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Decentralized Computation Offloading GameFor Mobile Cloud Computing

Xu Chen, Member, IEEE

Abstract—Mobile cloud computing is envisioned as a promising approach to augment computation capabilities of mobile devices foremerging resource-hungry mobile applications. In this paper, we propose a game theoretic approach for achieving efficient computationoffloading for mobile cloud computing. We formulate the decentralized computation offloading decision making problem among mobiledevice users as a decentralized computation offloading game. We analyze the structural property of the game and show that thegame always admits a Nash equilibrium. We then design a decentralized computation offloading mechanism that can achieve a Nashequilibrium of the game and quantify its efficiency ratio over the centralized optimal solution. Numerical results demonstrate that theproposed mechanism can achieve efficient computation offloading performance and scale well as the system size increases.

Index Terms—Mobile cloud computing, decentralized computation offloading, game theory.

✦

1 INTRODUCTION

As smart-phones are gaining enormous popularity, moreand more new mobile applications such as face recog-nition, natural language processing, interactive gaming,and augmented reality are emerging and attract greatattention [1], [2]. This kind of mobile applications aretypically resource-hungry, demanding intensive compu-tation and high energy consumption. Due to the physicalsize constraint, however, mobile devices are in gen-eral resource-constrained, having limited computationresources and limited battery life. The tension betweenresource-hungry applications and resource-constrainedmobile devices hence poses a significant challenge forthe future mobile platform development [3].

Mobile cloud computing is envisioned as a promisingapproach to address such a challenge. As illustrated inFigure 1, mobile cloud computing can augment the ca-pabilities of mobile devices for resource-hungry applica-tions, by offloading the computation via wireless accessto the resource-rich cloud infrastructure such as Ama-zon Elastic Compute Cloud (EC2) and Windows AzureServices Platform. In the cloud, each mobile device isassociated with a cloud clone, which runs on a virtualmachine (VM) that can execute mobile applications onbehalf of the mobile device1 [5], [6].

Although the cloud based approach can significantlyaugment computation capability of mobile device users,the task of developing a comprehensive and reliablemobile cloud computing system remains challenging.A key challenge is how to achieve an efficient com-

The author is with School of Electrical, Computer and Energy Engineering,Arizona State University, Tempe, USA. Email: xchen179@asu.edu. The paperhas been accepted by IEEE Transactions on Parallel and Distributed Systems.

1. In this study we focus on the mobile application services (e.g.,remote application execution) of the cloud. However, the cloud canalso provide a number of other services [4], such as platform services(e.g., storage and file backup services).

Fig. 1. An illustration of mobile cloud computing

putation offloading coordination among mobile deviceusers. One critical factor of affecting the performanceof mobile cloud computing is the wireless access effi-ciency [7]. If too many mobile device users choose tooffload the computation to the cloud via wireless accesssimultaneously, they may generate severe interferenceto each other, which would reduce the data rates forcomputation data transmission. This hence can lead tolow energy efficiency for computation offloading andlong data transmission time. In this case, it would notbe beneficial for the mobile device users to offloadcomputation to the cloud.

In this paper, we adopt a game theoretic approachto address such a challenge. Game theory is a use-ful framework for designing decentralized mechanisms,such that the mobile device users in the system canself-organize into the mutually satisfactory computa-tion offloading decisions. The self-organizing feature canadd autonomics into mobile cloud computing systemand help to ease the heavy burden of complex cen-tralized management (e.g., information collection frommassive mobile device users and computation offloadingscheduling) by the cloud. Moreover, as different mobiledevices are usually owned by different individuals andthey may pursue different interests, game theory is apowerful tool to analyze the interactions among multi-

ple mobile device users who act in their own interestsand devise incentive compatible computation offloadingmechanisms such that no mobile user has the incentiveto deviate unilaterally.

Specifically, we model the decentralized computationoffloading decision making problem among mobile de-vice users for mobile cloud computing as a decentralizedcomputation offloading game. We then propose a de-centralized computation offloading mechanism that canachieve the Nash equilibrium of the game. The mainresults and contributions of this paper are as follows:

• Decentralized computation offloading game formulation:We formulate the decentralized computation of-floading decision making problem among multiplemobile device users as a decentralized computa-tion offloading game, by taking into account bothcommunication and computation aspects of mobilecloud computing.

• Analysis of game structure: We analyze the decentral-ized computation offloading game in both homoge-nous and heterogeneous wireless access cases. Forthe homogenous case, we show that the game ad-mits the beneficial cloud computing group structure,which guarantees the existence of Nash equilibrium.For the more general heterogeneous case, we showthat the game is a potential game, and hence admitsthe finite improvement property and possesses aNash equilibrium.

• Decentralized mechanism for achieving Nash equilib-rium: We devise a decentralized computation of-floading mechanism such that mobile device usersmake decisions locally, which can significantly re-duce the controlling and signaling overhead of thecloud. We show that the mechanism can achieve aNash equilibrium of the decentralized computationoffloading game. We further quantify the price ofanarchy, i.e., the efficiency ratio of the mechanismover the centralized optimal solution. Numericalresults demonstrate that the proposed mechanismcan achieve efficient computation offloading perfor-mance and scale well as the system size increases.

The rest of the paper is organized as follows. Wefirst discuss related work in Section 2, and introducethe system model in Section 3. We then propose thedecentralized computation offloading game and developthe decentralized computation offloading mechanism inSections 4 and 5, respectively. We present the numericalresults in Section 6 and finally conclude in Section 7.

2 RELATED WORK

Most previous work has investigated the efficient com-putation offloading mechanism design from the per-spective of a single mobile device user [6]–[9], [9]–[15].Rudenko et al. in [12] demonstrated by experimentsthat significant energy can be saved by computationoffloading. Gonzalo et al. in [13] developed an adaptiveoffloading algorithm based on both the execution history

of applications and the current system conditions. Xianet al. in [14] introduced an efficient timeout scheme forcomputation offloading to increase the energy efficiencyon mobile devices. Rahimi et al. in [16] proposed a2-tier cloud architecture to improve both performanceand scalability of mobile cloud computing. Huang etal. in [11] proposed a Lyapunov optimization based dy-namic offloading algorithm to improve the mobile cloudcomputing performance while meeting the applicationexecution time. Barbera et al. in [7] showed by realisticmeasurements that the wireless access plays a key rolein affecting the performance of mobile cloud comput-ing. Wolski et al. in [15] proposed a prediction baseddecision making framework for determining when anoffloaded computation will outperform local executionon the mobile device. Wen et al. in [6] presented anefficient offloading policy by jointly configuring the clockfrequency in the mobile device and scheduling the datatransmission to minimize the energy consumption.

To the best of our knowledge, only a few works haveaddressed the computation offloading problem underthe setting of multiple mobile device users [17]–[19].Yang et al. in [17] studied the scenario that multipleusers share the wireless network bandwidth, and solvedthe problem of maximizing the mobile cloud computingperformance by a centralized heuristic genetic algorithm.Rahimi et al. in [18] took into consideration user mobilityinformation and proposed a centralized greedy schemeto solve the computation offloading problem with mul-tiple mobile users. Barbarossa et al. in [19] proposed acentralized scheduling algorithm to jointly optimize thecommunication and computation resource allocationsamong multiple users with the latency requirements.The centralized computation offloading schemes aboverequires that all the mobile device users submit theirown information (e.g., wireless channel gain and the sizeof computation tasks) to a centralized entity (e.g., thecloud), which will determine the offloading schedule ac-cordingly. Along a different line, in this paper we adoptthe game theoretic approach and devise a decentralizedmechanism wherein each mobile device user makes thecomputation offloading decision locally. This can helpto reduce the controlling and signaling overhead of thecloud.

3 SYSTEM MODEL

In this section, we introduce the system model of mobilecloud computing. We consider a set of N = {1, 2, ..., N}collocated mobile device users and each of which hasa computationally intensive and delay sensitive taskto be completed. There exists a wireless access base-station s, through which the mobile device users canoffload the computation to the cloud (e.g., Amazon EC2or Microsoft Azure). Similar to many previous studiesin mobile cloud computing [6], [17], [19] and mobilenetworking [20]–[22], to enable tractable analysis and getuseful insights, we consider a quasi-static scenario where

the set of mobile device users N remains unchangedduring a computation offloading period (e.g., withinseveral seconds), while may change across different pe-riods2. The general case that mobile users may departand leave dynamically within a computation offloadingperiod will be considered in a future work. Since boththe communication and computation aspects play a keyrole in mobile cloud computing, we next introduce thecommunication and computation models in details.

3.1 Communication Model

We first introduce the communication model for wirelessaccess. The wireless access base-station s can be either aWiFi access point, or a Femtocell network access point[23], or a macrocell base-station in cellular networksthat manages the uplink/downlink communications ofmobile device users. We denote an ∈ {0, 1} as thecomputation offloading decision of mobile device usern. Specifically, we have an = 1 if user n chooses tooffload the computation to the cloud via wireless access.We have an = 0 if user n decides to compute its tasklocally on the mobile device. Given the decision profilea = (a1, a2, ..., aN ) of all the mobile device users, we cancompute the uplink data rate for computation offloadingof mobile device user n as [24]

Rn(a) = W log2

(

1 +PnHn,s

ωn +∑

m∈N\{n}:am=1 PmHm,s

)

.

(1)Here W is the channel bandwidth and Pn is user n’stransmission power which is determined by the wirelessaccess base-station according to some power controlalgorithms such as [25], [26]. Further, Hn,s denotes thechannel gain between the mobile device user n and thebase-station, and ωn = ω0

n + ω1n denotes the background

interference power including the noise power ω0n and the

interference power ω1n from other mobile device users

who carry out wireless transmission but do not involvein the mobile cloud computing.

From the communication model in (1), we see thatif too many mobile device users choose to offload thecomputation via wireless access simultaneously, theymay incur severe interference, leading to low data rates.As we discuss latter, this would negatively affect theperformance of mobile cloud computing.

3.2 Computation Model

We then introduce the computation model. We considerthat each mobile device user n has a computation taskIn , (Bn, Dn) that can be computed either locally on themobile device or remotely on the cloud via computation

2. This assumption holds for many applications such as face recogni-tion and natural language processing, in which the size of computationinput data is not large and hence the computation offloading can befinished in a smaller time scale (e.g., within several seconds) than thetime scale of users’ mobility.

offloading. Here Bn denotes the size of computation in-put data (e.g., the program codes and input parameters)involving in the computation task In and Dn denotes thetotal number of CPU cycles required to accomplish thecomputation task In. A mobile device user n can applythe methods in [3], [5], [17] to obtain the information ofBn and Dn. We next discuss the computation overheadin terms of both energy consumption and processingtime for both local and cloud computing approaches.

3.2.1 Local Computing

For the local computing approach, a mobile device usern executes its computation task In locally on the mobiledevice. Let F l

n be the computation capability (i.e., CPUcycles per second) of mobile device user n. Here weallow that different mobile devices may have differentcomputation capability. The computation execution timeof the task In by local computing is then given as

T ln =

Dn

F ln

. (2)

For the computational energy, we have that

Eln = νnDn, (3)

where νn is the coefficient denoting the consumed en-ergy per CPU cycle. According to the realistic measure-ments in [6], [27], we can set νn = 10−11(F l

n)2.

According to (2) and (3), we can then compute theoverhead of the local computing approach in terms ofcomputational time and energy as

Z ln = γT

n Tln + γE

n Eln, (4)

where 0 ≤ γTn , γ

En ≤ 1 denote the weights of com-

putational time and energy for mobile device user n’sdecision making, respectively. To provide rich modelingflexibility and meet user-specific demands, we allow thatdifferent users can choose different weighting parame-ters in the decision making. For example, when a useris at a low battery state, the user would like to putmore weight on energy consumption (i.e., a larger γE

n )in the decision making, in order to save more energy.When a user is running some application that is sensitiveto the delay (e.g., video streaming), then the user canput more weight on the processing time (i.e., a largerγTn ), in order to reduce the delay. Note that the weights

could be dynamic if a user runs different applications orhas different policies/demands at different computationoffloading periods. For ease of exposition, in this paperwe assume that the weights of a user are fixed within onecomputation offloading period, while can be changed indifferent periods.

3.2.2 Cloud Computing

For the cloud computing approach, a mobile device usern will offload its computation task In to the cloud andthe cloud will execute the computation task on behalf ofthe mobile device user.

For the computation offloading, a mobile device usern would incur the extra overhead in terms of time andenergy for transmitting the computation input data tothe cloud via wireless access. According to the com-munication model in Section 3.1, we can compute thetransmission time and energy of mobile device user n

for offloading the input data of size Bn as, respectively,

T cn,off(a) =

Bn

Rn(a), (5)

and

Ecn(a) =

PnBn

Rn(a). (6)

After the offloading, the cloud will execute the com-putation task In. Let F c

n be the computation capability(i.e., CPU cycles per second) assigned to user n by thecloud. The execution time of the task In of mobile deviceuser n on the cloud can be then given as

T cn,exe =

Dn

F cn

. (7)

According to (5), (6), and (7), we can compute theoverhead of the cloud computing approach in terms ofprocessing time and energy as

Zcn(a) = γT

n

(

T cn,off(a) + T c

n,exe

)

+ γEn Ec

n(a). (8)

Similar to many studies such as [10]–[14], we neglectthe time overhead for the cloud to send the computationoutcome back to the mobile device user, due to the factthat for many applications (e.g., face recognition), thesize of the computation outcome in general is muchsmaller than the size of computation input data includ-ing the mobile system settings, program codes and inputparameters.

According to the communication and computationmodels above, we see that the computation offloadingdecisions a among the mobile device users are cou-pled. If too many mobile device users simultaneouslychoose to offload the computation task to the cloudvia wireless access, they may incur severe interferenceand this would lead to a low data rate. When thedata rate Rn(a) of a mobile device user n is low, itwould consume high energy in the wireless access foroffloading the computation input data to cloud andincur long transmission time as well. In this case, itwould be more beneficial for the user to compute thetask locally on the mobile device to avoid the longprocessing time and high energy consumption by thecloud computing approach. In the following sections,we will adopt a game theoretic approach to address theissue of how to achieve efficient computation offloadingdecision makings among the mobile device users.

4 DECENTRALIZED COMPUTATION OFFLOAD-ING GAME

In this section, we develop a game theoretic approachfor achieving efficient computation offloading decision

makings among the mobile device users. The primaryrationale of adopting the game theoretic approach is thatthe mobile devices are owned by different individualsand they may pursue different interests. Game theoryis a powerful framework to analyze the interactionsamong multiple mobile device users who act in theirown interests and devise incentive compatible compu-tation offloading mechanisms such that no user has theincentive to deviate unilaterally. Moreover, by leveragingthe intelligence of each individual mobile device user,game theory is a useful tool for devising decentralizedmechanisms with low complexity, such that the users canself-organize into a mutually satisfactory solution. Thiscan help to ease the heavy burden of complex centralizedmanagement by the cloud and reduce the controlling andsignaling overhead between the cloud and mobile deviceusers.

4.1 Game Formulation

We consider the decentralized computation offloadingdecision making problem among the mobile device userswithin a computation offloading period. Let a−n =(a1, ..., an−1, an+1, ..., aN ) be computation offloading de-cisions by all other users except user n. Given otherusers’ decisions a−n, user n would like to select a properdecision an ∈ {0, 1} (i.e., local computing or cloudcomputing) to minimize its computation overhead interms of energy consumption and processing time, i.e.,

minan∈{0,1}

Vn(an, a−n), ∀n ∈ N .

According to (4) and (8), we can obtain the overheadfunction of mobile device user n as

Vn(an, a−n) =

{

Z ln, if an = 0,

Zcn(a), if an = 1.

(9)

We then formulate the problem above as a strategicgame Γ = (N , {An}n∈N , {Vn}n∈N ), where the set ofmobile device users N is the set of players, An , {0, 1} isthe set of strategies for user n, and the overhead functionVn(an, a−n) of each user n is the cost function to beminimized by player n. In the sequel, we call the gameΓ as the decentralized computation offloading game. Wenow introduce the concept of Nash equilibrium [28].

Definition 1. A strategy profile a∗ = (a∗1, ..., a

∗N) is a Nash

equilibrium of the decentralized computation offloading gameif at the equilibrium a

∗, no player can further reduce itsoverhead by unilaterally changing its strategy, i.e.,

Vn(a∗n, a

∗−n) ≤ Vn(an, a

∗−n), ∀an ∈ An, n ∈ N . (10)

The Nash equilibrium has the nice self-stability prop-erty such that the users at the equilibrium can achievea mutually satisfactory solution and no user has theincentive to deviate. This property is very important tothe decentralized computation offloading problem, sincethe mobile devices are owned by different individualsand they may act in their own interests.

4.2 Game Property

We then study the existence of Nash equilibrium of thedecentralized computation offloading game. To proceed,we first introduce an important concept of best response[28].

Definition 2. Given the strategies a−n of the other players,player n’s strategy a∗n ∈ An is a best response if

Vn(a∗n, a−n) ≤ Vn(an, a−n), ∀an ∈ An. (11)

According to (10) and (11), we see that at the Nashequilibrium all the users play the best response strate-gies towards each other. Based on the concept of bestresponse, we have the following observation for thedecentralized computation offloading game.

Lemma 1. Given the strategies a−n of other mobile deviceusers in the decentralized computation offloading game, thebest response of a user n is given as the following thresholdstrategy

a∗n =

{

1, if∑

m∈N\{n}:am=1 PmHm,s ≤ Ln,

0, otherwise,

where the threshold

Ln =PnHn,s

2

(γTn +γE

n Pn)Bn

W(γTn Tl

n+γEn El

n−γTn Tc

n,exe) − 1

− ωn.

The proof is given in Section 8.1 of the separatesupplementary file. According to Lemma 1, we see thatwhen the received interference

∑

m∈N\{n}:am=1 PmHm,s

is lower enough, it is beneficial for user n to offload thecomputation to the cloud. Otherwise, the user n shouldcompute the task on the mobile device locally. Sincethe wireless access plays a critical role in mobile cloudcomputing, we next discuss the existence of Nash equi-librium of the the decentralized computation offloadinggame in both homogeneous and heterogeneous wirelessaccess cases.

4.2.1 Homogeneous Wireless Access Case

We first consider the case that users’ wireless accessis homogenous, i.e., PmHm,s = PnHn,s = K, for anyn,m ∈ N . This can correspond to the scenario that allthe mobile device users experience the similar channelcondition and are assigned with the same transmissionpower by the base-station. However, different users mayhave different thresholds Ln, i.e., they are heterogeneousin terms of computation capabilities and tasks.

For the homogenous wireless access case, without lossof generality, we can order the set N of mobile deviceusers so that L1

K≥ L2

K≥ ... ≥ LN

K. Based on this, we have

the following useful observation.

Lemma 2. For the decentralized computation offloading gamewith homogenous wireless access, if there exists a non-emptybeneficial cloud computing group of mobile device users S ⊆

Algorithm 1 Algorithm for finding beneficial cloud com-puting group

1: Input: the set of ordered mobile device users withL1

K≥ L2

K≥ ... ≥ LN

Kand L1

K≥ 0.

2: Output: a beneficial cloud computing group S.3: set S = {1}.4: for t = 2 to N do5: set S = S ∪ {t}6: if |S| > Lt

K+ 1 then

7: stop and go to return.8: else set S = S.9: end if

10: end for11: return S.

N such that

|S| ≤Li

K+ 1, ∀i ∈ S, (12)

and further if S ⊂ N ,

|S| >Lj

K, ∀j ∈ N\S, (13)

then the strategy profile wherein users i ∈ S play the strategyai = 1 and the other users j ∈ N\S play the strategy aj = 0is a Nash equilibrium.

The proof is given in Section 8.2 of the separatesupplementary file. For example, for a set of 4 users with(

L1

K, L2

K, L3

K, L4

K

)

= (5, 4, 3, 2), the beneficial cloud com-puting group is S = {1, 2, 3}. In general, when L1

K≥ 0,

we can construct the beneficial cloud computing groupby using Algorithm 1. Thus, we have the followingresult.

Theorem 1. The decentralized computation offloading gamewith homogenous wireless access always has a Nash equi-librium. More specifically, when L1

K< 0, all users n ∈ N

playing the strategy an = 0 is a Nash equilibrium. WhenL1

K≥ 0, we can construct a beneficial cloud computing group

S 6= ∅ by Algorithm 1 such that the strategy profile whereinusers i ∈ S play the strategy ai = 1 and the other usersj ∈ N\S play the strategy aj = 0 is a Nash equilibrium.

The proof is given in Section 8.3 of the separate sup-plementary file. Since the computational complexity ofordering operation (e.g., quicksort algorithm) is typicallyO(N logN) and the construction procedure in Algorithm1 involves at most N operations (with each operation ofthe complexity of O(1)), the beneficial cloud computinggroup construction algorithm has a low computationalcomplexity of O(N logN). This implies that we cancompute the Nash equilibrium of the decentralized com-putation offloading game in the homogenous wirelessaccess case in a fast manner.

4.2.2 General Wireless Access CaseWe next consider the general case including the casethat users’ wireless access can be heterogeneous, i.e.,

PmHm,s 6= PnHn,s. Since mobile device users may havedifferent transmission power Pn, channel gain Hn,s andthresholds Ln, the analysis based on the beneficial cloudcomputing group in the homogenous case can not applyhere. We hence resort to a power tool of potential game[29].

Definition 3. A game is called a potential game if it admitsa potential function Φ(a) such that for every n ∈ N , a−n ∈∏

i6=n Ai, and a′

n, an ∈ An, if

Vn(a′

n, a−n) < Vn(an, a−n), (14)

we haveΦ(a

′

n, a−n) < Φ(an, a−n). (15)

Definition 4. The event where a player n changes to an actiona

′

n from the action an is a better response update if and onlyif its cost function is decreased, i.e.,

Vn(a′

n, a−n) < Vn(an, a−n). (16)

An appealing property of the potential game is that itadmits the finite improvement property, such that anyasynchronous better response update process (i.e., nomore than one player updates the strategy at any giventime) must be finite and leads to a Nash equilibrium [29].Here the potential function to a game has the same spiritas the Lyapunov function to a dynamical system. If adynamic system is shown to have a Lyapunov function,then the system has a stable point. Similarly, if a gameadmits a potential function, the game must have a Nashequilibrium.

We now prove the existence of Nash equilibrium of thegeneral decentralized computation offloading game byshowing that the game is a potential game. Specifically,we define the potential function as

Φ(a) =1

2

N∑

n=1

∑

m 6=n

PnHn,sPmHm,sI{an=1}I{am=1}

+

N∑

n=1

PnHn,sLnI{an=0}, (17)

where I{A} is the indicator function such as I{A} = 1 ifthe event A is true and I{A} = 0 otherwise.

Theorem 2. The general decentralized computation offload-ing game is a potential game with the potential function asgiven in (17), and hence always has a Nash equilibrium andthe finite improvement property.

The proof is given in Section 8.4 of the separate supple-mentary file. Theorem 2 implies that any asynchronousbetter response update process is guaranteed to reacha Nash equilibrium within a finite number of iterations.This motivates the algorithm design in following Section5.

5 DECENTRALIZED COMPUTATION OFFLOAD-ING MECHANISM

In this section we propose a decentralized computationoffloading mechanism in Algorithm 2 for achieving theNash equilibrium of the decentralized computation of-floading game.

5.1 Mechanism Design

The motivation of using the decentralized computationoffloading mechanism is to coordinate mobile deviceusers to achieve a mutually satisfactory decision mak-ing, prior to the computation task execution. The keyidea of the mechanism design is to utilize the finiteimprovement property of the decentralized computationoffloading game and let one mobile device user improveits computation offloading decision at a time. Specifi-cally, by using the clock signal from the wireless accessbase-station for synchronization, we consider a slottedtime structure for the computation offloading decisionupdate. Each decision slot t consists the following twoparts:

• Interference Measurement: Each mobile device usern locally measures the received interference µn(t) =∑

m∈N\{n}:am(t)=1 PmHm,s generated by other userswho currently choose the decisions of offloading thecomputation tasks to the cloud via wireless access.To facilitate the interference measurement, for ex-ample, the users m who choose decisions am(t) = 1at the current slot will transmit some pilot signals tothe base-station. And each mobile device user canthen enquire its received interference µn(t) from thebase-station.

• Decision Update Contention: We exploit the finiteimprovement property of the game by having onemobile device user carry out a decision update ateach decision slot. We let users who can improvetheir computation performance compete for the de-cision update opportunity in a decentralized man-ner. More specifically, according to Lemma 1, eachmobile device user n first computes its set of bestresponse update based on the measured interferenceµn(t) as

∆n(t) , {a∗n : Vn(a∗n, a−n(t)) < Vn(an(t), a−n(t))}

=

{1}, if an(t) = 0 and µn(t) ≤ Ln,

{0}, if an(t) = 1 and µn(t) > Ln,

∅, otherwise.

The best response here is similar to the steepest de-scent direction selection to reduce user’s overhead.Then, if ∆n(t) 6= ∅ (i.e., user n can improve), usern will contend for the decision update opportunity.Otherwise, user n will not contend and adhere tothe current decision at next decision slot, i.e., an(t+1) = an(t). For the decision update contention, forexample, we can adopt the random backoff-basedmechanism by setting the time length of decision

update contention as τ∗. Each contending user n

first generates a backoff time value τn according tothe uniform distribution over [0, τ∗] and countdownuntil the backoff timer expires. When the timerexpires, if the user has not received any request-to-update (RTU) message from other mobile deviceusers yet, the user will update its decision for thenext slot as an(t + 1) ∈ ∆n(t) and then broadcasta RTU message to all users to indicate that it winsthe decision update contention. For other users, onhearing the RTU message, they will not update theirdecisions and will choose the same decisions at nextslot, i.e., an(t+ 1) = an(t).

According to the finite improvement property in The-orem 2, the mechanism will converge to a Nash equilib-rium of the decentralized computation offloading gamewithin finite number of decision slots. In practice, wecan implement that the computation offloading decisionupdate process terminates when no RTU messages arebroadcasted for multiple consecutive decision slots (i.e.,no decision update can be further carried out by anyusers). Then each mobile device user n executes thecomputation task according to the decision an obtainedat the last decision slot by the mechanism. Due to theproperty of Nash equilibrium, no user has the incentiveto deviate from the achieved decisions. This is veryimportant to the decentralized computation offloadingproblem, since the mobile devices are owned by differentindividuals and they may act in their own interests.By following the decentralized computation offloadingmechanism, the users adopt the best response to improvetheir decision makings and eventually self-organize intoa mutually satisfactory solution (i.e., Nash equilibrium).

We then analyze the computational complexity of thealgorithm. In each iteration, N mobile users will executethe operations in Lines 5 − 15. Since the operationsin Lines 5 − 15 only involve some basic arithmeticalcalculations, the computational complexity in each it-eration is O(N). Suppose that it takes C iterations forthe algorithm to converge. Then the total computationalcomplexity of the algorithm is O(CN). Numerical resultsin Section 6 show that the number of iterations C forconvergence increases linearly with the number of usersN . This demonstrates that the decentralized computationoffloading mechanism can converge in a fast manner inpractice.

5.2 Performance Analysis

We then discuss the efficiency of Nash equilibrium by thedecentralized computation offloading mechanism. Notethat the decentralized computation offloading game mayhave multiple Nash equilibria, and the proposed de-centralized computation offloading mechanism will ran-domly select one Nash equilibrium (since a random useris chosen for decision update). Following the definitionof price of anarchy (PoA) in game theory [30], wewill quantify the efficiency ratio of the worst-case Nash

Algorithm 2 Decentralized computation offloadingmechanism

1: initialization:2: each mobile device user n chooses the computation

decision an(0) = 1.3: end initialization

4: repeat for each user n and each decision slot t inparallel:

5: measure the interference µn(t).6: compute the best response set ∆n(t).7: if ∆n(t) 6= ∅ then8: contend for the decision update opportunity.9: if win the decision update contention then

10: choose the decision an(t + 1) ∈ ∆n(t) fornext slot.

11: broadcast the RTU message to other users.12: else choose the original decision an(t + 1) =

an(t) for next slot.13: end if14: else choose the original decision an(t+1) = an(t)

for next slot.15: end if16: until no RTU messages are broadcasted for M con-

secutive slots

equilibrium over the centralized optimal solution. LetΥ be the set of Nash equilibria of the decentralizedcomputation offloading game. Then the PoA is definedas

PoA =maxa∈Υ

∑

n∈N Vn(a)

mina∈

∏

Nn=1

An

∑

n∈N Vn(a),

which is lower bounded by 1. A larger PoA impliesthat the set of Nash equilibrium is less efficient (in theworst-case sense) using the centralized optimum as a

benchmark. Let Zcn =

(γTn +γE

n Pn)Bn

W log2

(

1+PnHn,s

ωn

) + γTn T

cn,exe. We

can show the following result.

Theorem 3. The PoA of the decentralized computation of-

floading game is at most∑

Nn=1

Zln

∑

Nn=1

min{Zln,Z

cn}

.

The proof is given in Section 8.5 of the separatesupplementary file. Intuitively, Theorem 3 indicates thatwhen users have lower cost of local computing (i.e.,Z ln is smaller), the Nash equilibrium is closer to the

centralized optimum and hence the PoA is lower. More-over, when the communication efficiency is higher (i.e.,PnHn,s is larger and hence Zc

n is larger), the performanceof Nash equilibrium can be improved. Numerical resultsin Section 6 demonstrate that the Nash equilibrium bythe decentralized computation offloading mechanism isefficient, with at most 10% performance loss, comparedwith the centralized optimal solution.

Fig. 2. Dynamics of user cost by the decentralizedcomputation offloading mechanism

6 NUMERICAL RESULTS

In this section, we evaluate the proposed decentral-ized computation offloading mechanism by numericalstudies. We first consider the mobile cloud computingscenario that N = 20 mobile device users are randomlyscattered over a 50m×50m region and the wireless accessbase-station is located in the center of the region. For thewireless access, we set the channel bandwidth W = 5MHz, the transmission power Pn = 100 mWatts, andthe background noise ωn = −100 dBm. According tothe physical interference model [24], we set the channelgain Hn,s = d−α

n,s , where dn,s is the distance betweenmobile device user n and the cloudlet and α = 4is the path loss factor. We set the decision weightsγTn = γE

n = 0.5. For the computation task, we use theface recognition application in [1], where the data sizefor the computation offloading Bn = 420 KB and thetotal number of CPU cycles Dn = 1000 Megacycles. TheCPU computational capability F l

n of a mobile device usern is randomly assigned from the set {0.5, 0.8, 1.0} GHzand the computational capability on the cloud F c

n = 100GHz [1].

We first show the dynamics of mobile device users’computation cost Vn(a) by the proposed decentralizedcomputation offloading mechanism in Figure 2. We seethat the mechanism can keep mobile users’ cost de-creasing and converge to an equilibrium. To verify thatthe convergent equilibrium is a Nash equilibrium, wefurther show the dynamics of the potential functionvalue Φ(a) of the decentralized computation offloadinggame in Figure 3. It demonstrates that the proposeddecentralized computation offloading mechanism canlead the potential function of the game to the minimumpoint, which is a Nash equilibrium according to theproperty of potential game.

To investigate the impact of computation size on de-centralized computation offloading, we then implementthe simulations with different number of CPU processingcycles Dn required for completing the computing task.Upon comparison, we also implement the local mobilecomputing solution such that all the mobile device userscompute their tasks locally on the mobile devices. The

results are shown in Figure 4. We see that the system-wide computing cost

∑

n∈N Vn(a) by decentralized com-putation offloading and local mobile computing solu-tions increases as the number of CPU processing cyclesDn increases. However, the system-wide computing cost∑

n∈N Vn(a) by decentralized computation offloadingincreases much slower than that of local mobile com-puting. This is because that as the number of CPUprocessing cycles Dn increases, more mobile device userschoose to utilize the cloud computing via computationoffloading to mitigate the heavy cost of local computing.

To evaluate the impact of communication data sizeon the decentralized computation offloading, we nextimplement the simulations with different data size forcomputation offloading Bn in Figure 5. We observethat the system-wide computing cost

∑

n∈N Vn(a) bydecentralized computation offloading as the data sizefor computation offloading Bn increases, due to thefact that a larger data size requires higher overheadfor computation offloading via wireless communication.Moreover, we see that the system-wide computing cost∑

n∈N Vn(a) by decentralized computation offloadingincreases slowly when the data size for computationoffloading Bn is large. This is because that the datasize for computation offloading Bn is large, more mobiledevice users choose to compute the tasks locally onthe mobile devices, in order to avoid the heavy cost ofcomputation offloading via wireless access.

To benchmark the performance of the decentral-ized computation offloading mechanism, we furtherimplement the system-wide computing cost mini-mization solution by centralized optimization, i.e.,maxa

∑

n∈N Vn(a). Notice that the centralized optimiza-tion solution requires the complete information of allmobile device users, such as the details of computingtasks, the transmission power, the channel gain, andthe CPU frequency of all mobile devices. While thedecentralized computation offloading mechanism onlyrequires each mobile device user to measure its receivedinterference and make the decision locally. We run ex-periments with the number of N = 10, 15, ..., 50 mobiledevice users being randomly scattered over the squarearea, respectively. We repeat each experiment 100 timesand show the average system-wide computing cost inFigure 6. We see that the system-wide computing costby all the computation offloading solutions increases asthe number of mobile device users N increases. Theproposed incentive compatible computation offloadingsolution can reduce up-to 33% and 38% computingcost over the solutions of all the users choosing thelocal computing and choosing the cloud computing, re-spectively. Compared with the centralized optimizationsolution, the performance loss of the decentralized com-putation offloading mechanism is less than 10% in allcases. This demonstrates the efficiency of the proposeddecentralized computation offloading mechanism. Wenext evaluate the convergence time of the decentralizedcomputation offloading mechanism. Figure 7 shows that

Fig. 3. Dynamics of potential functionby the decentralized computation of-floading mechanism

Fig. 4. System-wide computing costwith different number of CPU pro-cessing cycles

Fig. 5. System-wide computing costwith different data size for the compu-tation offloading

Fig. 6. Average system-wide comput-ing cost

Fig. 7. Number of iterations bydecentralized computation offloadingmechanism

Fig. 8. Number of controlling and sig-naling messages by the centralizedoptimal and decentralized computa-tion offloading mechanisms

the average convergence time increases linearly with thenumber of mobile device users N . This shows that thedecentralized computation offloading mechanism scaleswell with the size of mobile device users. This is criticalsince computing the centralized optimal computation of-floading solution involves solving the integer program-ming problem (i.e., the decision variables an ∈ {0, 1})and the computational complexity grows exponentiallyas the number of mobile device users N increases.

To evaluate the controlling and signaling overheadreduction by the decentralized computation offloadingmechanism, we further show the number of controllingand signaling messages exchanged among the mobileusers and between the users and the cloud in Figure8. It demonstrates that the decentralized computationoffloading mechanism can reduce the number of con-trolling and signaling messages by at least 89% over thecentralized optimal computation offloading scheme inall cases. This is because that for the decentralized com-putation offloading mechanism, a mobile user wouldexchange messages (for interference measurement anddecision update announcement) only when it updatesits computation decision. While for the centralized opti-mal computation offloading scheme, each mobile userneeds to report all its local parameters to the cloud,

including the transmission power, the channel gain, thebackground interference power, the local computationcapability, and many other parameters. Moreover, insome application scenarios, due to privacy concernssome mobile users may be sensitive to the revealing oftheir local parameters and hence do not have the incen-tive to participate in the centralized optimal computationoffloading scheme. While the decentralized computationoffloading mechanism does not have this issue sinceeach mobile user can make the computation offloadingdecision locally without exposing its local parameters.

7 CONCLUSION

In this paper, we consider the computation offloadingdecision making problem among mobile device users formobile cloud computing and propose as a decentralizedcomputation offloading game formulation. We show thatthe game always admits a Nash equilibrium for bothcases of homogenous and heterogenous wireless access.We also design a decentralized computation offloadingmechanism that can achieve a Nash equilibrium of thegame and further quantify its price of anarchy. Numeri-cal results demonstrate that the proposed mechanism isefficient and scales well as the system size increases.

For the future work, we are going to consider the moregeneral case that mobile users may depart and leavedynamically within a computation offloading period.In this case, the user mobility patterns might play animportant role in the problem formulation.

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8 APPENDIX

8.1 Proof of Lemma 1

According to (4), (8) and (9), we obtain that

Vn(a) =Zcn(a)an + Z l

n(1− an)

=(

γTn

(

T cn,off(a) + T c

n,exe

)

+ γEn Ec

n(a))

an

+(

γTn T

ln + γE

n Eln

)

(1− an)

=

(

(

γTn + γE

n Pn

)

Bn

Rn(a)+ γT

n Tcn,exe

)

an

+(

γTn T

ln + γE

n Eln

)

(1− an).

When user n’s best response a∗n = 1, by Definition 2, wehave that

Vn(1, a∗−n) ≤ Vn(0, a

∗−n),

which implies that(

γTn + γE

n Pn

)

Bn

Rn(a)+ γT

n Tcn,exe ≤ γT

n Tln + γE

n Eln.

That is,

Rn(a) ≥

(

γTn + γE

n Pn

)

Bn

γTn T

ln + γE

n Eln − γT

n Tcn,exe

.

According to (1), we then have that∑

m∈N\{n}:am=1

PmHm,s

≤Ln ,PnHn,s

2

(γTn +γE

n Pn)Bn

W(γTn Tl

n+γEn El

n−γTn Tc

n,exe) − 1

− ωn.

Similarly, we can analyze the case when user n’s bestresponse a∗n = 0.

8.2 Proof of Lemma 2

According to (12), for a user i ∈ S, its received interfer-ence

∑

m∈N\{n}:am=1

PmHm,s = (|S| − 1)K

≤ Li.

Then it follows from Lemma 1 that playing the strategyai = 1 is a best response. Similarly, for a user j ∈ N\S,we can show that playing the strategy aj = 0 is abest response. Since all the users play the best responsetowards each other, the proposed strategy profile is aNash equilibrium.

8.3 Proof of Theorem 1

Given the set of ordered mobile device users with L1

K≥

L2

K≥ ... ≥ LN

K, if L1

K< 0, it is easy to check that the

beneficial cloud computing group S = ∅. In this case,all users n ∈ N playing the strategy an = 0 is a bestresponse and hence a Nash equilibrium.

If L1

K≥ 0, we can construct a beneficial cloud comput-

ing group S 6= ∅ by Algorithm 1. At the first step, we set

S = {1}. It is easy to check that |S| = 1 ≤ L1

K+ 1, which

satisfies the condition in (12). At the step 2 ≤ t ≤ N , wedefine that S = S∪{t}. If |S| = t ≤ Lt

K+1, we then set that

S = S and continue to the next step. If |S| = t > Lt

K+ 1,

we can stop and obtain the beneficial cloud computinggroup as S = {1, .., t − 1}. This is because that: 1) Ssatisfies that |S| ≤ Lt−1

K+1 (otherwise we cannot proceed

to the step t), which implies that the condition in (12)is satisfied, i.e., |S| ≤ Lt−1

K+ 1 ≤ ... ≤ L1

K+ 1; 2)

since |S| = t > Lt

K+ 1, we have |S| = t − 1 > Lt

K,

which implies that the condition in (13) is satisfied, i.e.,|S| > Lt

K≥ ... ≥ LN

K. Note that the total number of

available steps of Algorithm 1 is bounded by N . Abeneficial cloud computing group S 6= ∅ hence mustcan be obtained.

8.4 Proof of Theorem 2

We first show that that Vk(1, a−k) < Vk(0, a−k) impliesΦ(1, a−k) < Φ(0, a−k) for a user k. For this case, ac-cording to (4), (8) and (9), the condition Vk(1, a−k) <

Vk(0, a−k) implies that∑

m 6=k

PmHm,sI{am=1} < Lk. (18)

Furthermore, according to (17), we know that

Φ(1, a−k)

=1

2PkHk,s

∑

m 6=k

PmHm,sI{am=1}

+1

2

∑

k 6=m

PmHm,sI{am=1}PkHk,s

+1

2

∑

n6=k

∑

m 6=n,k

PnHn,sPmHm,sI{an=1}I{am=1}

+∑

n6=k

PnHn,sLnI{an=0},

and

Φ(0, a−k)

=1

2

∑

n6=k

∑

m 6=n,k

PnHn,sPmHm,sI{an=1}I{am=1}

+PkHk,sLk +∑

n6=k

PnHn,sLnI{an=0},

which implies that

Φ(1, a−k)− Φ(0, a−k)

=1

2PkHk,s

∑

m 6=k

PmHm,sI{am=1}

+1

2

∑

k 6=m

PmHm,sI{am=1}PkHk,s

−PkHk,sLn

= PkHk,s

∑

m 6=k

PmHm,sI{am=1}

−PkHk,sLk. (19)

Combining (18) and (19), we have that

Φ(1, a−k) < Φ(0, a−k).

Similarly, for the case that Vk(0, a−k) < Vk(1, a−k) for auser k, we can also show that Φ(0, a−k) < Φ(1, a−k).

8.5 Proof of Theorem 3

Let a ∈ Υ be an arbitrary Nash equilibrium. We musthave that Vn(a) ≤ Z l

n. Otherwise, if Vn(a) > Z ln, the

mobile device user n can always improve by choosingan = 0 and experiencing a cost of Z l

n, which contradictswith the fact that a is a Nash equilibrium. Thus, we have

that∑

n∈N Vn(a)) ≤∑N

n=1 Zln.

For an arbitrary computation offloading decision pro-

file a = (an, a−n) ∈∏N

n=1 An, if an = 0, we haveVn(a) = Z l

n. If an = 1, we have that

Rn(a) = W log2

(

1 +PnHn,s

ωn +∑

m∈N\{n}:am=1 PmHm,s

)

≤ W log2

(

1 +PnHn,s

ωn

)

,

which implies that

Zcn(a) =

(

γTn + γE

n Pn

)

Bn

Rn(a)+ γT

n Tcn,exe

≤

(

γTn + γE

n Pn

)

Bn

W log2

(

1 +PnHn,s

ωn

) + γTn T

cn,exe

= Zcn.

Thus, we know that Vn(a) ≥ min{Z ln, Z

cn} and

∑

n∈N Vn(a) ≥∑N

n=1 min{Z ln, Z

cn}. Then it follows that

PoA =maxa∈Υ

∑

n∈N Vn(a)

mina∈

∏

Nn=1

An

∑

n∈N Vn(a)

≤

∑Nn=1 Z

ln

∑Nn=1 min{Z l

n, Zcn}

.