Decentralized Computation Offloading and Resource Allocation …networking.khu.ac.kr/layouts/net/publications/data/2018... · 2019-01-09 · Q.-V. Pham et al.: Decentralized Computation

Received October 18, 2018, accepted November 11, 2018, date of publication November 23, 2018,date of current version December 27, 2018.

Digital Object Identifier 10.1109/ACCESS.2018.2882800

Decentralized Computation Offloading andResource Allocation for Mobile-EdgeComputing: A Matching Game ApproachQUOC-VIET PHAM 1, TUAN LEANH 1, NGUYEN H. TRAN 1,3, (Senior Member, IEEE),BANG JU PARK4, AND CHOONG SEON HONG 11Department of Computer Science and Engineering, Kyung Hee University, Seoul 17104, South Korea2ICT Convergence Center, Changwon National University, Changwon 51140, South Korea3School of Computer Science, The University of Sydney, Sydney, NSW 2006, Australia4Department of Electronic Engineering, Gachon University, Seongnam 13120, South Korea

Corresponding author: Choong Seon Hong ([email protected])

This work was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea Government (MSIT) underGrant NRF-2017R1A2A2A05000995.

ABSTRACT In this paper, we propose an optimization framework of computation offloading and resourceallocation for mobile-edge computing with multiple servers. Concretely, we aim to minimize the system-wide computation overhead by jointly optimizing the individual computation decisions, transmit power ofthe users, and computation resource at the servers. The crux of the problem lies in the combinatorial natureof multi-user offloading decisions, the complexity of the optimization objective, and the existence of inter-cell interference. To overcome these difficulties, we adopt a suboptimal approach by splitting the originalproblem into two parts: 1) computation offloading decision and 2) joint resource allocation. To enabledistributed computation offloading, two matching algorithms are investigated. Moreover, the transmit powerof offloading users is found using a bisection method with approximate inter-cell interference, and thecomputation resources allocated to offloading users is achieved via the duality approach. Simulation resultsvalidate that the proposed framework can significantly improve the percentage of offloading users and reducethe system overhead with respect to the existing schemes. Our results also show that the proposed frameworkperforms close to the centralized heuristic algorithm with a small optimality gap.

INDEX TERMS Heterogeneous networks, matching theory, mobile edge computing, resource allocation.

I. INTRODUCTIONWith the radically increasing popularity of mobile terminalssuch as smart phones and tablet computers, a wide-rangeof mobile applications are constantly emerging, includingreal-time online gaming, augmented reality, natural languageprocessing, and ultra-high-definition video streaming. Theseapplications have stringent requirements of real-time commu-nication, high energy efficiency, and intensive computation.However, mobile devices are usually constrained with limitedbattery capacity and computation capability. To tackle theseissues, mobile-edge computing (MEC) has been developedby the European Telecommunications Standards Institute(ETSI) as a supplement to mobile cloud computing. Theidea behind the MEC concept is to move the cloud servicesand functions to the network edges, thus enabling the execu-tions of computation heavy tasks in close proximity to endusers.

The emergence of MEC provides a promising solutionfor resource-limited users that can migrate a part or allof the intensive computations to powerful MEC servers atthe network edges. However, computation offloading mayincur additional overhead in terms of energy consumptionand latency, e.g., the local execution only suffers fromthe locally processing delay whereas the remote executionlatency includes the transmission delay from the users tothe MEC server, the remotely processing delay at the MECserver, and the response delay from the MEC server back tothe users. Optimally determining the offloading decisions isan important issue in MEC networks [1], [2]. In the pres-ence of multiple users, the MEC server must be able tosimultaneously execute multiple computation tasks and thescarce wireless channel needs to be shared among multipleusers. As opposed to the resourceful cloud, the MEC serverusually has finite resources. Therefore, the joint optimization

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VOLUME 6, 2018

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Q.-V. Pham et al.: Decentralized Computation Offloading and Resource Allocation for MEC

of offloading decisions and resource allocation is consideredas one of the most important problems in MEC networksto improve the MEC network performance [1]. The last fewyears have seen the enormous amount of attention for afore-mentioned problems including offloading decisions in single-user systems [3]–[5] and in multi-user systems [6]–[12],joint optimization of communication and computationresource [13], [14], and joint offloading decision and resourceallocation [11], [12], [15], all in MEC networks with a singleserver, i.e., single-server MEC networks.

The densification of heterogeneous networks (HetNets)provides a promising solution for future networks to increasethe spectral-and-energy efficiency and enhance the coveragefor indoor and edge users [16], [17]. In HetNets, variouskinds of small cells with different characteristics can bedeployed by the network operators and/or users, e.g., micro-cell, femtocell, and picocell. With the concept of net-work densification and recent advancements in computinghardware, there may exist multiple MEC servers, whereeach one is connected to and collocated with a smallcelleNB (SeNB). In MEC HetNets, a user can either locally han-dle the computation or send a request to one among multipleMEC servers for computation offloading. To fully reap thebenefits of multi-server MEC HetNets, numerous technicalchallenges must be addressed. First, beside offloading deci-sions, one must answer the question ‘‘which server does auser offload to?’’, i.e., user association. In general, user asso-ciation is coupled with subchannel allocation, i.e., ‘‘whichsubchannel does a user utilize to offload the task’’, and powerallocation, which greatly complicates the joint optimizationof computation offloading and resource allocation in multi-serverMECHetNets. Second, since simultaneous offloadingsfrom multiple users can result in severe co-channel inter-cell interference, the offloading rates of users, which offloadto different MEC servers but on the same subchannel, arecoupled and/or small, thus making the resource allocationproblem very hard to solve and requiring efficient schemesfor interference management. In contrast, one common set-ting from existing studies in single-MEC networks based onorthogonal frequency-division multiple access (OFDMA) isthat both intra-cell and inter-cell interference are ignored.

Unlike conventional multi-cell networks, where MECservers are typically deployed at fixed-location macrocelleNBs (MeNBs), MEC servers are collocated at randomly-deployed SeNBs in HetNets without network predimension-ing and planning. This strongly affects the user associationsince an MEC server is able to serve multiple users withdifferent channel qualities and a user is under the servicecoverage of multipleMEC servers with different computationcapabilities. Due to the dynamics and unplanned deploymentof HetNets and the possibility of missing a central entity,the centralized optimization of both computation offloadingand resource allocation is not always efficient in the dis-tributed networks, for example, wireless sensor networks andInternet of Things (IoT). Although the centralized approachcan achieve the optimal solution and high performance for

the resource optimization problem and can fulfill variousquality-of-service (QoS) requirements of offloading users, allusers need to report their channel qualities to the centralizedentity, e.g., MEC server, and the centralized server is lonelyresponsible for accomplishing the optimization for the entirenetwork [18]. Therefore, the centralized optimization hasthe weakness of high computational complexity and hugereporting overhead.Moreover, theremay not exist a dedicatedbackhaul for information exchange between the MEC serversand even if there is, the wireless backhaul in 5G HetNetswould be congested due to the high burden of huge datasharing [19]. All of these points call for efficient schemesof distributed computation offloading multi-server MECHet-Nets. Most of existing works in MEC focus on solutions forsingle-server MEC homogeneous networks [10], [14], [15],single-server MEC HetNets [12], [13], [20], and differentaspects in multiple-server MEC networks, for example, com-putation offloading [21], [22], computation offloading andserver provisioning [23], [24], and centralized task offloadingand resource allocation [25].

More recently, some distributed schemes for computationoffloading in single-server MEC networks were proposedusing game theory [10], [26] and game-theoretic machinelearning [27]. Nevertheless, none of the existing works dis-cussed the joint optimization of distributed offloading deci-sions and resource allocation for multi-server MEC HetNets.Further, it is worth noticing that game-theoretic approachespresent some shortcomings [28]. For instance, finding thebest response of a user usually requires the knowledge ofother players’ actions, thus limiting the distributed imple-mentations. Next, analyzing the tractability of equilibrium inthe game-theoretic methods requires the specific structure inthe objective function, which is however not always feasi-ble in some practical scenarios. Matching theory, a subfieldof economics for designing low-complexity and distributedalgorithms in wireless networks [29], [30], is a promisingsolution to overcome some limitations of game-theoreticapproaches and centralized optimizations. There are severalbenefits to apply matching theory for resource allocation inwireless networks, e.g., the ability to characterize interactionsbetween heterogeneous players and define the general pref-erences and the flexibility to reflect different optimizationobjectives [28], [31].

In this paper, we develop an efficient framework of dis-tributed computation offloading and resource (computationand communication) allocation to minimize the system-wide computation overhead in multi-server MEC HetNets.Here, computation offloading pertains to finding the solu-tion for the user association and subchannel assignment andresource allocation relates to the transmit power allocationof offloading users and computation resource allocation atthe MEC servers. The considered problem faces difficultiescaused by the combinatorial nature of multi-user offload-ing decisions, the complexity of the optimization objective,and the existence of inter-cell interference among offloadingusers, i.e., an NP-hard combinatorial problem. In a nutshell,

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the main contributions and features of our work are summa-rized as follows:• In terms of the system model and problem formulation,we consider a network scenario with multiple SeNBscollocated with the corresponding MEC servers andmultiple users and define the objective function as thesystem-wide computation overhead. Then, an optimiza-tion problem is formulated subject to constraints on theMEC server and subchannel selections, maximum trans-mit power of mobile devices, and maximum computa-tion resources at the MEC servers. To solve the problem,we adopt a suboptimal approach by splitting the originalproblem into two parts: (i) computation offloading deci-sion, which includes user association and subchannelassignment, and (ii) joint resource allocation, which isfurther decomposed into the transmit power of offload-ing users and computation resource allocation at theMEC servers.

• In terms of the mathematical framework, matchingtheory is adopted to provide the distributed computa-tion offloading decision. Concretely, we employ twomatching games to design algorithms for user asso-ciation and subchannel assignment. Accordingly, adecentralized approach is investigated to determine theoffloading decision. With the proposed computationoffloading scheme, (i) users decide to offload theircomputation tasks only if they benefit from compu-tation offloading and remote execution and (ii) usersand servers make the offloading decision in a dis-tributed and autonomous fashion. Moreover, we approx-imate the inter-cell interference and find the transmitpower of offloading users using a bisection method,and then solve the computation resource allocationproblem via the KKT optimality conditions. Detailedanalyses of the matching stability, convergence, andalgorithmic complexity of the proposed algorithm arealso provided.

• In terms of the performance evaluation, we validatethe performance of the proposed algorithm throughextensive numerical experiments. Furthermore, wecompare our proposed algorithmwith four existing solu-tions: local computing only, offloading only, the heuris-tic offloading decision algorithm (HODA) proposedin [15], and the heuristic joint task offloading andresource allocation (hJTORA) proposed in [25]. Thesimulation results illustrate that our proposed algorithmcan achieve significant performance improvement interms of the number of offloading users and the system-wide computation overhead and can perform close to thecentralized heuristic algorithm with a small optimalitygap.

Our paper is organized as follows. The system modeland optimization problem are explained in Section II.In Section III, we propose to decompose the underlyingproblem into two subproblems and obtain the compu-tation offloading decisions using the matching games.

FIGURE 1. An MEC system with two SeNBs, each with an MEC server, andthree users, each with a distinct application.

Simulation results are presented and discussed in Section IV.Finally, conclusions and future works are drawn in Section V.

II. SYSTEM MODEL AND PROBLEM FORMULATIONA. NETWORK MODELWe consider a multi-user multi-server MEC HetNet as illus-trated in Fig. 1, where each MEC server is collocated withan SeNB.1 In general, MEC servers deployed by the net-work operator have a moderate computing capability andhave wireless channel connections to users through thecorresponding SeNBs. Small cells operate in an overlaidmanner, i.e., each small cell is able to reuse the wholespectrum of the macro cell and interference among smallcells exists. To enable tractable analysis and obtain usefulinsights, we employ a quasi-static network scenario whereusers remain unchanged during the computation offloadingperiod while they change across different periods. The gen-eral scenario, where users leave or join dynamically duringthe offloading period, is not our focus in this paper and canbe considered as future work.

Denote by M = {1, 2, . . . ,M} the set of SeNBs andbym the index of themth SeNB. The set of users is denoted byN = {1, 2, . . . ,N } and n is referred to as the nth user. In thispaper, we use OFDMA as the multiple access scheme in theuplink. Assume that there are S subchannels in a small cell,then the set of subchannels is denoted by S = {1, 2, . . . , S}and s is referred to as the sth subchannel. Each user is assignedto at most one subchannel and a subchannel is assigned toat most one user. The general case that a user is assigned tomultiple subchannels and a subchannel is assigned tomultipleusers will be considered in future work.

B. COMMUNICATION MODELFor every small cell, an MEC server is collocated withthe corresponding SeNB; therefore, a user can offloadits computation task to the MEC server via the SeNB.We define the offloading decision profile as A ={asnm|n ∈ N ,m ∈M, s ∈ S

}. Specifically, asnm = 1 if the

user n offloads its computation task to the MEC server mon the subchannel s, and asnm = 0 otherwise. Since each

1From that point, we use ‘‘MEC server’’ when referring to the relatedconcepts of computation resource and computation offloading, while using‘‘SeNB’’ whenmentioning interference, radio resource, and user association.

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computation task can be either computed locally or remotely,i.e., binary offloading, we have the following constraint:∑

m∈M

∑s∈S

asnm ≤ 1, ∀n ∈ N . (1)

Each SeNB assigns a subchannel to at most one user, so thefollowing constraint must be satisfied:∑

n∈Nasnm ≤ 1,∀m ∈M, s ∈ S. (2)

Moreover, since each server is collocated with a SeNB, whichis usually restricted by hardware limitations [32], the numberof users offloading to a server should be constrained by∑

n∈N

∑s∈S

asnm ≤ qm, ∀m ∈M, (3)

where qm is called a quota, which represents the maximumnumber of users the SeNB m can serve. In other words,each server can accommodate at most qm users to shareits CPU.

Given the offloading decision profile A, the transmissionrate from the user n to the server m over the subchannels is defined as Rsnm (A

s,P) = Bs log2(1+ 0snm (A

s,P)),

where Bs is the bandwidth of the subchannel s, As ={asnm|n ∈ N ,m ∈M

}is the offloading decision profile

on the subchannel s, and 0snm is the received signal-to-interference-plus-noise ratio (SINR) from the user n at theserver m using the subchannel s, which can be written as

0snm(As,P

)=

psnhsnm

n0 +∑

j 6=m∑

k∈Njaskmp

skh

skm. (4)

Here, n0 is the noise power which is identical for all users,Nj is the set of users that offload their computation tasks tothe MEC server j, P =

{p1, . . . , pn, . . . , pN

}is the transmit

power vector of all users, pn = {p1n, . . . , p

Sn} is the transmit

power vector of the user n with psn being the transmit power(inWatts) of the user n on subchannel s, and hsnm is the channelgain from the user n to the SeNB m on the subchannel s.The second term of the denominator in (4) is the inter-cellinterference from users offloading to other servers using thesame subchannel s. Correspondingly, the data rate from theuser n to the server m is Rnm (A,P) =

∑s∈S a

snmR

snm (A

s,P).

C. COMPUTATION MODEL OF MOBILE DEVICESEach user n has a computation task In = {αn, βn} [1], [10],where αn is the computation input data size (in bits) andβn is the number of CPU cycles required to complete thetask In (i.e., computationworkload/intensity). Binary offload-ing is considered in our work. The extension to considerpartial offloading is highly promising.

We denote f ln as the computation capability (in CPU cyclesper second) of the user n, where the superscript l standsfor local. Due to heterogeneous mobile devices, users canhave different computation capabilities. Let t ln be the com-pletion time of the task In by the user n, which can becomputed as t ln =

βnf ln. To compute the energy consump-

tion E ln (in Joules) of the user when the task is exe-cuted locally, we adopt the model in [1], [15], and [11].

Specifically, E ln can be derived as E ln = κnβn(f ln)2, where

κn is a coefficient relating to the chip’s hardware archi-tecture. According to the measurements in [11], we setkn = 5 × 10−27. It is worth noting that t ln and E ln dependon unique features of the user n and the running application;therefore, they can be computed in advance.

The local computation overhead in terms of the compu-tational time and energy consumption is computed as Z ln =λtnt

ln + λenE

ln, where λ

tn ∈ [0, 1] and λen + λtn = 1 are

respectively weighted parameters of the computational timeand energy consumption of the user n. Similar to heteroge-neous users, different users may have different values of λtnand λen. The weighted parameters can affect the offloadingdecisions of users. Consider a network scenario with threeusers as an example, where the first user with a latency-sensitive application sets λtn = 1 and λen = 0, the seconduser with an energy-hungry application and low battery statecan set λtn = 0 and λen = 1, and the third user can set0 < λtn, λ

en < 1 if it takes both computational time and energy

consumption into consideration of the offloading decision.

D. COMPUTATION MODEL OF MEC SERVERSTo offload the computation task, a user incurs extra overheadin terms of the energy consumption and latency. The extraoverhead in time is composed of the transmission time fromthe users to the server, the execution time at the server, andthe response time of the computational result back to theuser. The extra overhead in energy consumption includes theenergy consumption for computation offloading, remote exe-cution, and transmission of the computational result back tothe user. Since our current work focuses on the user perspec-tive and since MEC servers are generally powered by cablepower supply [10], [11], we ignore the energy consumptionfor remote execution. Moreover, the computational result isrelatively small compared to the input data, so the responsetime and consumed energy consumption are neglected.

The time and energy costs for offloading the computationtask In are, respectively, computed as

toffn (A,P) =∑

m∈MαnanmRnm

, ∀n ∈ N , (5)

where anm =∑

s∈S asnm, and

Eoffn (A,P) = pntoffn =

pnζnαn∑

m∈ManmRnm

, ∀n ∈ N ,

(6)

where pn = pTn 1 with XT being the normal transpose X ,ζn is the power amplifier efficiency of the user n. The execu-tion time of the computation task In is given by texen (A,Fn) =∑

m∈Manmβnfnm

,∀n ∈ N , where fnm is the computation capa-bility (in CPU cycles per second) that is assigned to the user nby the server m. Here, F = {Fn|n ∈ N } is the computationresource profile, where Fn = {fnm|∀m ∈M} is the computa-tion resource vector of the user n.

Since each MEC server typically has the limited computa-tion capability, the total computation resources assigned to

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offloading users cannot exceed its maximum computationcapability f max

m , i.e., the constraint,∑

n∈N fnm ≤ f maxm ,

∀m ∈M, must be satisfied. Similar to the local computationoverhead, the remote computation overhead can be computedas Z rn (A,P,Fn) = λ

tn(toffn + t

exen)+ λenE

offn .

E. PROBLEM FORMULATIONSince our focus is to minimize the system-wide com-putation overhead in terms of the computational timeand energy consumption, the objective function is definedas Z (A,P,F) =

∑n∈N Zn (A,P,Fn), where Zn is given by

Zn =(1−

∑m∈M anm

)Z ln +

(∑m∈M anm

)Z rn ,∀n ∈ N .

For a given offloading decision profile A, powerallocation P, and computation resource F, as well as theobjective function Z (·), we have the optimization formula-tion problem of joint computation offloading decision andresource allocation (OPT-JCORA) as follows:

min{A,P,F}

Z (A,P,F)

s.t. C1: 0 < psn ≤ pmaxn , ∀s ∈ S, ∀n ∈ Noff

C2: asnm = {0, 1} , ∀n ∈ N , m ∈M, s ∈ SC3:

∑m∈M

∑s∈S

asnm ≤ 1, ∀n ∈ N

C4:∑

n∈Nasnm ≤ 1, ∀m ∈M, s ∈ S

C5:∑

n∈N

∑s∈S

asnm ≤ qm, ∀m ∈MC6: fnm > 0, ∀n ∈ Nm, ∀m ∈MC7:

∑n∈N

fnm ≤ f maxm , ∀m ∈M, (7)

where pmaxn is the maximum transmit power of the user n,

and Noff =⋃

m∈MNm is the set of offloading users that arenot able to compute their tasks locally. In the optimizationformulation (7), if n /∈ Noff, pn = 0, i.e., the user n executesthe task locally. In addition, if n /∈ Nm, fnm = 0, i.e., theMEC server m does not assign any computation resource tothe user n. The constraint C1 makes sure that the transmitpower of the user n does not exceed the maximum value.The constraint C2 denotes that A is a binary vector. Theconstraints C3, C4, and C5 guarantee that a computationtask is executed either locally or remotely, a subchannel isassigned to at most one user, and the maximum numberof users qm can offload to an MEC server m, as explainedin (1), (2), and (3), respectively. The constraint C6 denotesthat the assigned computation resource from an MEC serverto a user is positive, and the constraint C7 makes sure thatthe total computation resource of an MEC server assigned tooffloading users does not exceed its maximum value f max

m .The considered problem (7) is difficult to solve due to the

following reasons:• There exist relationships among A, P, and F. Further,the data rate of users in the denominator in (5) and (6)makes the objective function highly complicated. Thus,the objective function Z (A,P,F) is not a convexfunction.

• There are three set of optimization variables: offload-ing decision A, power allocation P, and computation

resource F. P and F are continuous variables whileA is a binary variable; therefore, the feasible solution setof the problem (7) is not convex.

To enable distributed computation offloading, in the nextsection, we propose to decompose the problem (7) into twoparts: computation offloading decision and joint resourceallocation.

III. PROPOSED ALGORITHMA. PROBLEM DECOMPOSITIONThe OPT-JCORA is a mixed-integer and non-linearoptimization problem since the offloading decision A is anbinary vector and P and F are continuous vectors. Moreover,the OPT-JCORA problem is NP-Hard [15], [33]. Hence,it is hard to obtain the global centralized optimal solution.It is observed that the resource constraints C1, C6, andC7 are decoupled from the offloading constraints C2, C3,C4, and C5. Therefore, we proceed with solving the prob-lem (7) by adopting a suboptimal approach and splitting itinto two parts: (i) computation offloading (CO) decision and(ii) joint computation and communication resource allocation(JCCRA). The CO subproblem is written as follows:

minA

Z (A)

s.t. C2, C3, C4, C5, (8)

where Z (A) is the objective value of the JCCRA subproblem,which is formulated as

minP,F

∑n∈Noff

Zn (A,P,Fn)

s.t. C1, C6, C7. (9)

By solving two subproblems (8) and (9) sequentially in eachiteration until convergence, the solution to the underlyingproblem (7) can be finally obtained. The proposed frameworkfor solving the OPT-JCORAproblem is summarized in Fig. 2.Next, we present our algorithm to solve the OPT-JCORA

problem by solving first the CO subproblem via twomatching games and then deriving the solution for theJCCRA subproblem. After that, we propose an algorithm forthe OPT-JCORA problem. Besides, we also provide detailedanalyses of the matching stability, convergence, and compu-tational complexity of the proposed algorithm.

B. COMPUTATION OFFLOADING AS A MATCHINGPROBLEMWe consider the offloading decision problem for a given Pand F by solving the following optimization problem:

minAZ (A)

s.t. C2, C3, C4, C5 (10)

psn = pmaxn /S, ∀n ∈ N (11)

fnm = f maxm /qm, ∀n ∈ N , m ∈M. (12)

In order to evaluate the average contribution of each user tothe total objective, the transmit power of each user is divided

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FIGURE 2. Proposed framework.

equally among all the subchannels and the total computationresources of each server is uniform among the maximum qmoffloading users, i.e., psn = pmax

n /S and fnm = f maxm /qm,

as illustrated (11) and (12), respectively. This approach hasbeen widely used in literature for scheduling, user associa-tion, and resource allocation in OFDMA networks [16], [34],[35], where simulation results showed that the achieved gainsare negligible compared to that of optimal power allocation.

The CO subproblem is a mixed integer nonlinear pro-gramming and an NP-Hard problem [15], [33]. To solve theCO problem, the objective (10) is boiled down to thefollowing:

Z (A) =∑n∈N

Z ln

+

∑n∈N

an

[ ∑m∈M

anm

(λtnαn + λ

enpnαnζ

−1n

Rnm+λtnβn

fnm

)− Z ln

],

(13)

where an =∑

m∈M anm, whose value denotes the user n’soffloading decision, i.e., an = 1 (an = 0) indicates thatthe user n offloads (executes locally). Recall that this paperconsiders binary offloading. To be executed remotely, usersmust benefit from computation offloading in terms of theexecution latency and/or energy consumption. From (13),the user n executes the task locally if the following conditionholds

ϒn :=

(λtnαn + λ

enp

minn αnζ

−1n

Rmaxnm

+λtnβn

f0

)− Z ln ≥ 0, (14)

where Rmaxnm = Bs log2

(1+ pnmaxm∈M,s∈S

{hsnm

}/n0

)and

f0 = maxm∈M{f maxm

}. Let Nloc = {n ∈ N |ϒn ≥ 0} be

the set of users that execute their tasks locally, and Npof =

{n ∈ N |ϒn < 0} be the set of users that potentially offloadtheir tasks to the MEC servers. To find an offloading solu-tion for the remaining users, we temporarily assume thatan = 1,∀n ∈ Npof, ignore the fixed parts, and rewrite the

objective function in (13) as follows:

Z (A) =|Npof|∑n=1

M∑m=1

anm

Znm︷︸︸︷

λtnαn+λenpnαnζ

−1n

S∑s=1

asnmBs log2(1+0snm

)+ λtnβnfnm

.(15)

In the case of offloading, each user needs to select theserver for the remote computation and the subchannel for thetask offloading. From (15), if Znm is given for all n ∈ Npof,

m ∈ M, each user n is required to choose anm,m ∈ Msuch that Z (A) is minimized. Further, if anm is given forthe user n, it is required to select asnm, s ∈ S such that theoffloading rate Rnm is maximized, thus minimizing Z (A).Therefore, the computation offloading decision problem cannow be further divided into two subproblems: 1) which MECserver does a user offload to and 2) which subchannel doesa user utilize to offload the task. The first subproblem findsthe matching X = [anm]|Npof|×M between

∣∣Npof∣∣ users

and M servers so as to minimize the overhead Znm. For agiven X , the second subproblem is to determine the matchingY =

[asnm

]|Npof|×M×S

between users in Nm,∀m ∈ M, thatoffload their tasks to the sameMEC server and subchannels Sin order to maximize the achievable offloading rate Rsnm.

In what follows, we propose distributed processes basedon a one-to-many matching game to find which MEC servera user offloads to, i.e, the user association, and basedon a one-to-one matching game to find which subchan-nel a user utilizes to offload the task, i.e., the subchannelassignment, [29], [30]. In the first matching game, there aretwo types of players: users and servers. The strategy of usersis to select the best MEC server to maximize the benefit ofcomputation offloading and the strategy of servers is to eitheraccept or reject the offloading requests from users. From theconstraints, each user can offload the task to at most oneMEC server according to C3 and each MEC server can exe-cute multiple tasks from users according to C5. The two typesof players in the second matching game are users and sub-channels. The strategy of users is to select the most preferredsubchannel to maximize the offloading rate, and that of thesubchannels is to either accept or reject the bids from users.In this second game, each user is assumed to offload on atmost one subchannel and each subchannel can be matchedwith at most one offloading user. These two matching gamesare inspired by the stable marriage problem: there are two setsof matching players (one for men and one for women), whoseek to be married to each other and a matching comprises(man, woman) pairs [28]. Similarly in our work, a matchingconsists of (user, server) pairs and (user, subchannel) pairs inthe first many-to-one matching game and in the second one-to-one matching game, respectively.

1) MATCHING GAME FOR USER ASSOCIATIONDefinition 1: Given two disjoint sets of players, M and

Npof, a one-to-many matching function 9 : Npof → M is

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defined such that for all n ∈ Npof and m ∈M1) 9(m) ⊆ Npof and |9(m)| ≤ qm;2) 9(n) ⊆M and |9(n)| ∈ {0, 1};3) m = 9(n)↔ n = 9(m).

This matching game is defined by a tuple(M,Npof,q

),

with q = {qm|∀m ∈ M} being the MEC servers’ quotavector. The first condition implies that each MEC server mcan execute at most qm computation tasks as in C5, the secondcondition indicates that each user can offload the task to atmost one server, and the third condition implies that if theuser n is matched with the MEC server m, then the server mis also matched with the user n. The output of this game is auser association mapping 9 between users and servers.Next, we define φn,UA(m) as the preference of the user n

for MEC server m and φm,UA(n) as the preference of MECserver m for the user n. We also define �n,UA and �m,UAas the preference relations of the user n and MEC server m,respectively. The notation m1 �n,UA m2 implies that theuser n prefers the MEC server m1 to m2, i.e., φn,UA(m1) >φn,UA(m2). Similarly, the notation n1 �m,UA n2 impliesthat the MEC server m prefers the user n1 to n2 if thecomputation overhead with n1 is smaller than that with n2,i.e., φm,UA(n1) < φm,UA(n2).Preference of the mobile user: For user association prob-

lems in wireless multicell networks, the average SINR overall subchannels is considered as one of the most commoncriteria [16], [35], [36]. The preference value of the user nwhen it offloads the task In to the MEC server m is defined as

φn,UA(m) = ϕUAα−1n log2

(1+

∑s∈S

0snm

)+ εUAβ

−1n fnm,

(16)

where ϕUA and εUA are two weighted parameters andfnm = f max

m /qm. Intuitively, the user n prefers to offloadto the server that provides the higher offloading rate andcomputation resource. Since we allow each server to accom-modate more than one user, multiple users may coexistand share computation resources when they offload to thesame server. Similar to [31], we assume that the coexistingusers share the server’s CPU rate equally with effect fromtheir computation workloads, i.e., each user assumes a shareof f max

m /βnqm. Thus, the preference of user is based on theaggregation of the transmission delay (1st component) andremote processing time (2nd component).

Preference of the MEC server:We define the preferencevalue of the MEC server m when it executes the computationtask from the user n as the computation overhead as follows:

φm,UA(n) =λtnαn + λ

enpnαnζ

−1n

Rnm+λtnβn

fnm, (17)

where Rnm = Bs log2(1+

∑s∈S 0

snm). We say that the

MEC serverm prefers the user n1 to n2 (when n1 and n1 selectthe sameMEC serverm) if the user n2 has lower computationoverhead than the user n2.

Now, we propose a distributed algorithm to find the match-ing between users and MEC servers while minimizing thecomputation overhead. The specific details of the proposedalgorithm are given in Alg. 1, with new notations introducedbelow. First, the list of potential MEC servers for each user nis defined as Mn and initialized as M, and the sets ofunmatched users is Nunmatched = Npof, the set of rejectedand requested users for each MEC server m are defined asempty setsN rej

m andN reqm , respectively. In addition, each user

constructs its preference over all potential servers accordingto Eq. (16).

Algorithm 1 Users - MEC Servers Matching Algorithm1: Initialization

1) M, Npof, bUAnm = 0,∀n ∈ Npof.2) Construct the preference for all users in Npof

via (16) and the preference list Mn = M,∀n ∈Npof, set Nunmatched = Npof, and initialize the listof requested usersN req

m = ∅ and the list of rejectedusers N rej

m = ∅,∀m ∈M.2: Find a stable matching 9∗

3: while∑

m∈M∑

n∈NpofbUAnm 6= 0 do

4: for n = 1 to |Nunmatched| do5: Find m = argmaxm∈Mn

φn,UA(m).6: Send a request to the serverm by setting bUAnm = 1.7: end for8: for m = 1 to M do9: Update N req

m ← {n : bUAnm = 1,∀n ∈ Npof}.10: Construct the preference according to (17).11: if |N req

m | ≤ qm then12: Update Nm← N req

m .13: else14: repeat15: Accept n = argmax

n∈�m,UA

∑n∈N req

m

φm,UA(n).

16: Update Nm← Nm ∪ n.17: until |Nm| = qm18: end if19: Update N rej

m ← {Nreqm \ Nm}.

20: Update Mn← {Mn \ m},∀n ∈ N rejm .

21: end for22: Update Nunmatched ← Nunmatched ∩ {N

rej1 ∪ . . . ∪

N rejM }}.

23: end while24: End of the algorithm: output is a stable matching 9∗.

Next, each user n decides the best MEC server, which hasthe largest preference among Mn, by using its preferencerelation in line 5 and sends a bit request bUAnm to the MECserver m. The bid function bUAnm is 1 if the user m offloadsto the MEC server m and 0 otherwise (line 6).After bidding of all users, each MEC server collects the

bid requests from users and updates the list of requestedusers (line 9) and constructs the preference over all requestedusers via Eq. (17). The MEC server m is able to accept all

75874 VOLUME 6, 2018


of the requested users if the number of requested users issmaller than its quota qm (lines 12), otherwise it will select qmamong

∣∣N reqm∣∣ requested users (lines 15 and 16). Unmatched

users are then inserted into the set of rejected users of eachMEC server. Meanwhile, each MEC server is removed fromthe preference list of its rejected users. Finally, the list ofunmatched users is updated (line 22). Once there is no furtherrequested user (condition checking at line 3), the algorithmstops.

If a stable matching does not exist, it is computation-ally difficult to find the solution. Fortunately, the outcomeof Alg. 1 is a stable matching 9∗. To explain how thematching Alg. 1 achieves a stable matching, we presentthe definitions of a blocking pair and a stable matching inDefinitions 2 and 3, respectively [30], [37].Definition 2 (Blocking Pair): The pair (m0, n0) is a block-

ing pair for the matching 9, only if m0 �n,UA m, m ∈ 9(n0)and n0 �m,UA n, n ∈ 9(m0), for m0 /∈ 9(n0) and n0 /∈

9(m0). In other words, there exists a partnership (m0, n0)such that m0 and n0 are not matched with each other underthe current matching 9 but prefer to be matched with eachother.Definition 3 (Stable Matching): A matching 9 is said to

be stable if it admits no blocking pair.Theorem 1: The matching9∗ produced by Alg. 1 is stable

and ensures a local optimal solution to the CO problem.Proof: For a given transmit power allocation, computa-

tion resource allocation, and subchannel assignment, the pref-erence of each user and MEC server is fixed. Therefore,Alg. 1 is known as the deferred acceptance algorithm inthe two-sided matching problem between users and MECservers, which guarantees a stable matching [29]. The firstpart is proved.

At each iteration t , the outcome of Alg. 1 maps to a user-server association X (t), which captures to the objective

Z(X (t)

)=

|Npof|∑n=1

M∑m=1

x(t)nm

(λtnαn + λ

enpnαnζ

−1n

Rnm+λtnβn

fnm

).

Assume there exists a blocking pair (m0, n0) at iteration tsuch that the preference of the MEC server m and user ncan be improved when (m0, n0) is added to the currentmatching 9. Accordingly, Z (X (t)) > Z (X (t+1)), i.e., thecomputation overhead is reduced. According to Definition 3,there is no blocking pair at the final matching. As a result,the matching algorithm converges to a local optimal solutionto the CO problem. The second part is proved. �

2) MATCHING GAME FOR SUBCHANNEL ASSIGNMENTAfter determining the user association mapping 9 or Nm,

m ∈M, the one-to-one matching game is defined to find thesubchannel assignment.2 It is shown in (15) that for a givenUA mapping (i.e., anm is given for all users) each user tries

2Here, we omit the subscript of MEC server m and consider the matchingfor the users in Nm and the set of subchannel S.

to maximize its transmission rate to the associated server inorder tominimize the computation overhead Z(A). Therefore,the subchannel assignment problem for each user can bedetermined by solving the following problem:

maxY

S∑s=1

[Rsnm = Bs log2

(1+ 0snm

)]s.t. asnm = {0, 1} , ∀n ∈ Nm, s ∈ S∑

s∈Sasnm ≤ 1, ∀n ∈ Nm∑

n∈Nmasnm ≤ 1, s ∈ S

psn = pmaxn /S, ∀n ∈ Nm. (18)

Definition 4: Given two disjoint sets of players,Nm andS,a one-to-one matching function� : Nm→ S is defined suchthat for all n ∈ Nm and s ∈ S1) �(s) ⊆ Nm and |�(s)| ∈ {0, 1};2) �(n) ⊆ S and |�(n)| ∈ {0, 1};3) n = �(s)↔ s = �(n).

The first two conditions ensure that each user can utilize atmost one subchannel and a subchannel is assigned to at mostone user and condition 3 implies that if the user n is matchedwith subchannel s, then subchannel s is also matched withthe user n. The outcome of this one-to-one matching gameis the association mapping � between the set of users Nmand the set of subchannels S. Similar to the matching def-inition for user association, we define φn,CA(s) as the pref-erence of the user n for subchannel s and φs,CA(n) as thepreference of subchannel s for the user n. The notations1 �n,CA s2 denotes that the user n prefers subchannels1 to s2, i.e., φn,CA(s1) > φn,CA(s1), and the notationn1 �s,CA n2 indicates that subchannel s prefers user n1 to n2,i.e., φs,CA(n1) > φs,CA(n2).Preference of the mobile user: After determining the

MEC server selection, the user m achieves the followingpreference when it accesses to the subchannel s

φn,CA(s) = Rsnm. (19)

The preference in (19) implies that 1) subchannel selectionof a user only affects the achievable offloading rate, which inturn determines the offloading time, as illustrated in (5), and2) each user prefers to offload over the subchannel that offersa higher offloading rate.Preference of the MEC server for subchannels: The

preference of the MEC server m on subchannel s to bematched with the user n can be written as

φs,CA(n) = ϕCARsnm −∑

m′∈M,m′ 6=mδsm′g

snm′p

sn, (20)

where ϕCA and δsm are two weighted coefficients. The pref-erence in (20) implies that the SeNB m assigns the sub-channel s to the user n so as to maximize the achievableoffloading rate and minimize the aggregated interference toother SeNBs.

A distributed algorithm is designed to allocate subchannelsof an SeNB to its associated users. The specific details ofthe algorithm are summarized in Alg. 2. First, we obtain the

VOLUME 6, 2018 75875


sets of users Nm from Alg. 1 and the subchannels S, andinitialize the set of unmatched users Nunmatched, the set ofpotential subchannels for each user Sn, the lists of requestedusers N req

s and rejected users N rejs for each subchannel s.

Each user n also constructs its preference over all potentialsubchannels Sn (step 3 in Initialization). Next, each user nselects the best subchannel s (line 5) and sends an accessrequest for subchannel s to SeNB m (line 6). Here, the bidvalue bCAns is set to 1 if the user n bids for the subchannel sand 0 otherwise. At SeNB m, the list of requested users toeach subchannel s is updated (line 9). Then, the MEC serverm selects the best user among the

∣∣N reqs∣∣ requested users for

each subchannel s (line 11) and assigns the subchannel s tothat user (line 12). After that, the list of unmatched usersfor each subchannel s, N rej

s , is updated (line 13) and eachsubchannel s is removed from the list of potential subchannelsof its rejected usersN rej

s (line 14). Based on the list of rejectedusers for all subchannels, the list of unmatched users is alsoupdated (line 16). The algorithm stops if there is no furtherbidding between users Nm and subchannels S (line 3).

Algorithm 2 Users - Subchannels Matching Algorithm1: Initialization

1) Nm, S, bCAns = 0,∀n ∈ Nm.2) Set Nunmatched = Nm, Sn = S,∀n ∈ Nm, the list

of requested usersN reqs = ∅ and the list of rejected

users N rejs = ∅,∀s ∈ S.

3) Construct the preference for all users in Nmvia (19).

2: Find a stable matching �∗

3: while∑s∈S

∑n∈Nm

bUAns 6= 0 do

4: for n = 1 to |Nunmatched| do5: Find s = argmaxs∈Sn φn,CA(s).6: Send a request to the serverm by setting bCAns = 1.7: end for8: for s = 1 to S do9: Update N req

s ← {n : bCAns = 1,∀n ∈ Nm}.10: Construct the preference via (20).11: Find n = argmaxn∈N req

sφs,CA(n).

12: Assign the subchannel s to the user n.13: Update N rej

s ← {Nreqs \ n}.

14: Update Sn← {Sn \ s},∀n ∈ N rejs .

15: end for16: Update Nunmatched ← Nunmatched ∩ {N

rej1 ∪ . . . ∪

N rejS }}.

17: end while18: End of the algorithm: outcome is a stable matching�∗.

Theorem 2: The matching�∗ generated by Alg. 2 is stableand can achieve a local maximum of the problem (18).

Proof: The definitions of a blocking pair and sta-ble matching for the matching problem between users andsubchannels are similar to those in Definition 2 and 3,respectively. For a given power allocation, computation

resource allocation, and association matching9∗, the match-ing in Alg. 2 has the nature of deferred acceptance. Thus,a stable matching �∗ can be found by Alg. 2. Note that theoutcome of Alg. 2 at each iteration t maps to a subchannelassignment Y (t) and the objective for a Y (t) is Rn(Y (t)) =∑S

s=1 Rsnm(Y

(t)). Moreover, the matching at iteration (t + 1)guarantees that Rn(Y (t)) ≤ Rn(Y (t+1)), i.e., the objectiveis monotonically improved during the matching process,thus reducing the computation overhead. Consequently,Theorem 2 is proved. �

C. JOINT CCRA SUBPROBLEMFor a given A, the objective function of the JCCRA subprob-lem can be rewritten as

minP,F

∑m∈M

∑n∈Nm

λtnαn + λenunpn

Rnm+

∑m∈M

∑n∈Nm

λtnβn

fnm

,(21)

where un = αn (ζn)−1. Observe from (21) that the first

term is for the transmit power allocation of the users andthe second term is for the computation resource allocationof the MEC servers. In addition, the constraints C1, C5, andC6 are decoupled in P and F. Therefore, it is possible tofurther decompose the JCCRA subproblem into two sub-problems of P and F. The two following subsections aredevoted to the optimization of transmit power of the usersand the computation resource allocation of the MEC servers,respectively.

1) TRANSMIT POWER ALLOCATION OF MOBILE USERSThe optimization of the transmit power P can be written as

minP

∑(m,n)∈Gs

λtnαn + λenunp

sn

log2(1+ psnhsnm

n0+I snm

)s.t. 0 < psn ≤ p

maxn , ∀n ∈ Gs, (22)

where Gs = {(m, n) | n ∈ 9(m), n = �(s),∀ ∈ Noff,

m ∈ M, s ∈ S}. In (22), we only consider users thatoffload to different MEC servers but on the same subchannel.Obviously, (22) is a non-linear fractional and non-convexproblem due to the sum-of-ratios form of the objectivefunction and the existence of inter-cell interference I snm =∑(m′,n′)∈Gs,n′ 6=n p

sn′h

sn′m among users in Gs. One poten-

tial approach to the sum-of-ratios problem (22) relies onits transformation into a parametric convex programmingproblem [17]. However, we find the approximate upperbound3 of I snm such that (22) can be decomposed into indi-vidual subproblems for different offloading users.

Suppose that the transmit power of the user n is obtainedby solving the following problem:

minpsn

λtnαnB−1s + λ

enunB

−1s psn

log2(1+ psnhsnm

n0+Is,0nm

)s.t. 0 < psn ≤ p

maxn , (23)

3Upper bound of I snm is due to the minimization problem (22).

75876 VOLUME 6, 2018


where I s,0nm =∑(m′,n′)∈Gs,n′ 6=n p

maxn′ hsn′m. The problem (23)

is still not easy to solve due to the fractional form of theobjective function. In the following however, we show thatthe objective function of (23) is quasi-convex and the solutionto (23) can be achieved using a bisection algorithm.Theorem 3: The objective function of (23) is quasiconvex.Proof: Let ηn(psn) =

λtnαnB−1s +λ

enunB

−1s psn

log2

(1+ psnh

snm

n0+Is,0nm

) , which is the

ratio of a linear function and a concave function, and itssublevel sets Sa = {psn ∈

(0 pmax

n]| ηn(psn) ≤ a},∀a ∈ R+.

The set Sa can be equally expressed as

Sa = {psn ∈(0 pmax

n]| λtnαnB

−1s + λ

enunB

−1s psn

− a log2

(1+

psnhsnm

n0 + Is,0nm

)≤ 0},∀a ∈ R+.

Let fn(psn) = λtnαnB

−1s + λ

enunB

−1s psn − a log2

(1+ psnh

snm

n0+Is,0nm

).

According to [38], Sa is a convex set if for any ρ1, ρ2 ∈ Saand any θ with 0 ≤ θ ≤ 1, we have

θρ1 + (1− θ) ρ2 ∈ Sa. (24)

The condition (24) holds when fn (θρ1 + (1− θ) ρ2) ≤ 0.Actually, fn(psn) is a convex function due to the subtractionof a linear function and a concave function. By definition,fn (θρ1 + (1− θ) ρ2) ≤ θ fn(ρ1) + (1− θ) fn(ρ2). Due toρ1, ρ2 ∈ Sa, we have fn(ρ1) ≤ 0 and fn(ρ2) ≤ 0. Therefore,the condition (24) holds, Sa is a convex set, and then ηn(psn)is a quasiconvex function. �One approach to quasiconvex optimization is a bisec-

tion algorithm, which solves a convex feasibility problemat each step [38]. However, for further reduction of com-plexity, we use the approach proposed in [15] to solve thequasiconvex problem (23). The basic idea is that the optimalsolution ps,∗n either lies at the border of the constraint or sat-isfies the constraint ∂ηn

(ps,∗n

)/∂psn = 0. We have

η′n(psn)= φn(psn)/

(log2

(1+ psnh

snm/

(n0 + I s,0nm

)))2, where

φn(psn) = λenunB

−1s log2

(1+

psnhsnm

n0 + Is,0nm

)−hsnmln 2

λtnαnB−1s + λ

enunB

−1s psn

n0 + Is,0nm + psnhsnm

. (25)

Moreover, the first-order derivative of (25) is expressed as

φ′n(psn) =

1ln 2

λtnαnB−1s + λ

enunB

−1s psn(

n0+Is,0nm

hsnm+ psn

)2 . (26)

From (25) and (26), we have φn(0) < 0 and φ′n(psn)> 0,

∀psn ∈(0 pmax

n], i.e., φn(·) is a monotonically increas-

ing function and diminishes at psn = 0. Therefore, itera-tively checking the condition φn(psn) ≤ 0, we can designan efficient bisection method as in Alg. 3. In each step,the interval is bisected, i.e., ps,tn =

(ps,un + p

s,ln)/2; therefore,

the number of iterations required for Alg. 3 to terminateis dlog2

(ps,un − p

s,ln)/εe.

Algorithm 3 Bisection Method for the QuasiconvexOptimization Problem1: Initialization2: Set the tolerance ε, ps,ln = 0, and ps,un = pmax

n .3: Compute φn

(ps,un

).

4: Find the optimal solution ps,∗n5: if φn

(ps,un

)≤ 0 then

6: ps,∗n = ps,un .7: else8: repeat9: Set ps,tn =

(ps,un + p

s,ln)/2.

10: if φn(ps,tn)≤ 0 then

11: ps,ln = ps,tn .12: else13: ps,un = ps,tn .14: end if15: until ps,un − p

s,ln ≤ ε

16: Set ps,∗n =(ps,un + p

s,ln)/2.

17: end if18: Output: the optimal solution ps,∗n .

Note that the output of Alg. 3 is the approximatesolution ps,∗n to the quasiconvex problem (23). After findingthe solution to (23) for all users in Gs, I snm can be approxi-mated as I snm =

∑n′∈Gs,n′ 6=n p

s,∗n′ h

sn′m. Replacing I

s,0nm in (23)

with I snm, we obtain approximation problems for the powerallocation of users. The transmit power of users is finallyachieved by solving the approximation problems via thebisection Alg. 3.

2) COMPUTATION RESOURCE ALLOCATIONOF MEC SERVERSThe computation resource allocationF is determined by solv-ing the following optimization problem (OPT-CRA):

minF

∑m∈M

∑n∈Nm

λtnβn

fnm

s.t. C6, C7. (27)

The problem (27) can be decomposed into M individualproblems, corresponding to M MEC servers. However, evenwith (27), we show that the optimal computation allocationof a single MEC server merely depends on the set of itsassociated users.Theorem 4: The OPT-CRA problem is a convex problem.Proof: It is clear that the feasible solution set of the

OPT-CRA is convex. The remaining task is to show theconvexity of the objective function. We have the followingderivatives

∂2

∂f 2nm

∑m∈M

∑n∈Nm

λtnβn

fnm

= 2λtnβnf 3nm

,

∀n ∈ Noff, m ∈M∂2

∂fnm∂fkj

∑m∈M

∑n∈Nm

λtnβn

fnm

= 0, ∀(m,m) 6= (k, j).

VOLUME 6, 2018 75877


Let ∇2gl(F) be the Hessian matrix. Then, with all v ∈RNoff , vT∇2gl(F)v =

∑n∈Noff

2v2nλtnβn/f

3nm ≥ 0, where the

equality happens only if λtn = 0, i.e., the user n is with anenergy-hungry application. Therefore, the Hessian matrix isa positive semidefinite matrix. We conclude that OPT-CRAis a convex optimization problem [38]. �Since OPT-CRA is a convex problem, the optimal solution

can be optimally achieved via the KKT optimalityconditions [39], [40]. In particular, the optimal computationresource for user n at MEC server m is calculated as follows:

f ∗nm =f maxm

√λtnβn∑

n∈Nm

√λtnβn

. (28)

Remark 1: It is revealed from (28) that the computationresource is determined by the weighted parameter λtn andcomputation workloads βn of all users. Here, λtn can beinterpreted as the importance level of computational time ofthe user n. Specifically, if all users have the same computationworkload, i.e., βn = βk ,∀n 6= k, n, k ∈ Nm, the larger thevalue of λtn is, the more the computation resource should beassigned to the user n by the corresponding MEC server m inorder to minimize the processing time.

D. ALGORITHM FOR JCORA PROBLEMIn this subsection, we propose a joint framework to findthe optimal solution to the underlying problem (7). Thedetails of the proposed algorithm are summarized in Alg. 4,which is referred to as JCORAMS (JCORA Multi-Server).In general, the proposed algorithm has three phases: thepre-computation offloading decision, computation offload-ing and resource allocation, and post-computation offload-ing decision. The purpose of the first phase is to filterout users who cannot benefit from computation offloading,i.e., those users who should execute their tasks locally, andto reduce the input dimension for the second phase, i.e., non-offloading users are not taken into account during the secondphase.

The second phase is further divided into four steps: user-server association, subchannel allocation, transmit powercontrol, and computation resource allocation.• User-server association: The users and MEC serversjoin a one-to-many matching via Alg. 1. The preferenceof a user over potential servers and the preference ofa server over the |Npof| users are calculated accordingto (16) and (17), respectively. The optimal user asso-ciation is obtained via Alg. 1, where a user sends thecomputation offloading proposal to the most preferredserver and a server accepts a number of preferred usersbased on its quota. Alg. 1 stops when every user iseither accepted by one server or rejected by all preferredservers.

• Subchannel allocation: After all users know their asso-ciated MEC servers, users offloading to the same MECserver join a one-to-one matching game. Each user inNm calculates the preference over its preferred subchan-nels according to (19) and the MEC server m computes

Algorithm 4 JCORAMS Algorithm1: Input:M, N , S .2: repeat3: Pre-computation offloading decision: each user n

computes ϒn and checks the offloading condi-tion (14)

to determine the offloading decision.4: Computation offloading and resource allocation5: Matching between users and MEC servers6: Each user calculates its preference via (16).7: Each server constructs its preference via (17).8: Obtain the optimal matching 9∗ via Alg. 1.9: Matching for subchannel allocation in SeNBs m ∈

M10: Each user has its preference over S subchannels.11: Each subchannel gets its preference over Nm users.12: Obtain the optimal matching �∗ via Alg. 2.13: Transmit power allocation of users in Gs,∀s ∈ S14: Each user in Gs finds ps,∗n and I snm.15: Solve (23) with I s,0nm = I snm for the user n.16: Computation resource allocation at MEC servers17: Computation resource is allocated via (28).18: Post-computation offloading decision19: For n ∈ Npof,m ∈ 9∗(n), s ∈ �∗(n), compute ϒ∗n

ϒ∗n =λtnαn + λ

enp∗nαnζ

−1n

Rs∗

nm∗+λtnβn

fnm∗− Z ln. (29)

20: Each user checks the offloading decision ϒ∗n > 0to

decide to offload (an = 1) or not offload (an = 0).21: MEC servers and users update the preference lists for

the next iteration.22: until ϒ∗n ≤ 0,∀n ∈ Npof for two consecutive

times or asnm = 0,∀n ∈ N ,m ∈M, s ∈ S.23: Output: the optimal solution

(A∗,P∗,F∗

).

its preference on all subchannels over its associatedusers according to (20). A user sends the proposalto the most preferred subchannel and a SeNB assignsa subchannel to the most preferred user, who hasthe highest preference among requested users, andrejects the proposals of other users on that subchannel.Alg. 2 terminates when there is no bidding between usersand subchannels.

• Transmit power control: Once two matching algorithmsfor user association and subchannel allocation terminate,the transmit power of offloading users is allocated. Notethat there may be S groups Gs and the transmit powerof users in the group Gs is achieved by solving the indi-vidual problem (23). The approximate transmit powerof a user in Gs is found via Alg. 3 by fixing the inter-cell interference at the maximum transmit power of theother users. After that, the inter-cell interference of userscan be well approximated, and these approximation

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problems are solved to find the transmit power of usersin Gs.

• Computation resource allocation: Computation resourceallocation at MEC servers is executed when Alg. 2 ter-minates. Each MEC server m allocates the computationresources to its associated users inNm according to (28).

The third phase acts as the second filter since we assumethat all of the users in Npof offload their tasks to theMEC servers. After the second phase in each iteration, it isnecessary to determine whether or not the users benefit fromcomputation offloadingwith resource allocation from the sec-ond phase. There are two scenarios for a user to not offloadthe computation task. The first is that the user is not acceptedby any server and/or subchannel and the other is that theuser does not benefit from computation offloading with thepreferred server and subchannel. In these cases, rejected usersin the previous iterations will be considered as new users inthe next iterations. A user will not join the next matchingsin the UA and CA steps if it is rejected in the previous itera-tions. Moreover, users that are served by an MEC server andbenefit from computation offloading in the previous iterationswill not be involved in the next matchings. However, theCCRA subproblem will consider both previously acceptedusers and currently proposed users.

To start a new iteration, at the end of the previous iteration,each user is required to update its potential servers and sub-channels and calculate the preferences for the next matchings.Similarly, each server is required to remove the rejected usersfrom the preference list, update the remaining quota and sub-channel availability, and calculate the preferences for the nextmatching processes. The system-wide computation overheaddecreases as more iterations are processed. The proposedalgorithm converges and terminates when either the matchingof two consecutive iterations remains unchanged or all usersdo not benefit from computation offloading, thus executingtheir tasks locally (line 22).

E. CONVERGENCE AND STABILITYIn order to analyze the convergence and stability of theproposed algorithm, let Gs, s ∈ S denote the group formedby Alg. 4 at the end of each iteration t and introduce thedefinition of group stable [16], [35]. Then, we show thatmatching games have group stability and the proposed algo-rithm converges to above group-stable points.Definition 5 (Group Stable): The group Gs,∀s ∈ S is

blocked by another groupG′s′ if two following conditions hold:1) No user inside Gs can leave it to join G′s′ .2) No user inside G′s′ can leave it to join Gs,

The group Gs is said to be group stable if it is not blocked byany group. In addition, matchings in the proposed algorithmare stable if and only if all groupsGs,∀s ∈ S are group stable.We prove that the group Gs created by Alg. 4 at the

end of an iteration t is group stable by contradiction. Nowassume that there is a user n inside Gs can leave it to joinanother group G′s′ . This implies that Gs formed at the end of

the CA phase is not stable. However, according to Theorem 2,Gs is stable and there is no another matching G′s′ that canprovide a higher preference for user n, i.e., φms,CA(n) >

φm′

s′,CA(n). Therefore, no user inside Gs can leave it to joinanother group. Similarly, assume that there is a user n′ outsideGs can join it. We consider two scenarios for this condition.First, if

(n, n′

)∈ Nm, n′ �ms,CA n. Nevertheless, since Gs is

stable at the CA phase, n ∈ �m(s) and the server m assignsthe subchannel s to n instead of n′. Thus the user n′ cannotjoin Gs. Second, if n ∈ Nm, n′ ∈ Nm′ , and m 6= m′, we haven ∈ �m(s) and n′ ∈ �m′ (s). Then, given that the user n′ haspositive computation offloading gain, the post-computationoffloading phase will assign n′ toNm′ and add

(n′,m′

)to Gs.

However, because Gs is stable at the CA phase, the post-computation offloading phase will not accept n′ to join Gs.For those two scenarios, no user outside Gs can join it. As aconsequence, we can conclude that each group Gs, s ∈ S isnot blocked by any other groups, thus making the matchingsof the proposed algorithms are stable.Theorem 5: Matchings from the UA and CA games are all

stable in each iteration of the proposed Alg. 4.Proof: The proof is similar to that in [16, Appendix A].

It is therefore omitted. �Given the stable outcomes from the UA and CA match-

ing games in each iteration, transmit power and computa-tion resource for accepted users are optimized. Then, usersare required to check the offloading conditions to decideto offload or not. This post-computation offloading pro-cess in the third phase of Alg. 4 can be considered as thematching between subchannels and pairs of users-servers.To observe the optimality property of this matching, we real-ize the definition ofweak Pareto optimality (PO) [41]. Denoteby Z (G) the system-wide computation overhead obtained bythe matching G. The matching G is weak PO if there is noother matching G′ with Z

(G′)� Z (G), which is strict for

one user [42].Theorem 6: The post-computation offloading phase

achieves a weak PO solution for the CCRA subproblem ineach iteration.

Proof: We consider a matching G obtained by the post-computation offloading phase in any iteration t and assumethat there is a unstable matching G′ that is better than G,i.e., Z

(G′)< Z (G). Since G′ is unstable, there exists at least

one blocking pair. Let the matching between the subchannels and the user-server (n,m) be a blocking pair. The reasonsbehind instability of G′ are:1) Zn > Z ln, an = 1, n ∈ 9(m), and n ∈ �m(s),2) Zn ≤ Z ln, an = 0, n ∈ 9(m), and n ∈ �m(s).

For case 1, the post-computation offloading phase will con-struct a new stable matching G by setting an = 0, i.e., the usern chooses to execute the task locally instead of offloading thecomputation task to the MEC server over the subchannel s.This decreases the computation overhead of the user n fromZ rn to Z ln (i.e., a quantity of

(Z rn − Z

ln)> 0), thus resulting

in Z (G) < Z(G′)since the computation overhead of other

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users is left unchanged in G′. For case 2, the post-computationoffloading phase will construct a new stable matching G byremoving the user n from the set of local users (an = 0) andthen assigning to the set of offloading users with the server m(an = 1, n ∈ Nm). Since for the user n, the local executionwith negative computation offloading gain is replaced by theremote execution with positive computation offloading gain,removing the user n from the set of local users can decreasethe system-wide computation overhead or Z (G) < Z (G′).

Hence, under both cases, the unstable matching G′ isreplaced by the new stable matching G. Accordingly, thereis no unstable matching G′ that can lower the system-widecomputation overhead. Based on the definition of weak PO,we can conclude that the post-computation offloading phaseproduces a stable outcome and a weak PO solution for theCCRA subproblem in each iteration. �Theorem 7: The JCORAMS algorithm outputs a group

stable Gs after a finite number of iterations and is guaranteedto converge.

Proof: The numbers of preference relations of the users,MEC servers, and subchannels, i.e., �n,UA, �m,UA, �n,CA,and �s,CA, in each iteration are finite since the numbers ofusers, MEC servers, and subchannels are finite. According toTheorems 5 and 6, matchings generated by the JCORAMSalgorithm in each iteration are proved to be all stable. Addi-tionally, since only rejected users in the previous iterationsare processed in the next iterations and no user is rejectedby the same server and subchannel more than once in theUA and CA phases, respectively, the number of preferencerelations reduces after each iteration. The gradual finitenessof the preference relations ensures the convergence of theproposed algorithm after a finite number of iterations. �

F. COMPLEXITY ANALYSISThe optimal offloading decision can be obtained by theexhaustive search. All possible combinations of users,servers, and subchannels are searched and the one that min-imizes the system-wide computation overhead is selectedas the optimal solution. The exhaustive search for possiblecombinations has a complexity O((MS)N ). For each com-bination, the bisection method is run to find the approx-imated interference for each user and then to find thesolutions for the approximation problems. Here, the com-plexity of achieving the transmit power for all users isO(2N log2

(pmaxn)/ε). The complexity of allocating com-

putation resources for users at M servers is O (M). Sinceeach user needs to check the offloading condition anddecides to offload if it benefits from computation offloading,the complexity of finding the computation offloading deci-sions is O (N ). Therefore, the complexity of the exhaustivesearch is O((MS)N

(M + N + 2N log2

(pmaxn)/ε)). It can be

seen that the complexity of the exhaustive search increasesexponentially with N , which would not be affordable forMEC systems with a massive number of users.

To analyze the computational complexity of the proposedalgorithm, we consider the worst case when the preferences

of all users for all servers as well as the preferences ofusers offloading to the same server for all subchannels arethe same. At the end of each iteration, the preference rela-tions used in the next iteration are updated. The computa-tional complexities to sort the preference list of a user and aserver using a standard sorting algorithm are O (M log (M))and O (N log (N )), respectively. Similarly, the complexityto sort the preferences for all users Nm and subchannels isO (NmS log (NmS)). The total lengths of the input preferencesin the UA phase and CA phase are, respectively, 2NM and2NmS. From [43], it can be shown that the stable matchingsin the UA and CA phases can be obtained with the compu-tational complexities of O (NM) and O (NmS) respectively,which are both linear in the input size. Given the stable out-puts from two matching games (i.e., computation offloadingdecisions are given), as aforementioned, the complexities ofthe JCCRA phase and the post-computation offloading phaseare also linear. Since the proposed algorithm terminates aftera finite number of iterations, the complexity of the proposedalgorithm is finally linear in the numbers of users, servers, andsubchannels. Compared with the exhaustive search, the pro-posed algorithm has a reasonable complexity and appearsefficient in dense MEC systems.

IV. NUMERICAL SIMULATIONA. SIMULATION SETTINGSIn order to evaluate our proposed algorithm, we use thefollowing simulation settings for all simulations. We firstconsider the scenario where 9 SeNBs are randomly deployedin a small indoor area of 250×250 m2 to serve 36 users. EachSeNB consists of 4 subchannels and has a quota of 4 users(qm = 4,∀m ∈ M). The bandwidth of each subchannel isBs = 5 MHz, each user has the maximum transmit powerpmaxn = 100 mW, and n0 = −100 dBm. The path-lossmodel is −140.7 − 36.7 log10 (d), where d (kilometers) isthe distance from the user to the serving SeNB. For thecomputation task, we adopt the face recognition applicationin [10], [15], and [44], where the computation input data sizeis αn = 420 KB and the total required number of CPUcycles is βn = 1000 Megacycles. The CPU computationcapability f ln of the user n is randomly assigned from the set{0.5, 0.8, 1.0} GHz [9], [10] and the computation capabilityof each MEC server f max

m = 4.0 GHz, ∀m ∈ M. Theweighted parameters of the computational time and energyconsumption are both 0.5, i.e., λtn = λen = 0.5,∀n ∈ N .Finally, we set the values of ϕUA, εUA, ϕCA, and δsm (∀m ∈M, s ∈ S) to 8×106, 0.2, 1, and 0.1, respectively. For all theresults, each plot is the average of 100 channel realizationsand in each realization, the user and MEC server locationsare uniformly distributed randomly.

B. SIMULATION RESULTSIn the following, we present the performance of our proposedapproach compared with several representative benchmarkmethods. For existing frameworks, the following solutionsare considered:

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1) Local computing only: there is no computationoffloading. All users perform computations locally,i.e., an = 0,∀n ∈ N .

2) Offloading only: all users offload their computationtasks to the MEC servers i.e., an = 1,∀n ∈ N .This is achieved by running our algorithm withoutthe pre-computation offloading decision and post-computation offloading decision steps. Note that whenthe number of users exceeds the system capacity,some requested users are rejected by the algorithm,i.e., ∃n ∈ N |an = 0.

3) HODA [15]: the offloading decision, transmit power,and computation resources are determined in each cellindependently.

To allow fair comparison between algorithms for singleMEC server and multiple MEC servers, the final results,i.e., percentage of offloading users and system-wide com-putation overhead, do not take into account the inter-cellinterference among offloading users.

In the first experiment, we vary the number of users from10 to 50 with a step deviation of 4 and examine the per-centage of offloading users. From Fig. 3a, the percentageof offloading users is relatively high4 when the number ofuser is small. However, the percentage of offloading usersgradually decreases when the number of users increases.This is reasonable since 1) each user might have a highprobability to associate with its preferred MEC server andoffload its computation task over a good subchannel and 2)small intercell interference makes users profit more fromcomputation offloading. In addition, when the number ofusers keeps increasing, each user needs to compete withthe others for using radio and computation resources, thuslowering the probabilities for each user to connect with itspreferred MEC server and subchannel. Due to the limitednumber of MEC servers, number of subchannels in each cell,and quota of each SeNB, a portion of requested users mustbe rejected by the proposed algorithm and they must executetheir tasks locally.

We also examine the performances in terms of thepercentage of offloading users and system-wide computa-tion overhead of our proposed approach and three comparedframeworks. It is observed from Fig. 3a that the percentageof offloading users in the local computing only method is 0while that of the offloading only approach is 1 and starts todecrease as N becomes smaller and greater than 36, respec-tively. This is due to the quota of each SeNB, the number ofsubchannels of each cell, and the number ofMEC servers, andhence the system capacity in terms of the number of admittedoffloading users is limited by M × min{q, S}. Fig. 3b alsoreveals that the performance of the offloading only algorithmbecomes worse than that for the local computing only methodwhen N gets larger. This is due to competition among users

4The percentage of offloading users should be 1 when the number ofusers is relatively small. However, user and MEC server locations are bothrandomly generated in each simulation realization, so a user may not offloadits computation task due to bad channel connections with MEC servers.

FIGURE 3. Comparison of JCORAMS and three baseline frameworksunder different numbers of users. (a) Percentage of offloading users.(b) Total computation overhead.

for the limited radio and computation resources. Comparedwith three baseline schemes, i.e., offloading only, local only,and HODA, our proposed algorithm can achieve better per-formance in terms of the percentage of offloading users andyield a lower computation overhead.

The second experiment compares the performances ofour proposed algorithm with the existing frameworks underdifferent computation task profiles. It is shown in Figs. 4a-4bthat when the input data size α is large enough (1 MB inthis case), the computation overhead of the offloading onlymethod can reach that of the local computing only scheme.It is therefore better to offload less computation tasks asα increases, i.e., a computation task with small data size ismore preferable to computation offloading than one with highdata size. The reason for this is that by increasing the inputdata size, the time cost and energy cost for offloading com-putation tasks become higher, as seen from Eqs. (5) and (6).This observation agrees with the performance linesof JCORAMS and HODA, where fewer users benefit fromcomputation offloading and the system-wide computationoverhead increases as α increases. From Figs. 4c-4d, we canobserve that the percentage of offloading users and thesystem-wide computation overhead increase with the com-putation workload β. This is reasonable since both the localcompletion time and remote execution time increase asβ increases; however, the computation capability of a useris often limited and an MEC server can offer offloading userswith higher computation capability, i.e., users therefore bene-fit from computation offloading if their computation tasks areexecuted by the MEC servers. From the above reasons, it isbetter to offload a computation task with small input data sizeand large computation intensity rather than one with largeinput data size and small computation intensity. Obviously,the proposed algorithm achieves the better performance thanthe baseline solutions.

Next, by varying the weighted parameter of the computa-tional time λt from 0.1 to 0.9 with a deviation step of 0.1 andsetting the weighted parameter of the energy consumption λeto 1 − λt , we further explore the performance comparison

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FIGURE 4. Comparison of our proposed approach and three alternativesunder different computation task profiles.(a) Percentage of offloadingusers. (b) Total computation overhead. (c) Percentage of offloading users.(d) Total computation overhead.

between our proposed algorithm and existing ones. It is worthnoting that λt and λe are the same for all users; however,the extension to different λt and λe for different users doesnot affect the comparison among algorithms. Selecting a userwith fl = 0.8 GHz as an example, we have t l = 1000 ×106/0.8 × 109 = 1.25 (seconds) and E l = 5 × 10−27 ×1000 × 106 ×

(0.8× 109

)2= 3.20 (Joules). It is obvious

that for local computing, the energy consumption is nearlythree times larger than the completion time. As a result, withthe increment of λt , the system-wide computation overheadby the local only and offloading only schemes decreases andincreases, respectively, as observed from Fig. 5b. In addition,there are fewer users that tend to offload their computationtasks to the MEC servers due to the lower local computationoverhead, and the percentage of offloading users reduces,as shown in Fig. 5a. Here, the system-wide computationoverheads by JCORAMS and HODA still increase and onlystart to decline when λt is large enough. The main reasonfor this is the dominance of the offloading and executiontime (toff + texe) over the offloading energy consumption(Eoff). Therefore, selection of the weighted parameters playsan important role in the achieved performances. Again, ourproposed algorithm is superior to the baseline solutions.

In the fourth experiment, we discuss the impacts of themaximum transmit power pmax

n on the performances of theconsidered approaches. From the Fig. 6, it is seen that

FIGURE 5. Comparison of our proposed approach and three alternativeframeworks under different weighted parameters. (a) Percentage ofoffloading users. (b) Total computation overhead.

FIGURE 6. Comparison of our proposed approach and three existingframeworks under variant maximum transmit power. (a) Percentage ofoffloading users. (b) Total computation overhead.

when pmaxn increases, the percentage of offloading users and

system-wide computation overhead increases and decreases,respectively, and all become saturated when pmax

n is suffi-ciently large. For example, withN = 36 and pmax

n = 0.55 (W)for all users, the percentage of offloading users is 100%,i.e., the performances of our proposed and the offloadingonly schemes are the same, and the computation overheadis 17.9. This is due to the fact that increasing the offloadingrates makes the time and energy costs for offloading thecomputation tasks smaller and as a consequence, there aremore users that tend to offload their computation tasks to theMEC servers.

The final experiment represents the number of offloadingusers and system-wide computation overhead for the differentalgorithms when the maximum computation capability f0 ofMEC servers varies from 0.5 GHz to 4 GHz. It is shownin Fig. 7a that the percentage of offloading users monoton-ically increases with f0, and the increasing rate graduallydecreases, i.e., increasing f0 from 0.5 GHz to 1.0 GHz makesmore users benefit from computation offloading than thatfrom 1.0 GHz to 1.5 GHz. The reason is that when thecomputation capability of MEC servers is small, the exe-cution time is high and so the remote computation over-head becomes higher than the local computation overhead.

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FIGURE 7. Performance of the proposed algorithm under differentcomputation capabilities of MEC servers. (a) Percentage of offloadingusers. (b) Total computation overhead.

At the same time, because more users benefit from com-putation offloading, the system-wide computation overheaddecreases. In order to evaluate the optimality of the pro-posed algorithm, we compare JCORAMS with the hJTORAalgorithm proposed in [25], where at each step, MEC serverand subchannel selections are heuristically found by solvingall possible resource allocation problems, and the algorithmends when there is no feasible way to increase the objectivevalue. Observe from Fig. 7b that at f0 = 3.5 GHz theproposed algorithm generates the total computation overheadof 63.579, which is close to that of hJTORA with the gapof 8.86%. Fig. 7b also depicts that when the inter-cell interfer-ence is taken into consideration, offloading all computationtasks to the MEC servers is very inefficient. This is due tothe fact that (i) the locations of users and MEC servers arerandomly distributed in each simulation realization, so someusers may have very bad connections to the MEC servers,and (ii) when the inter-cell interference exists and becomesevere, the offloading rates of offloading users are relativelylow and the offloading time becomesmuch higher. In this sce-nario users, with the bad connections and severe interference,should locally handle their computations, while the other sendrequests to the MEC servers for computation offloading. As aresult, a joint optimization of offloading decision, resourceallocation, and interference management is highly needed toimprove the network performance, which is clearly demon-strated by the comparison between our proposed algorithmand hJTORA with the local and offloading only schemesin Fig. 7.

V. CONCLUSION AND FUTURE WORKIn this paper, we have proposed an optimization problemfor jointly determining the computation offloading decisionand allocating the transmit power of users and computationresources at the MEC servers. Our proposed framework isdifferent from existing ones in that 1) we have considered aHetNet with multiple MEC servers, where a user is possiblyunder the service coverage of multiple MEC servers and2) we have proposed a decentralized computation offloading

scheme based on the matching theory. The simulation resultsvalidated that the proposed algorithm can achieve better per-formances than three alternative frameworks. In addition,our proposed algorithm could perform close to that of thecentralized exhaustive search with the small optimality gap.

A joint framework of resource allocation and server selec-tion in collocation edge computing systems is currently underinvestigation of our ongoing work. Moreover, we will takeinto account the effects of computation offloading to the qual-ity of service of macrocell users. Finally, we will consider ofhierarchical MEC systems for differentiated applications ofusers where users with latency-sensitive applications offloadtheir tasks to the first tier at small cells while users withlatency-tolerant applications offload their tasks to the secondMEC server tier at macrocells.

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QUOC-VIET PHAM (M’18) received theB.S. degree in electronics and telecommu-nications engineering from the Hanoi Uni-versity of Science and Technology, Vietnam,in 2013, and the M.S. and Ph.D. degreesin telecommunications engineering from InjeUniversity, South Korea, in 2015 and 2017,respectively. From 2017 to 2018, he was a Post-Doctoral Researcher with Kyung Hee Univer-sity, South Korea. He is currently a Research

Professor with the ICT Convergence Center, Changwon National University,South Korea. His research interests include network optimization, mobileedge/cloud computing, and resource allocation for wireless networks. Hereceived the Best Ph.D. Thesis Award in Engineering from Inje Universityin 2017.

TUAN LEANH received the B.Eng. and M.Eng.degrees in electronic and telecommunication engi-neering from the Hanoi University of Technology,Vietnam, and the Ph.D degree from Kyung HeeUniversity, Seoul, South Korea, in 2007, 2010, and2017, respectively. He was a Member of Tech-nical Staff with the Network Operations Center,Vietnam Telecoms National Co., Vietnam Postsand Telecommunications Group from 2007 to2011. Since 2017, he has been a Post-Doctoral

Researcher with the Department of Computer Science and Engineering,KyungHeeUniversity. His research interests include queueing, optimization,control, and game theory to design, analyze, and optimize the resource allo-cation in communication networks, including cognitive radio network, wire-less network virtualization, URLLC, mobile-edge computing, and vehicularcommunications.

NGUYEN H. TRAN (S’10–M’11–SM’18)received the B.S. degree in electrical and computerengineering from the Ho ChiMinh City Universityof Technology in 2005 and the Ph.D. degree inelectrical and computer engineering from KyungHee University in 2011. Since 2012, he has beenan Assistant Professor with the Department ofComputer Science and Engineering, Kyung HeeUniversity. His research interest is applying ana-lytic techniques of optimization, game theory, and

stochastic modeling to cutting-edge applications, such as cloud and mobile-edge computing, datacenters, heterogeneous wireless networks, and big datafor networks. He received the Best KHU Thesis Award in engineeringin 2011 and several best paper awards, including the IEEE ICC 2016,the APNOMS 2016, and the IEEE ICCS 2016. He is an Editor of the IEEETRANSACTIONS ON GREEN COMMUNICATIONS AND NETWORKING.

75884 VOLUME 6, 2018


BANG JU PARK received the B.S. and M.S.degrees in electronic engineering from Kyung HeeUniversity, South Korea, in 1986 and 1988, respec-tively, and the Ph.D. degree in radio engineeringfrom the Korea Institute of Science and Technol-ogy, Kyung Hee University, South Korea, in 2006.From 1988 to 2012, he was a Science Reporterwith JoongAng Daily, South Korea. From 2008 to2012, he was a Visiting Professor with the Col-lege of Bio-Nano Engineering, Gachon University,

South Korea. In 2013, he moved to Gachon University, where he is currentlya Full Professor with the Department of Electronic Engineering and theDirector of the Institute of Bio and Mechatronics. His research interestsinclude future Internet and radio manipulation for biomedical instrumentalapplications.

CHOONG SEON HONG received the B.S.and M.S. degrees in electronic engineering fromKyung Hee University, Seoul, South Korea,in 1983 and 1985, respectively, and the Ph.D.degree from Keio University in 1997. In 1988,he joined Korea Telecom (KT), where he wasinvolved in broadband networks as a Member ofTechnical Staff. In 1993, he joined Keio Univer-sity, Japan. He was with the TelecommunicationsNetwork Laboratory, KT, as a Senior Member of

Technical Staff and the Director of Networking Research Team, until 1999.Since 1999, he has been a Professor with the Department of ComputerEngineering, Kyung Hee University. His research interests include futureInternet, ad hoc networks, network management, and network security. Heis a member of the ACM, the IEICE, the IPSJ, the KIISE, the KICS,the KIPS, and the OSIA. He has served as the General Chair, the TPCChair/Member, or an Organizing Committee Member for international con-ferences, such as NOMS, IM, APNOMS, E2EMON, CCNC, ADSN, ICPP,DIM, WISA, BcN, TINA, SAINT, and ICOIN. He is currently an AssociateEditor of the IEEE TRANSACTIONS ON NETWORK AND SERVICE MANAGEMENT,the International Journal of Network Management, and the IEEE JOURNALOF COMMUNICATIONS AND NETWORKS, and an Associate Technical Editor of theIEEE Communications Magazine.

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Decentralized Computation Offloading and Resource Allocation …networking.khu.ac.kr/layouts/net/publications/data/2018... · 2019-01-09 · Q.-V. Pham et al.: Decentralized Computation