Computation Rate Maximization in UAV-Enabled Wireless Powered Mobile-Edge Computing Systems

Computation Rate Maximization in U A V -Enabled W ireless Po wered Mobile-Edge Computing Systems Fuhui Zhou, Member , IEEE , Y ongpeng W u, Senior Member , IEEE , Rose Qingyang Hu, Senior Member , IEEE , and Y i Qian, Senior Member , IEEE Abstract —Mobile edge computing (MEC) and wir eless power transfer (WPT) are tw o promising techniques to enhance the computation capability and to prolong the operational time of low-power wireless devices that are ubiquitous in Internet of Things. Howev er , the computation performance and the harvested ener gy are significantly impacted by the sev ere pr op- agation loss. In order to address this issue, an unmanned aerial vehicle (U A V)-enabled MEC wireless powered system is studied in this paper . The computation rate maximization problems in a U A V -enabled MEC wireless powered system are inv estigated under both partial and binary computation offloading modes, subject to the energy harv esting causal constraint and the U A V’ s speed constraint. These problems are non-con vex and challenging to solve. A two-stage algorithm and a three-stage alternati ve algorithm are respectively proposed for solving the formulated problems. The closed-form expressions for the optimal central processing unit frequencies, user offloading time, and user transmit power are derived. The optimal selection scheme on whether users choose to locally compute or offload computation tasks is proposed for the binary computation offloading mode. Simulation results show that our proposed resource allocation schemes outperforms other benchmark schemes. The results also demonstrate that the pr oposed schemes con verge fast and hav e low computational complexity . Index T erms —Mobile-edge computing, wireless power transfer , unmanned aerial vehicle-enabled, resource allocation, binary Manuscript received January 4, 2018; revised May 1, 2018 and accepted June 4, 2018. Date of publication ****; date of current version ****. The research of F . Zhou was supported in part by the Natural Science Foundation of China under Grant 61701214, in part by the Y oung Natural Science Foundation of Jiangxi Province under Grant 20171BAB212002, in part by The Open Foundation of The State Ke y Laboratory of Integrated Services Networks under Grant ISN19-08, and in part by The Postdoctoral Science Foundation of Jiangxi Province under Grant 2017M610400, Grant 2017KY04 and Grant 2017RC17. The research of Y . W u was supported by the Natural Science F oundation of China under Grant 61701301 and in part by Y oung Elite Scientist Sponsorship Program by CAST . The research of Prof. R. Q. Hu was supported in part by the National Science Foundation under Grants EECS- 1308006, NeTS-1423348, EARS-1547312 and the Natural Science F oundation of China under Grant 61728104. The research of Prof. Y . Qian was supported by the National Science Foundation under Grants EECS-1307580, NeTS- 1423408 and EARS-1547330. The corresponding author is Y ongpeng W u. F . Zhou is with the Department of Electrical and Computer Engineering as a Research Fellow at Utah State University , U.S.A. F . Zhou is also with the School of Information Engineering, Nanchang University , P . R. China, 330031. He is also with State Ke y Laboratory of Inte grated Services Networks, Xidian University , Xian, 710071, P . R. China (e-mail: zhoufuhui@ieee.org). Y . W u is with Shanghai Key Laboratory of Navigation and Location Based Services, Shanghai Jiao T ong University , Minhang, 200240, China (Email:yongpeng.wu2016@gmail.com). R. Q. Hu is with the Department of Electrical and Computer Engineering, Utah State University , USA. (e-mail: rose.hu@usu.edu). Y . Qian is with the Department of Electrical and Computer Engineer- ing, Univ ersity of Nebraska-Lincoln, Omaha, NE 68182, USA. (E-mail: yqian2@unl.edu). computation offloading, partial computation offloading. I . I N T RO D U C T I O N T HE Internet of Things (IoT) has been widely dev eloped with the unprecedented proliferation of mobile devices, such as smart phones, cloud-based mobile sensors, tablet computers and wearable devices, which facilitates the real- ization of smart en vironment (e.g. smart city , smart home, smart transportation, etc.) [1]. IoT enables mobile users to experience intelligent applications (e.g., automatic navigation, face recognition, unmanned driving, etc.) and to enjoy di verse services with high quality of service (QoS) such as mobile online gaming, augmented reality , etc. These services normally require a massive number of size-constrained and low-po wer mobile devices to perform computation-intensiv e and latency- sensitiv e tasks [2]. Howe ver , it is challenging for mobile devices to perform these services due to their low computing capability and finite battery lifetime. Mobile edge computing (MEC) and wireless power transfer (WPT) have been deemed two promising technologies to tackle the above mentioned challenges [2]-[4]. Recently , MEC has receiv ed an ever -increasing level of attention from industry and academia since it can significantly improve the compu- tation capability of mobile devices in a cost-effecti ve and energy-sa ving manner [2]. It enables mobile devices to offload partial or all of their computation-intensi ve tasks to MEC servers that locate at the edge of the wireless network, such as cellular base stations (BSs) and access points (APs). Different from the con ventional cloud computing, MEC servers are deployed in a close proximity to end users. Thus, MEC has the potential to provide low-latenc y services, to save energy for mobile users, and to achiev e high security [2]. Up to now , there are a number of leading companies (e.g., IBM, Intel, and Huawei) that hav e identified MEC as a promising technique for the future wireless communication networks. In general, MEC has two operation modes, namely , partial and binary computation offloading. In the first mode, the computation task can be partitioned into two parts, and one part is locally ex ecuted while the other part is offloaded to the MEC servers for computing [5]-[9]. For the second mode, computation tasks cannot be partitioned. Thus they can be either executed locally or completely offloaded [10]. On the other hand, WPT can provide low-po wer mobile devices with sustainable and cost-effecti ve energy supply by using radio-frequency (RF) signals [3]. It facilitates a perpetual operation and enables users to hav e high QoE, especially in the case that mobile devices do not have suf ficient battery energy for offloading task or taking the services when the battery energy is exhausted. Compared to the con ventional energy harvesting techniques, such as solar or wind charging, WPT is more attractive since it can provide a controllable and stable power supply [4]. It is en visioned that the computation performance can be significantly impro ved by inte grating WPT into MEC networks [11]-[16]. Howe ver , the harvested power le vel can be significantly degraded by the se vere prop- agation loss. Recently , an unmanned aerial vehicle (U A V)- enabled WPT architecture has been proposed to improv e the energy transfer efficiency [17]-[20]. It utilizes an unmanned aerial vehicle (U A V) as an energy transmitter for powering the ground mobile users. It was shown that the harvested power level can be greatly improved due to the fact that there is a high possibility that short-distance line-of-sight (LoS) energy transmit links exist [17]-[20]. Moreover , the computation performance can also be improv ed by using the U A V -assisted MEC architecture [21]-[25]. Furthermore, UA V - assisted architectures can provide flexible deployment and low operational costs, and are particularly helpful in the situations that the con ventional communication systems are destroyed by natural disasters [26]-[32]. Motiv ated by the above mentioned reasons, a UA V -enabled and wireless po wered MEC network is studied in this paper . In order to maximize the achiev able computation rate, the communication and computation resources and the trajectory of the U A V are jointly optimized under both partial and binary computation offloading modes. T o the authors’ best knowl- edge, this is the first work that considers the U A V -enabled wireless po wered MEC network and studies the computation rate maximization problems in this type of network. A. Related W ork and Motivation In wireless powered MEC systems, it is of great importance to design resource allocation schemes so as to efficiently exploit ener gy , communication, and computation resources and improv e the computation performance. Resource allocation problems hav e been extensi vely in vestigated in the conv en- tional MEC networks [5]-[10] and also in MEC networks rely- ing on energy harvesting [11]-[16]. Recently , efforts have also been dedicated to designing resource allocation and trajectory schemes in UA V -enabled wireless powered communications network [17]-[20] and U A V -assisted MEC networks [21]-[25]. These contributions are summarized as follows. In MEC networks, the communication and computation resources and the selection of the offloading mode were jointly optimized to achiev e the objectiv e of the system design, e.g., the users’ consumption energy minimization [5], [6], the rev enue maximization [7], the maximum cost minimization [8], etc. Specifically , in [5], the total energy of all users in a multi-cell MEC network was minimized by jointly optimizing the user transmit precoding matrices and the central processing unit (CPU) frequencies of the MEC server allocated to each user . It was shown that the performance achieved by jointly optimizing the communication and computation resources is superior to that obtained by optimizing these resources sep- arately . The authors in [6] extended the energy minimization problem into the multi-user MEC systems with time-division multiple access (TDMA) and orthogonal frequency-di vision multiple access (OFDMA), respectively . It was proved that the optimal offloading policy has a threshold-based structure, which is related to the channel state information (CSI) [6]. Particularly , mobile users offload their computation tasks when the channel condition is strong; otherwise, they can locally ex ecute the computation tasks. In [7], the revenue of the wireless cellular netw orks with MEC was maximized by jointly designing the computation of floading decision, resource allocation, and content caching strategy . The works in [5]- [7] focused on optimizing a single objecti ve, which ov er - emphasizes the importance of one metric and may not achiev e a good tradeoff among multiple metrics. Recently , the authors in [8] and [9] studied the fairness and multi-objectiv e opti- mization problem in MEC networks. It was shown that there exist multiple tradeoffs in MEC systems, such as the tradeoff between the total computation rate and the fairness among users. Dif ferent from the works in [5]-[9], MEC systems with the binary computation offloading mode were considered and the optimal resource allocation strategy was designed to minimize the consumption energy in [10]. Energy harvesting was not considered in the MEC systems [5]-[10]. Recently , the authors in [11]-[16] hav e studied the resource allocation problem in various MEC systems relying on energy harvesting. In [11] and [12], The reinforcement learning and L yapunov optimization theory were used to design resource allocation schemes in MEC systems relying on the con ventional energy harvesting techniques. Different from [11] and [12], the resource allocation problems were studied in wireless powered MEC systems [13]-[16]. Specif- ically , the authors in [13] proposed an energy-efficient com- puting frame work in which the ener gy consumed for local computing and task offloading is from the harvested energy . The consumed energy was minimized by jointly optimizing the CPU frequenc y and the mode selection. In [14], the energy minimization problem was extended into a multi-input single-out wireless po wered MEC system, and the offloading time, the offloading bits, the CPU frequency and the energy beamforming were jointly optimized. Unlike [14], energy efficienc y was defined and maximized in a full-duplex wireless powered MEC system by jointly optimizing the transmission power , offloaded bits, computation energy consumption, time slots for computation offloading and energy transfer [15]. In contrast to the work in [13]-[15], the computation bits were maximized in a wireless po wered MEC system under the binary computation of floading mode [16]. T wo sub-optimal algorithms based on the alternating direction method were proposed to solve the combinatorial programming problem. The proposed algorithms actually did not provide the optimal selection scheme for the user operation mode. Although WPT has been exploited to improve the com- putation performance of MEC systems [13]-[16], the energy harvested by using WPT can be significantly degraded by the se vere propagation loss. The energy con version ef ficiency is low when the distance between the energy transmitter and the harvesting users is large. In order to tackle this challenge, the authors in [17]-[20] proposed a U A V -enabled wireless powered architecture where a U A V transmits energy to the harvesting users. Due to the high possibility of ha ving line-of-sight (LoS) air-to-ground energy harvesting links, the harvesting energy can be significantly improved by using this architecture. Moreo ver , it w as sho wn that the harv esting energy can be further improved by optimizing the trajectory of the U A V [18]-[20]. Thus, it is en visioned that the application of the U A V -enabled architecture into wireless powered MEC systems is promising and valuable to be studied [26]. Ho we ver , to the authors’ best knowledge, few in vestigations hav e focused on this area. Recently , the UA V -enabled MEC systems hav e been studied and their resource allocation schemes hav e been proposed [21]-[25]. In [21], the U A V -enabled MEC architecture was first proposed and the computation performance was improved by using U A V . The authors in [22] proposed a new caching U A V framework to help small cells to offload traf fic. It was shown that the throughput can be greatly improved while the ov erload of wireless backhaul can be significantly reduced. In order to further improve the computation performance, the authors in [23] and [24] designed a resource allocation scheme that jointly optimizes the CPU frequenc y and the trajectory of the U A V . In [25], a theoretical game method was applied to design a resource allocation scheme for the UA V - enabled MEC system and the existence of Nash Equilibrium was demonstrated. Although resource allocation problems have been well stud- ied in MEC systems [5]-[10], MEC systems relying on energy harvesting [11]-[16] and UA V -enabled MEC systems [21]- [25], few in vestigations hav e been conducted for designing resource allocation schemes in the UA V -enabled wireless pow- ered MEC systems. Moreover , resource allocation schemes proposed in the above-mentioned works are inappropriate to U A V -enabled MEC wireless po wered systems since the com- putation performance not only depends on the optimization of energy , communication and computation resources, b ut also relies on the design of the U A V trajectory . Furthermore, the application of UA V into wireless powered MEC systems has the potential to enhance the user computation capability since it can improve the energy con version efficienc y and task offloading efficiency [33], [34]. Thus, in order to improv e the computation performance and provide mobile users with high QoE, it is of great importance and w orthiness to study resource allocation problems in UA V -enabled wireless powered MEC systems. Howe ver , these problems are indeed challenging to tackle. The reasons are from two aspects. On one hand, there exists dependence among different variables (e.g., the CPU frequency , the task offloading time and the variables related to the trajectory of the U A V), which mak es the problems non-con vex. On the other hand, when the binary computation offloading mode is applied, the resource allocation problems in U A V -enabled wireless powered MEC systems hav e binary variables related to the selection of either local computation or offloading tasks. It makes the problem a mixed integer non- con vex optimization problem. B. Contributions and Organization In contrast to [5]-[16], this paper studies the resource allocation problem in U A V -enabled wireless powered MEC systems, where a UA V transmits energy signals to char ge multiple mobile users and provides computation services for them. Although the computation performance is limited by the flight time of the U A V , it is worth studying U A V - enabled wireless powered MEC systems since these systems are promising in en vironments such as mountains and desert areas, where no terrestrial wireless infrastructures exist, and in en vironments where the terrestrial wireless infrastructures are destroyed due to the natural disasters [33], [34]. Thus, in this paper, the weighted sum computation bits of all users are maximized under both partial and binary computation offloading modes. The main contributions of this work are summarized as follows: 1) It is the first time that the resource allocation framew ork is formulated in U A V -enabled MEC wireless po wered systems under both partial and binary computation of- floading modes. The weighted sum computation bits are maximized by jointly optimizing the CPU frequencies, the offloading times and the transmit po wers of users as well as the UA V trajectory . Under the partial computa- tion of floading mode, a two-stage alternative algorithm is proposed to solve the non-conv ex and challenging computation bits maximization problem. The closed- form expressions for the optimal CPU frequencies, the offloading times and the transmit powers of users are deriv ed for any giv en trajectories. 2) Under the binary computation of floading mode, the weighted sum computation bits maximization problem is a mixed integer non-con ve x optimization problem, for which a three-stage alternative algorithm is proposed. The optimal selection scheme on whether users choose to locally compute or of fload tasks is deri ved in a closed- form expression for a gi ven trajectory . The structure for the optimal selection scheme sho ws that whether users choose to locally compute or offload their tasks to the U A V for computing depends on the tradeoff between the achiev able computation rate and the operation cost. Moreov er , the trajectory of the UA V is optimized by us- ing the successiv e conv ex approximation (SCA) method under both partial and binary computation of floading modes. 3) The simulation results show that the computation per- formance obtained by using the proposed resource allo- cation scheme is better than these achiev ed by using the disjoint optimization schemes. Moreov er , it only takes sev eral iterations for the proposed alternati ve algorithms to conv erge. Furthermore, simulation results verify that the priority and fairness of users can be improv ed by U s e r 1 U s e r 2 U s e M U s e m E n e r g y h a r v e s t i n g L o c a l c o m p u t i n g W i r e l e s s pow e r e d l i nk C om put a t i on of f l oa di ng l i nk U s e m x y z Fig. 1: The system model. using the weight vector . Additionally , it is shown that the total computation bits increase with the number of users. The remainder of this paper is organized as follows. Section II giv es the system model. The resource allocation problem is formulated under the partial computation offloading mode in Section III. Section IV formulates the resource allocation problem under the binary computation offloading mode. Sim- ulation results are presented in Section V . Finally , our paper is concluded in Section VI. I I . S Y S T E M M O D E L A U A V -enabled wireless powered MEC system is consid- ered in Fig. 1, where an RF ener gy transmitter and an MEC server are implemented in U A V . The UA V transmits energy to M users and provides MEC services for these users. Each user has an energy harvesting circuit and can store energy for its operation. The U A V has an on-board communication circuit and an on-board computing processor . So does each user . The computing processor of each user is an on-chip micro- processor that has low computing capability and can locally ex ecute simple tasks. The U A V has a powerful processor that can perform computation-intensiv e tasks [21]-[25]. Similar to [13]-[16], each user can simultaneously perform energy harvesting, local computing and computation of floading while the U A V can simultaneously transmit energy and perform computation. In this paper , all devices are equipped with a single antenna. W ithout loss of generality , a three-dimensional (3D) Eu- clidean coordinate is adopted. Each user’s location is fixed on the ground. The location of the m th ground user is denoted by q m , where q m = [ x m , y m ] , m ∈ M and M = { 1 , 2 , · · · , M } . Boldface lower case letters represent vectors and boldface upper case letters represent matrices. x m and y m are the hori- zontal plane coordinates of the m th ground user . It is assumed that user positions are kno wn to the U A V for designing the trajectory [18]-[20]. A finite time horizon with duration T is considered. During T , the U A V flies at the same altitude lev el denoted by H ( H > 0 ). In practice, the fixed altitude is the minimum altitude that is appropriate to the work terrain and can avoid building without the requirement of frequent aircraft descending and ascending. A block fading channel model is applied, i.e., during each T , the channel remains static. For the ease of exposition, the finite time T is discretized into N equal time slots, denoted by n = 1 , 2 , · · · , N . At the n th slot, it is assumed that the horizontal plane coordinate of the U A V is q u [ n ] = [ x u [ n ] , y u [ n ]] . Similar to [27]-[32], it is assumed that the wireless channel between the UA V and each user is dominated by LOS. Thus, the channel power gain between the U A V and the m th user, denoted by h m [ n ] , can be giv en as h m [ n ] = β 0 d − 2 m,n = β 0 H 2 + k q u [ n ] − q m k 2 , m ∈ M , n ∈ N , (1) where β 0 is the channel power gain at a reference distance d 0 = 1 m; d m,n is the horizontal plane distance between the U A V and the m th user at the n th slot, n ∈ N , N = { 1 , 2 , · · · , N } ; k·k denotes its Euclidean norm. The details for the UA V -enabled wireless powered MEC system are presented under partial and binary computation of floading modes in the following, respectiv ely . A. P artial Computation Offloading Mode Under the partial computation offloading mode, the com- putation task of each user can be partitioned into two parts, one for local computing and one for offloading to the U A V . The energy consumed for local computing and task offloading comes from the harvested energy . In this paper , in order to shed meaningful insights into the design of a UA V -enabled wireless powered MEC system, similar to [4], [13]-[16], the linear energy harvesting model is applied. Thus, the harvested energy E m [ n ] at the m th user during n time slots is giv en as E m [ n ] = n X i =1 T η 0 h m [ i ] P 0 N , m ∈ M , n ∈ N , (2) where η 0 denotes the energy conservation efficienc y , 0 < η 0 ≤ 1 and P 0 is the transmit power of the UA V . In this paper, the UA V employs a constant power transmission [18]-[20]. The details for the operation of each user under the partial computation offloading mode are presented as follows. 1) Local Computation: Similar to [14]-[16], the energy harvesting circuit, the communication circuit, and the compu- tation unit are all separate. Thus, each user can simultaneously perform energy harvesting, local computing, and computation offloading. Let C denote the number of CPU cycles required for computing one bit of ra w data at each user . In order to efficiently use the harv ested energy , each user adopts a dynamic voltage and frequency scaling technique and then can adaptiv ely control the energy consumed for performing local computation by adjusting the CPU frequency during each time slot [14]-[16]. The CPU frequency of the m th user during T T h e f i r s t s l o t T h e s e c o n d s l o t . . . T h e t h s l o t n U A V U s e r M O f f l o a d i n g . . . U s e r 1 U A V D o w n l o a d U A V   1 1 t 0  0    1 M t O f f l o a d i n g U s e r 1 U A V . . . T h e t h s l o t N D o w n l o a d U s e r M . . . . . . . . . T N Fig. 2: The TDMA protocol for multiuser computation of fload- ing. the n th slot is denoted by f m [ n ] with a unit of cycles per second. Thus, the total computation bits ex ecuted at the m th user during n slots and the total consumed energy at the m th user during n slots are respectiv ely given as n P k =1 T f m [ k ] N C and n P k =1 γ c f 3 m [ k ] [14]-[16], where γ c is the effecti ve capacitance coefficient of the processor’ s chip at the m th user , n ∈ N , m ∈ M . Note that γ c is dependent of the chip architecture of the m th user . 2) Computation Offloading: In order to av oid interference among users during the offloading process, a TDMA protocol shown in Fig. 2 is applied. Specifically , each time slot consists of three stages, namely , the offloading stage, the computation stage, and the downloading stage. In the of floading stage, M users of fload their respective computation task one by one during each slot. Let t m [ n ] × T / N (0 ≤ t m [ n ] ≤ 1) denote the duration in which the m th user offloads its computation task to the U A V at the n th slot, n ∈ N , m ∈ M . Similar to [16], the computation task of the m th user to be offload is composed of raw data and communication overhead, such as the encryption and packer header . Let ν m R m [ n ] denote the total number of bits that the m th user of floads to the U A V during the n th slot, where R m [ n ] is the number of raw data to be computed at the U A V and ν m indicates the communication ov erhead included in the offloading task. Thus, one has R m [ n ] ≤ B T t m [ n ] ν m N log 2  1 + h m [ n ] P m [ n ] σ 2 0  , n ∈ N , m ∈ M , (3) where B is the communication bandwidth; P m [ n ] is the transmit po wer of the m th user at the n th slot and σ 2 0 denotes the noise power at the m th user . After all users offload their computation tasks at the n th slot, the U A V performs computing task and sends the com- puting results back to all the users. Similar to [14]-[16], the computation time and the downloading time of the UA V are neglected since the UA V has a much stronger computation capability than the users and the number of the bits related to the computation result is very small. Since the total offloading time of all users does not exceed the duration of one time slot, one has M X m =1 t m [ n ] ≤ 1 , n ∈ N . (4) Since the energy consumed for local computing and task offloading comes from the harvested energy , the follo wing energy harvesting causal constraint should be satisfied. T N n X k =1  γ c f 3 m [ k ] + t m [ k ] P m [ k ]  ≤ η 0 T N n X k =1 h m [ k ] P 0 , n ∈ N , m ∈ M . (5) Under the partial computation offloading mode, the total computation bits R m of the m th user is giv en as R m = N X n =1 T f m [ n ] N C + B T t m [ n ] ν m N log 2  1 + h m [ n ] P m [ n ] σ 2 0  , m ∈ M . (6) B. Binary Computation Offloading Mode Under the binary computation offloading mode, the compu- tation task cannot be partitioned. All the users need to choose to either locally compute the task completely or offload the entire task. This case can be widely experienced in practice. For example, in order to improve the estimation accuracy , the raw data samples that are correlated need to be jointly computed altogether [10], [16]. Let M 0 and M 1 denote the set of users that choose to perform local computation and the set of users that choose to perform task offloading, respectively . Thus, M = M 0 ∪ M 1 and M 0 ∩ M 1 = Θ , where Θ denotes the null set. 1) Users Choosing to P erform Local Computing: In this case, a user in M 0 exploits all the harvested ener gy to perform local computing. Thus, the total computation rate of the i th user denoted by R L i can be giv en as R L i = N X n =1 T f i [ n ] N C , i ∈ M 0 . (7) And the energy harvesting causal constraint for a user in M 0 can be giv en as T N n X k =1 γ c f 3 i [ k ] ≤ η 0 T N n X k =1 h i [ k ] P 0 , n ∈ N , i ∈ M i . (8) 2) Users Choosing to P erform T ask Offloading: Each user in M 1 exploits all the harvested energy to perform task of- floading. The TDMA protocol is applied to av oid interference among these users during the offloading process. Since the total offloading time of all users in M 1 at the n th slot cannot exceed the duration of a time slot, one has X j ∈M 1 t j [ n ] ≤ 1 , n ∈ N . (9) Let R O j denote the total computation rate of the j th user in the set M 1 . Then, one has R O j = N X n =1 B T t j [ n ] ν j N log 2  1 + h j [ n ] P j [ n ] σ 2 0  , j ∈ M 1 . (10) The energy harvesting causal constraint for a user in M 1 can be giv en as T N n X k =1 t j [ k ] P j [ k ] ≤ η 0 T N n X k =1 h j [ k ] P 0 , n ∈ N , j ∈ M 1 . (11) Sections III and IV will respecti vely formulate the compu- tation rate maximization problem for the partial and binary computation offloading modes. I I I . R E S O U R C E A L L O C A T I O N U N D E R T H E P A RT I A L C O M P U T A T I O N O FFL OA D I N G M O D E In this section, the resource allocation problem is studied under the partial computation of floading mode. The weighted sum computation bits are maximized by jointly optimizing the CPU frequencies, the offloading times and the transmit powers of users as well as the trajectory of the UA V . In order to tackle this non-con vex problem, a two-stage alternati ve algorithm is proposed. A. Resour ce Allocation Problem F ormulation Under the partial computation of floading mode, the weighted sum computation bits maximization problem in the U A V -enabled wireless powered MEC system is formulated as P 1 , P 1 : max f m [ n ] ,P m [ n ] , q u [ n ] ,t m [ n ] M X m =1 w m × " N X n =1 T f m [ n ] N C + B T t m [ n ] ν m N log 2  1 + h m [ n ] P m [ n ] σ 2 0  # (12a) s.t. C 1 : f m [ n ] ≥ 0 , P m [ n ] ≥ 0 , m ∈ M , n ∈ N , (12b) C 2 : T N n X k =1  γ c f 3 m [ k ] + t m [ k ] P m [ k ]  ≤ η 0 T N n X k =1 h m [ k ] P 0 m ∈ M , n ∈ N , (12c) C 3 : M X m =1 t m [ n ] ≤ 1 , n ∈ N , (12d) C 4 : k q u [ n + 1] − q u [ n ] k 2 ≤ V max T N , n ∈ N , (12e) C 5 : q u [1] = q 0 , q u [ N + 1] = q F , (12f) where V max denotes the maximum speed of the U A V in the unit of meter per second; q 0 and q F are the initial and final horizontal locations of the U A V , respectiv ely . In (12) , w m denotes the weight of the m th user , which takes the priority and the fairness among users into consideration. C 1 is the CPU frequency constraint and the computation of floading power constraint imposed on each user; C 2 represents the energy harvesting causal constraint; C 3 is the time constraint that the total time of all users offloading the computation bits cannot exceed the duration of each time slot; C 4 and C 5 are the speed constraint and the initial and final horizontal location constraint of the U A V , respectiv ely . P 1 is non-con vex since there exist non-linear couplings among the variables, f m [ n ] , P m [ n ] , q u [ n ] , t m [ n ] and the objectiv e function is non-conca ve with respect to the trajectory of the U A V . In order to solve it, a two-stage alternative optimization algorithm is proposed. The details for the algorithm are presented as follows. B. T wo-Stage Alternative Optimization Algorithm Let z m [ n ] = t m [ n ] P m [ n ] , n ∈ N . For a given trajectory , P 1 can be transformed into P 2 . P 2 : max f m [ n ] ,z m [ n ] ,t m [ n ] M X m =1 w m × " N X n =1 T f m [ n ] N C + B T t m [ n ] ν m N log 2  1 + h m [ n ] z m [ n ] t m [ n ] σ 2 0  # (13a) s.t. C 1 , C 3 , (13b) C 5 : T N n X k =1  γ c f 3 m [ k ] + z m [ k ]  ≤ η 0 T N n X k =1 h m [ k ] P 0 , m ∈ M , n ∈ N . (13c) It is easy to prove that P 2 is con vex and can be solv ed by using the Lagrange duality method [35], based on which the optimal solutions for the CPU frequency and the transmit power can be derived. Let f opt m [ n ] and P opt m [ n ] denote the optimal CPU frequency and transmit power of the m th user at the n th time slot, respectively , where m ∈ M and n ∈ N . By solving P 2 , Theorem 1 can be stated as follows. Theor em 1: For a given trajectory q u [ n ] , the optimal CPU frequency and transmit power of users can be respectiv ely expressed as f opt m [ n ] = v u u u t w m 3 C γ c N P k = n λ m,k , (14a) P opt m [ n ] =        0 , if t m [ n ] = 0 ,   w m B ν m ln 2 N P k = n λ m,k − σ 2 0 h m [ n ]   + , otherwise, (14b) where λ m,n ≥ 0 is the dual variable associated with the constraint C 2 ; [ a ] + = max ( a, 0) and max ( a, 0) denotes the bigger value of a and 0 . Pr oof: See Appendix A. Remark 1: It can be seen from Theorem 1 that users choose to offload their computation tasks only when the channel state information between users and the UA V is stronger than a threshold, namely , h m [ n ] ≥  σ 2 0 ν m ln 2 N P k = n λ m,k  / ( w m B ) . This indicates that the user chooses to perform local com- putation when the horizontal distance between the user and the UA V is larger than β 0 w m B σ 2 0 ν m ln 2 N P k = n λ m,k − H 2 . Moreover , it can be seen that the larger the weight is, the higher the chance for the user to chooses to offload its computation task. Furthermore, users prefers to offload their computation task when the local computation frequency is very large, namely , f opt m [ n ] ≥ q σ 2 0 ν m ln 2 3 C γ c B h m [ n ] . Theor em 2: If there exists a time slot that f opt m [ n ] = 0 , the equation f opt m [ k ] = 0 must hold, 0 ≤ k ≤ n . Pr oof: Since λ m,n is the dual v ariable and λ m,n ≥ 0 , from Theorem 1 f opt m [ n ] increases with n . Thus, if there exists a time slot n so that f opt m [ n ] = 0 , one must hav e f opt m [ k ] = 0 , for 0 ≤ k ≤ n . Theorem 2 is proved. Remark 2: Theorem 2 indicates that the user CPU frequency increases with the time slot index. This means that the number of computation bits obtained by local computing increases with the time slot index. Moreov er , the user CPU frequency increases with the weight assigned to that user since more resources are allocated to the user with a higher weight. Theor em 3: For a giv en trajectory q u [ n ] , the optimal user offloading time can be obtained by solving the following equation. log 2  1 + h m [ n ] z m [ n ] σ 2 0 t m [ n ]  − h m [ n ] z m [ n ] ln 2 { σ 2 0 t m [ n ] + h m [ n ] z m [ n ] } − ν m N α n B T = 0 . (15) Remark 3: Theorem 3 can be readily prov ed based on the proof for Theorem 1. Thus this proof is omitted for the sake of saving space. Moreov er , (15) can be solved by using the bisection method [35]. The v alues of the dual v ariables are needed in order to obtain the optimal CPU frequency , the optimal transmit power and the optimal offloading time for all users. The subgradient method in Lemma 1 can be used to tackle this problem [36]. Lemma 1: The subgradient method for obtaining the dual variables is giv en as λ m,n ( l + 1) = [ λ m,n ( l ) − θ ( l ) ∆ λ m,n ( l )] + , m ∈ M , n ∈ N (16a) α n ( l + 1) = [ α n ( l ) − ϑ ( l ) ∆ α n ( l )] + , n ∈ N , (16b) where l denotes the iteration index; θ ( l ) and ϑ ( l ) represent the iterative steps at the l th iteration. In (16) , ∆ λ m,n ( l ) and ∆ α n ( l ) are the corresponding subgradients, gi ven as ∆ λ m,n ( l ) = η 0 T N n X k =1 h m [ k ] P 0 − T N n X k =1 h γ c  f l,opt m [ k ]  3 + z m l,opt [ k ] i , (17a) ∆ α n ( l ) = 1 − M X m =1 t l,opt m [ n ] , n ∈ N , (17b) where f l,opt m [ n ] , z m l,opt [ n ] , and t l,opt m [ n ] denote the optimal solutions at the l th iterations. According to [35], the subgra- dient guarantees to con verge to the optimal value with a very small error range. C. T rajectory Optimization For any giv en CPU frequency , transmit po wer , and of fload- ing time of users, the trajectory optimization problem can be formulated as P 3 . P 3 : max q u [ n ] M X m =1 w m ×   N X n =1 B T t m [ n ] ν m N log 2   1 + β 0 P m [ n ] σ 2 0  H 2 + k q u [ n ] − q m k 2      (18a) s.t. C 2 : T N n X k =1  γ c f 3 m [ k ] + t m [ k ] P m [ k ]  ≤ η 0 T N n X k =1 β 0 P 0 H 2 + k q u [ k ] − q m k 2 , m ∈ M , n ∈ N (18b) C 4 and C 5 . (18c) Since C 2 is non-conv ex and the objective function is non- concav e with respect to q u [ n ] , P 3 is non-conv ex and we use the SCA technique to solve the optimization problem. The obtained solutions can be guaranteed to satisfy the Karush- Kuhn-T ucker (KKT) conditions of P 3 [27]. By using the SCA technique, Theorem 4 is giv en as follo ws. Theor em 4: For any local trajectory q u, [ n ] , n ∈ N at the  th iteration, one has n X i =1 P 0 β 0 H 2 + k q u [ i ] − q m k 2 ≥ P 0 β 0 h m [ n ] , (19a) h m [ n ] = n X i =1      H 2 + 2 k q u, [ i ] − q m k 2 − k q u [ i ] − q m k 2  H 2 + k q u, [ i ] − q m k 2  2      (19b) where the equality holds when q u [ n ] = q u, [ n ] . Pr oof: Let f ( z ) = a b + z , where a and b are positi ve constants, and z ≥ 0 . Since f ( z ) is con vex with respect to z , the following inequality can be obtained: a b + z ≥ a b + z 0 − a ( b + z 0 ) 2 ( z − z 0 ) , (20) where z 0 is a gi ven local point. By using (20) , Theorem 4 is prov ed. In order to tackle the objective function of P 3 , Lemma 2 is giv en as follows. Lemma 2: [27] Using the SCA method, the follo wing inequality can be obtained, log 2   1 + β 0 P m [ n ] σ 2 0  H 2 + k q u [ n ] − q m k 2    ≥ y m, ( { q u [ n ] } ) , (21a) y m, ( { q u [ n ] } ) = log 2   1 + β 0 P m [ n ] σ 2 0  H 2 + k q u, [ n ] − q m k 2    − β 0 P m [ n ] log 2 e  σ 2 0 H 2 + β 0 P m [ n ] + σ 2 0 k q u, [ n ] k 2   H 2 + k q u, [ n ] k 2  ×  k q u [ n ] k 2 − k q u, [ n ] k 2  , (21b) where the equality holds when q u [ n ] = q u, [ n ] . T ABLE I: T wo-stage alternativ e optimization algorithm Algorithm 1 : The two-stage alternativ e optimization algorithm 1: Setting: P 0 , T , N , V max , q 0 , q F , and the tolerance errors ξ , ξ 1 ; 2: Initialization: The iterative number i = 1 , λ i m,n , α i n and q i u [ n ] ; 3: Repeat 1: calculate f opt,i m [ n ] and P opt,i m [ n ] using Theorem 1 for given q i u [ n ] ; use the bisection method to solve (20) and obtain t i,opt m [ n ] ; update λ i m,n and α i n using the subgradient algorithm; initialize the iterative number j = 1 ; Repeat 2: solve P 4 by using CVX for the given f opt,i m [ n ] , P opt,i m [ n ] and t i,opt m [ n ] ; update j = j + 1 , and q j u [ n ] ; if N P n =1    q j u [ n ] − q j − 1 u [ n ]    ≤ ξ q i u [ n ] = q j u [ n ] ; break; end end Repeat 2 update the iterative number i = i + 1 ; if   R i − R i − 1   ≤ ξ 1 break; end end Repeat 1 4: Obtain solutions: f opt m [ n ] , P opt m [ n ] and t opt m [ n ] and q opt u [ n ] . Using Theorem 4 and Lemma 2, P 3 can be solved by iterativ ely solving the approximate problem P 4 , giv en as P 4 : max q u [ n ] M X m =1 w m " N X n =1 B T t m [ n ] y m, ( { q u [ n ] } ) ν m N # (22a) s.t. C 4 and C 5 , (22b) n X k =1  γ c f 3 m [ k ] + t m [ k ] P m [ k ]  ≤ η 0 P 0 β 0 h m [ n ] , m ∈ M , n ∈ N . (22c) It can be seen that P 4 is con vex and can be readily solved by using CVX [4]. By solving P 2 and P 4 , a two-stage alternative optimization algorithm denoted by Algorithm 1 is further dev eloped to solve P 1 . The details for Algorithm 1 can be found in T able I. In T able I, R i denotes the value of the objectiv e function of P 1 at the i th iteration. I V . R E S O U R C E A L L O C A T I O N I N B I N A RY C O M P U T A T I O N O FFL OA D I N G M O D E In this section, the weighted sum computation bits max- imization problem is studied in the U A V -enabled wireless powered MEC system under the binary computation offloading mode. The CPU frequencies of the users that choose to perform local computation, the of floading times, the transmit powers of users that choose to perform task offloading, the trajectory of the U A V , and the mode selection are jointly optimized to maximize the weighted sum computation bits of all users. The formulated problem is a mixed integer non-con vex optimization problem, for which a three-stage alternativ e optimization problem is proposed. A. Resour ce Allocation Problem F ormulation Under the binary computation of floading mode, the weighted sum computation bit maximization problem subject to the energy harvesting causal constraints, the UA V speed and position constraints is formulated as P 5 , P 5 : max f i [ n ] ,P j [ n ] ,q [ n ] , t j [ n ] , M 0 , M 1 X i ∈M 0 N X n =1 w i f i [ n ] T C N + X j ∈M 1 w j B T ν j N N X n =1 t j [ n ] log 2  1 + h j [ n ] P j [ n ] σ 2 0  (23a) s.t. T N n X k =1 γ c f 3 i [ k ] ≤ η 0 T N n X k =1 h i [ k ] P 0 , n ∈ N , i ∈ M 0 , (23b) T N n X k =1 t j [ k ] P j [ k ] ≤ η 0 T N n X k =1 h j [ k ] P 0 , n ∈ N , j ∈ M 1 , (23c) X j ∈M 1 t j [ n ] ≤ 1 , n ∈ N , (23d) M = M 0 ∪ M 1 , M 0 ∩ M 1 = Θ , (23e) f i [ n ] ≥ 0 , P j [ n ] ≥ 0 , i ∈ M 0 , j ∈ M 1 , (23f) C 4 and C 5 . (23g) (23b) and (23c) are the energy harvesting causal constraints imposed on these users who choose to perform local computa- tion and on these users who choose to perform task offloading, respectiv ely; (23d) is the of floading time constraint during each slot and (23e) is the user operation selection constraint. In P 5 there exist close couplings among different optimiza- tion variables. Furthermore, the binary user operation mode selection makes P 5 a mixed integer programming problem. The exhaustiv e search method leads to a prohibitiv ely high computational complexity , especially when there exist a large number of users. Motiv ated by how we solve P 1 , P 5 has a similar structure as P 1 when the operation modes of users are determined. Thus, the optimal CPU frequency , transmit power , and offloading time of users can be obtained by using the same method as the one used for P 1 and the trajectory optimization for the UA V can also be achie ved by using the SCA method. As such, a three-stage alternative optimization algorithm is proposed based on the two-stage Algorithm 1. The details for the algorithm are presented as follows. B. Thr ee-Stage Alternative Optimization Algorithm In order to efficiently solve P 5 , a binary variable denoted by ρ m is introduced, where ρ m ∈ { 0 , 1 } and m ∈ M . ρ m = 0 indicates that the m th user performs local computation mode while ρ m = 1 means that the m th user performs task offloading. Moreov er , the user operation selection indicator variable ρ m is relaxed as a sharing factor ρ m ∈ [0 , 1] . Thus, P 5 can be rewritten as P 6 : max f m [ n ] ,P n [ n ] , q [ n ] , t m [ n ] ,ρ m M X m =1 N X n =1 w m  (1 − ρ m ) f m [ n ] T C N + B T t m [ n ] ρ m ν m N log 2  1 + h m [ n ] P m [ n ] σ 2 0  (24a) s.t. (1 − ρ m ) T N n X k =1 γ c f 3 m [ k ] + ρ m T N n X k =1 t m [ k ] P m [ k ] ≤ η 0 T N n X k =1 h m [ k ] P 0 , m ∈ M , (24b) M X m =1 ρ m t m [ n ] ≤ 1 , n ∈ N , (24c) f m [ n ] ≥ 0 , P m [ n ] ≥ 0 , n ∈ N , m ∈ M , (24d) C 4 and C 5 . (24e) Even by relaxing the binary variable ρ m , P 6 is still dif ficult to solve as there exist couplings among different variables. For any gi ven ρ m and the trajectory of the UA V , P 6 has a similar structure as P 1 . Thus, using the same techniques applied to P 1 , the optimal CPU frequency , transmit power and offloading time of users for a given ρ m and the UA V trajectory can be obtained. It is easy to verify that the optimal CPU frequency , transmit power and offloading time of users for a giv en trajectory have the same forms gi ven by Theorem 1 and Theorem 3. Theor em 5: For any given f m [ n ] , P m [ n ] , t m [ n ] and q u [ n ] , the user operation selection scheme can be obtained by ρ opt m =  0 if G 1 ≥ G 2 , 1 otherwise; (25a) G 1 = N X n =1 ( w m f m [ n ] C − υ m,n n X k =1 γ c f 3 m [ k ] ) , (25b) G 2 = N X n =1  B t m [ n ] ν m log 2  1 + h m [ n ] P m [ n ] σ 2 0  − υ m,n n X k =1 t m [ k ] P m [ k ] − N T ε n t m [ n ] ) , (25c) where υ m,n ≥ 0 and ε n ≥ 0 are the dual v ariables associated with the constraints giv en by (24b) and (24c) , respectiv ely . Pr oof: See Appendix B. Remark 4: Theorem 5 indicates that the user operation selection scheme depends on the tradeoff between the achie v- able computation rate and the operation cost. If the tradeoff of the user achiev ed by local computing is better than that obtained by task of floading, the user chooses to perform local computing; otherwise, the user chooses to offload its computation tasks to the U A V for computing. Finally , the trajectory optimization for any gi ven ρ m , f m [ n ] , P m [ n ] and t m [ n ] can be obtained by solving P 7 , giv en as P 7 : max q u [ n ] M X m =1 w m ρ m " N X n =1 B T t m [ n ] y m, ( { q u [ n ] } ) ν m N # (26a) s.t. C 4 and C 5 , (26b) (1 − ρ m ) n X k =1 γ c f 3 m [ k ] + ρ m n X k =1 t m [ k ] P m [ k ] ≤ η 0 P 0 β 0 h m [ n ] , m ∈ M , n ∈ N , (26c) where h m [ n ] and y j ( { q u [ n ] } ) are giv en by (19b) and (21b) , respectiv ely . P 7 is conv ex and can be efficiently solved by using CVX [4]. Based on Theorem 1, Theorem 5 and the so- lutions of P 7 , a three-stage alternativ e optimization algorithm denoted by Algorithm 2 is proposed to solve P 5 . The details for Algorithm 2 are presented in T able 2. In T able 2, R l and R i denote the v alue of the objectiv e function of P 5 at the l th and i iteration, respectiv ely . C. Complexity Analysis The complexity of Algorithm 1 comes from four aspects. The first aspect is from the computation of the CPU frequency and the offloading po wer . The second aspect is from the bisection method for obtaining the offloading time. The third aspect is from the subgradient method for computing the dual variables. The fourth aspect comes from the application of CVX for solving P 4 . Let L 1 and L 2 denote the num- ber of iterations required for the outer loop and the inner loop of Algorithm 1, respectively . Let ` 1 and ` 2 denote the tolerance error for the bisection method and the subgradi- ent method, respecti vely . Thus, according to the works in [35], [38] and [39], the total complexity of Algorithm 1 is O  L 1  2 M N + M log 2 ( ` 1 /T ) + 1 /` 2 2 + L 2 N 3  and O ( · ) is the big- O notation [35]. The complexity of Algorithm 2 comes from fiv e aspects. Four aspects are the same as these of Algorithm 1. The fifth aspect is from the computation of the operation selection indicator variable ρ m . Let L 1 , L 2 and L 3 denote the number of iterations required for the first, second and third loop of Algorithm 2, respectively . Similar to the complexity analysis for Algorithm 1, the total complexity of Algorithm 2 is O  L 1 L 2  2 M N + M + M log 2 ( ` 1 /T ) + 1 /` 2 2 + L 3 N 3  . T ABLE II: Three-stage alternativ e optimization algorithm Algorithm 2 : The three-stage alternative optimization algorithm 1: Setting: P 0 , T , N , V max , q 0 , q F , and the tolerance errors ξ , ξ 1 and ξ 2 ; 2: Initialization: The iterative number i = 1 , υ i m,n and ε i n , and q i u [ n ] ; 3: Repeat 1: initialize the iterative number l = 1 and ρ l m ; Repeat 2: calculate f opt,i m [ n ] and P opt,i m [ n ] using Theorem 1 for given q i u [ n ] and ρ opt,l m ; use the bisection method to solve (20) and obtain t i,opt m [ n ] ; update υ i m,n and ε i n using the subgradient algorithm; calculate ρ opt,l m using Theorem 5 and update l = l + 1 ; if   R l − R l − 1   ≤ ξ break; end initialize the iterative number j = 1 ; Repeat 3: solve P 7 by using CVX for the given f opt,i m [ n ] , P opt,i m [ n ] , t i,opt m [ n ] and ρ opt,l m ; update j = j + 1 , and q j u [ n ] ; if N P n =1    q j u [ n ] − q j − 1 u [ n ]    ≤ ξ q i u [ n ] = q j u [ n ] ; break; end end Repeat 3 update the iterative number i = i + 1 ; if   R i − R i − 1   ≤ ξ 1 break; end end Repeat 2 end Repeat 1 4: Obtain solutions: f opt m [ n ] , P opt m [ n ] and t opt m [ n ] , ρ opt m and q opt u [ n ] . V . S I M U L ATI O N R E S U LT S In this section, simulation results are presented to compare the performance of our proposed designs with that of other benchmark schemes. The con vergence performance of the proposed algorithms is also ev aluated. The simulation settings are based on the works in [7], [14], [16] and [23]. The positions of users are set as: q 1 = [0 , 0] , q 2 = [0 , 10] , q 3 = [10 , 10] , q 4 = [10 , 0] . The detailed settings are giv en in T able III. The weight vector of each user [ w 1 w 2 w 3 w 4 ] is set as [0 . 1 0 . 4 0 . 3 0 . 2] . Fig. 3 shows the U A V trajectory under different schemes with T = 2 seconds. The U A V transmit po wer is set as P 0 = 0 . 1 W . In the constant speed scenario, the UA V flies straight with a constant speed from the initial position to the final position. In the semi-circle scenario, the UA V flies along the trajectory that is a semi-circle with its diameter being k q F − q 0 k . The trajectory of the offloading mode is obtained by using Algorithm 1 for the partial computation of floading mode and the trajectory of the binary mode is obtained by using Algorithm 2 for the binary computation of floading mode. It can be seen from the trajectories of our proposed schemes the U A V is always close to user 2 and user 3, irrespectiv e of T ABLE III: Simulation Parameters Parameters Notation T ypical V alues Numbers of Users M 4 The height of the UA V H 10 m The time length of the UA V flying T 2 sec Numbers of CPU cycles C 10 3 cycles/bit Energy conv ersation efficiency η 0 0 . 8 Communication bandwidth B 40 MHz The receiver noise power σ 2 0 10 − 9 W The number of time slots N 50 The effectiv e switched capacitance γ c 10 − 28 The channel power gain β 0 − 50 dB The tolerance error ξ , ξ 1 10 − 4 The initial position of the UA V q 0 [0 , 0] The final position of the UA V q F [10 , 0] The maximum speed of the UA V V max 20 m/s 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 x (m) y(m) The offloading mode A constant speed A semi−circle User 2 (0,10) User 4 (10,0) User 1 (0,0) User 3 (10,10) The binary mode Fig. 3: The trajectory of the UA V under different schemes with T = 2 seconds. the operation modes. The reason is that the weights of user 2 and user 3 are larger than these of user 1 and user 4. Thus, the UA V needs to fly close to user 2 and user 3 so as to provide more ener gy to them. This indicates that the priority and the fairness among users can be obtained by using the weight vector . Fig. 4 shows the weighted sum computation bits of all users versus the transmit power of the U A V under different schemes. The optimal local computing is the mode that all users only perform local computing while the optimal offloading mode is that all users only perform task offloading. And the trajectory of the U A V is jointly optimized under these two benchmark schemes. The results under the binary mode and the partial offloading mode are obtained by using Algorithm 2 and Algorithm 1, respectively . In Fig. 4 the weighted sum computation bits achiev ed under the partial offloading mode is the largest among these obtained by other schemes. The reason is that all the users can dynamically select the operation mode based on the quality of the channel state information under the partial computation offloading mode. Moreover , the optimal offloading mode outperforms the optimal local computing. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 1.5 2 2.5 3 3.5 4 4.5 x 10 7 The transmit power of the UAV (W) The weighted sum computation bits of all users (bits) Optimal local computing Optimal offloading The binary mode The partial offloading mode Fig. 4: The weighted sum computation bits of all users versus the transmit power of the UA V under dif ferent schemes. This result is consistent with the results obtained in [13]. Furthermore, the weighted sum computation bits of all users increase with the U A V transmit power . It can be explained by the fact that the harvesting energy increases with the transmit power of the UA V . Thus, users hav e more energy to perform local commutating or task offloading. Fig. 5 shows the weighted sum computation bits of all the users versus the transmit power of the U A V under different trajectories with the partial computation offloading mode and the binary computation offloading mode. As shown in Fig. 5, the weighted sum computation bits of all the users achiev ed by using our proposed schemes are larger than that obtained by using the trajectory with a constant speed and than that obtained by using the semi-circle trajectory , irrespective of the operation modes. This indicates that the optimization of the trajectory of the UA V can improv e the weighted sum computa- tion bits. It also verifies that our proposed resource allocation scheme outperforms the disjoint optimization schemes. Fig. 6 shows the total computation bits of each user under different operation modes. The transmit power of the U A V is set as P 0 = 0 . 1 W . The total computation bits of user 2 and user 3 are higher than those of user 1 and user 4. The reason is that the weights of user 2 and user 3 are larger than those of user 1 and user 4. Thus, the resource allocation scheme should consider the priority of user 2 and user 3. This further verifies that the application of the weight vector can improve the priority and also the fairness of users. Fig. 7 is gi ven to v erify the efficiency of our proposed Algorithm 1 and Algorithm 2. The transmit po wer of the U A V is giv en as 0 . 1 W or 0 . 2 W . The results sho w that Algorithm 1 and Algorithm 2 only need se veral iterations to con verge. This indicates that the proposed Algorithm 1 and Algorithm 2 are computationally ef fecti ve and have a f ast con vergence rate. It can also be seen that the weighted sum computation bits of all the users achiev ed under the partial 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 1.5 2 2.5 3 3.5 4 4.5 x 10 7 The transmit power of the UAV (W) The weighted sum computation bits of all users (bits) The partial offloading mode The partial offloading mode with the semi−circle trajectory The partial offloading mode with a constant speed 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 1.5 2 2.5 3 3.5 4 4.5 x 10 7 The transmit power of the UAV (W) The weighted sum computation bits of all users (bits) The binary mode with joint optimization The binary mode with the semi−circle trajectory The binary mode with a constant speed (a) (b) Fig. 5: (a) The weighted sum computation bits of all users versus the transmit po wer of the U A V under dif ferent trajec- tories with the partial computation offloading mode; (b) The weighted sum computation bits of all users versus the transmit power of the U A V under different trajectories with the binary computation offloading mode. computation offloading mode are larger than those obtained under the binary computation offloading mode. The reason is that users can simultaneously perform local computing and task offloading when the channel state information is strong under the partial computation of floading mode. Howe ver , users can only perform either local computing or task offloading in the binary of floading mode e ven when the channel state infor- mation is strong. The computation performance is improv ed by the flexible selection of the operation mode based on the channel state information. User 1 User 2 User 3 User 4 0 1 2 3 4 5 6 7 x 10 6 Users The total compuation bits of each user (Bits) The binary mode The partial offloading mode Fig. 6: The total computation bits of each user under different operation modes with P 0 = 0 . 1 W . 2 4 6 8 10 12 14 16 18 20 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 x 10 7 The number of iterations The weighted sum computation bits of all users (bits) P 0 =0.2 W, Algorithm 1 P 0 =0.2 W, Algorithm 2 P 0 =0.1 W, Algorithm 1 P 0 =0.1 W, Algorithm 2 Fig. 7: The weighted sum computation bits of all users versus the number of iterations required by using Algorithms 1 and 2 under different transmit powers of the UA V and different operation modes. Fig. 8 shows the weighted sum computation bits of all users versus the number of users under different operation modes. The transmit power of the U A V is set as P 0 = 0 . 2 W or P 0 = 0 . 4 W . In Fig. 8 the weighted sum computation bits of all users increase with the number of users. The reason is that more users can exploit the harvesting energy to perform local computing and computation offloading. It is also observed that the growth rate decreases with the increase of the number of users. The reason is that the of floading time allocated for each user decreases with the increase of the number of users since the total offloading time is limited by T . T able IV is giv en to e v aluate the run times of Algorithm 1 and Algorithm 2 sho wn in the top of the next page. The run times are obtained by using a computer with 64-bit Intel(R) 2 3 4 5 6 7 8 9 10 11 12 1 1.5 2 2.5 3 3.5 4 x 10 7 The number of users The weighted sum computation bits of all users (bits) The partial offloading mode, P 0 =0.4 W The binary mode, P 0 =0.4 W The partial offloading mode, P 0 =0.2 W The binary mode, P 0 =0.2 W Fig. 8: The weighted sum computation bits of all users versus the number of users under different transmit powers of the U A V and different operation modes. Core(TM) i7-4790 CPU, 8 GB RAM. From T able IV we can see that the required run time of Algorithm 1 is smaller than that of Algorithm 2. This indicates that the complexity of Algorithm 1 is lower than that of Algorithm 2. It can be verified by the complexity analysis presented in Subsection C of Section IV . Moreover , the effect of the number of time slots on the run time is larger than that of the number of users. The reason is that the complexity of these two algorithms mainly depends on the number of time slots. This can also be verified by the complexity analysis. V I . C O N C L U S I O N S The resource allocation problems were studied for UA V - enabled wireless powered MEC systems under both the par- tial and binary computation offloading modes. The weighted sum computation rates of users were maximized by jointly optimizing the CPU frequencies, the user offloading times, the user transmit powers, and the U A V trajectory T wo alter- nativ e algorithms were proposed to solve these challenging problems. The closed-form expressions for the optimal CPU frequencies, user offloading times, and user transmit power were derived. Moreov er, the optimal selection scheme whether users choose to locally compute or offload tasks was proposed for the binary computation of floading mode. It was shown that the performance achiev ed by using our proposed resource allocation scheme is superior to these obtained by using the disjoint optimization schemes. Simulation results also verified the ef ficiency of our proposed alternativ e algorithms and our theoretical analysis. The exploitation of U A V to improve the ener gy con versation efficienc y and the computation performance was studied in this paper . Ho wever , the computation performance is also limited by the flight time of the U A V . It is interesting to exploit multiple antennas techniques to tackle this challenge. This will be in vestigated in our future work. T ABLE IV : Comparison of the required run time of Algorithm 1 with that of Algorithm 2 (s) ` ` ` ` ` ` ` ` ` Algorithms ( N , M ) (50 , 2) (50 , 4) (50 , 8) (60 , 2) (60 , 4) (60 , 8) (70 , 2) (70 , 4) (70 , 8) Algorithm 1 43.72 104.54 186.38 154.74 198.65 235.85 224.74 291.53 352.72 Algorithm 2 89.35 167.17 265.46 223.19 275.42 321.87 308.56 388.92 468.39 A P P E N D I X A P RO O F O F T H E O R E M 1 Let λ m,n and α n denote the dual variables associated with the constraint C 2 and C 3 , respectively , where λ m,n ≥ 0 and α n ≥ 0 . Then, the Lagrangian of P 2 can be given by (27) at the tope of this page, where Ξ denotes a collection of all the primal and dual variables related to P 2 . Let µ m,n = N P k = n λ m,k and g m [ k ] = η 0 h m [ k ] P 0 − γ c f 3 m [ k ] − z m [ k ] . Then, the Lagrangian function L (Ξ) can be re written by (28) at the tope of this page. And the Lagrangian dual function of P 2 can be presented as g ( λ m,n , α n ) = max 0 ≤ f m [ n ] L (Ξ) . (29) Based on (29) , the optimal solutions of P 2 can be obtained by solving its dual problem, giv en as min λ m,n ,α n g ( λ m,n , α n ) . (30) It can be seen from (30) that the dual problem can be decoupled into M independent optimization problems, giv en by (31) at the tope of the next page. Thus, let the deriv ation of (31b) with respect to f m [ n ] and z m [ n ] be zero, one has T w m N C − 3 T γ c f 2 m [ k ] N N X k = n λ m,k = 0 , (32a) w m B T t m [ n ] ν m N ln 2 h m [ n ] σ 2 0 t m [ n ] + h m [ n ] z m [ n ] − T N N X k = n λ m,k =0 . (32b) Note that z m [ k ] = t m [ k ] P m [ k ] and P m [ k ] ≥ 0 . Moreover , the case that t m [ n ] = 0 can be identified as P m [ n ] = 0 . Thus, based on (32) , Theorem 1 is proved. The proof for Theorem 1 is complete. A P P E N D I X B P RO O F O F T H E O R E M 5 Let υ m,n and ε n denote the dual variables with respect to the constraints giv en by (24b) and (24c) , respectively , where υ m,n ≥ 0 and ε n ≥ 0 . Then, for any giv en f m [ n ] , P m [ n ] , t m [ n ] and q u [ n ] , the Lagrangian of P 6 can be expressed by (33) at the tope of the next page, where Ξ 1 denotes a collection of all the primal and dual variables related to P 6 . Ξ 2 denotes a collection of υ m,n , α n , f m [ n ] , z m [ n ] , t m [ n ] and ρ m . Using the same techniques that are used for the proof of Theorem 1, for any giv en f m [ n ] , z m [ n ] , t m [ n ] and q u [ n ] , P 6 can be solved by solving M independent optimization problems, giv en by (34) at the tope of the next page, where ` m [ n ] = η 0 h m [ n ] P 0 − (1 − ρ m ) γ c f 3 m [ n ] − ρ m z m [ n ] and $ m,n = N P k = n υ m,k . 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L (Ξ) = M X m =1 w m " N X n =1 f m [ n ] C T N + B T t m [ n ] ν m N log 2  1 + h m [ n ] z m [ n ] σ 2 0 t m [ n ]  # + M X m =1 N X n =1 λ m,n ( η 0 T N n X k =1 h m [ k ] P 0 − T N n X k =1  γ c f 3 m [ k ] + z m [ k ]  ) + N X n =1 α n ( 1 − M X m =1 t m [ n ] ) , (27) L (Ξ) = M X m =1 N X n =1 w m  T N f m [ n ] C + B T t m [ n ] ν m N log 2  1 + h m [ n ] z m [ n ] σ 2 0 t m [ n ]  + µ m,n g m [ k ] + α n M − α n t m [ n ] . (28) max λ m,n ,α n ,f m [ n ] ≥ 0 L m ( λ m,n , α n , f m [ n ] , z m [ n ] , t m [ n ]) (31a) L m ( λ m,n , α n , f m [ n ] , z m [ n ] , t m [ n ]) (31b) = N X n =1  w m  T N f m [ n ] C + B T t m [ n ] ν m N log 2  1 + h m [ n ] z m [ n ] σ 2 0 t m [ n ]  + µ m,n g m [ n ] + α n M − α n t m [ n ]  . (31c) L 1 (Ξ 1 ) = M X m =1 w m " (1 − ρ m ) N X n =1 f m [ n ] C T N + B T ρ m t m [ n ] ν m N log 2  1 + h m [ n ] z m [ n ] t m [ n ] σ 2 0  # + M X m =1 N X n =1 υ m,n ( η 0 T N n X k =1 h m [ k ] P 0 − T N n X k =1  (1 − ρ m ) γ c f 3 m [ k ] + ρ m z m [ k ]  ) + N X n =1 ε n ( 1 − M X m =1 ρ m t m [ n ] ) , (33) max υ m,n ,ε n ,f m [ n ] ≥ 0 L 1 m (Ξ 2 ) (34a) L 1 m (Ξ 2 ) = N X n =1 w m  T (1 − ρ m ) f m [ n ] N C + B T ρ m t m [ n ] ν m N log 2  1 + h m [ n ] z m [ n ] t m [ n ] σ 2 0  + N X n =1 $ m,n ` m [ n ] + ε n M − ε n t m [ n ] , (34b) ∂ L 1 m (Ξ 2 ) ∂ ρ opt m    < 0 , ρ opt m = 0 , = 0 , 0 < ρ opt m < 1 , > 0 , ρ opt m = 1; m ∈ M (35a) ∂ L 1 m (Ξ 2 ) ∂ ρ opt m = ( N X n =1 − w m f m [ n ] C T N + B T t m [ n ] ν m N log 2  1 + h m [ n ] z m [ n ] t m [ n ] σ 2 0  ) + N X n =1 υ m,n ( − T N n X k =1  − γ c f 3 m [ k ] + z m [ k ]  ) − N X n =1 ε n t m [ n ] . (35b) [17] H. W ang, J. W ang, G. Ding, L. W ang, T . A. Tsiftsis, P . K. 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Qi, “Energy-efficient optimal power allocation for fading cognitive radio channels: Ergodic capacity , outage capacity and minimum-rate capacity , ” IEEE T rans. W ireless Commun. , vol. 15, no. 4, pp. 2741-2755, Apr . 2016. [37] F . Zhou, Z. Li, J. Cheng, Q. Li, and J. Si, “Robust max-min fairness resource allocation in sensing-based wideband cognitiv e radio with SWIPT : Imperfect channel sensing, ” IEEE Syst. J. , to be published, 2017. [38] S. Bubeck, “Con vex optimization: Algorithms and complexity , ” In F oundations and Tr ends in Machine Learning , vol. 8, no. 3, pp. 231-357, 2015. https://arxiv .org/abs/1405.4980 [39] C. Gutierrez, F . Gutierrez, M.C. Rivara, “Complexity on the bisection method, ” Theor etical Computer Science , vol. 382, pp. 131-138, 2007. Fuhui Zhou received the Ph. D. degree from Xidian Univ ersity , Xian, China, in 2016. He is an associate Professor with School of Information Engineering, Nanchang University . He is now a Research Fellow at Utah State Univ ersity . He has worked as an international visiting Ph. D student of the University of British Columbia from 2015 to 2016. His research interests focus on cognitive radio, green communi- cations, edge computing, machine learning, NOMA, physical layer security , and resource allocation. He has published more than 40 papers, including IEEE Journal of Selected Areas in Communications, IEEE T ransactions on W ireless Communications, IEEE W ireless Communications, IEEE Network, IEEE GLOBECOM, etc. He has served as T echnical Program Committee (TPC) member for many International conferences, such as IEEE GLOBECOM, IEEE ICC, etc. He serves as an Associate Editor of IEEE Access. Y ongpeng W u (S’08–M’13–SM’17) received the B.S. degree in telecommunication engineering from W uhan Uni versity , W uhan, China, in July 2007, the Ph.D. degree in communication and signal pro- cessing with the National Mobile Communications Research Laboratory , Southeast Univ ersity , Nanjing, China, in November 2013. Dr . Wu is currently a T enure-Track Associate Professor with the Department of Electronic Engi- neering, Shanghai Jiao T ong University , China. Pre- viously , he was senior research fellow with Institute for Communications Engineering, T echnical University of Munich, Germany and the Humboldt research fellow and the senior research fellow with Institute for Digital Communications, Uni versity Erlangen-N ¨ u rnberg, German y . During his doctoral studies, he conducted cooperative research at the Department of Electrical Engineering, Missouri Univ ersity of Science and T echnology , USA. His research interests include massive MIMO/MIMO systems, physical layer security , signal processing for wireless communications, and multivariate statistical theory . Dr . Wu was awarded the IEEE Student Tra vel Grants for IEEE Inter- national Conference on Communications (ICC) 2010, the Alexander von Humboldt Fellowship in 2014, the T ravel Grants for IEEE Communication Theory W orkshop 2016, and the Excellent Doctoral Thesis A wards of China Communications Society 2016. He was an Exemplary Reviewer of the IEEE T ransactions on Communications in 2015, 2016. He is the lead guest editor for the upcoming special issue “Physical Layer Security for 5G Wireless Networks” of the IEEE Journal on Selected Areas in Communications. He is currently an editor of the IEEE Access and IEEE Communications Letters. He has been a TPC member of various conferences, including Globecom, ICC, VTC, and PIMRC, etc. Rose Qingyang Hu is a Professor of Electri- cal and Computer Engineering Department at Utah State University . She received her B.S. degree from Univ ersity of Science and T echnology of China, her M.S. degree from New Y ork University , and her Ph.D. degree from the University of Kansas. She has more than 10 years of R&D experience with Nortel, Blackberry and Intel as a technical manager , a senior wireless system architect, and a senior research scientist, activ ely participating in industrial 3G/4G technology development, standard- ization, system level simulation and performance ev aluation. Her current research interests include next-generation wireless communications, wireless system design and optimization, green radios, Internet of Things, Cloud computing/fog computing, multimedia QoS/QoE, wireless system modeling and performance analysis. She has published over 180 papers in top IEEE journals and conferences and holds numerous patents in her research areas. Prof. Hu is an IEEE Communications Society Distinguished Lecturer Class 2015-2018 and the recipient of Best Paper A wards from IEEE Globecom 2012, IEEE ICC 2015, IEEE VTC Spring 2016, and IEEE ICC 2016. Y i Qian received a Ph.D. degree in electrical engi- neering from Clemson Univ ersity . He is a professor in the Department of Electrical and Computer En- gineering, University of Nebraska-Lincoln (UNL). Prior to joining UNL, he worked in the telecommu- nications industry , academia, and the government. Some of his previous professional positions include serving as a senior member of scientific staff and a technical advisor at Nortel Networks, a senior sys- tems engineer and a technical advisor at sev eral start- up companies, an assistant professor at University of Puerto Rico at Mayaguez, and a senior researcher at National Institute of Standards and T echnology . His research interests include information assur- ance and network security , network design, network modeling, simulation and performance analysis for next generation wireless networks, wireless ad-hoc and sensor networks, vehicular networks, smart grid communication networks, broadband satellite networks, optical networks, high-speed networks and the Internet. Prof. Y i Qian is a member of ACM and a senior member of IEEE. He was the Chair of IEEE Communications Society T echnical Committee for Communications and Information Security from January 1, 2014 to December 31, 2015. He is a Distinguished Lecturer for IEEE V ehicular T echnology Society and IEEE Communications Society . He is serving on the editorial boards for several international journals and magazines, including serving as the Associate Editor-in-Chief for IEEE Wireless Communications Magazine. He is the T echnical Program Chair for IEEE International Conference on Communications (ICC) 2018.