1

A Multi-User Mobile Computation Offloading andTransmission Scheduling Mechanism for

Delay-Sensitive ApplicationsChangyan Yi, Member, IEEE , Jun Cai, Senior Member, IEEE , and Zhou Su, Member, IEEE

Abstract—In this paper, a mobile edge computing framework with multi-user computation offloading and transmission scheduling fordelay-sensitive applications is studied. In the considered model, computation tasks are generated randomly at mobile users along thetime. For each task, the mobile user can choose to either process it locally or offload it via the uplink transmission to the edge for cloudcomputing. To efficiently manage the system, the network regulator is required to employ a network-wide optimal scheme forcomputation offloading and transmission scheduling while guaranteeing that all mobile users would like to follow (as they may naturallybehave strategically for benefiting themselves). By considering tradeoffs between local and edge computing, wireless features andnoncooperative game interactions among mobile users, we formulate a mechanism design problem to jointly determine a computationoffloading scheme, a transmission scheduling discipline and a pricing rule. A queueing model is built to analytically describe thepacket-level network dynamics. Based on this, we propose a novel mechanism, which can maximize the network social welfare (i.e.,the network-wide performance), while achieving a game equilibrium among strategic mobile users. Theoretical and simulation resultsexamine the performance of our proposed mechanism, and demonstrate its superiority over the counterparts.

Index Terms—Mobile dge computing, computation offloading, transmission scheduling, delay sensitive, mechanism design, game.

F

1 INTRODUCTION

DUE to the increasing popularity of smart mobile de-vices (e.g., smart phones and tablets), a variety of

novel mobile applications, such as natural language pro-cessing, face recognition, interactive gaming, augmentedreality and healthcare monitoring, are developed and re-cently attracting great interests [1]. These mobile applica-tions commonly consume intensive computation resourcesand tremendously high energy, which prevent them frombeing executed locally by most resource-constrained mobiledevices. To overcome these computation and power limita-tions on mobile devices and at the same time achieve highnetwork efficiency, mobile cloud computing is envisionedas a promising paradigm, which allows mobile devices tofully/partially offload their computation tasks to resource-rich cloud infrastructures (such as Amazon EC2, GoogleCompute Engine and Microsoft Azure) [2]. However, sincetraditional public clouds are usually located far away frommobile devices/users, the data exchange via the wide areanetwork may result in a considerably long latency. Tofurther overcome this drawback, mobile edge computinghas been proposed as an alternative solution. By enablingcloud computing capabilities at the edges of ubiquitousradio access networks (e.g., 3G/4G macro-cell or small-cell base stations) that are close to mobile users, mobile

• C. Yi and J. Cai are with the Department of Electrical and ComputerEngineering, University of Manitoba, Winnipeg, MB, Canada, R3T 5V6.(E-mail: changyan.yi@umanitoba.ca, jun.cai@umanitoba.ca).

• Z. Su is with the School of Mechatronic Engineering and Automation,Shanghai University, China. (E-mail: zhousu@ieee.org).

edge computing can provide pervasive, prompt and agilecomputation services at anytime and anywhere [3], [4].

For mobile edge computing, offloading computationtasks must involve wireless communications between mo-bile users and the edge cloud, and thus its performancehighly depends on the wireless access efficiency [5]. Due toinherently limited radio resources [6], if the wireless accessamong multiple mobile users for computation offloading isnot well coordinated, the wireless network capacity may bequickly strained by the overwhelmed offloading tasks, lead-ing to low transmission efficiency (e.g., a long data trans-mission delay) and eventually resulting in dissatisfactionson mobile edge computing services. This prompted somerecent research efforts [7]–[10] in studying both computationoffloading and radio resource allocation for mobile edgecomputing. Nevertheless, there are still some critical issueswhich are closely related to practical implementations andhave not been well addressed:

i) In practice, mobile applications, such as live streamingapplications, may generate a stream of computationtasks, which arrive randomly at mobile users alongthe time [11], [12]. Such potential network uncertaintiesrequests a dynamic management of the system withlong-term performance guarantees.

ii) Naturally, different mobile users may run computation-intensive applications with different delay sensitivities.For instance, augmented reality services normally re-quire delays less than 100 ms, while 4K live videos cantolerate up to 500 ms delays [13]. Thus, a proper delay-sensitive scheduling mechanism should be adoptedto meet the heterogeneous quality-of-service (QoS) re-quirements of mobile users in computation offloading.

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iii) Most importantly, human/device intelligence allowsmobile users to behave strategically and selfishly, eventhough they are rational [14]–[16]. As a result, mobileusers may pursue different self-interests by selectingtheir own computation offloading strategies [17], [18].Obviously, such strategic behaviors may affect the wire-less traffic volume in computation offloading and theoverall system performance [19]. Since transmissionresources are limited and shared among mobile users,this naturally leads to a noncooperative competitions atuser ends [20] and motivates the need of consideringa game-theoretic decision making process [7], [21]–[23]. Therefore, in order to guarantee efficiency androbustness, mobile edge computing system should bedesigned for not only maximizing the overall networkperformance, but also regulating the strategic and non-cooperative interactions (i.e., the game behaviors) ofsmart mobile users in a desired manner, i.e., making theoutcome of the network-wide optimal solution achievean equilibrium such that all mobile users are satisfied.

However, addressing all these features raises new chal-lenges in the management of mobile edge computing sys-tems. Specifically,

a. It is very difficult to explicitly analyze the complicatedrelationship between dynamics of the control system andQoS of mobile users’ delay-sensitive computation tasks,and thus a packet-level queueing modeling is required.

b. Since both computation offloading decisions and wire-less transmission scheduling contribute to the overallperformance of mobile edge computing systems, theyhave to be jointly determined with the consideration oftheir mutual effects, which results in a complicated jointoptimization problem.

c. The designed mechanism should be capable of induc-ing mobile users to voluntarily participate in the sys-tem following the network-wide optimal management.However, by considering the fact that mobile users maystrategically deviate the optimal solution unilaterally forpotentially maximizing their own interests, this requestsus to integrate the regulation of mobile users’ noncoop-erative game behaviors into the network-wide optimiza-tion. Obviously, more constraints/requirements relatedto individuals have to be introduced and included, whichfurther increase the complexity of the original optimiza-tion problem.

To tackle these difficulties, in this paper, we proposea novel mechanism which can efficiently manage multi-user computation offloading and transmission schedulingfor delay-sensitive applications in mobile edge computing.In this work, we consider random arrivals of heterogeneousdelay-sensitive computation tasks at different mobile users.Upon receiving a task, the associated mobile user can strate-gically decide to either process it locally or offload it via theuplink cellular transmission to the edge (i.e., the wirelessaccess point) for cloud computing. Since offloading willincur delay costs and cloud service charges, while runningcomputations locally may consume a large amount of en-ergy, there always exists a tradeoff among energy consump-tion, delay performance and payment cost for mobile users.The packet level operation of such management frame-

work is first modeled as a dynamic priority queue, wherethe heterogeneity of mobile users in terms of their delaysensitivities in running different applications is taken intoaccount. Based on this formulated queueing model, we thendesign a mechanism to jointly determine the computationoffloading scheme, the transmission scheduling disciplineand the pricing rule. Theoretical analyses show that ourproposed mechanism can reach an equilibrium which notonly maximize the network social welfare1, but also preventindividual mobile users from deviating the network-wideoptimal solution unilaterally.

The main contributions of this paper are in the following.

• Joint computation offloading and wireless transmis-sion scheduling for edge computing with delay-sensitive applications are formulated as a mechanismdesign problem upon a dynamic queueing model.

• A delay-dependent prioritized transmission schedul-ing discipline is developed to minimize the total de-lay cost of all mobile users for any given computationoffloading decision.

• With the objective of maximizing the network socialwelfare, an optimal computation offloading schemeis derived through the convex optimization.

• An appropriate pricing rule is designed to make thenetwork-wide optimal computation offloading andtransmission scheduling scheme achieve an equilib-rium of the noncooperative strategy making processamong individual mobile users.

• Theoretical analyses and simulation results examinethe performance of our proposed mechanism, anddemonstrate its feasibility and superiority comparedto the counterparts.

The rest of this paper is organized as follows: Section2 gives a brief review of related works and highlights thenovelties of this work. Section 3 describes the consideredsystem model and the problem formulation. In Section 4, anefficient multi-user computation offloading and transmis-sion scheduling mechanism for delay-sensitive applicationsin mobile edge computing is proposed and analyzed. Sim-ulation results are presented in Section 5, followed by thediscussions of conclusion and future work in Section 6.

2 RELATED WORK

Mobile edge computing, as a key enabling technology forInternet-of-things (IoT) and 5G, has drawn a lot of researchattentions recently. For instance, Chen et al. in [23] formu-lated a decentralized computation offloading game, whereeach mobile user could choose to either fully offload itscomputing task to the edge cloud or process it locally forminimizing the overhead. In [24], Dinh et al. proposedan optimization approach for determining computation of-floading and CPU frequency of a mobile user with theobjective of minimizing both its task execution time andenergy consumption. These works focused on the compu-tation offloading process only, while ignoring the resourceallocation in wireless access networks.

1. From the network point of view, this is a common objective, as it isequivalent to studying how network resource can be fully utilized andoptimized to maximize the overall system performance.

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Some recent works have been dedicated in studyingjoint radio-and-computational resource management [8]–[10]. Specifically, Wang et al. in [8] presented a decentralizedmethod to jointly optimize computation offloading, spec-trum allocation and content caching in wireless cellular net-works with the mobile edge computing capability. Sardellittiet al. in [9] developed an iterative algorithm based on thesuccessive convex approximation for minimizing the totalenergy consumption of mobile users under delay and powerbudgets. You et al. in [10] introduced an optimal threshold-based policy for managing offloading data volumes andchannel access opportunities in time-division multiple ac-cess (TDMA) mobile edge computing systems. However,all these works employed a common assumption that thenetwork scenario was quasi-static, i.e., the set of computingtasks was assumed to remain unchanged regardless of thepotential temporal dynamics.

A few works have started to analyze mobile edge com-puting in dynamic network scenarios with random taskarrivals. For example, Kwak et al. in [12] investigated theenergy-delay tradeoff for mobile edge computing with vari-ous types of tasks that were generated randomly at the userend. The minimization of long-term average task executioncosts was discussed in [25] and [26], where the formeroptimized the computation offloading by using Markovdecision process (MDP) and the latter jointly controlledthe local CPU frequency, modulation scheme and data rateunder a semi-MDP framework. Besides, using queueing the-ory to analyze the performance of mobile edge computinghas been initiated in [27], [28]. Because of the complicatedtemporal correlation in dynamic operations, most of theseworks were restricted to a single-user case only.

Mechanism design integrating queueing scheduling is anovel technique for optimally managing dynamic systemswith multi-user interactions [29], [30]. Its recent applicationsinclude power allocations for wireless networks with dy-namic energy harvesting [31] and delay-sensitive transmis-sion managements in remote healthcare networks [32]–[34].However, mechanisms developed in these works cannot beapplied for solving the problem in this paper, because noneof them can capture the variable inputs of queueing systems(due to the potential computation offloading), and mostof them relied on the assumptions of specific distributions(e.g., exponential and poisson) in the queueing modeling.

In summary, different from all existing works in theliterature, this paper studies a joint optimization of com-putation offloading and transmission scheduling for mobileedge computing systems with delay-sensitive applicationsby considering generally distributed network dynamics andmultiple heterogeneous strategic mobile users.

3 SYSTEM MODEL AND PROBLEM FORMULATION

In this section, the network model of multi-user compu-tation offloading and transmission scheduling for mobileedge computing with delay-sensitive applications is firstdescribed. Then, utility functions and strategies of mobileusers are defined. After that, a mechanism design problem,which jointly considers computation offloading and trans-missions scheduling, is formulated. For convenience, Table1 lists some important notations used in this paper.

TABLE 1IMPORTANT NOTATIONS IN THIS PAPER

Symbol MeaningN set of mobile usersM number of cellular channelsλi arrival rate of computation tasks at mobile user iλlocali local processing rate of mobile user iλoffloadi computation offloading rate of mobile user iSi size of mobile user i’s computation tasksRi uplink transmission rate of mobile user iE[Ui] expected utility of mobile user iCi expected cost of mobile user iTi experienced delay of user i in offloading each task1/µi mean transmission time for one task of mobile user iρi offloading traffic intensity of mobile user iLi workload of mobile user i in uplink transmissionsL total workload in uplink transmissionsai(τ) amount of tasks in mobile user i’s buffer at time τWi(·) delay cost function of mobile user iVi(·) utility gain of user i in computation offloadingY (·) network operating costSW network social welfareβi computation offloading ratio of mobile user iπi service charge on mobile user iζ transmission scheduling disciplineQ(·) formulated queueing management system

3.1 Network Model

Consider a group of mobile users, denoted by N , who arerunning computation-intensive and delay-sensitive applica-tions2. Each mobile user has a stream of computation tasksthat are required to be completed as soon as possible. Thereexists a wireless access point through which mobile userscan offload their computation tasks to the edge cloud. Inpractice, the wireless access point can be a 3G/4G macro-cellor small-cell base station, which manages wireless transmis-sions for all associated mobile users over M homogeneouscellular channels3 by using OFDMA. Note that similar to[11], [36], we particularly focus on computation offloadingand uplink transmission scheduling for mobile users, andignore the communication overhead for the edge cloud tosend computation outcomes back to mobile users. This isbecause the size of computation outcomes is commonlynegligible compared to that of computation tasks (eachof which may consist of various mobile system settings,programs and input parameters).

For each mobile user i, ∀i ∈ N , let the dynamic arrivalof its computation tasks be a generally distributed randomprocess with an average rate λi, and consider that eachtask has a random size of Si with a finite mean E[Si]and requires a random number of CPU cycles Zi with afinite mean E[Zi]. Upon receiving each computation task,the mobile user can choose to either process it locally or

2. Delay-sensitive applications can be described by three potentialfeatures: i) there is a maximum tolerable delay requirement for each ofits computation task; ii) its computation task suffers a cost (e.g., a loss ofvalue) which increases with the delay before it has been completed (i.e.,a delay cost); and iii) the combination of i) and ii). Practical examplesinclude augmented reality services which normally require delays lessthan 100 ms [13], and mobile gaming where the quality of experience(QoE) commonly decreases with the increase of delay [35].

3. This means that i) each channel has the same bandwidth; and ii)all channels experience the same fading effect.

4

...

Channel 1

Channel 2

Channel M

Transmission

scheduling

Mobile User 1

Mobile User 2

Mobile User N

Local processing unit

Local processing unit

Local processing unit

λ1

λN

λ2

λ1β1

λ2β2

λNβN

λ2(1− β2)

λ1(1− β1)

λN (1− βN )

Delay-sensitive

computation task

...

Mobile edge

Telecom cloud

Access point

Fig. 1. An illustration of computation offloading and uplink transmissionscheduling for mobile edge computing.

offload it to the edge cloud via the uplink cellular transmis-sion. The network regulator has to determine a long-termnetwork-preferable scheme for joint computation offloadingand uplink transmission scheduling while guaranteeing thatall individual mobile users would like to follow. Denoteβ = (β1, β2, . . . , βN ) as the computation offloading scheme,where βi ∈ [0, 1] indicates the long-term offloading ratio(or equivalently the offloading probability for each com-putation task) of mobile user i,∀i ∈ N . Then, the aver-age arrival rates of computation tasks for offloading andlocal processing can be calculated as λoffload

i = βiλi andλlocali = (1 − βi)λi, respectively. If a mobile user decides

to offload a computation task to the edge cloud (i.e., notto process it locally) while cannot immediately deliver itdue to the channel shortage in uplink cellular transmissions,this computation task will be temporarily stored in thebuffer of the associated mobile user until it is scheduled fortransmission. By considering the fact that the resources forwireless transmissions (i.e., radio channels) are much moreconstrained than those for higher-layer traffic managementand computing [5], [23], we limit our focus on wirelesstransmission delays only and omit the detailed discussionson other potential latencies caused by task processing inlocal/edge cloud computing4. Furthermore, we do not con-sider buffer overflow5. Then, the management frameworkof computation offloading and transmission scheduling formobile edge computing can be modeled as a queueingsystem, as shown in Fig. 1.

With a proper scheduling, the uplink transmission rate

4. It is worth noting that this does not mean that the task processingtime is ignored in the proposed analytical framework. As will beshown in Section 3.2, the task processing time at local CPU is indirectlyconsidered in (3) for calculating the local processing cost, and the taskprocessing time at the cloud can be included by the formulation ofthe general delay cost function W (·). Therefore, the detailed study onthe task processing can be easily integrated, and the assumption ofneglecting the processing time of task computing is not necessary.

5. The storages of current mobile devices are in hundred gigabytes sothat the size of each user’s buffer can be considerably large. Moreover,with the capability of local processing, the traffic flow can be wellbalanced, and thus buffer overflow barely exists.

of each mobile user i,∀i ∈ N , on one cellular channel canbe expressed as

ri = B · log2

(1 +|hi|2 · Pi · d−ηi

σ2

), (1)

where B and σ2 denote the channel bandwidth and thevariance of additive Gaussian noise, respectively; Pi is thepre-determined transmission power of mobile user i; |hi|2captures the Rayleigh fading effect and follows an exponen-tial distribution with a unity mean; d−ηi signifies the pathloss effect, where di specifies the distance from mobile useri to the wireless access point and η ≥ 2 is the path loss ex-ponent. Consider that mobile users are roaming within thecell according to a random mobility pattern, e.g., followinga random waypoint model [37]. Then, the distance betweenmobile user i,∀i ∈ N , and the wireless access point can bemodeled as a random variable Di. Due to the randomnessof both |hi|2 and di, the uplink transmission rate of mobileuser i,∀i ∈ N , can be represented by a generally distributedrandom variable Ri with a probability density function(PDF) fRi

(·) and a finite mean E[Ri]. Similar definitions canalso be found in [38] and [39].

3.2 Utility Functions and Strategies of Mobile UsersSince mobile users are rational and potentially selfish [7],[17], their individual utility functions and strategies haveto be carefully taken into account in the management ofcomputation offloading and uplink transmission schedul-ing. Given the computation offloading and transmissionscheduling scheme, the long-term expected utility of eachmobile user i,∀i ∈ N , can be represented as

E[Ui] = λi · E[vi]− Ci(λlocali , λoffload

i ), (2)

where E[vi] is the mean valuation of completing onecomputation task for mobile user i, so that λiE[vi] indi-cates the long-term expected valuation gained by mobileuser i (including both local and edge cloud computing);Ci(λ

locali , λoffload

i ) denotes the total expected cost of mobileuser i, which consists of a processing cost for local comput-ing C local

i , a contract cost for edge cloud computing Cconti ,

an uplink transmission cost C txi and a delay cost Cdelay

i .Following the similar definitions in [8], [24], we discuss allthese cost terms in detail as follows.

For each mobile user i,∀i ∈ N , its computation tasks canbe executed by either the local or the edge cloud processingunits. Local processing will cause energy consumption onmobile user i, which can be computed as

C locali = λlocal

i E[Zi]ϕi = (1− βi)λiE[Zi]ϕi, (3)

where E[Zi] denotes the average number of CPU cyclesrequired by one computation task of mobile user i, and ϕispecifies the average energy cost per CPU cycle, which canbe obtained by existing measurement approaches [40]. Thus,C locali denotes the mean cost per unit of time for mobile user

i in local processing. In contrast, for computation tasks thatare offloaded by mobile user i to the edge cloud, a contractcost exists (to compensate the edge in consuming cloudcomputing resources) and can be calculated as

Cconti = λoffload

i πconti = βiλiπ

conti , (4)

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where πconti is the average contract cost for one computation

task, and Cconti represents the mean contract cost per unit of

time for mobile user i in offloading computation tasks.

Besides, to offload computation tasks to the edge cloud,each mobile user i, ∀i ∈ N , has to rely on the uplink cellulartransmission, which results in another cost, denoted by C tx

i .Intuitively, C tx

i should include transmission energy cost andtransmission service charge, i.e.,

C txi = λoffload

i (E txi + πi) = βiλi(E tx

i + πtxi ), (5)

where E txi and πtx

i stand for the mean transmission energycost and transmission service charge for one computationtask of mobile user i, respectively. To be more specific, E tx

i iscomputed as

E txi = E

[SiRi

]Piψi =

(E[Si]

∫ ∞−∞

1

rfRi(r)dr

)Piψi, (6)

where E[Si/Ri], Pi and ψi are mobile user i’s average uplinktransmission time for one computation task, pre-determinedtransmission power, and cost coefficient of transmissionenergy, respectively. By substituting (6) into (5), we have

C txi = βiλi

((E[Si]

∫ ∞−∞

1

rfRi

(r)dr

)Piψi + πtx

i

). (7)

In addition, since mobile users are running differentdelay-sensitive applications, there exists a specific delaycost, i.e., Cdelay

i , on each mobile user i,∀i ∈ N , due to thedelay in uplink transmissions for computation offloading.Without loss of generality, we define

Cdelayi = λoffload

i E[Wi(Ti)] = βiλiE[Wi(Ti)], (8)

where Ti is a random variable which models the time du-ration from a computation task arriving at mobile user i tobeing received by the wireless access point (i.e., Ti describesthe total delay that an offloaded computation task will expe-rience in uplink transmissions). Obviously, Ti is determinedby the uplink transmission scheduling and the volumn ofoffloaded traffics λoffload = (λoffload

1 , λoffload2 , . . . , λoffload

N ) orequivalently the computation offloading scheme β. Wi(·)denotes a general delay cost function which is defined tobe increasing and convex with respect to Ti. Clearly, Wi(·)well depicts the intuition in practice that mobile users arecommonly less sensitive to a small delay, but their concernsgrow dramatically when the delay continuously increasesand becomes very large [41], [42], e.g., approaching a certaindeadline. The physical meaning of Cdelay

i can be interpretedas the cost resulted by mobile user i’s dissatisfaction on thequality of experience in running its delay-sensitive applica-tion and the energy cost in buffering the computation tasksin uplink offloading.

After defining all cost terms as in (3)–(8), the expectedutility of each mobile user i,∀i ∈ N , can be written as

E[Ui] = λiE[vi]− (C locali + Ccont

i + C txi + C

delayi ). (9)

To further characterize its properties and simplify equa-tions, let us define

Vi(βi)=λiE[vi]− C locali − βiλiE tx

i

=λiE[vi]−(1−βi)λiE[Zi]ϕi−βiλiE txi , ∀i ∈ N .

(10)

Intuitively, we must have E[Zi]ϕi > E txi , i.e., the local pro-

cessing cost is larger than the associated uplink transmissionenergy cost. Otherwise, mobile user i will never offload itscomputation tasks because doing so will definitely result ina utility loss. In addition, there should be a lower bound onE[vi] such that E[vi] ≥ E[Zi]ϕi, i.e., the value of completinga computation task is larger or equal to its local processingcost. Otherwise, mobile user i will not be willing to runsuch application because it is too computation-intensive anddoing so may bring a high risk of suffering a utility loss. Byignoring these trivial cases, we can show that Vi(βi) mustbe an increasing function with respect to βi.

With (10), the utility function (9) for each mobile useri,∀i ∈ N , can now be rewritten as

E[Ui] = Vi(βi)− Cconti − βiλiπtx

i − Cdelayi

= Vi(βi)− βiλi(πconti + πtx

i )− βiλiE[Wi(Ti)].(11)

Note that the utility function shown above is derivedbased on the assumption that all mobile users will follow thedesigned computation offloading and transmission schedul-ing management. However, as indicated previously, sincemobile users are rational and potentially selfish, thoughthey are not able to manipulate the uplink transmissionscheduling (because it is centrally controlled by the networkregulator), they can strategically determine their own com-putation offloading ratios, i.e., β̃ = {β̃1, β̃2, . . . , β̃N}, whichmay be different from the network-wide optimal computa-tion offloading scheme β = {β1, β2, . . . , βN}, if and only ifthey can benefit from such behavior (i.e., improve their ownutilities). Obviously, such strategic behaviors from mobileusers will change the traffic volume in uplink transmissionsfor computation offloading. Considering that the uplinkspectrum resources are limited and shared among multiplemobile users, any individual strategic behavior from a mo-bile user on varying its offloading ratio can directly affectthe performance of the uplink transmission scheduling,which may further trigger frequent changes of offloadingstrategies from all mobile users, leading to system instabil-ity. Thus, in order to ensure the robustness and achieve amutually satisfactory equilibrium, our designed mechanismhas to satisfy the following essential requirements.

• Incentive compatibility: For each mobile user i,∀i ∈ N ,its expected utility should be maximized when β̃i =βi, i.e.,

E[Ui(β̃i)] ≤ E[Ui(βi)], ∀β̃i 6= βi, (12)

where E[Ui(β̃i)] indicates the expected utility of mo-bile user i with computation offloading strategy β̃iand can be computed by substituting β̃i into (9).

• Individual rationality: For each mobile user i,∀i ∈ N ,its expected utility should always be non-negativeif it follows the network-determined computationoffloading scheme β, i.e.,

E[Ui(βi)] ≥ 0. (13)

By satisfying both incentive compatibility (12) and in-dividual rationality (13), all mobile users will be willingto participate in the system and manage their computationoffloading strictly following the designed mechanism.

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3.3 Problem Formulation

The network social welfare (SW) of the considered man-agement framework for delay-sensitive mobile computationoffloading and transmission scheduling is defined as

SW = E[Ureg] +N∑i=1

E[Ui], (14)

where E[Ureg] and∑Ni=1 E[Ui] are the utilities of the network

regulator and all mobile users, respectively.Here, the utility of the network regulator E[Ureg] can be

mathematically expressed as

E[Ureg] =N∑i=1

Cconti +

N∑i=1

βiλiπtxi − Y (λoffload)

=

(N∑i=1

βiλi(πconti + πtx

i )

)− Y (λoffload),

(15)

where∑Ni=1 C

conti and

∑Ni=1 βiλiπ

txi are the charges col-

lected from mobile users for their computation offloadingand uplink transmissions, respectively; Y (λoffload) denotesthe network operating cost in providing cloud computationand wireless spectrum access. Following the convention in[43], [44], we assume that Y (λoffload) is a known increasingand convex function with respect to the data traffic λoffload.

Substituting (15) and (11) into (14) yields

SW =N∑i=1

Vi(βi)−N∑i=1

βiλiE[Wi(Ti)]− Y (λoffload). (16)

In order to maximize the social welfare SW , the net-work regulator has to carefully determine the computa-tion offloading scheme β = (β1, β2, . . . , βN )6, the uplinktransmission scheduling discipline ζ and the pricing ruleπ = (π1, π2, . . . , πN ), where πi = πcont

i + πtxi ,∀i ∈ N . Note

that it is not necessary to differentiate πconti and πtx

i for eachmobile user i,∀i ∈ N , because both of them are servicecharges paid to the network regulator.

In summary, the problem of designing an efficient mech-anism for joint delay-sensitive computation offloading andtransmission scheduling can be formulated as

[β, ζ,π] = arg max SW (17)s.t.,0 ≤ βi ≤ 1, ∀i ∈ N , (18)

πi ≥ 0, ∀i ∈ N , (19)

βi=arg maxβ̃i

(Vi(β̃i)−β̃iλiπi−β̃iλiE[Wi(Ti)]

), (20)

Vi(βi)− βiλiπi − βiλiE[Wi(Ti)] ≥ 0, ∀i ∈ N , (21)

λoffload = (β1λ1, β2λ2, . . . , βNλN ), (22)

(T1, T2, . . . , TN ) ∈ Q(λoffload, ζ), (23)

where constraints (18) and (19) show the ranges of eachmobile user’s computation offloading ratio and servicecharge, respectively; constraints (20) and (21) indicate the

6. This centrally derived computation offloading scheme can be con-sidered as a suggestion to mobile users. Although the computationoffloading ratio is actually determined strategically by each mobileuser, we will show that, under the designed mechanism, the individualoptimal decision will be exactly the same as the network-wide optimalmanagement scheme.

requirements for incentive compatibility and individual ra-tionality, respectively, and are derived by substituting (11)into (12) and (13); constraint (22) illustrates the traffic forcomputation offloading, which also equals the traffic foruplink transmissions; and constraint (23) states that theexperienced delay of each mobile user for offloading acomputation task, i.e., Ti,∀i ∈ N , is determined by thequeueing management system (as depicted in Fig. 1), de-noted by Q(·). Obviously, solving this problem directly isvery challenging because i) Ti,∀i ∈ N , is an endogenousfactor of the underlaying queueing system and it dependson the input λoffload and the scheduling discipline ζ (whichare both decision-dependent); ii) ζ is not a simple decisionvariable (or vector) but a complicated management policyof the queue; iii) to guarantee both incentive compatibilityand individual rationality, π may need to be devised as afunction with respect to the queueing dynamics. Therefore,in the following, we will propose a novel approach to designthe mechanism, i.e., [β, ζ,π], which can meet all theserequirements. Note that the payment term π is cancelled outin the network social welfare SW in (16), and thus it is notincluded in the objective function (17) but only included inconstraints (20) and (21). This motivates us to decouple theoriginal problem to a management problem for determining[β, ζ] and a pricing design problem for determining π, andsuch decoupling will not affect the network-wide optimality,if we can ensure that the determination of π is based on theoptimal β and ζ.

4 MULTI-USER COMPUTATION OFFLOADING ANDTRANSMISSION SCHEDULING MECHANISM

In this section, an efficient multi-user mobile computa-tion offloading and transmission scheduling mechanismfor delay-sensitive applications in mobile edge computing(named as MOTM) is proposed. We first study the optimaltransmission scheduling discipline ζ for minimizing thetotal delay cost of all mobile users given a fixed computationoffloading decision. Then, based on the derived relationshipbetween the transmission scheduling and the computationoffloading, we formulate a convex optimization problem todetermine the optimal computation offloading scheme βfor maximizing the network social welfare. After that, anappropriate pricing rule π is devised which can regulatethe noncooperative game behaviors of mobile users. Finally,we summarize the proposed MOTM and prove its feasibilityand optimality.

4.1 Delay-Sensitive Transmission Scheduling

From (16), we can observe that the first and the third termsof the network social welfare (SW) only depend on mobileusers’ computation offloading ratios β (or the offloadedtraffics) regardless of any other decisions. This implies thatthese two terms become constants under a given computa-tion offloading scheme. In addition, as explained that thepayment terms have been cancelled out in SW , and thedesign of the pricing rule is mainly for guaranteeing incen-tive compatibility and individual rationality (i.e., constraints(20) and (21)). Thus, by assuming a given computationoffloading scheme, termed as β̃, and ignoring constraints

7

(20) and (21), the original problem formulated in (17)–(23)can be relaxed as

ζ = arg maxSW

= arg maxN∑i=1

Vi(β̃i)−N∑i=1

β̃iλiE[Wi(Ti)]− Y (β̃)

= arg minN∑i=1

β̃iλiE[Wi(Ti)] (24)

s.t., λoffload = (β̃1λ1, β̃2λ2, . . . , β̃NλN ), (25)

(T1, T2, . . . , TN ) ∈ Q(λoffload, ζ), (26)

which turns out to be a pure queueing scheduling problemwith the objective of minimizing the total delay cost ofmobile users. Obviously, after solving this problem, we canobtain the optimal transmission scheduling discipline ζ andas well as the minimum delay cost under the given β̃.

It is worth noting that, different from traditional delay-sensitive queueing scheduling problems [45], [46] whichcommonly limited their focuses on linear delay costs, in thispaper, we consider a more practical scenario where mobileusers are allowed to have heterogeneous and nonlinear de-lay cost functions, i.e., Wi(·),∀i ∈ N . This greatly increasesthe difficulty in finding the optimal ζ because the nonlineardelay sensitivities of mobile users cannot be reflected bythe simple fixed and static priorities. Moreover, since it isrequired to derive the delay distributions of the queueingsystem, i.e., the distribution of Ti,∀i ∈ N , to quantifyE[Wi(Ti)], the mean analysis [47], a widely used methodunder linear delay cost settings, is no longer applicable. Inthe following, a novel transmission scheduling discipline isthus proposed and analyzed.

Before the analysis, we first introduce some new nota-tions. Denote 1/µi = E[Si/Ri] as the mean uplink transmis-sion time for one computation task of mobile user i,∀i ∈ N ,and ρi = λoffload

i /µi = β̃iλi

µias its individual offloading

traffic intensity (i.e., a ratio). Besides, define Li = ρiTi andL =

∑i∈N Li as the workload of each mobile user i and

the total workload waiting for edge cloud computing (oruplink transmissions), respectively. In addition, describe themarginal delay cost of each mobile user i as W ′i (t) = ∂Wi(t)

∂t ,where Wi(·) stands for the nonlinear delay cost function,and t is the realization of its random delay Ti. Furthermore,let 1/µ =

∑i∈N E[Si]∑i∈N E[Ri]

and ρ =∑i∈N ρi.

We then build the transmission scheduling discipline ζaccording to a delay-dependent dynamic priority rankingrule as follows.

Transmission scheduling discipline ζ: Denote ai(τ) asthe amount of tasks that mobile user i,∀i ∈ N , has buffered upto the time instant τ . Then, whenever a channel becomes free attime τ , it will be scheduled for serving the uplink transmissionsof mobile user i with the highest priority index Ii(τ), which isdefined as

i = arg maxi∈N

Ii(τ) = arg maxi∈N

W ′i

(ai(τ)

β̃iλi

)µi. (27)

An illustration example of ζ: Consider that there aretwo mobile users (MU1 and MU2) and one uplink chan-nel available at time τ . Assume that the offloading task

rates of MU1 and MU2 are 2 tasks/sec. and 3 tasks/sec.,respectively. The mean transmission time for one task ofMU1 and MU2 are set as 1/µ1 = 0.1 sec. and 1/µ2 = 0.2sec., respectively. Define the delay cost functions of MU1and MU2 as W1(T1) = T 2

1 and W2(T2) = 2T 22 , so that

W ′1(t) = 2t and W ′2(t) = 4t. If at time τ , the amount oftasks temporarily stored in the buffers of MU1 and MU2are a1(τ) = 5 and a2(τ) = 3, then according to (27), thepriority indices of MU1 and MU2 can be calculated as

I1(τ) = W ′1(5/2)× 0.1 = 2× (5/2)× 0.1 = 0.5, (28)I2(τ) = W ′2(3/3)× 0.2 = 4× (3/3)× 0.2 = 0.8. (29)

Since I2(τ) > I1(τ), MU2 will be scheduled for transmis-sion on the available channel starting from time τ .

In the following theorem, we prove that ζ is asymptot-ically optimal in minimizing the total delay cost of mobileusers and derive the corresponding delay distributions.

Theorem 1. Given the computation offloading decision β̃ and byapplying the proposed transmission scheduling discipline ζ,

(a) the total delay cost of all mobile users in the system, i.e.,∑Ni=1 β̃iλiE[Wi(Ti)], can be asymptotically minimized, as

the traffic intensity ρ tends to 1;(b) the delay distribution of each individual, i.e., the cumulative

distribution function (CDF) of Ti,∀i ∈ N , can be approxi-mated as

FTi(t|β̃) ≈ FL(g−1i (ρit)|β̃). (30)

Here, FL(·) is the CDF of the total workload L, which can beexpressed by Brownian approximation [48] as

FL(`|β̃) ≈ 1− ρe−γ` with γ =ρ

µ(1− ρ)

κ2a + κ2s2

, (31)

where κa and κs are the coefficients of variation of theoffloading task interarrival time and transmission time,respectively. g−1i (·) denotes the inverse function of gi(·)which maps the scheduling discipline ζ to the allocationof the workload over each mobile user i,∀i ∈ N , andg(·) = (g1(·), g2(·), · · · , gN (·)) can be obtained by solvingthe linear-constrained convex minimization problem:

g(L) = arg mingi(L)=Li,∀i∈N

N∑i=1

β̃iλiE[Wi(Ti)] (32)

s.t.,∑N

i=1Li = L, (33)

Li = ρiTi, Li ≥ 0, ∀i ∈ N . (34)

Proof: Intuitively, the total workload L of the queue-ing system is independent of the scheduling discipline,when the traffic intensity ρ tends to 1 (i.e., the queueingsystem approaches the heavy traffic limit). However, thescheduling discipline does impact how the total workload Lis distributed among different mobile users. From [49], weknow that as ρ tends to 1, the ratio between the workloadof each mobile user i and the total workload, denoted byLi

L ,∀i ∈ N , becomes a constant.Thus, the queueing scheduling problem for determining

ζ shown in (24)–(26) can be transformed to the problem ofallocating L over mobile users, i.e., the determination ofg(L) = (g1(L), g2(L), · · · , gN (L)), with the aim of mini-mizing their total delay cost. This transformed problem can

8

be formulated as illustrated in (32)–(34). Since the objectivefunction is convex (because Wi(·),∀i ∈ N , is defined as aconvex function) and all constraints are linear, this problemcan be easily solved by checking the first-order conditions:

β̃iλiρi

W ′i

(Liρi

)−ξi =

β̃jλjρi

W ′j

(Ljρj

)−ξj , ∀i, j ∈ N , (35)

with Liξi = 0 and∑Ni=1 Li = L, where ξi ≥ 0 is the

Lagrange multiplier with respect to the non-negativity con-straint on Li,∀i ∈ N .

Furthermore, the instantaneous workload of each mobileuser at any time τ can be computed as

Li(τ) =ai(τ)

µi, ∀i ∈ N ,∀τ, (36)

where ai(τ) is the amount of tasks that have been stored inthe buffer of mobile user i up to time τ . Then, we have

Li(τ)

ρi=ai(τ)

µi

µi

β̃iλi=ai(τ)

β̃iλi, ∀i ∈ N ,∀τ. (37)

Substituting (37) into (35), we can observe that the pro-posed scheduling discipline ζ based on (27) actually triesto distribute the workloads of mobile users to satisfy thefirst-order conditions (35) at any time τ . Therefore, ζ isasymptotically optimal in minimizing the total delay costof mobile users, and Theorem 1 (a) is proved.

With the asymptotically optimal solution of g(L) =(g1(L), g2(L), · · · , gN (L)), we can derive the CDF of eachmobile user’s delay Ti,∀i ∈ N , as

FTi(t|β̃) = Prob.(Ti ≤ t|β̃)

= Prob.(Li ≤ ρit|β̃)

≈ Prob.(gi(L) ≤ ρit|β̃)

= FL(g−1i (ρit)|β̃),

(38)

where FL(`|β̃) is the CDF of the total workload L and canbe directly expressed by Brownian approximation [48], asshown in (31). This completes the proof of Theorem 1 (b).

Theorem 1 indicates that the proposed transmissionscheduling discipline ζ can asymptotically minimize the to-tal delay cost of all mobile users, for any given computationoffloading decision β̃. Although the optimality of ζ relies onthe heavy traffic approximation (i.e., ρ tends to 1), since it iswidely known that the ever-increasing mobile data traffic israpidly straining the capacity of existing cellular networks[6], it can be expected that ζ will actually achieve a goodperformance under practical wireless settings.

Moreover, with the CDF of Ti,∀i ∈ N , as derived inTheorem 1 (b), the minimum total delay cost of all mobileusers can be calculated as

minN∑i=1

β̃iλiE[Wi(Ti)]=N∑i=1

β̃iλi

∫ ∞−∞

Wi(t)dFTi(t|β̃). (39)

Note that (39) specifies a closed-form relationship betweenthe computation offloading decision β̃ and the minimizedtotal delay cost of all mobile users under the proposedtransmission scheduling discipline ζ. This relationship willbe used in Section 4.2 to determine the optimal computationoffloading scheme.

4.2 Optimal Computation Offloading Scheme

With the (asymptotically) optimal transmission schedulingdiscipline ζ devised in Section 4.1, we are now able todetermine the optimal computation offloading scheme β bysolving

β = arg maxSW

= arg maxN∑i=1

Vi(βi)−N∑i=1

βiλiEζβ[Wi(Ti)]− Y (β) (40)

s.t., 0 ≤ βi ≤ 1, ∀i ∈ N , (41)

where∑Ni=1 βiλiE

ζβ[Wi(Ti)] denotes the minimized total

delay cost of mobile users resulted by the optimal transmis-sion scheduling discipline ζ with respect to β. Obviously,this problem (i.e., (40)–(41)) is a subproblem of the originalone formulated in (17)–(23) without taking into account thepricing rule and individual utilities of mobile users. In thefollowing, we present some important characteristics of thisproblem and the corresponding solution.

Theorem 2. There always exists a unique optimal solution of βwhich maximizes the network social welfare SW .

Proof: From (41), we can obviously see that the fea-sible set of the optimal computation offloading ratios isnonempty and compact. Moreover, i) Vi(βi) is linear andincreasing with respect to βi,∀i ∈ N , as shown in (10); ii)∑Ni=1 βiλiE

ζβ[Wi(Ti)] is convex and increasing with respect

to βi,∀i ∈ N , because queueing delays are naturally convexand increasing with respect to the traffic load, and the wait-ing cost Wi(·) is defined as an increasing convex functionwith respect to βi,∀i ∈ N ; and iii) Y (β) is ordinarily convexand increasing with respect to βi,∀i ∈ N , as explainedin Section 3.3. All these together imply that the objectivefunction (40) is concave with respect to the offloading ratiosβi,∀i ∈ N . Therefore, the problem formulated in (40)–(41),is a concave maximization problem with a nonempty andcompact feasible set, so that there always exists a uniqueoptimal solution of β.

Theorem 3. Let β∗ = (β∗1 , · · · , β∗N ) be the network-wideoptimal computation offloading scheme, i.e., β∗ maximizes thenetwork social welfare SW , then we must have

∂Vi(βi)

∂βi=λiEζ

β∗ [Wi(Ti)] +N∑j=1

β∗j λj∂Eζ

β∗ [Wj(Tj)]

∂βi

+∂Y (β∗)

∂βi+ ωi − νi, ∀i ∈ N ,

(42)

and 0 ≤ β∗i ≤ 1, ∀i ∈ N , (43)ωiβ∗i = 0, ∀i ∈ N , (44)

νi(1− β∗i ) = 0, ∀i ∈ N , (45)

where ωi ≥ 0 and νi ≥ 0 are Lagrange multipliers with respectto constraints βi ≥ 0 and βi ≤ 1, respectively.

Proof: It has been proved in Thoerem 2 that thesocial welfare maximization problem formulated in (40)–(41), is a standard concave maximization problem. Thus, byapplying Karush-Kuhn-Tucker (KKT) optimality conditions[50] along with some simple mathematical manipulations,we can obtain (42)–(45).

9

Theorem 2 indicates the existence and uniqueness of theoptimal solution for (40)–(41), and hence existing software-based optimization tools [51] can be directly employed tonumerically determine the optimal computation offloadingscheme. Besides, Theorem 3 illustrates the necessary andsufficient optimality conditions which will be used in Sec-tion 4.3 to theoretically verify the network-wide optimalityof the designed pricing rule in regulating the strategic andnoncooperative behaviors of mobile users.

4.3 Design of the Pricing RuleAfter deriving the optimal joint computation offloadingand transmission scheduling scheme, denoted by [β, ζ], asshown in Sections 4.1 and 4.2, our remaining problem is todetermine the pricing rule π based on the optimal β andζ to meet incentive compatibility and individual rationality(i.e., constraints (20) and (21)). Since the network optimum(i.e., the maximization of the network social welfare SW) isnot affected by the designed pricing rule, the main purposeof designing π is to induce all smart mobile users to followthe determined [β, ζ].

Naturally, as rational and selfish entities, mobile usersmay deviate from the management decisions made by thenetwork regulator, if and only if they can benefit from suchbehaviors. Particularly, each mobile user i,∀i ∈ N , canstrategically and independently determine its self-preferredcomputation offloading ratio βi ∈ [0, 1] with the objective ofmaximizing its own expected utility, which has been definedin Section 3 and is rewritten here as

Ui(βi) = Vi(βi)− βiλiπi − βiλiEζβ[Wi(Ti)], (46)

where Vi(βi) is the value gained by mobile user i fromcomputation offloading, calculated by (10); πi is the unde-termined service charge on mobile user i for each offloadedcomputation task; and Eζ

β[Wi(Ti)] stands for the averagedelay cost of one computation task of mobile user i whenthe optimal uplink transmission scheduling discipline ζ isadopted. Note that, since the transmission scheduling isexecuted centrally by the network regulator, mobile usersare not able to manipulate ζ, even though they are strategic7.

To maximize the individual utility Ui(βi),∀i ∈ N , smartmobile users will compete with each other by adjusting theirown computation offloading ratios, i.e., βi,∀i ∈ N , so as topossibly increase their valuation gains, while at the sametime lower their service charges and waiting delay costsfrom computation offloading and uplink transmissions.This will obviously result in a distributed decision makingproblem, or more specifically, a noncooperative game on aqueueing system with discipline ζ. Formally, this game canbe represented as

G , {N ,B, {Ui(βi,β−i)}i∈N }, (47)

where mobile users in the setN act as players; B denotes thestrategy set of mobile users’ computation offloading ratiosβi,∀i ∈ N ; and Ui(βi,β−i) is the utility function of mobileuser i,∀i ∈ N , in terms of its strategy βi given that thestrategy of all other mobile users is β−i = β\βi. The Nashequilibrium (NE) of game G can be defined as follows.

7. In this paper, we do not consider malicious attacks and assumethat all mobile users are trustworthy.

Definition 1 (NE). A strategy profile βe = (βe1 , βe2 · · · , βeN ) is

a Nash equilibrium of game G if for every mobile user i,∀i ∈ N ,we have

Ui(βei ,β

e−i) ≥ Ui(βi,βe−i), ∀βi ∈ [0, 1]. (48)

It is obvious that the NE of game G largely depends onthe pricing rule π. Therefore, in order to guarantee thatall mobile users will be willing to self-manage their com-putation offloading according to the network-wide optimalscheme so that both their own utilities and the social welfareSW can be maximized, it is clear that π should be designedwith the goal of making the network-wide optimal solutionbe an NE of game G, i.e.,

β∗i = βei , ∀i ∈ N , (49)

where βei and β∗i are the outputs (i.e., computation offload-ing ratio on mobile user i,∀i ∈ N ) of the NE of game G andthe network-wide optimal computation offloading problem(40)–(41), respectively.

Designed pricing rule π∗: Let β∗ denote the network-wideoptimal computation offloading scheme (as devised in Section 4.2),then the optimal pricing rule π∗ = (π∗1 , π

∗2 , · · · , π∗N ) for mobile

users in computation offloading and transmission scheduling isdefined as

π∗i (βi) =1

λiαi +

1

βiλiθi, ∀i ∈ N , (50)

where

αi =∂Y (β∗)

∂βi+

N∑j=1,j 6=i

β∗j λj∂Eζ

β∗ [Wj(Tj)]

∂βi, (51)

θi ≤ θ̄i = V (β∗i )− β∗i λiEζβ∗ [Wi(Ti)]− αiβ∗i . (52)

Note that αi and θi are coefficients that are independent of βi anddetermined by β∗ only.

Next, we prove that, by employing the designed pricingrule π∗, the NE of game G always exists and its equilib-rium conditions are satisfied by the network-wide optimalcomputation offloading scheme β∗.

Theorem 4. The game G , {N ,B, {Ui(βi,β−i)}i∈N } withthe designed pricing rule, i.e., π , π∗, has at least one NE.

Proof: Since the strategies of mobile users are theirself-determined computation offloading ratios which mustbe within the range of [0, 1], the strategy set B of game Gcan be expressed in the form of Cartesian product as

B =N∏i=1

[0, 1] ⊂ RN , (53)

which is obviously nonempty, convex and compact.Substituting the pricing rule (50)–(52) into the utility

function (46) of mobile user i,∀i ∈ N , and taking the first-order derivative with respect to βi, we have

∂Ui∂βi

=∂

∂βi

(Vi(βi)− βiλiEζ

β[Wi(Ti)]− βiαi − θi)

=∂Vi(βi)

∂βi−λiEζ

β[Wi(Ti)]−βiλi∂Eζ

β[Wi(Ti)]

∂βi−αi.

(54)

As shown in (10), Vi(βi) is linear with βi. Thus, the second-

10

order derivative of Ui,∀i ∈ N , can be calculated based on(54) as

∂2Ui∂β2

i

= −2λi∂Eζ

β[Wi(Ti)]

∂βi− βiλi

∂2Eζβ[Wi(Ti)]

∂β2i

. (55)

Recall that the waiting cost Wi(·) is defined as an increasingconvex function, and the queueing delay Ti is naturallyincreasing and convex with respect to βi, so that

∂Eζβ[Wi(Ti)]

∂βi≥ 0,

∂2Eζβ[Wi(Ti)]

∂β2i

≥ 0. (56)

Substituting (56) into (55) yields ∂2Ui

∂β2i≤ 0, which means that

Ui is concave with respect to βi.In conclusion, since the strategy set is nonempty, convex

and compact, and the utility functions are continuous andconcave, there exists at least one NE in game G.

Theorem 5. With the adoption of the designed pricing rule,i.e., π , π∗, the network-wide optimal computation offloadingscheme β∗ is an NE of game G, namely

β∗ = βe. (57)

Proof: Since the NE of game G exists (as proved inTheorem 4), the strategy of each mobile user i,∀i ∈ Nwill converge to the equilibrium βei , such that its individualutility is maximized, i.e.,

βei = arg maxUi

= arg maxVi(βi)− βiλiπi − βiλiEζβ[Wi(Ti)] (58)

s.t., 0 ≤ βi ≤ 1, (59)

Substituting the pricing rule (50)–(52) into (58) and ap-plying the KKT conditions, we can show that βei ,∀i ∈ N ,must satisfy

∂Vi(βi)

∂βi=λiEζ

βe [Wi(Ti)] + βei λi∂Eζ

βe [Wi(Ti)]

∂βi

+∂

∂βi

(βei λi(

1

λiαi +

1

βei λiθi)

)+ ωi − νi

=λiEζβe [Wi(Ti)] +

N∑j=1

βejλj∂Eζ

βe [Wj(Tj)]

∂βi

+∂Y (βe)

∂βi+ ωi − νi,

(60)

and 0 ≤ βei ≤ 1, (61)ωiβ

ei = 0, (62)

νi(1− βei ) = 0, (63)

where ωi ≥ 0 and νi ≥ 0 are Lagrange multipliers.Obviously, the equilibrium conditions (60)–(63) for each

mobile user i,∀i ∈ N , are exactly the same as the optimalityconditions (42)–(45) for deriving β∗. This means that β∗,which satisfies (42)–(45), must also satisfy (60)–(63), ∀i ∈ N ,and thus we have β∗ = βe.

Theorems 4 and 5 indicate that the NE of game G canbe obtained by directly solving the optimization problemdefined in (40)–(41), rather than running the distributeddecision making process among mobile users. In addition,it is not necessary to further prove the uniqueness of NE

or guarantee a fast convergence speed of game G, becausein practice, the network regulator can always centrally de-termine β∗ in advance and then induce all mobile users tofollow it by adopting the designed pricing rule π, or in otherwords, no iterative strategy adaption is needed.

4.4 Summary of MOTMIn summary, the proposed multi-user mobile computa-tion offloading and transmission scheduling mechanism(MOTM) consists of the transmission scheduling disciplineζ built based on the delay-dependent dynamic priorityorder (27), the computation offloading scheme β derivedfrom the optimization problem (40)–(41) and the pricing ruleπ , π∗ designed in (50)–(52) to make β , β∗ = βe. Inthe following, we show that MOTM can indeed meet allrequirements or constraints imposed by the original prob-lem formulated in Section 3.3, namely, incentive compatibility,individual rationality and optimality (i.e., network social welfaremaximization).

Theorem 6 (Incentive compatibility). All mobile users willbe willing to manage their computation offloading according tothe proposed MOTM, denoted by [β, ζ,π], because doing so canmaximize their own utilities, i.e.,

βi = arg maxβ̃i∈[0,1]

(Vi(β̃i)−β̃iλiπi−β̃iλiEζ

β[Wi(Ti)]). (64)

Proof: It has already been proved in Theorem 5that by determining the computation offloading scheme asβ , β∗ along with the optimal transmission schedulingdiscipline ζ and the correspondingly designed pricing ruleπ, the NE conditions for the distributed strategy makingprocess among mobile users can be satisfied, leading toβ , β∗ = βe. Since NE is ordinarily defined as theequilibrium such that all mobile users can maximize theirown utilities, MOTM, which reaches the NE, obviouslyguarantees (64), and thus is incentive-compatible.

Theorem 7 (Individual rationality). With the implementationof the proposed MOTM, denoted by [β, ζ,π], the expected utilityof each mobile user i,∀i ∈ N , is always non-negative, i.e.,

E[Ui] = Vi(βi)− βiλiπi − βiλiEζβ[Wi(Ti)] ≥ 0. (65)

Proof: Substituting the designed pricing rule π, i.e.,(50)–(52), into the expected utility of each mobile user i,∀i ∈N , we have

E[Ui] =Vi(βi)−βiλiEζβ[Wi(Ti)]−βiλiπi

=Vi(βi)−βiλiEζβ[Wi(Ti)]−βiαi−θi

≥Vi(βi)−βiλiEζβ[Wi(Ti)]−βiαi− θ̄i (66)

=Vi(βi)−βiλiEζβ[Wi(Ti)]−Vi(βi)+βiλiEζ

β[Wi(Ti)]

= 0,

which completes the proof.

Theorem 8 (Optimality). The proposed MOTM, denoted by[β, ζ,π], can maximize the network social welfare SW in thejoint management of delay-sensitive computation offloading andtransmission scheduling.

Proof: The reasons why the social welfare SW can bemaximized by the proposed mechanism MOTM, denoted

11

by [β, ζ,π], are twofold: i) the computation offloading andtransmission scheduling decisions, i.e., β and ζ, have beenjointly optimized with the objective of maximizing the socialwelfare SW , as shown in Sections 4.1 and 4.2; and ii) it hasbeen proved in Theorems 6 and 7 that all mobile users arewilling to follow the network-wide optimal managementdecisions [β, ζ] under the specifically designed pricing ruleπ. Therefore, the proposed mechanism is robust and thenetwork optimality (i.e., the maximization of network socialwelfare SW) determined by [β, ζ] can be maintained.

5 SIMULATION RESULTS

In this section, simulations are conducted to evaluate theperformance of the proposed mechanism in mobile com-putation offloading and transmission scheduling for delay-sensitive applications. All results are obtained by usingMonte Carlo simulations over 100 runs with various systemparameters.

5.1 Simulation Settings

Consider a MATLAB-based simulation environment fora mobile edge computing system with N mobile usersrunning delay-sensitive computational applications and Mchannels dedicated for computation offloading, where Nand M are in ranges from 50 to 100 and 5 to 15, respectively.According to the 4G cellular network characteristics [52], thetransmission power of each mobile user is set as 100 mWand the uplink transmission rate is determined randomlyfrom 2 to 5 Mbps. Based on the configurations of mobileedge computing assisted video gaming applications [12],[53], the average values of the packet size and requiredCPU cycles of a computation task are approximated as 500Kb and 1000 Megacycles, respectively. For different mobileusers, their computation tasks may be heterogeneous, andthus we further allow the actual values of the packet sizeand required CPU cycles of each computation task fromeach mobile user to be taken randomly within [400, 600]Kb and [800, 1200] Megacycles, respectively. In addition,assume that the arrival rate of computation tasks at eachmobile user i, ∀i ∈ N , is selected within [2, 5] per minutesand define the delay cost function as Wi(Ti) = εiT

2i , where

εi reflects the marginal delay sensitivity of each mobile useri and is chosen randomly over [1, 6]. Besides, for simplicity,let the mean valuation of one computation task be 1, thenetwork operation cost be 0, and the energy cost coefficientsϕi = ψi = 1. Similar settings have also been employed in[7], [8], [54]. Note that some parameters may be varied fordifferent evaluation purposes.

5.2 Performance Evaluations

To show the performance of our proposed transmissionscheduling discipline ζ in minimizing the total delay costof all mobile users, Fig. 2 compares it with two exist-ing scheduling disciplines, i.e., the first-come first-serve(FCFS) scheduling [55] which treats all uplink transmis-sions equally without considering their heterogeneous delaysensitivities, and the fixed-priority scheduling [33] whichmaintains a fixed transmission order based on mobile users’

Number of mobile users50 60 70 80 90 100

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Fixed-priority scheduling

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Fig. 2. Performance comparison of the different scheduling disciplines.

marginal delay sensitivities but regardless of their experi-enced delays in computation offloading. It is intuitive thatthe total waiting costs increase with the number of mobileusers for all three scheduling disciplines. This is becausewhen the number of mobile users is larger, the queueingsystem is more congested so that waiting delays becomelonger. Besides, the FCFS scheduling leads to the highestdelay costs due to ignoring potential priorities among mo-bile users indicated by their different delay sensitivities. Thefixed-priority scheduling outperforms FCFS because of theconsideration of mobile users’ marginal delay sensitivities.More importantly, it is shown that the proposed delay-sensitive transmission scheduling discipline ζ achieves thebest performance in minimizing the total delay cost. Thisis because the proposed scheduling discipline ζ well bal-ance the waiting costs of all mobile users in computationoffloading by taking into account both their marginal delaysensitivities and experienced delays.

Fig. 3 investigates the delay cost of a mobile user withdifferent marginal delay sensitivities, when the proposedmechanism, MOTM, is applied. It can be observed that themobile user’s delay cost first increases and then decreaseswith the increase of its marginal delay sensitivity. Accordingto the definition, the delay cost of a mobile user is a functionof both its marginal delay sensitivity and experienced delay.Thus, the waiting cost is considerably small for two extremecases: i) when the marginal delay sensitivity tends to zero,i.e., the mobile user is running an application which doesnot care about any delay so that no delay cost is introduced;ii) when the marginal delay sensitivity is extremely large,i.e., the mobile user is highly sensitive to the delay so that itscomputation tasks will be mostly processed locally (to avoidpotential transmission and scheduling delays) or offloadedwith the highest priority via the uplink transmission. Fur-thermore, since the queueing system is more congested witha larger number of mobile users N , each mobile user willexperience a longer delay and hence suffer a higher delaycost when N increases, as shown in Fig. 3.

Fig. 4 illustrates the computation offloading ratio of amobile user with different marginal delay sensitivities inthe proposed MOTM. It can be seen from this figure that a

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Marginal delay sensitivity1 2 3 4 5 6

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Fig. 3. Delay cost of a mobile user in MOTM.

Marginal delay sensitivity1 2 3 4 5 6

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Fig. 4. Computation offloading ratio of a mobile user in MOTM.

mobile user’s computation offloading ratio decreases withthe increase of its marginal delay sensitivity. Naturally,mobile edge computing introduces a longer delay than localprocessing due to the wireless transmissions between themobile user and the edge cloud. Thus, the mobile user witha higher delay sensitivity would prefer to process morecomputation tasks locally and correspondingly decreaseits computation offloading ratio. Besides, we can see thatsuch decreasing trend is more obvious when the marginaldelay sensitivity gets larger. This is because to maximizethe individual utility, the proposed MOTM can well balancethe local energy consumption and the offloading delay costof each mobile user. As a result, when the delay cost of amobile user becomes dominant (as its marginal delay sensi-tivity increases), its computation offloading ratio decreasessignificantly. In addition, since it is intuitive that a largernumber of channels (i.e., a larger M ) always leads to abetter scheduling performance in computation offloading(i.e., a shorter delay), Fig. 5 also shows that given the samemarginal delay sensitivity, the offloading ratio of a mobileuser increases with M .

Fig. 5 examines the incentive compatibility of the pro-

Computation offloading ratio0 0.2 0.4 0.6 0.8 1

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Optimal offloading ratios User's best strategies (NE)

Fig. 5. Optimal offloading strategies of mobile users in MOTM.

posed MOTM by showing the utility of a mobile user withdifferent computation offloading ratios. In the consideredsystem, each mobile user may strategically deviate thenetwork-wide optimal management scheme by choosing itsown computation offloading ratio for potentially increasingits individual utility. The trend of curves in Fig. 5 indicatesthat the utility of a mobile user first increases with thecomputation offloading ratio. This is because increasing theoffloading ratio can lower the local processing cost andexploit more benefits from edge computing so that a higherutility can be obtained. However, after a certain point (i.e.,the offloading ratio determined by the optimal managementscheme as presented in Fig. 4), since the cost of computationoffloading (including the induced delay cost and servicecharge) becomes dominant, the utility decreases. Obviously,the computation offloading ratio which can maximize theindividual utility is the best strategy (i.e., NE) that will beadopted by the mobile user. Therefore, Fig. 5 reveals that thenetwork-wide optimal solution also reaches the NE of eachmobile user, which proves Theorem 6.

In Figs. 6 and 7, the superiority of applying the proposedMOTM in multi-user delay-sensitive mobile edge comput-ing system is demonstrated. For comparison purpose, thelocal computing scheme without enabling edge computingcapability (local computing), the decentralized game basedcomputation offloading scheme (DCO) [7] and the energy-efficient centralized computation offloading scheme (ECO)[10] are simulated as benchmarks. DCO manages the com-putation offloading and uplink wireless transmissions byrunning a distributed decision making process among mo-bile users, while ECO centrally optimizes the joint resourceallocations in mobile edge computing by taking into accountthe heterogeneities of users’ priorities, local processing costsand channel conditions. However, both of them rely on theassumption of quasi-static network scenarios, so that theirmanagement decisions are made myopically by ignoring theimpacts of random task arrivals and buffering delays.

Fig. 6 compares the network social welfare (i.e., SW)produced by different computation offloading and transmis-sion scheduling management schemes with respect to thenumber of mobile users. Intuitively, SW increases with the

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Fig. 6. Comparison on SW with different number of mobile users.

number of mobile users because more users implies morecomputation tasks and higher total valuations of task pro-cessing (including both local and edge computing). Besides,it is shown that local computing results in the lowest SW .This is because without the help of edge computing, mobileusers have to suffer tremendously high energy consump-tions in utilizing local CPU resources. In contrast, due to theemployment of edge computing, SW is obviously higherfor all other schemes (i.e., DCO, ECO and the proposedMOTM), in which computation offloading is enabled. Fur-thermore, ECO outperforms DCO because of the consider-ation of global optimality in the centralized managementand the presumption that mobile users will never behaveselfishly and strategically. Moreover, we can see that the pro-posed MOTM achieves the best performance in Fig. 6. This isbecause MOTM can maximize SW by considering the long-term tradeoffs of the system in a stochastic network sce-nario, and can also guarantee that no individual mobile userhas the incentive to deviate the optimal solution. We furthercompare these four management schemes with respect tothe number of channels in Fig. 7, and similar observationsas in Fig. 6 can be obtained. It is worth noting that sincemore channels indicates a better scheduling performance incomputation offloading, SW increases with the number ofchannels for all management schemes, except for the localcomputing scheme. This is because local computing doesnot require the support of wireless transmissions, so that itsperformance is independent of the number of channels.

6 CONCLUSION AND FUTURE WORK

In this paper, the joint computation offloading and trans-mission scheduling for delay-sensitive applications in mo-bile edge computing has been studied. To characterize thedynamic management of the system with potential networkuncertainties, a queueing model is formulated. By consid-ering tradeoffs between local and edge computing, wirelessfeatures and noncooperative game behaviors of smart mo-bile users, we propose a novel mechanism, namely MOTM,to jointly determine the computation offloading scheme,the transmission scheduling discipline and the pricing rule.Both theoretical analyses and simulation results show that

Number of channels5 10 15

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Fig. 7. Comparison on SW with different number of channels.

our proposed mechanism can guarantee that no individualmobile user has the incentive to strategically deviate thenetwork-wide optimal management and can largely im-prove the social welfare compared to the counterparts.

In the future work, we will further integrate the cloudcomputing management in the designed mechanism. Specif-ically, if the cloud computing resource is limited and theoffloaded tasks are overwhelmed, a congestion delay at thecloud may happen and will also contribute to the total com-putation offloading delay. In this case, the overall systemmay need to be formulated as a tandem queueing modelwith a multi-server multi-class priority queue for uplinktransmission scheduling followed by another multi-servermulti-class priority queue for cloud computing service. Thismotivates us to analyze the joint distribution of two queue-ing delays and develop an algorithm for jointly determiningthe optimal priority-aware disciplines for both queues.

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