uav-assisted wireless powered cooperative mobile edge

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/JIOT.2019.2958975, IEEE Internet ofThings Journal

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UAV-assisted Wireless Powered Cooperative MobileEdge Computing: Joint Offloading, CPU Control

and Trajectory OptimizationYuan Liu, Ke Xiong, Member, IEEE, Qiang Ni, Senior Member, IEEEPingyi Fan, Senior Member, IEEE, Khaled Ben Letaief, Fellow, IEEE

Abstract—This paper investigates the UAV-enabled wirelesspowered cooperative mobile edge computing (MEC) system,where a UAV installed with an energy transmitter (ET) andan MEC server provides both energy and computing services tosensor devices (SDs). The active SDs desire to complete their com-puting tasks with the assistance of the UAV and their neighboringidle SDs that have no computing task. An optimization problemis formulated to minimize the total required energy of UAV byjointly optimizing the CPU frequencies, the offloading amount,the transmit power and the UAV’s trajectory. To tackle thenon-convex problem, a successive convex approximation (SCA)-based algorithm is designed. Since it may be with relativelyhigh computational complexity, as an alternative, a decompositionand iteration (DAI)-based algorithm is also proposed. Simulationresults show that both proposed algorithms converge withinseveral iterations, and the DAI-based algorithm achieves thesimilar minimal required energy and optimized trajectory withthe SCA-based one. Moreover, for a relatively large amount ofdata, the SCA-based algorithm should be adopted to find anoptimal solution, while for a relatively small amount of data, theDAI-based algorithm is a better choice to achieve the smallercomputing energy consumption. It also shows that the trajectoryoptimization plays a dominant factor in minimizing the totalrequired energy of the system and optimizing acceleration has agreat effect on the required energy of the UAV. Additionally, byjointly optimizing the UAV’s CPU frequencies and the amountof bits offloaded to UAV, the minimal required energy forcomputing can be greatly reduced compared to other schemes.And, by leveraging the computing resources of idle SDs, theUAV’s computing energy can also be greatly reduced.

Index Terms—UAV communication, wireless power transfer,mobile edge computing, computation offloading, trajectory de-sign.

Manuscript received xxx, revised xxx, and accepted xxx. Date of publicationJune xxx; date of current version xxx. This work was supported in part by theFundamental Research Funds for the Central Universities (No. 2018YJS040),by the General Program of the National Natural Science Foundation ofChina (NSFC) under grant no. 61671051, by the National key research anddevelopment program under Grant 2016YFE0200900, by the Beijing naturalHaidian joint fund under Grant L172020, and also in part by the major projectsof Beijing Municipal Science and Technology Commission under grant no.Z181100003218010. (Corresponding author: K. Xiong ([email protected])).

Y. Liu and K. Xiong are with School of Computer and InformationTechnology, Beijing Jiaotong University, and also with the Beijing KeyLaboratory of Traffic Data Analysis and Mining, Beijing Jiaotong University,Beijing 100044, China.

Q. Ni is with the School of Computing and Communications and DataScience Institute, Lancaster University, England. e-mail: [email protected].

P. Y. Fan is with the Department of Electronic Engineering, TsinghuaUniversity, Beijing, R.P. China, 100084. e-mail: [email protected].

K. B. Letaief is with the School of Engineering, Hong Kong University ofScience & Technology (HKUST), China. e-mail: [email protected].

I. INTRODUCTION

A. Background

Recent advancements in Internet of Things (IoT) havearoused abundant new applications, including intelligent graz-ing, autonomous control and environmental monitoring [1]-[4]. In IoT systems, a large number of sensor devices (SDs)need to be deployed to collect environmental data. The SDsare usually powered by batteries with limited energy capacity,which thus are required to be replaced or recharged up period-ically. In large-scale IoTs and rugged environment, frequentlyreplacing and recharging batteries may bring huge labour cost.In order to power the low-power SDs in a self-sustainable wayand avoid replacing batteries manually, radio-frequency (RF)signal based wireless power transfer (WPT), also referred toas RF-based energy harvesting (EH) has been widely regardedas a promising solution [5], [6]. It is reported that currentRF-based EH has already been capable of transferring powerabout several milliWatts at a distance of up to tens of meters[7]. Thus, it is suitable to power low-power devices in IoTsand wireless sensor networks (WSNs) [8].

On the other hand, massive amounts of collected data isrequired to be online computed in real-time IoT systems.Due to the limited communication, computation, and storagecapabilities, it is difficult for SDs to complete latency-sensitivecomputation tasks with their own computing resources. Totackle this issue, mobile edge computing (MEC) has emergedas an effective solution, which is able to offer intensive compu-tation services at the network edge with relatively light accessburden and low transmission delay. With MEC employed,by offloading computation intensive tasks from SDs to MECservers, SDs’ computation capabilities are supplemented [9].

As both RF-based EH and MEC benefit IoT systems, MECassisted wireless network design with RF-based EH has beenpaid increasing attention, see e.g., [8] - [11]. However, in mostexisting works, only fixed energy transmitters (ETs) and MECservers were considered. As each fixed ET and MEC serverhas limited service coverage radius, in order to cover largerarea and serve more SDs, more fixed ETs and MEC servershave to be deployed, which results in high cost. Moreover,with fixed ETs and MEC servers, the SDs located far awayfrom the servers may always not be served well due to theirrelatively weak wireless links.

Fortunately, wireless communication with unmanned aerialvehicles (UAVs) is a promising technology to compensate



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for the shortcoming of the fixed ETs and MEC serversmentioned above. UAV-enabled wireless communication canachieve ubiquitous coverage in rural or remote areas withoutinfrastructures or with insufficient terrestrial infrastructures.It is worth noting that the line-of-sight (LoS) channels enableUAVs to have their signal coverage over a much larger numberof SDs as compared to the terrestrial communications. Besides,UAV could be dispatched to fly closer to SDs for establishingstrong communication links. As a result, when the UAV fliesover the SD, the transmit power required by the SD to sendthe data to the UAV can be greatly reduced and hence thenetwork lifetime can be prolonged.

B. Related Work

Recently, UAV-assisted wireless communication has attract-ed increasing interests, owing to UAVs’ merits such as on-demand operations, flexible deployment, controllable mobility,and superior link quality [12]. UAVs can be used as aerialbase stations [13] - [15] and mobile relays [16], [17] toassist terrestrial communication infrastructures for informationdissemination, and it can be used to collect data in IoToutdoors [18] as well. When UAVs are equipped with large-capacity batteries, they can also serve as mobile ETs to chargeSDs. In [19], the UAV was deployed with WPT to power SDs,where the sum energy received by all energy receivers wasmaximized. In [20], a UAV-enabled wireless powered com-munication network (WPCN) was studied, where the uplinkcommon throughput among all ground users was maximized.When UAVs are installed with computing processors, they canprovide computing services as MEC servers. In [21], the UAVwas used as an MEC apparatus to help complete computingtasks, where the UAV trajectory, the ratio of offloading tasks,and the user scheduling variables were optimized to minimizethe maximum delay among all users. In [22], the UAV acted asa relay to assist the users in computing or further offload tasksto the AP for computing, where the UAV trajectory, the com-putation resource scheduling, and bandwidth allocation wereoptimized to minimize the weighted sum energy consumptionof the UAV and users.

As a matter of fact, UAVs can be equipped with both energysources and computing components to provide energy supplyand at the same time enhance the computing capability in IoTs[23]. With UAVs acting as flying WPT and MEC servers, theaforementioned shortcomings of deploying fixed ET and MECservers can be greatly eliminated. Therefore, a few recentworks began to study UAV-assisted wireless powered MECnetworks. In [24], a UAV-enabled wireless powered MECnetwork was investigated without considering the propulsionenergy requirement of the UAV, where the achievable compu-tation rate was maximized with the WPT constraints. In [25],the UAV provides users with energy supply and computationoffloading, where the total energy consumed at UAV wasminimized, while in UAV’s propulsion energy consumptionmodel, only UAV’s velocity on the required energy was takeninto account.

C. Motivation and Contributions

This paper also studies the UAV-assisted wireless poweredMEC networks, where a UAV equipped with an ET and anMEC server charges SDs and provides computing service toactive SDs.1 The UAV’s trajectory, the offloading bits and thetransmit power as well as CPU frequencies are jointly opti-mized. The main differences between our work and existingones are summarized as follows.

1) In most existing works, see e.g., [21], [22], [25], althoughUAV’s trajectory was optimized, in their UAV’s propulsion-related energy consumption model, only the effect of UAV’svelocity on the required energy was taken into account, wherehowever, the effect of acceleration on UAV’s required energywas neglected. That is, in their works, the UAV was ideallyassumed to be able to change its velocity arbitrarily withoutenergy consumption. In practice, the change of UAV’s velocityalso consumes energy. Therefore, in this paper, a generalpropulsion energy model is considered, where the effects ofboth UAV’s velocity and acceleration are taken into account.

2) In most existing works, see e.g., [20], [24], the trans-mitted signals from UAV were only used to charge activeSDs. However, in practice, due to the broadcast nature ofwireless links, the transmitted signals from UAV can alsobe received by idle SDs.2 As the idle SDs also have somecomputing resources, in order to fully utilize the transferredenergy of UAV and the computing resources of idle SDs, inour work, neighboring idle SDs are allowed to act as helpers tocooperatively assist active SDs to complete computation taskswith the harvested energy.

3) In some existing work, see e.g., [21], [26], only thenumber of bits offloaded to the UAV in each time slot wasoptimized, where the UAV CPU frequencies were fixed asconstants, implying that the UAV worked at its maximum CPUfrequency to process the received data. Such configurationcan’t achieve energy saving. In this paper, the received dataat the UAV is not required to be processed completely, andit is allowed to be computed in the subsequent time slots forachieving a better performance. Therefore, besides the amountof bits offloaded to UAV in each time slot, the data processedby the UAV in each time slot is also with significance to beoptimized. Thus, the amount of bits offloaded to the UAVand the UAV’s CPU frequencies in each time slot are jointlyoptimized to achieve a computational equilibrium over thetime period. By doing so, much less energy is required forcomputing compared to existing schemes with fixed CPUfrequency.

The main contributions of this paper are summarized asfollows.

1) For the UAV-assisted wireless powered cooperative MECsystem, an optimization problem is formulated to minimizethe total required energy of UAV via jointly optimizing thenumber of offloading bits, the CPU frequencies, the transmitpower at the active SDs and the UAV’s trajectory, subjectto active SDs’ computing task constraints, the information-causality constraints, energy harvesting causality constraints

1The active SDs are the SDs with data required to process.2The idle SDs are the SDs without data to process.



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and UAV’s trajectory constraints.

2) Since the optimization problem is non-convex, whichis difficult to handle, to make it solvable, some auxiliaryoptimization variables are introduced and the first-order Taylorexpansion is applied to transform the problem into convex.Then, a successive convex approximation (SCA)-based algo-rithm is designed to efficiently solve the optimization problem.Since the SCA-based algorithm requires to optimize a seriesof optimization variables and search the optimal solution itera-tively with updated trajectory variables in each iteration, whichmay be with a relatively high computational complexity. Thus,as an alternative, a decomposition and iteration (DAI)-basedalgorithm is also presented, which optimizes the offloadingamount, CPU frequencies and trajectory variables separatelyand iteratively with relatively low complexity.

3) The worst-case computational complexities of the twopresented algorithms are analyzed by using the interior-pointmethod (IPM) theory [27]. The computational complexity of

SCA-based algorithm is about O(N152 ), and the computational

complexity of DAI-based algorithm is about O(N112 ), where

N is the number of time slots.

4) Simulation results show that the proposed algorithmsconverge within several iterations, and the longer the flyingtime, the more iterations are required to reach the con-vergence. Moreover, both presented algorithms achieve thesimilar minimal required energy and the optimized trajectories,and they obtain significant performance gain compared toother benchmarks. It also can be seen that both algorithmsare feasible when the computation bits are relatively small, butonly the SCA-based algorithm is feasible with higher requiredcomputing energy when the computation bits are relativelylarge. That is, for a relatively large amount of data, the SCA-based algorithm should be adopted, while for a relativelysmall amount of data, the DAI-based algorithm is a betterchoice. It is shown that the propulsion-related energy occupiesa dominant part of the total required energy, and trajecto-ry design plays an important role in UAV-enabled wirelesscommunication system. Besides, the acceleration has a greateffect on the required energy of the UAV, and optimizingboth the UAV’s CPU frequency and the offloading bits cangreatly reduce the required energy for computing at UAV.Particularly, with the increment of active SDs’ computingtask, the gain obtained by optimizing the CPU frequenciesdecreases, but the gain obtained by optimizing the offloadingbits increases. Additionally, with the assistance of idle SDs,the UAV’s computing energy also can be greatly reduced.

The remainder of this paper is organized as follows. Insection II, a UAV-enabled wireless powered cooperative MECsystem is introduced. In section III, an optimization problemfor minimizing UAV’s total required energy is formulated.In section IV, two algorithms are designed to tackle thenon-convex problem, and their complexities are analysed.In section V, simulation results are provided. Finally, weconclude the paper in section VI.

Information flow

Energy flow

Fig. 1: UAV-aided wireless powered cooperative MEC system.

II. SYSTEM MODEL

A. Network Model

A UAV-enabled wireless powered cooperative MEC systemis considered as shown in Fig. 1, where a UAV equippedwith an ET and an MEC server charges SDs and providescomputing service for active SDs. For a given period T , onlya part of SDs are active with data to be processed, and allactive SDs’ tasks are allowed to be accomplished with thehelp of UAV. In order to fully utilize the transferred energy ofUAV, the broadcast nature of wireless links and the computingresources of idle SDs that with no computing task, neighboringidle SDs are aroused to cooperatively participate in the taskcomputing for the active SDs.

Active SDs and idle SDs are located on ground, andthe UAV flies horizontally at a fixed altitude H of severalmeters. A three-dimensional (3D) Cartesian coordinate systemis employed to describe the positions of SDs and UAV. Forthe active SD m, its position is [xm, ym, 0]

T , and for the idleSD k, its position is [xk, yk, 0]

T , where m = 1, ...,M andk = 1, ...,K with M and K being the number of active SDsand idle SDs, respectively. To efficiently design the flyingtrajectory, T is discretized into N time slots with equallyinterval of T

N . Let n be the n-th time slot, where n = 1, ..., N .As the time slot is relatively small enough, UAV’s location intime slot n can be roughly considered to be unchanged anddenoted as [x[n], y[n],H]T .

B. Channel Model

As all SDs are distributed on ground, for simplicity, thepositions of the m-th active SD and the k-th idle SD on thehorizontal plane (i.e., the 2-D plane) are qm = [xm, ym]T andwk = [xk, yk]

T . Similarly, the UAV trajectory projected onthe horizontal plane at time slot n is described by qu[n] =[x[n], y[n]]T . qinitialu = qu[0] and qfinalu = qu[N ] are definedas the initial and final locations of the UAV, respectively.Moreover, the velocity and acceleration of the UAV in timeslot n are v[n] = [vx[n], vy[n]]

T and a[n] = [ax[n], ay[n]]T .

Following [13], the relationships among the UAV’s location,



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velocity and acceleration are expressed by qu[n+ 1] = qu[n] + v[n] TN +1

2a[n] TN

2, (1a)

v[n+ 1] = v[n] + a[n] TN . (1b)

In terms of the coordinates, the distance between UAV andthe m-th active SD at time slot n is given by

du,m[n] =√∥qu[n]− qm∥2 +H2, (2)

and that between UAV and the k-th idle SD at time slot n is

du,k[n] =√∥qu[n]− wk∥2 +H2. (3)

The distance between active SD m and idle SD k is

dm,k[n] = ∥qm − wk∥. (4)

In general, UAV is deployed in scenarios outdoors. Thus,similar to existing works [20], [21], it is assumed that thecommunication channel from UAV to a SD is dominated bythe line-of-sight (LoS) link, which is therefore modeled bythe free-space path loss model. As a result, the channel powergain from UAV to the m-th active SD and k-th idle SD attime slot n can be respectively expressed by

hu,m[n] =β0

du,m[n]2=

β0∥qu[n]− qm∥2 +H2

, (5a)

hu,k[n] =β0

du,k[n]2=

β0∥qu[n]− wk∥2 +H2

, (5b)

where β0 denotes the channel power gain at the referencedistance of one meter. Both distance-dependent path loss effectand small-scale fading effect are taken into account for thechannel between an active SD and an idle SD, so the channelgain of the link between the m-th active SD and the k-th idleSD is given by

hm,k = φζm,k[n]β0(dm,k)−α, (6)

where φ is a constant determined by system parameters [28],ζ denotes the exponentially distributed random variable withunit mean accounting for Rayleigh fading, and α is the pathloss exponent. In (6), β0 has the same physical meaning asthat in (5).3

C. Energy Harvesting ModelFollowing the EH model presented in [29], [30], the harvest-

ed energy at the m-th active SD and the k-th idle SD duringn time slots are respectively expressed by Em[n] =

∑n

i=1ηPuhu,m[i] TN , (7a)

Ek[n] =∑n

i=1ηPuhu,k[i]

TN , (7b)

where 0 ≤ η ≤ 1 is the energy conversion efficiency ofconverting the received RF signals into direct current (DC)signals in energy harvesting, and Pu is UAV’s transmit power.

3 For both the air to ground link and ground to ground link, the end-to-endinformation transmissions at a distance of one meter are mainly dominated bythe LoS links. Therefore, the β0 of the air to ground channel in (5) and the β0

of the terrestrial channel in (6) can be considered to be the same. Moreover,the transmission from the air to ground is dominated by LoS links, whichis therefore modeled by the free-space path loss model. But, for the groundto ground link, besides the path loss effect, the small fading effect causedby multi-path propagation also exists. Therefore, to make it much closer topractice, in (6) the small scale fading is also taken into account by ζ.

slot slot ... slot... slot

...

2n

1n

1m

2m M

Nn

T

... UAV1k

2k K

Fig. 2: TDMA protocol for active SDs computation offloading.

D. Offloading Model

In order to avoid the co-channel interference between theWPT and information transmission, WPT and computationtask offloading are implemented over orthogonal frequencybands. In each time slot, all active SDs may offload theirdata to the UAV and their nearby idle SDs for computing. Inorder to avoid inter SDs interference, time-division multipleaccess (TDMA) protocol is employed for multiple active SDsoffloading computation tasks to the UAV and their nearby idleSDs. The time frame structure is shown in Fig. 2, where eachtime slot is further divided into M sub-slots, and each sub-slotis with time interval of T

(NM) . As there are (K +1) potentialhelpers (i.e., K idle SDs and one UAV), the m-th active SDcan offload its computing tasks to the (K+1) helpers through(K + 1) information flows. In order to avoid the interferencebetween information flows, each sub-slot is equally dividedinto (K + 1) small time slices, and in each time slice onlyone information flow is transmitted in term of TDMA manner.Thus, each time slice is with time interval of δo = T

(NM(K+1)) .

Let Lm,u[n] and Lm,k[n] be the computation bits that them-th active SD offloads to UAV and the k-th idle SD at timeslot n, respectively. In order to successfully upload the bitsto UAV and neighboring idle SDs, the achievable informationrate from the m-th active SD to UAV and the k-th idle SDshould satisfy that

δoB log2

(1 +

pm,u[n]hu,m[n]

σ2

)≥ Lm,u[n], (8a)

and

δoB log2

(1 +

pm,k[n]hm,k[n]

σ2

)≥ Lm,k[n], (8b)

where B is the communication bandwidth and σ2 is noisepower. Thus, the transmit power at the m-th active SD tooffload Lm,u[n] and Lm,k[n] bits respectively satisfies that

pm,u[n] ≥σ2(2

Lm,u[n]

Bδo − 1)

hu,m[n], (9a)

pm,k[n] ≥σ2

(2

Lm,k[n]

Bδo − 1

)hm,k[n]

. (9b)

As a result, the required energy of the m-th active SD tooffload tasks to UAV and k-th idle SD is respectively expressedby {

Eoffm,u[n] = pm,u[n]δo, (10a)

Eoffm,k[n] = pm,k[n]δo. (10b)



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E. Computing Model

Once the m-th active SD, the k-th idle SD and the UAV areassigned with computing data, they may perform computingoperations respectively. According to [21], [25], the requiredcomputation energy at the m-th active SD, the k-th idle SDand the UAV are respectively expressed by

Ecompm [n] = γcfm[n]3 T

N , (11a)

Ecompk [n] = γcfk[n]

3 TN , (11b)

Ecompu [n] = γcfu[n]

3 TN , (11c)

where γc denotes the effective CPU switch capacitance, fm[n],fk[n] and fu[n] respectively denote CPU frequencies forexecuting computation with a unit of cycles per second atthe m-th active SD, the k-th idle SD and UAV. Thus, theinformation bits can be computed at the m-th active SD, thek-th idle SD and the UAV during the first n time slots arerespectively given by

Rlocalm [n] =

∑n

i=1

fm[i]

C1

T

N, (12a)

Rlocalk [n] =

∑n

i=1

fk[i]

C2

T

N, (12b)

Rlocalu [n] =

∑n

i=1

fu[i]

C3

T

N, (12c)

where C1, C2 and C3 are CPU cycles required for executingone input-bit of computation tasks at active SDs, idle SDs andUAV respectively.

F. The required Energy at UAV

The required energy of UAV is in general composed of threemain components, i.e., the required energy for communication,the required energy for computing and the required propulsionenergy. Particularly, the required energy for communicationover T is Ecom

u = TPu in Joule (J). Following (11c),the required energy for computing over T is ETcomp

u =∑Nn=1E

compu [n]. Moreover, the required propulsion energy of

fixed-wing UAV over T is given by

Eflyu =

T

N

N∑n=1

[c1 ∥v[n]∥3 +

c2∥v[n]∥

(1 +

∥a[n]∥2

g2

)],

(13)which depends on UAV’s flying speed and acceleration [31].In (13), v[n] and a[n] are UAV’s velocity and acceleration attime slot n, c1 and c2 are two constant parameters related toaerodynamics, and g is gravitational acceleration with nominalvalue 9.8 m/s2. Thus, the total required energy at the UAV is

Θ = Eflyu + ETcomp

u + Ecomu . (14)

III. PROBLEM FORMULATION

In order to minimize the total required energy at the UAV,such that active SDs’ computation tasks can be completedwithin T , an optimization problem is formulated via jointlythe transmit power, the CPU frequencies, and UAV’s trajec-tory, velocity and acceleration. Let F = {fm[n], fu[n], fk[n]}denote the CPU frequency vector of active SDs, UAV and

idle SDs, L = {Lm,u[n], Lm,k[n]} denote the computation bitsoffloading vector to UAV and idle SDs, Q = {qu[n], v[n], a[n]}be UAV’s trajectory, velocity and acceleration, and P ={pm,u[n]} be the transmit power vector for offloading com-putation bits to UAV. The optimization problem can be math-ematically given by

P1 : minF,L,Q,P

Θ

s.t.∑n

i=1Ecomp

m [i] +∑n

i=1Eoff

m,u[i]

+∑n

i=1

∑K

k=1Eoff

m,k[i] ≤ Em[n], (15a)∑n

i=1γcf

3k [i]

TN ≤ Ek[n], (15b)

n∑i=2

fk[i]

C2

T

N≤

n−1∑i=1

M∑m=1

Lm,k[i], ∀ n ∈ N1 (15c)

∑n

i=2

fu[i]

C3

T

N≤

n−1∑i=1

M∑m=1

Lm,u[i], ∀ n ∈ N1 (15d)

∑N

n=2

fk[n]

C2

T

N=∑N−1

n=1

∑M

m=1Lm,k[n], (15e)∑N

n=2

fu[n]

C3

T

N=∑N−1

n=1

∑M

m=1Lm,u[n], (15f)

Rlocalm [N ] +

∑N−1

n=1

∑K

k=1Lm,k[n]+∑N−1

n=1Lm,u[n] = Rm, (15g)

δoB log2

(1 +

pm,u[n]hu,m[n]

σ2

)≥ Lm,u[n], (15h)

fk[1] = 0, fu[1] = 0,

Lm,k[N ] = 0, Lm,u[N ] = 0, (15i)

qu[n+ 1] = qu[n] + v[n] TN +1

2a[n] TN

2, (15j)

v[n+ 1] = v[n] + a[n] TN , (15k)∥v[n]∥ ≤ Vmax, ∥a[n]∥ ≤ amax, (15l)

qinitialu = qfinalu , (15m)fm[n] ≥ 0, fk[n] ≥ 0, fu[n] ≥ 0,

Lm,k[n] ≥ 0, Lm,u[n] ≥ 0, (15n)

where Rm denotes the number of computation bits of them-th active SD. Vmax and amax are the maximum flyingspeed and acceleration of UAV. and N1 is the set of {1, ...,N−1}. (15a) is the energy causal constraint of the active SDs,which describes that the required energy for local computingand offloading of the m-th active SD is limited by harvestedenergy. (15b) is the energy casual constraint of idle SDs,which means that the required energy to help computing taskscan’t exceed their harvested energy. Constraints (15c) and(15d) indicate that the total number of computation bits atUAV and the k-th idle SD in the first n time slots can’texceed the offloading computation bits of the active SDs inthe first (n− 1) time slots. That is, UAV and idle SDs start toexecute computing at the n-th time slot only when active SDsfinish offloading computation bits at the (n− 1)-th time slot.Constraints (15e) and (15f) ensure that the total computationbits processed by UAV and idle SDs should be equal to theoffloading computation bits of active SDs. Constraint (15g)



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indicates the amount of computing task for each active SD.(15h) implies that the achievable information rate between them-th SD and UAV at slot n should exceed the computationbits that the m-th active SD offloads to UAV. (15i) means thatUAV and idle SDs don’t execute computation task in the firstslot and all active SDs don’t offload computation tasks in thelast slot. (15j) - (15m) represent UAV’s trajectory constraints,including the maximum velocity and acceleration constraints,and the initial and final positions constraints.

Due to the non-convexity of the objective function and thecoupling variables of {pm,u[n]} and {qu[n]} in constraints(15a), (15b) and (15h), problem P1 is non-convex, whichis difficult to tackle. Therefore, we shall design efficientalgorithms to solve it.

IV. SOLUTION METHODS AND COMPLEXITY ANALYSIS

This section describes the two proposed solution methods,i.e., the SCA-based algorithm and the DAI-based algorithm.The detail information of the two proposed algorithms andtheir complexity analysis are presented as follows.

A. SCA-based algorithm

In this section, we successively solve a convex approxima-tion counterpart of the non-convex problem P1 by employ-ing the SCA approach. Firstly, in order to handle the non-convexity of the Efly

u term in objective function of problemP1, by introducing slack variables τ [n] such that

τ [n] ≥ 0, (16a)

and∥v[n]∥2 ≥ τ [n]2, (16b)

then, Eflyu can be relaxed to be

Eflyu =

T

N

N∑n=1

[c1 ∥v[n]∥3 +

c2τ [n]

(1 +

∥a[n]∥2

g2

)]. (17)

According to (16b), it could be inferred that Eflyu ≥ Efly

u .Therefore, Efly

u can be regarded as an upper bound of Eflyu .

Eflyu is jointly convex w.r.t. v[n] and τ [n]. Thus, by taking

place of Eflyu with Efly

u , the objective function of problem P1becomes convex. However, the new constraint (16b) is non-convex. We thus use first-order Taylor expansion to deal withit. That is, for a given feasible point vr[n], it satisfies that

∥v[n]∥2 ≥ ∥vr[n]∥2 + 2vTr [n](v[n]− vr[n]) , ψlb(v[n]),(18)

where the equality holds at the point v[n] = vr[n] [32]. Then,constraint (16b) can be approximated by

ψlb(v[n]) ≥ τ [n]2, (19)

which is convex since ψlb(v[n]) is linear w.r.t. v[n]. Moreover,constraints (15a), (15b) and (15h) are still non-convex w.r.t.qu[n] and pm,u[n], so we design a SCA-based algorithm totackle them, which is operated in an iterative manner.

Specifically, in the r-th iteration of the SCA-based algorith-m, we handle the non-convex constraints (15a) and (15b) interms of the following Theorem 1.

Theorem 1: Let q(r)u [n] be the UAV’s position at the r-th

iteration. The non-convex terms Em[n] and Ek[n] in (15a) and(15b) satisfy that Em[n] ≥

∑n

i=1ηPuβ0hu,m[i] TN , Em[n], (20a)

Ek[n] ≥∑n

i=1ηPuβ0hu,k[i]

TN , Ek[n], (20b)

where

hu,m[i] =H2 + 2∥q(r)u [i]− qm∥2 − ∥qu[i]− qm∥2(

H2 + ∥q(r)u [i]− qm∥2)2 , (21)

and

hu,k[i] =H2 + 2∥q(r)u [i]− wk∥2 − ∥qu[i]− wk∥2(

H2 + ∥q(r)u [i]− wk∥2)2 . (22)

The equalities of (20a) and (20b) hold, when qu[n] = q(r)u [n].

Proof: For the function with the form of f(x) = ab+x ,

where x ≥ 0, and a, b > 0, it is convex w.r.t. x. Therefore, itsfirst-order Taylor expansion at x0 ≥ 0 satisfies that

a

b+ x≥ a

b+ x0− a (x− x0)

(b+ x0)2=ab+ 2ax0 − ax

(b+ x0)2. (23)

By letting a = ηPuβ0TN , b = H2, x0 = ∥q(r)u [i] − wk∥2 and

x = ∥qu[i]−wk∥2, (20a) and (20b) can be obtained, and thenTheorem 1 is proved.

For the non-convex constraint (15h) w.r.t. pm,u[n] and qu[n],by introducing a slack variable ym,u[n], it can be transformedto be

δoB(Am,u[n]− log2(ym,u[n] +H2)) ≥ Lm,u[n], (24)

where

Am,u[n] = log2(ym,u[n] +H2 + pm,u[n]γ), (25)

and∥qu[n]− qm∥2 ≤ ym,u[n], (26)

with γ = β0

σ2 being the reference signal-to-noise ratio (SNR).(24) is still non-convex because its second term of the lefthand side is a convex function of ym,u[n]. We tackle (24) byusing the following Lemma 1.

Lemma 1: Let y(r)m,u[n] = ∥q(r)u [n] − qm∥, and

ϕlb(ym,u[n]) = − log2(y(r)m,u[n] + H2) − ym,u[n]−y(r)

m,u[n]

(y(r)m,u[n]+H2) ln 2

,(24) is transformed to be

δoB(Am,u[n] + ϕlb(ym,u[n])) ≥ Lm,u[n]. (27)

Proof: Note that − log2(1 + x) is convex w.r.t. x, forx ≥ −1. Thus, the global linear lower bound of − log2(1+x)is derived by − log2(1 + x) ≥ − log2(1 + x)− x−x

(1+x) ln 2 . By

letting x =ym,u[n]

H2 , (27) is obtained, and then Lemma 1 isproved.

By taking place of Eflyu with Efly

u , Em[n] with Em[n], Ek[n]with Ek[n], and (15h) with (26) and (27), problem P1 can betransformed to be

PA-1 : minF,L,Q,P,

τ [n],ym,u[n]

Eflyu + ETcomp

u [n] + Ecomu



7

s.t. (15c)− (15g), (15i)− (15n), (28a)∑n

i=1Ecomp

m [i] +∑n

i=1Eoff

m,u[i]+∑n

i=1

∑K

k=1Eoff

m,k[i] ≤ Em[n], (28b)∑n

i=1

TN γcf

3k [i] ≤ Ek[n], (28c)

δoB(Am,u[n] + ϕlb(ym,u[n])) ≥ Lm,u[n], (28d)

τ [n] ≥ 0, ψlb(v[n]) ≥ τ [n]2, (28e)

∥qu[n]− qm∥2 ≤ ym,u[n]. (28f)

All constraints and the objective function of problem PA-1 areconvex, so the original non-convex problem P1 can be solvedby iteratively solving problem PA-1 by updating q

(r)u [n] and

v(r)[n] with the SCA manner. For clarity, the presented SCA-based algorithm is summarized in Algorithm 1.

Algorithm 1 SCA-based algorithm for solving P1

1: Set iteration index r = 1, and iteration tolerance ξ ≥ 0;2: Initialize UAV’s trajectory q

(r)u [n] and velocity v(r)[n];

3: Initialize the objective function E(r)ob = 0;

4: repeat5: Solve problem PA-1 for any given {q(r)u [n], v(r)[n]} and

obtain its optimal solutions {q∗u[n], v∗[n], E∗ob};

6: Update q(r+1)u [n] = q∗u[n], v(r+1)[n] = v∗[n],

E(r+1)ob = E∗

ob;7: Update the iterative number r = r + 1;8: until the stopping criterion |E(r+1)

ob − E(r)ob | ≤ ξ is met;

9: Obtain optimal solutions: q∗u[n], v∗[n], a∗[n],f∗m[n], f∗u [n],

f∗k [n], L∗m,k[n], L

∗m,u[n], p

∗m,u[n], τ

∗[n], y∗m,u[n].

B. Complexity analysis of the SCA-based algorithm

The convex problem PA-1 formulated by SCA-based algo-rithm involves linear inequalities second order cone (SOC)constraints. According to [27], the SOC constraints dominateits complexity. PA-1 has (M + K + 4N) SOC constraints,in which (M + K) SOCs with dimension of (2n + 1),and 3N SOCs with dimension of 3, and N SOCs withdimension of 2, where n = 1, ..., N . Besides, PA-1 has(4MN + 5N + KN(M + 1)) variables. So, in term of[27], the complexity to solve PA-1 can be given by O(n1 ·

√2(M +K + 4N) · (4(M +K)(N + 1)3 + 31N + n21)

) with n1 = O(4MN + 5N + KMN + KN). As problemPA-1 has to be solved iteratively with I1 times, the complexityof SCA-based approach for solving P1 is about I1O(n1 ·√2(M +K + 4N) · (4(M +K)(N + 1)3 + 31N + n21)).

C. DAI-based Algorithm

This section presents another algorithm based on DAI tosolve problem P1. Specifically, P1 is firstly decomposed intotwo subproblems to optimize CPU frequencies, the numberof offloading bits {fm[n], fu[n], fk[n], Lm,k[n], Lm,u[n]}, andUAV’s trajectory {qu[n], v[n], a[n]}, separately. And then, asuboptimal solution is obtained by solving the two subprob-lems in an iterative manner until the algorithm converges.

1) Optimizing {fm[n], fu[n], fk[n], Lm,k[n], Lm,u[n]} withgiven {qu[n], v[n], a[n]}: For given UAV’s trajectory, problemP1 can be simplified to be PB-1, i.e.,

PB-1 : minF,L

ETcompu

s.t. (15a)− (15g), (15i), (15n).

PB-1 is convex, which can be solved by using the Lagrangiandual method [32]. Particularly, with the Lagrangian dualmethod, we obtained some explicit results on the optimalsolution to PB-1, which is described in Theorem 2.

Theorem 2: For given q(r)u [n], the optimal solution to prob-

lem PB-1 can be given by

f∗m[n] =

√γm

3γcC1

∑Nj=n νm,j

, (29a)

f∗k [n] =

0, n = 1√√√√λk,N −∑N−1

j=n λk,j

3γcC2

∑N−1j=n µk,j

, n = 2, ..., N − 1,√λk,N

3γcC2µk,N, n = N,

(29b)

f∗u [n] =

0, n = 1√θN −

∑N−1j=n θj

3γcC3, n = 2, ..., N − 1√

θN3γcC3

, n = N

(29c)

L∗m,k[n] = (29d)

Bδo log2

(1 +

Bδohm,k[n](∑N−1

j=n+1 λk,j + γm − λk,N )

σ2 ln 2∑N

j=n νm,j

)L∗m,u[n] = (29e)

Bδo log2

1 +Bδohu,m[n]

(∑N−1j=n+1 θj + γm − θN

)σ2 ln 2

∑Nj=n νm,j

,

where νm,n ≥ 0, µk,n ≥ 0, λk,n ≥ 0, θn ≥ 0, γm ≥ 0 are thedual variables corresponding to constraints {(15a)− (15g)}.

Proof: See Appendix A.Moreover, we also obtain the following theoretical results

for better understanding the system design.Corollary 1: The optimal f∗m[n], f∗k [n] and f∗u [n] to prob-

lem P1 increase with the increment of time slot n.Proof: In (29a), (29b) and (29c), the dual variables

{νm,n, µk,n, λk,n, θn, γm} are positive constants, and as nincreases, the denominator of (29a) decreases. Thus, f∗m[n]increases as n increases. Similarly, as n increases, the denom-inator of (29b) decreases and the numerator of (29b) increases.Thus, f∗k [n] increases as n increases. The numerator of (29c)increases as n increases, leading to the increment of f∗u [n].

Corollary 1 implies that as time slot n increases, theaccumulated energy of the active SDs increases, which canbe used to execute local computing and offloading. Thus, thetotal amount of data of local computing increases, the totalamount of data offloaded to the idle SDs and UAV increase.



8

In this case, the active SDs, idle SDs and UAV need to increasetheir CPU frequencies to complete their computing tasks.

Corollary 2: If there exist a time slot n such that f∗m[n] =0, f∗k [n] = 0, and f∗u [n] = 0, then f∗m[i] = 0, f∗k [i] = 0 andf∗u [i] = 0 for all i = 0, ..., n− 1.

Proof: Following Corollary 1, one can see that all f∗m[n],f∗k [n] and f∗u [n] are non-negative and non-decreasing functionsw.r.t n. Thus, 0 ≤ f∗m[i] ≤ f∗m[n], 0 ≤ f∗k [i] ≤ f∗k [n] and0 ≤ f∗u [i] ≤ f∗u [n], when i ≤ n. As a result, if f∗m[n] = 0,f∗k [n] = 0 and f∗u [n] = 0, it also holds that f∗m[i] = 0, f∗k [i] =0 and f∗u [i] = 0 for all i = 0, ..., n− 1.

Corollary 3: L∗m,k[n] and L∗

m,u[n] increase with the incre-ment of hm,k[n] and hm,u[n].

Proof: From (29d) and (29e), it is seen that L∗m,k[n] and

L∗m,u[n] are logarithmic functions w.r.t. hm,k[n] and hm,u[n]

respectively, which are non-decreasing. Thus, as hm,k[n] andhm,u[n] increase, L∗

m,k[n] and L∗m,u[n] increase.

Corollary 3 indicates that with the decrement of the distancebetween active SDs and idle SDs, and that between activeSDs and the UAV, the channel gains are improved and thenumber of computation bits that offloading to idle SDs andUAV increases.

2) Optimizing {qu[n], v[n], a[n]} with given{fm[n], fu[n], fk[n], Lm,k[n], Lm,u[n]}: For given CPUfrequencies and the number of offloading bits, problem P1can be simplified to be PB-2, i.e.,

PB-2 : minQ,τ [n]

Eflyu

s.t. (28b), (28c), (28e), (15j)− (15m).

PB-2 is convex and can be directly solved with CVX [32].By solving PB-1 and PB-2 alternately, a suboptimal solutionto P1 can be achieved when the convergence condition issatisfied. For clarity, the DAI-based algorithm is summarizedin Algorithm 2.

Algorithm 2 DAI-based algorithm for solving problem P1

1: set j = 0, r = 0, and iteration tolerance ξ1 and ξ2;2: Initialize UAV’s trajectory q

(r)u [n] and velocity v(r)[n];

3: Initialize the objective function E(j)ob = 0;

4: repeat5: Obtain fm[n], fu[n], fk[n],Lm,k[n], Lm,u[n] by solving

problem PB-1 for any given {q(r)u [n], v(r)[n]};6: repeat7: Obtain q∗u[n], v∗[n] by solving problem PB-2 for

given fm[n], fu[n], fk[n], Lm,k[n], Lm,u[n];8: Update q

(r+1)u [n] = q∗u[n], v(r+1)[n] = v∗[n],

a(r+1)[n] = a∗[n];9: Update the iterative number r = r + 1;

10: until stopping criterion∑N

n=1 ∥q(r+1)u [n] − q

(r)u [n]∥2

≤ ξ1 is met;11: Obtain E(j+1)

ob for given q∗u[n], v∗[n] and a∗[n];

12: Update the iterative number j = j + 1;13: until the stopping criterion |E(j+1)

ob − E(j)ob | ≤ ξ2 is met;

14: Obtain optimal solutions: q∗u[n], v∗[n], a∗[n], τ∗[n], f∗m[n],

f∗u [n], f∗k [n], L

∗m,k[n], L

∗m,u[n].

D. Complexity analysis of DAI-based algorithm

Both PB-1 and PB-2 involve linear inequalities and SOCconstraints. According to [27], the SOC constraints dominatetheir complexity. The complexity of dealing with problem PB-1can be negligible as it is solved with closed-form solutions.PB-2 has (M +K + 3N) SOC constraints, in which (M+K)SOCs with dimension of (2n + 1), N SOCs with dimensionof 2, and 2N SOCs with dimension of 3, where n = 1, ..., N .Besides, it involves 4N variables. So, the complexity ofsolving PB-2 can be given by O(n2 ·

√2(M +K + 3N) ·

(4(M + K)(N + 1)3 + 22N + n22)) with n2 = O(4N). Asproblem PB-1 and PB-2 have to be solved I2I3 times iteratively,the complexity of DAI-based algorithm for solving P1 is aboutI2I3O(n2 ·

√2(M +K + 3N) ·(4(M+K)(N+1)3+22N+

n22)).In order to compare the complexities of the two proposed

algorithms, without loss of generality, we let M = βN andK = γN , and then the complexities of the two algorithmsfor solving P1 are summarized in Table I, which shows thatthe computational complexity of the DAI-based algorithm islower than that of the SCA-based one when β = γ = 1.

V. SIMULATION RESULTS

This section provides some simulation results to discuss theperformance of the two presented algorithms and the effects ofdifferent parameters on system performance. A UAV-assistedwireless powered cooperative MEC system is simulated, wherethe SDs are randomly distributed within an area of 100 ×100 m2. The positions of active SDs are q1 =[10,10], q2=[2,80], q3 =[80,100], q4 =[100,20], and the positions of idleSDs are w1 =[40,40] and w2 =[60,60]. The simulation settingsare based on the works in [16], [24], [33], and the detailedparameter settings are summarized in Table II.

TABLE II: Simulation Parameters

Parameters Notation ValuesNumber of active SDs M 4Number of idle SDs K 2The height of the UAV H 10 mThe transmit power of UAV Pu 30 dBmCommunication bandwidth B 40 MHzThe system noise power σ2

0 10−9 WattUAV’s maximum speed Vmax 20 m/sUAV’s maximum acceleration amax 5 m/s2

The channel power gain β0 -20 dBEnergy harvesting efficiency η 0.8Number of CPU cycles C1, C2, C3 100Effective switched capacitance γc 10−24

The flying time of UAV T 10 sThe number of time slots N 50Constant related to aerodynamics c1 9.26Constant related to aerodynamics c2 2250The gravitational acceleration g 9.8 m/s2

The precisions ξ, ξ1, ξ2 10−5

Initial and final positions of UAV qinitialu = qfinalu [50,25]

Fig. 3 plots the optimized UAV trajectories of the proposedtwo algorithms for different period T . It shows that both theSCA-based algorithm and DAI-based algorithm achieve thesimilar optimized trajectories of UAV. Moreover, the inflightrange of UAV is smaller when T is shorter, e.g., T = 8s. As



9

TABLE I: Complexity analysis for the proposed algorithms

Algorithms Complexity Order

SCA-based Algorithm I1O(n1 ·

√2(M +K + 4N) · (4(M +K)(N + 1)3 + 31N + n2

1))≈ N

152

DAI-based Algorithm I2I3O(n2 ·

√2(M +K + 3N) · (4(M +K)(N + 1)3 + 22N + n2

2))≈ N

112

n1 = O(4MN + 5N +KMN +KN), n2 = O(4N)

Fig. 3: Optimized UAV trajectories of the proposed algorithmsfor different T .

0 2 4 6 8 10 12 14

Number of iterations

2000

4000

6000

8000

10000

12000

The

min

imal

req

uire

d en

ergy

of U

AV

(Jo

ule)

T = 20 s (SCA)T = 20 s (DAI)T = 15 s (SCA)T = 15 s (DAI)T = 10 s (SCA)T = 10 s (DAI)

6 8 10

2800

3400

Fig. 4: The minimal required energy of UAV versus thenumber of iterations for different T .

T increases, the UAV enlarges its turning radius and exploitsits mobility to adaptively adjust its trajectory to minimize itspropulsion energy consumption.

Fig. 4 shows the achieved minimal required energy andthe convergence behaviors of the two proposed algorithms fordifferent T . It is seen that the DAI-based algorithm achievesthe similar minimal required energy of UAV with the SCA-based one. And both algorithms converge well within severaliterations, e.g., 8 iterations. Moreover, the larger T , the moretotal required energy of UAV. Because, in UAV-assisted com-munication systems, propulsion-related energy is the dominantpart of total required energy. When the flying time increases,UAV requires more propulsion energy to maintain its aloft andsupport its mobility.

Fig. 5 compares UAV’s computation energy requirementversus the computation bits of each active SD of the twopresented algorithms with T = 10s and T = 15s. It is shownthat the larger the time period T is, the smaller requiredcomputing energy is. Because when T is larger, the more

104 1.5x104 2x104 2.5x104 3x104 3.5x104 4x104 4.5x104

The computation bits of each active SD (bits)

0

1

2

3

4

The

com

putin

g en

ergy

of U

AV

(Jo

ule)

10-7

SCA-based algorithm (T=10s)DAI-based algorithm (T=10s)SCA-based algorithm (T=15s)DAI-based algorithm (T=15s)

T=10s T=15s

Fig. 5: UAV’s computing energy requirement versus eachactive SD’s computation bits.

energy SDs can accumulate for local computing, and thesmaller the amount of bits required to be offloaded to UAV.Moreover, it also can be seen that both algorithms are feasiblewhen the computation bits are relatively small, but only theSCA-based algorithm is feasible when the computation bitsare relatively large. For example, when the computation bitsexceed 2.5×104 bits with T = 10s, and when the computationbits exceed 3.5 × 104 bits with T = 15s, the DAI-basedalgorithm is infeasible. The results from Fig. 4 and Fig. 5indicate that propulsion-related energy is the dominant part oftotal required energy, and under the same parameter settings,both proposed algorithms can achieves the similar minimalrequired propulsion-related energy of UAV. Besides, the DAI-based algorithm achieves lower required computing energyof the UAV than the SCA-based one. For a relatively largeamount of data, the SCA-based algorithm is feasible whichshould be adopted while with higher required computingenergy, and for a relatively small amount of data, the DAI-based algorithm is a better choice for achieving lower requiredcomputing energy.

Fig. 6 shows the optimized UAV trajectories in differentscenarios. Three scenarios are simulated, where the active SDsare located at left side of distributed area as shown in Fig. 6(a),at both sides of the distributed area as shown in Fig. 6(b) andat the right side of distributed area shown in Fig. 6(c). FromFig. 6, one can see that UAV’s trajectories are heavily relianton the locations of active SDs. The UAV tends to fly closeto the active SDs, so that the active SDs can harvest enoughenergy to complete their computation tasks. When the activeSDs are evenly distributed on both sides of the distributionarea, UAV’s trajectories is almost symmetrical about the activeSDs’ positions, as shown in Fig. 6(b).

Fig. 7 depicts the number of iterations versus the con-vergence precision of the proposed SCA-based algorithm fordifferent T . It can be observed that for a given period T , to



10

20 60x(m)

20

80y(

m)

idle SD2

active SD4

active SD3

active SD2 idle SD1

active SD1

(a)

20 80x(m)20

50

80

y(m

)

active SD1

active SD2

idle SD1

idle SD2

active SD3

active SD4

(b)

40 80x(m)20

80

y(m

)

active SD1

idle SD1

idle SD2

active SD4

active SD3

active SD2

(c)Fig. 6: Optimized UAV’s trajectories of three situations with different locations of active SDs.

10-1 10-3 10-5 10-6 10-7

Convergence precision

0

5

10

15

20

25

Num

ber

of it

erat

ions

T = 8 sT = 10 sT = 13 sT = 15 sT = 20 s

Fig. 7: The number of iterations versus convergence precisionfor different T .

achieve higher precision, more iterations are required. More-over, with the increment of T , the convergence performancedecreases.

Fig. 8(a) depicts the optimized trajectories of UAV obtainedby our proposed SCA-based algorithm and the benchmarkone, i.e., UAV trajectory design without optimizing UAV’sacceleration. Particularly, in the benchmark algorithm, theacceleration of UAV is neglected, and the speed of UAV iskept to be unchanged over T . It is observed that withoutoptimizing the acceleration, the UAV has to exploit its mobilityto move closer to active SDs, yielding a longer trajectory.The acceleration obtained from the trajectory optimized by thebenchmark algorithm shows that the acceleration of each timeslot is actually very large. Thus, by employing the benchmarkalgorithm to optimize UAV’s trajectory will generate morepropulsion energy than the proposed algorithm as shown inFig. 8(b).

Fig. 9 compares the minimal required energy of UAVachieved by our proposed trajectory design with other fixedtrajectory designs, i.e., diamond trajectory and circular trajec-tory.4 It is observed that as T increases, the minimal requiredenergy of UAV increases. Compared to the fixed trajectory

4In general, when the initial and final positions are different, the directlink is widely considered as a benchmark trajectory, see, e.g., [25], [34].While when the initial and final positions are same, some regular geometricshapes, such as diamond and circular trajectories are intuitively employed asbenchmark trajectories for performance comparison, see, e.g., [14], [21].

0 20 40 60 80 100x(m)

0

20

40

60

80

100

y(m

)

our proposedbenchmark

active SD4

active SD2

active SD3

idle SD2

active SD1

idle SD1

(a)

10 11 12 13 14 15Time period(s)

0

0.5

1

1.5

2

2.5

3

3.5

The

req

uire

d en

ergy

of U

AV

(Jo

ule)

104

our proposedbenchmark

10 15

3000

(b)Fig. 8: The optimized trajectories of UAV and the minimalrequired energy of UAV with different trajectory designs.

0 20 40 60 80 100

x(m)

0

50

100

150

y(m

)

Diamond trajectoryCircle trajectoryProposed trajectory

active SD2

active SD1

idle SD1

active SD4

idle SD2

active SD3

(a)

10 11 12 13 14 15

Time period (s)

2800

2900

3000

3100

3200

3300

3400

3500

3600

3700

The

min

imal

req

uire

d en

ergy

of U

AV

(Jo

ule)

Diamond trajectoryCircle trajectoryProposed trajectory

(b)Fig. 9: The minimal required energy of UAV versus period Twith different trajectory designs.

designs, our proposed trajectory design always require theleast energy since our proposed algorithm fully exploits theadvantages of trajectory optimization. It also demonstrates thattrajectory optimization plays a very important role in UAV-enabled wireless communication systems.

Fig. 10 depicts the performance gain brought by optimizingdifferent variables with our proposed algorithm, where inScheme I, all variables are jointly optimized. In Scheme II, theCPU frequencies and UAV’s trajectory are jointly optimizedwith fixed offloading bits to UAV, and in Scheme III, theoffloading bits and UAV’s trajectory are jointly optimized with



11

104 1.5x104 2x104 2.5x104 3x104 3.5x104 4x104 4.5x104


0

1

2

3

4

5

6

7

8

9T

he c

ompu

ting

ener

gy o

f UA

V (

Joul

e)

10-7

Scheme I

Scheme II

Scheme III

The gain obtained by optimizing the offloading bits

The gain obtained by optimizing CPU frequencies

Fig. 10: The UAV’s computing energy versus each active SD’scomputation bits for different optimization schemes.

0 5 10 15 20 25 30 35 40 45 50

Time slot index

0

200

400

600

800

1000

1200

1400

activ

e S

Ds'

bit

allo

catio

n (b

its)

active SD1-UAVactive SD2-UAVactive SD3-UAVactive SD4-UAVactive SD1-localactive SD2-localactive SD3-localactive SD4-local

Fig. 11: Optimized bit allocation of local computing andoffloading to UAV with T = 15s.

fixed UAV’s CPU frequencies. The performance gap betweenScheme I and Scheme II demonstrates the gain brought byoptimizing the offloading bits to UAV, and the performancegap between Scheme I and Scheme III demonstrates the gainbrought by optimizing UAV’s CPU frequency. It shows that thejoint optimization of UAV’s CPU frequency and the numberof bits offloaded to UAV in each time slot can achieve muchless energy for computing. As the active SDs’s computationbits increases, the gain obtained by optimizing offloadingbits increases, and the gain obtained by optimizing CPUfrequency decreases until the optimized CPU frequencies reachthe maximum computing capacity of UAV.

Fig. 11 shows the optimized offloading allocations of localcomputing and UAV offloading. It is observed that in ourconsidered system, a large number of bits are offloaded toUAV at the beginning, and less bits are offloaded to UAV later,which is much different from the conclusion in [35]. Because[35] focused on minimizing users’ communication energy, butour work aimed at minimizing UAV’s total required energy inwhich the propulsion energy consumption is of the dominantpart. Moreover, the active SDs prefer to process equal numberof bits in each time slot, because active SDs’ CPU frequencieshave to remain stable for saving computation energy, which isconsistent with [31].

Fig. 12 plots UAV’s computing energy requirement versusthe total number of computation bits of active SDs with

104 1.5x104 2x104 2.5x104 3x104 3.5x104


0

0.5

1

1.5

2

2.5

3

3.5

4

The

com

putin

g en

ergy

of U

AV

(Jo

ule)

10-7

K=0K=1K=2K=3

Fig. 12: UAV’s computing energy requirement versus eachactive SD’s computation bits.

0 1 2 3 4Number of idle SDs

1

2

3

4

5

6

7

8

Com

puta

tion

ener

gy (

Joul

e)

10-9

(a)

0 1 2 3 4

Number of idle SDs

20

30

40

50

60

70

80

90

100

110

Con

verg

ence

tim

e (s

)

(b)Fig. 13: UAV’s computation energy and the convergence timeversus the number of idle SDs.

different number of idle SDs. As the total number of com-putation bits of active SDs increases, the computing energyrequirement of UAV increases. Moreover, UAV’s computingenergy requirement is greatly reduced with the assistance ofidle SDs. More helping idle SDs, less computing energy ofUAV is required.

Fig. 13(a) and Fig. 13(b) depict UAV’s computation energyrequirement and the convergence time versus the number ofidle SDs, respectively. From Fig. 13(a), UAV’s computationenergy requirement decreases as the number of idle SDsincreases, since the more idle SDs, the more computationresources can be used, which is consistent with Fig. 12. But thedecreasing rate is gradually slowing down with the incrementof the number of idle SDs. From Fig. 13(b), as the numberof idle SDs increases, the convergence time of the algorithmgrows. Because when the number of idle SDs increases, thenumber of variables that need to be jointly optimized increases,resulting in a longer time to converge.

VI. CONCLUSION

This paper studied the joint optimization of the CPUfrequencies, the offloading bits, the transmit power and theUAV’s trajectory of the UAV-enabled wireless powered coop-erative MEC system. An optimization problem was formulatedto minimize the required energy of UAV. The SCA-basedalgorithm and the DAI-based algorithm were proposed to



12

tackle the non-convex problem. Theoretical analysis showsthat the DAI-based algorithm has lower computational com-plexity than the SCA-based one. Simulation results show thatboth algorithms converge within several iterations and theyachieve the similar minimal required energy and the opti-mized trajectory. The proposed algorithms obtain significantperformance gain compared to other benchmarks, which indi-cates that the propulsion-related energy occupies the dominantpart of the total required energy and trajectory design playsan important role in UAV-enabled wireless communicationsystem. Additionally, the jointly optimization of the UAV’sCPU frequency and the offloading bits can achieve much lessenergy for computing of UAV. And, with the help of idle SDs,UAV’s computing energy requirement can be greatly reduced.However, as the number of idle SDs increases, the number ofoptimization variables increases, resulting in a longer time toconverge.

APPENDIX ATHE PROOF OF THEOREM 2

PB-1 is a convex problem which can be solved by Lagrangiandual method. Let νm,n, µk,n, λk,n, θn, γm denote the dualvariables w.r.t. constraints (15a) - (15g), and Ξ denotes acollection containing all the optimization variables and dualvariables related to PB-1. The Lagrangian of PB-1 can beexpressed by ζ (Ξ), i.e.,

ζ (Ξ) =N∑

n=2

T

Nγcfu[n]

3 +M∑

m=1

N∑n=1

νm,n

{n∑

i=1

T

Nγcfm[i]3

+∑n

i=1

σ2(2

Lm,u[i]

Bδo − 1)

hu,m[i]+∑n

i=1

σ2

(2

Lm,k[i]

Bδo − 1

)hm,k[i]

−∑n

i=1

T

NηPuhu,m[i]

}+

K∑k=1

N∑n=1

µk,n

{n∑

i=1

T

Nγcfk[i]

3 −n∑

i=1

T

NηPuhu,k[i]

}+

K∑k=1

N−1∑n=2

λk,n

{n∑

i=2

fk[i]

C2

T

N−

n−1∑i=1

M∑m=1

Lm,k[i]

}+

K∑k=1

λk,N

{N∑i=2

fk[i]

C2

T

N−

N−1∑i=1

M∑m=1

Lm,k[i]

}+

N−1∑n=2

θn

{n∑

i=2

fu[i]

C3

T

N−

n−1∑i=1

M∑m=1

Lm,u[i]

}+

θN

{N∑i=2

fu[i]

C3

T

N−∑N−1

i=1

M∑m=1

Lm,u[i]

}+

M∑m=1

γm{N∑

n=1

fm[n]

C1

T

N+

N−1∑n=1

K∑k=1

Lm,k[n] +

N−1∑n=1

Lm,u[n]−Rm

}Thus, the derivations of ζ (Ξ) w.r.t.{fm[n], fk[n], fu[n], Lm,k[n], Lm,u[n]} can be given by

∂ζ

∂fm[n]= 3

T

NγcC1

∑N

j=nνm,jfm[n]2 − T

Nγm,

∂ζ

∂fk[n]=

0, n = 1

3T

Nγcfk[n]

2∑N−1

j=nµk,j +

T

N

∑N−1j=n λk,j − λk,N

C2,

n = 2, ..., N − 1

3T

Nγcµk,Nfk[n]

2 − T

N

λk,NC2

, n = N

∂ζ

∂fu[n]=

0, n = 1

3T

Nγcfu[n]

2 +T

N

∑N−1j=n θj − θN

C3, n = 2, ..., N − 1

3T

Nγcfu[n]

2 − T

N

θNC3

, n = N

∂ζ

∂Lm,k[n]= (2

Lm,k[n]

Bδo − 1)−

Bδo

(∑N−1j=n+1 λk,j − λk,N + γm

)hm,k[n]

σ2 ln2∑N

j=n νm,j

,

∂ζ

∂Lm,u[n]= (2

Lm,u[n]

Bδo − 1)−

Bδo

(∑N−1j=n+1 θj − θN + γm

)hu,m[n]

σ2 ln2∑N

j=n νm,j

.

Apply the Karush-Kuhn-Tucker (KKT) conditions and let∂ζ

∂fm[n] ,∂ζ

∂fk[n], ∂ζ

∂fu[n], ∂ζ

∂Lm,k[n], ∂ζ

∂Lm,u[n]be equal to zero,

we can obtain the corresponding optimal solutions given inTheorem 2 with some straightforward calculations. Thus, theproof for Theorem 2 is completed.

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Yuan Liu received the B.S. degree from Collegeof Computer and Information Technology, LiaoningNormal University, Dalian, China, in 2017. She iscurrently pursuing the Ph.D. degree with the Schoolof Computer and Information Technology, BeijingJiaotong University, Beijing, China. Her current re-search interests include UAV communications, ener-gy harvesting in wireless communication networks,wireless sensor networks.

Ke Xiong (M’14) received the B.S. and Ph.D.degrees from Beijing Jiao Tong University (BJTU),Beijing, China, in 2004 and 2010, respectively. From2010 to 2013, he was a Post-Doctoral Research Fel-low with the Department of Electrical Engineering,Tsinghua University, Beijing. Since 2013, he hasbeen a Lecturer and an Associate Professor of BJTU,where he is currently a Full Professor with theSchool of Computer and Information Technology.From 2015 to 2016, he was a Visiting Scholar at theUniversity of Maryland at College Park. His current

research interests include wireless cooperative networks, wireless powerednetworks, and network information theory.

He has published more than 100 academic papers in referred journalsand conferences. He is a member of China Computer Federation (CCF)and also a senior member of the Chinese Institute of Electronics (CIE). Dr.Xiong serves as an Associate Editor-in Chief of the Chinese journal NewIndustrialization Strategy, and the editor of Computer Engineering & Software.In 2017, he served as the leading editor of the Special issue “Recent Advancesin Wireless Powered Communication Networks” for EURASIP Journal onWireless Communications and Networking. Currently, he also serves as areviewer more than 15 international journals including IEEE Transactions onSignal Processing, IEEE Transactions on Wireless Communications, IEEETransactions on Communications, IEEE Transactions on Vehicular Technolo-gy, IEEE Communication Letters, IEEE Signal Processing Letters and IEEEWireless Communication Letters. He also served as a Session Chair for IEEEGLOBECOM2012, IET ICWMMN2013, IEEE ICC2013, ACM MOMM2014and the Publicity and Publication Chair for IEEE HMWC2014, as well as theTPC co-chair of IET ICWMMN2017 and IET ICWMMN2019.

Qiang Ni (M’04-SM’08) received the B.Sc., M.Sc.,and Ph.D. degrees from the Huazhong University ofScience and Technology, China, all in engineering.He is currently a Professor and the Head of the Com-munication Systems Group, School of Computingand Communications, Lancaster University, Lancast-er, U.K. His research interests include the area offuture generation communications and networking,including green communications and networking,millimeter-wave wireless communications, cognitiveradio network systems, non-orthogonal multiple ac-

cess (NOMA), heterogeneous networks, 5G and 6G, SDN, cloud networks,energy harvesting, wireless information and power transfer, IoTs, cyberphysical systems, AI and machine learning, big data analytics, and vehicularnetworks. He has authored or co-authored over 200 papers in these areas. Hewas an IEEE 802.11 Wireless Standard Working Group Voting Member anda contributor to the IEEE Wireless Standards.



14

Pingyi Fan (M’03-SM’09) received the B.S andM.S. degrees from the Department of Mathematicsof Hebei University in 1985 and Nankai Universityin 1990, respectively, received his Ph.D degree fromthe Department of Electronic Engineering, TsinghuaUniversity, Beijing, China in 1994. He is a professorof department of EE of Tsinghua University cur-rently. From Aug. 1997 to March. 1998, he visitedHong Kong University of Science and Technology asResearch Associate. From May. 1998 to Oct. 1999,he visited University of Delaware, USA, as research

fellow. In March. 2005, he visited NICT of Japan as visiting Professor. FromJune. 2005 to May 2014, he visited Hong Kong University of Science andTechnology for many times and From July 2011 to Sept. 2011, he is a visitingprofessor of Institute of Network Coding, Chinese University of Hong Kong.

Dr. Fan is a senior member of IEEE and an oversea member of IEICE. Hehas attended to organize many international conferences including as Generalco-Chair of IEEE VTS HMWC2014, TPC co-Chair of IEEE InternationalConference on Wireless Communications, Networking and Information Se-curity (WCNIS 2010) and TPC member of IEEE ICC, Globecom, WCNC,VTC, Inforcom etc. He has served as an editor of IEEE Transactions onWireless Communications, Inderscience International Journal of Ad Hoc andUbiquitous Computing and Wiley Journal of Wireless Communication andMobile Computing. He is also a reviewer of more than 30 internationalJournals including 20 IEEE Journals and 8 EURASIP Journals. He hasreceived some academic awards, including the IEEE Globecom’14 Best PaperAward, IEEE WCNC’08 Best Paper Award, ACM IWCMC’10 Best PaperAward and IEEE ComSoc Excellent Editor Award for IEEE Transactionson Wireless Communications in 2009. His main research interests includeB5G technology in wireless communications such as MIMO, OFDMA, etc.,Network Coding, Network Information Theory and Big Data Analysis etc.

Khaled Ben Letaief (S’85-M’86-SM’97-F’03) Dr.Letaief received his Ph.D. from Purdue University,USA. He is Chair Professor and Provost of HBKU,a newly established research-intensive university inQatar. He has served as HKUST Dean of Engi-neering from 2009 to 2015. Under his leadership,HKUST School of Engineering has not only trans-formed its education and scope and produced veryhigh caliber scholarship, it has also actively pursuedknowledge transfer and societal engagement in broadcontexts. It has also dazzled in international rank-

ings.Dr. Letaief is a world-renowned leader in wireless communications and

networks. In these areas, he has over 500 journal and conference papers andgiven invited keynote talks as well as courses all over the world. He has made6 major contributions to IEEE Standards along with 13 patents. He is thefounding Editor-in-Chief of IEEE Transactions on Wireless Communicationsand was instrumental in organizing many IEEE flagship conferences as wellas serving IEEE in many leadership positions, including IEEE ComSocVice-President for Technical Activities and IEEE ComSoc Vice-President forConferences.

He is recipient of 6 Teaching awards and 12 IEEE Best Paper awards. Heis ISI Highly Cited Researcher, IEEE Fellow, HKIE Fellow, and recipient of2007 IEEE Joseph LoCicero Award; 2009 IEEE Marconi Prize Award; 2010Purdue Outstanding Electrical and Computer Engineer Award, 2011 IEEEHarold Sobol Award, and 2011 IEEE Wireless Communications TechnicalCommittee Recognition Award.

uav-assisted wireless powered cooperative mobile edge

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