum CE under the maxmin fairness criterion is achieved when .
Proof: Let denote the optimal solution of . denotes the maximum CE achieved under the maxmin fairness criterion. It is not difficult to prove that and . It is assumed that . Let denote another solution of satisfying , , , and . denotes the corresponding maximum CE. When is not large enough to achieve the maximum output power , one has since . It is quite evident that satisfies all the constraints of . Since and , one has . This contradicts the assumption that is the optimal solution. Thus, . When is large to achieve the maximum output power , since , one has and thus . Thus, is also the optimal solution. Theorem 1 is proved.
Remark 2
It can be seen from Theorem 1 that the maximum CE achieved under the maxmin fairness criterion increases with the transmission power of the power station in the wireless powered MEC networks with TDMA. If the transmission power level of the power station is not large enough for achieving the maximum harvested power of the user, the CE can be increased by increasing the transmission power of the power station.
Motivated by the Dinkelbach’s method [26], Lemma 1 is given to transform to a tractable problem.
Lemma 1
The optimal solution of can be obtained if and only if the following equation holds.
(8a)  
(8b) 
where and denote the maximum CE and optimality, respectively. The proof can be readily obtained from the generalized fractional programming theory [36].
Based on Lemma 1, can be solved by solving a parameter problem, denoted by , given as
(9a)  
(9b) 
Here is a nonnegative parameter. Although is more tractable, it is still nonconvex and has coupling among optimization variables. Auxiliary variables are further introduced, where . Using the auxiliary variables and Theorem 1, can be equivalently expressed as
(10a)  
(10b)  
(10c)  
(10d)  
(10e) 
Lemma 2
is convex and can be efficiently solved by using the convex optimization tool [37].
Proof: Firstly, it is evident that the objective function and the constraints , , of satisfy the conditions of a convex problem since the objective function is linear and the constraints , , are linear inequality constraints. For the constraint given by , is a linear function with respect to and is the perspective of , which is a concave function of . Since the perspective operation preserves convexity [37], is concave with respect to and . Thus, it is easy to obtain that the constraint given by is a convex constraint. For the constraint , the right side is a linear function with respect to and the left side is a linear function in regard to , and . Moreover, since the local CPU frequency is nonnegative, is a convex function with respect to when . Thus, the constraint given by is also a convex constraint. Using the same analysis method for the constraint given by , it is easy to prove that the constraint given by is also a convex constraint. Thus, it is proved that is convex.
In this paper, in order to gain more meaningful insights, the optimal solutions are obtained in closed forms by using the Lagrange duality method [36]. Towards that end, let and denote the optimal local computation frequency and the optimal offloading power of the th user, , respectively. By solving , Theorem 2 can be stated as follows.
Theorem 2
In the wireless powered MEC systems with TDMA, the optimal local computation frequency and the optimal offloading power of the th user for maximizing the CE under the maxmin fairness criterion have the following mathematical expressions:
(11a)  
(11b) 
where , and are the dual variables corresponding to the constraints given by , , and , respectively.
Proof:
See Appendix A.
Remark 3
It can be seen from Theorem 2 that the th user chooses to offload its computation task only when the channel between the th user and the MEC server is good enough, namely, . Moreover, the local computation frequency decreases with the increase of , which is related to the CE. It indicates that the users prefer to offload their computation task to the MEC server in order to improve the CE. Furthermore, it can be seen that the users prefer to offload their computation task when the local CPU frequency is too high, namely, . Additionally, when , the optimal offloading power is zero. It means that the users only perform local computation in order to maximize their CE.
By solving , Theorem 3 is stated to clarify the characteristic of the EH time and .
Theorem 3
For the given , , and , in order to maximize the Lagrangian of , the optimal EH time and computation offloading time need to satisfy the following equations.
(12a)  
(12b)  
(12c) 
(13a)  
(13b) 
where is the solution of the following equation.
(14) 
Proof:
See Appendix B.
By solving , Theorem 4 can be stated to obtain the maximum CE denoted by .
Theorem 4
In the wireless powered MEC systems with TDMA under the partial computation offloading mode, the maximum CE under the maxmin fairness criterion is given as
(15a)  
(15b) 
Proof:
Since is convex and the Slater’s conditions are satisfied, the Lagrangian of should be upperbound with respect to . When , the Lagrangian decreases with . Thus, the maximum of Lagrangian is achieved with since . When , the maximum of Lagrangian is obtained when the optimal solution is achieved.
Finally, an iterative algorithm denoted by Algorithm 1 is given to obtain the maximum CE. Specifically, when , the optimal solution is obtained, where and denote the iterative number and the optimal solution achieved at the th iteration, respectively. Otherwise, an optimal solution is adopted with an error tolerance . In other words, the maximum CE is obtained when , where denotes the absolute operator. The details of Algorithm 1 are given in Table I.
Remark 4
By using Theorem 1 and introducing auxiliary variables , it is seen that is a generalized fractional programming problem [36]. Moreover, Algorithm 1 is proposed for solving based on the Dinkelbach’s method. Thus, according to [36], Algorithm 1 can converge when updating . The detail proof for the convergence can be seen in [36].
Algorithm 1: The iterative algorithm for 
1) Input settings: 
the error tolerance , and , 
the maximum iteration number . 
2) Initialization: 
EE and the iteration index . 
3) Optimization: 
for n=1:N 
solve by using CVX for the given ; 
obtain the solution ; 
if 
the maximum CE is obtained; 
break; 
else 
update and ; 
end 
end 
IiiB Binary Offloading Mode
IiiB1 Problem Formulation
Under the binary offloading mode, when the TDMA protocol is applied, the CE maximization problem under the maxmin fairness criterion can be formulated as .
(16a)  
(16b)  
(16c)  
(16d)  
(16e)  
(16f)  
(16g) 
where the constraints given by and are the requirements of the minimum computed bits. The constraints given by and are the EH causal constraints. is challenging to solve. Moreover, it is impractical to use the exhaustive search method for determining the operational mode selection due to the extremely high complexity, especially when the number of users is large.
IiiB2 Alternative Optimization Algorithm
In order to solve , let indicate that the th user performs local computation mode and mean that the th user performs task offloading, where . Moreover, is relaxed as a continuous sharing factor . Thus, can be expressed as
(17a)  
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