I Introduction
Wireless powered mobile edge computing (MEC) has attracted growing research interests to support selfsustainable computation for massive lowpower wireless devices, by combining emerging radiofrequency (RF) signal based wireless power transfer (WPT)[1, 2] and MEC [3, 4, 5] techniques into a joint design[6, 7, 8]. By deploying hybrid access points (APs) each with triple roles of energy transmitter, information transceiver, and MEC server, this technique can provide continuous wireless energy supply for end users, such that they can rely on the harvested energy for local computing and computation offloading. As compared with conventional MEC systems with fixed battery supplies at users, the wireless powered MEC can significantly enhance the costefficiency and sustainability of future Internetofthings (IoT) networks, by e.g., avoiding frequent battery replacement at users.
The design of wireless powered MEC systems encounters various new technical challenges due to the involvement of WPT, computation, and communication. In order to optimize the system performance, it is crucial to perform multiresource allocation to achieve optimal balance between the wireless energy supply from the AP versus the computation and communication energy demand at these users. In the literature, there have been several prior works investigating joint WPT, communication, and computation resource allocation in wireless powered MEC systems, under different system setups with one single user [6], multiple users [7, 8], and user cooperation [9, 10], respectively. These works [6, 7, 8, 9, 10] focused on oneshot optimization over a particular time slot by assuming unchanged wireless channels and static task models at users.
In practice, however, due to the randomness of computation traffics, the task arrival rates at users may fluctuate substantially over time and space. On the other hand, due to the randomness of wireless channels, the wireless energy harvested from the AP may also change significantly over time among different users. Therefore, it is an important yet challenging problem to jointly design the WPT at the AP and the computation and communication resource allocation at the users, in order to properly control the users’ harvested energy to well match their dynamic computation requirements in an energyefficient manner. Notice that in the literature, some prior works [11, 12, 13] investigated joint communication and computation management in MEC systems with dynamic task arrivals over time, while other works (see, e.g., [15, 14] and the references therein) considered dynamic energy management with random energy arrivals for energy harvesting based systems. However, there still lacks a joint design of the WPT at the AP and the communication/computation energy demand at the users for wireless powered MEC systems, by taking into account both energy and task dynamics over time and space. This thus motivates our study in this work.
This paper considers a wireless powered multiuser MEC system, where a multiantenna AP employs the energy beamforming to charge a number of distributed users with dynamic task arrivals, and each user utilizes the harvested energy to execute the computation tasks via local computing and computation offloading. Suppose that the downlink WPT from the AP to the users and the uplink computation offloading are operated simultaneously over orthogonal frequency bands, and frequency division multiple access (FDMA) protocol is employed to enable simultaneous task offloading from multiple users to the AP. We focus on joint computation and wireless resource allocation over a finite time horizon consisting of multiple slots. Under this setup, our objective is to minimize the system energy consumption including the AP’s transmission energy and the MEC server’s computing energy over the whole horizon, by jointly optimizing the transmit energy beamforming at the AP, as well as the local computing and task offloading strategies at the users over different slots.
To further characterize the fundamental performance limit, we assume that the channel state information (CSI) and the task arrival information are perfectly known apriori at the AP. Accordingly, we adopt the offline optimization approach to solve the energy minimization problem.^{1}^{1}1Notice that the offline optimization approach has been widely adopted in the energy harvesting based wireless communication [14] and the energy management in smart grids [15]. The obtained optimal offline solution can serve as the performance upper bound for any practical online designs, and can also inspire practical online designs in the case when the CSI and the task arrival information are casually known. We leave the online optimization for the wireless powered MEC systems to our future work. Leveraging the Lagrange duality method, we obtain wellstructured optimal solution to the jointWPTMEC design problem. It is shown that in order to maximize the system energy efficiency, the number of task inputbits at each user and the AP are monotonically increasing over time, and different users’ offloading strategies are adapted according to both the wireless channel conditions and the task load at the AP. Numerical results demonstrate the benefit of the proposed joint design over other benchmark schemes without such joint design.
Ii System Model and Problem Formulation
We consider a wireless powered multiuser MEC system as shown in Fig. 1, where an antenna AP (integrated with an MEC server) employs WPT to charge a set of singleantenna users, and provides cloudlike computation capability to enable their computationintensive applications. We consider a finite time horizon with duration , which is divided into slots each with duration . Let denote the set of the slots. For each user , the computation tasks arrive at the beginning of each slot , with denoting the corresponding number of task inputbits. In order to enable remote computing at the AP, at each slot , the users need to offload the computation tasks to the AP and download the computation results of previously executed tasks from the AP. It is assumed that each user is subject to a task completion deadline of , i.e., each user needs to successfully execute their respective tasks before the end of this time horizon. As commonly adopted in the literature[7, 6, 5, 8], where the computation results are generally much smaller than the task inputbits, we focus on the time and energy consumed by task offloading from the users to the AP and ignore those consumed by computation result downloading. In the following, we first introduce the task execution at the users via local computing and computation offloading, and then present the WPT and the remote task execution at the AP.
Iia Task Execution and Computation Offloading at Users
During each slot , each user can execute its tasks via local computing and computation offloading. Let and denote the number of task inputbits for user ’s local computing and offloading, respectively, which are design variables to be optimized later. In this case, we first have the following task causality constraints, such that at each slot , the cumulative number of task inputbits executed until that slot (i.e., ) cannot exceed that having already arrived at user (i.e., ), i.e.,
(1) 
In addition, we have the computation deadline constraints at the users, i.e., each user needs to accomplish the task execution by the end of the last slot , which is expressed as
(2) 
IiA1 Local Computing at Users
First, we consider the local computing at user for executing task inputbits at slot , for which CPU cycles in total are required, where denotes the number of required central processing unit (CPU) cycles for executing one task inputbit at user . Note that the value depends on the type of applications and the CPU architectures of user . In order to maximize the energy efficiency for local computing, each user can apply dynamic voltage and frequency scaling (DVFS) technique by adjusting the CPU frequency as during each slot [7]. As a result, in slot , the energy consumption for local computing at user is
(3) 
where is the effective switched capacitance coefficient of user ’s CPU chip architecture.
IiA2 Computation Offloading from Users to AP
In order to avoid cochannel interference, we adopt the FDMA protocol for the users to simultaneously offload their task inputbits to the AP for remote server’s computing, where each user is allocated with an identical system bandwidth of . For each slot , let denote the channel power gain from user to the AP. Assuming the maximalratio combining (MRC) based receiver at the AP, the transmission energy consumption at user for offloading task inputbits within slot is[7]
(4) 
where is the additive white Gaussian noise (AWGN) power at the AP receiver.
IiB WPT and Remote Task Execution at AP
The AP needs to charge the users via downlink energy beamforming and remotely execute the offloaded computation tasks for the users.
IiB1 WPT from AP to Users
First, consider the downlink energy beamforming from the AP to the users. At slot , let denote the energybearing transmit signal and the transmit energy covariance matrix of the AP, where denotes the expectation operation and the superscript
represents the conjugate transpose of a matrix or vector. For ease of analysis, we consider that the input RF power at each user
is within the linear regime of the rectifier.^{2}^{2}2In practice, our proposed wireless powered MEC system design is extendable to the scenario with nonlinear energy harvesting models at each user, for which the transmit waveform optimization needs to be considered jointly with the energy beamforming design at the AP[2]. Then the amount of energy harvested by user at slot is[1](5) 
where denotes the downlink channel vector from the AP to user at slot , denotes the constant energy harvesting efficiency of user , and denotes the trace of a matrix.
IiB2 Remote Computing at AP
Next, we discuss the remote execution at the AP. During slot , the AP needs to execute the users’ offloaded tasks by its integrated MEC server. Let denote the number of task inputbits computed by the AP within slot . In practice, at each slot , the MEC server can only execute the task inputbits offloaded from the users at the previous slots. Hence, at each slot , the number of task inputbits cumulatively executed by the AP (i.e., ) cannot exceed that cumulatively offloaded from all the users until the previous slot (i.e., ). As in (IIA), the task causality constraints at the AP are given by
(6) 
In addition, the AP needs to complete all the task execution by the last slot . As a result, as in (2), the computation deadline constraint for the AP is expressed as
(7) 
Based on (2), (6), and (7), the number of task inputbits offloaded by user at slot must satisfy , as there is no time left for the AP to further perform computation before the deadline.
In order to remotely compute the tasks in an energyefficient manner, the AP employs the DVFS technique by adjusting the CPU frequency as during each slot , where denotes the number of required CPU cycles for executing one task inputbit by the MEC server. In this case, as in (3), the total amount of computation energy consumption at the AP across the slot horizon is expressed as
(8) 
where is the capacitance coefficient specified by the MEC server’s CPU architecture.
IiC Problem Formulation
Note that each user is powered by the energy beamforming from the AP to achieve selfsustainable computation. In this case, at each slot , the cumulatively consumed energy at each user up to that slot cannot exceed that cumulatively harvested from the AP. As a result, we obtain the energy causality constraints on users’ battery storage:
(9) 
We are interested in an energyefficient jointWPTMEC design to minimize the system energy consumption subject to the energy/task causality and computation deadline constraints. Specifically, we aim to minimize the energy consumption of the AP, i.e., , while ensuring the successful task execution before the deadline, by jointly optimizing the energy transmit covariance matrices , the number of task inputbits for remote execution at the AP, and the number of task inputbits and for users’ local computation and computation offloading, respectively. As a result, the energy minimization problem is formulated as
(10a)  
(10b)  
(10c)  
(10d) 
In particular, we consider offline optimization for problem (10) by assuming that are perfectly known apriori at the AP, in order to characterize the fundamental performance upper bound and inspire practical online designs in future work. Under this assumption, problem (10) is a convex optimization problem and can thus be solved by standard convex optimization techniques such as the interiorpoint method[16]. However, to reveal more design insights, we apply the Lagrange duality method to obtain a wellstructured optimal solution to problem (10) in Section III.
Iii Optimal Solution
As problem (10) is convex and satisfies Slater’s condition[16], strong duality holds between problem (10) and its dual problem. Let , , , , and denote the Lagrange multipliers associated with the constraints in (IIA), (2), (6), (7), and (9), , , respectively. By defining , , and , where the superscript denotes the transpose of a vector, the partial Lagrangian of problem (10) is given by
(11) 
where for any . Accordingly, the dual function is expressed as
(12) 
In order to obtain the dual problem of problem (10), we first establish the following lemma.
Lemma iii.1
In order for the dual function to be lower bounded from below, it must hold that
(13a)  
(13b) 
Proof:
See Appendix A.
Based on Lemma III.1, the dual problem of problem (10) is then expressed as
(14a)  
(14b)  
(14c)  
(14d) 
Denote as the feasible solution set of for problem (14). Then, we solve problem (10) optimally by solving its dual problem (14) equivalently.
Iiia Evaluating Dual Function
Under any given , by removing the irrelevant constant terms in (III), the optimization problem in (III) can be decomposed into the following independent subproblems for different time slots and users.
(15)  
(16)  
(17)  
(18) 
Let , , , and , , , denote the optimal solutions to the subproblems in (15), (16), (17), and (18), respectively. We establish the following lemmas.
Lemma iii.2
The optimal solution to problem (15) is given by , , where denotes the null space of a matrix.
Proof:
Lemma iii.3
Proof:
See Appendix B.
Lemma iii.4
The optimal solution to problem (17) is given by
(20) 
Lemma iii.5
The optimal solution to problem (18) is given by, ,
(21) 
Note that Lemmas III.4 and III.5 can be are similarly proved as for Lemma III.3, and thus we omit the proofs for brevity. Based on Lemma III.2, it follows that is not unique if is rank deficient. For simplicity, we choose , , for problem (15) without loss of optimality for obtaining the dual function only. With (19)–(21) and , , can be obtained under any given set .
IiiB Obtaining Optimal Dual
Next, we maximize over to solve the dual problem (14). Note that the dual function is always concave but not necessarily differentiable. Therefore, problem (14) is convex, and can thus be solved by subgradient based methods such as the ellipsoid method [16], in which the subgradient of is with respect to . Let denote the obtained optimal dual solution to problem (14).
IiiC Finding Optimal Primal
With obtained, it remains to find the optimal primal solution to problem (10). Since , and are the unique optimal solution to problems (16), (17), and (18), respectively, the optimal , , and to problem (10) can be directly obtained by replacing with the optimal dual in (19), (20), and (21), respectively.
On the other hand, the optimal solution to problem (10) cannot be obtained from Lemma III.2 alone, and , , are even not feasible for problem (10). Therefore, an additional procedure is required to obtain the optimal . With obtained, the optimal to problem (10) can be obtained by solving the following semidefinite program (SDP) via convex solvers (e.g., CVX toolbox [16]):
(22a)  
(22b) 
By combining together with , we finally obtain the optimal solution to problem (10). In summary, Algorithm 1 for solving problem (10) is presented in Table I.
To gain essential design insights, based on Lemmas III.3 and III.4, we have Proposition III.1 for the optimal executed task inputbits allocated to the users’ local computing and the AP’s remote computing as follows.
Proposition iii.1
For any user and the AP (), the optimal number of executed task inputbits is monotonically increasing over time, i.e.,
(23) 
Proof:
See Appendix C.
Remark iii.1
Proposition III.1 indicates the monotonically increasing feature of the optimal number of executed task inputbits for both the users and the AP. This can be intuitively understood as follows. Notice that the computation energy consumption functions in (3) and (8) for each slot are convex with respect to the number of task inputbits. Therefore, in order to reduce the energy consumption, it is desirable for the users and the AP to evenly distribute the computation tasks as far as possible. Due to the task causality constraints in (IIA) and (6), it is evident that the computation load at each of the users and the AP should monotonically increase over time, as more tasks will be accumulated over time. This feature is reminiscent of the monotonically increasing power allocation in energy harvesting based wireless communications due to the energy causality constraints (see, e.g., [14])..
Remark iii.2
Based on (21) in Lemma III.5, in slot , the optimal number of offloaded task inputbits for each user should be adapted according to the task load at the AP. Specifically, under the case when the channels for both WPT and computation offloading remain unchanged between user and the AP, it holds that if , . This implies that, when the penalty in energy consumption for the AP to violate the task causality constraint at slot is smaller than that for user , user should offload more task inputbits at the next slot , . In addition, it is intuitively expected that is propositional to the offloading channel power gain from user to the AP at slot .
Iv Numerical Results
In this section, we provide numerical results to evaluate the performance of the proposed algorithm, where and are set with each slot duration being second. For comparison, we consider the following four benchmark schemes.

Local computing only: Each user accomplishes its computation tasks by only local computing. This scheme corresponds to solving problem (10) by setting and , , .

Full offloading: Each user accomplishes its computation tasks by fully offloading them to the AP. This scheme corresponds to solving problem (10) by setting , , .

Myopic design: In each slot , both user and the AP accomplish their task inputbits of and , respectively, i.e., we have and , , . In this scheme, the system energy minimization can be implemented independently over each individual slot, as investigated in [7].

Separate WPTMEC design: This scheme separately designs the energy beamforming for WPT and the users’ computation offloading for MEC. First, the sumenergy consumption of the users is minimized subject to their individual computation latency constraints[4]. The AP then designs its energy beamforming with energy minimization under the given energy demands at users.
In the simulation, the system parameters are set as follows, unless stated otherwise. We set , CPU cycles/bit, , , , the receiver noise power Watt, and the system bandwidth for offloading MHz. We also consider a distancedependent Rayleigh fading channel model [7] with the channel power gain at a unit of reference distance set as dB. The number of task inputbits for user at slot
is set as a random variable following the uniform distribution of
with and inputbits unless stated otherwise. The numerical results are obtained by averaging over randomized channel realizations and randomized task realizations.Fig. 2 shows the average energy consumption per slot at the AP versus the user number , where the distances between the AP and the users are identical with meters, . It is observed that the proposed design achieves the lowest average energy consumption among all the five schemes, and its gain over the benchmark schemes becomes more significant when increases. At small value of (e.g., ), the benchmark myopicdesign and separateWPTMECdesign schemes are observed to achieve nearoptimal performance close to that achieved by the proposed design. It is also observed that the separateWPTMECdesign performs inferior to the myopicdesign scheme at large values. This validates the importance of the proposed joint design for energy saving in multiuser scenarios. Both the localcomputingonly and fulloffloading schemes are observed to perform inferior to the other two benchmark schemes. When , the fulloffloading scheme outperforms the localcomputingonly scheme, and the reverse is true when becomes larger. This is due to the fact that the energy consumption for offloading increases faster (exponentially) than that for local computing (cubically).
Fig. 3 shows the average energy consumption at the AP per slot versus the maximum number of task inputbits , where and meters, . It is observed that the proposed design outperforms the other benchmark schemes. As increases, the performance gain achieved by the proposed design becomes more substantial. The myopicdesign scheme is observed to achieve a near optimal performance close to the proposed design at small values (e.g., ). Similar performance trends for the benchmark schemes are observed as in Fig. 2.
V Conclusion
This paper studied the optimal resource allocation for a wireless powered multiuser MEC system to minimize the energy consumption at the AP subject to the computation latency and energy harvesting constraints at users, by jointly optimizing the AP’s energy beamforming for WPT and remote execution, as well as each user’s local computing/offloading. Using the Lagrange duality method, we developed an efficient algorithm to obtain a wellstructured optimal solution, where the optimal number of task inputbits at the AP and users is monotonically increasing over time. Numerical results demonstrated the merit of our proposed jointWPTMEC design compared to other benchmark schemes without such joint design. In future work, we will extend the results to other setups when the energy/task arrival information and the CSI are only causally known or even apriori unknown at the AP. For such cases, our offline optimization will serve as a performance upper bound, and the obtained structured solution may motivate online algorithms. In general, how to obtain optimal solution for such new scenarios will be a challenging problem that requires further investigation.
a Proof of Lemma iii.1
First, we prove (13a). Suppose that the matrix is not semidefinite positive for some . In this case, by letting , , where and , it can be shown from (III) that . Thus the matrix must hold for all for to be bounded from below. Next, we consider (13b). Suppose that for some and . In this case, by letting , , and , it can be shown from (III) that . Thus, the fact cannot be true for to be bounded from below. Lemma III.1 is now proved.
B Proof of Lemma iii.3
Introducing the Lagrange multiplier associated with the constraint , the Lagrangian of problem (16) is and the dual problem of problem (16) is then
(24) 
Note that problem (16) is convex and satisfies Slater’s condition. Therefore, the strong duality holds between problems (16) and (24). Let and be the optimal solutions for problems (16) and (24), respectively. The KarushKuhnTucker (KKT) optimal conditions are then given by
(25a)  
(25b) 
where (25a) denotes that the gradient of must vanish at and the first equality in (25b) is the complementary slackness condition. Note that acts as a slack variable in (25a) and it can then be eliminated. Together with , we obtain , .
C Proof of Proposition iii.1
First, consider the case with . Based on Lemma III.3, , we have
(26)  
(27) 
where the inequality follows from the fact that for any . It thus holds that .
Next, we consider the case with . Similarly, from Lemma III.4, , it follows that
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