Non-cooperative game approach for task offloading in edge clouds
aa r X i v : . [ c s . G T ] D ec JOURNAL OF L A TEX CLASS FILES, VOL. , NO. , SEPTEMBER 1
Non-cooperative game approach for taskoffloading in edge clouds
Bo Yang,
Member, IEEE , Zhiyong Li,
Member, IEEE , and Wenbin Liu
Abstract —Task offloading provides a promising way to enhance the capability of the mobile terminal (also called terminal user) that isdistributed on network edge and communicates edge clouds with wireless. Generally, there are multiple edge cloud nodes with distinctprocessing capability in a geographic area, which can offer computing service for various terminal users. Furthermore, the terminalusers are competitive and selfish, i.e., each user takes into account only maximizing her own profit, while conducting task offloadingstrategies. In this paper, we focus on the resource management optimization for edge clouds, and formulate the problem of resourcecompetition among terminal users as a non-cooperative game, in which the terminal user who acts as the player always pursues theminimization of the expected response time for her tasks by optimizing allocation strategies. We present the utility function of the userwith queuing theory, and then prove the existence of Nash equilibrium for the formulated game. Using the concept of Nash bargainingsolution to calculate the optimal task offloading scheme for the user, we propose a distributed task offloading algorithm with lowcomputation complexity. The results of simulated experiments demonstrate that our method can quickly reach the Nash equilibriumpoint, and deliver satisfying performance at the expected response time of the user’s tasks.
Index Terms —Edge cloud, expected response time, game theory, Nash bargaining solution, task offloading ✦ NTRODUCTION W I th the development of high-powered mobile ter-minals and 5G communication technologies, mobilecomputing has received extensive attention. In this scenario,the applications of the mobile user often are offloaded tothe cloud due to the capability constraint of the terminal.For example, to quickly respond the computing-intensivespeech recognition, Google Voice Search and Apple Sirialways request computing resources from cloud data centers[1]. However, they might suffer the risk of uncertain delaycaused by the network while the user’s tasks are transmittedto the remote cloud data center via multiple network nodes.To address the problem, the edge cloud computing (ECC)model has been emerged. Unlike traditional cloud comput-ing models, it transfers a number of computing resourcesfrom the cloud to network edges closest to the user, allowingtime-sensitive tasks to be processed nearby and avoiding theuncertainty of transmission delay [2].Due to the advantage of the proximate access for themobile user, ECC has been widely agreed to be a keytechnology for the mobile computing. However, comparedto traditional large-scale cloud computing, an ECC node isusually equipped with only small or medium size comput-ing resources because of the limit of the physical space.Hence, it cannot meet all users when the number of userrequests is relatively large. In fact, a mobile user can obtain • Bo Yang and Wenbin Liu are with the College of Information andManagement of Hunan University of Finance and Economics, Changsha,China, 410205.E-mail: { bo yang, zhiyong.li } @hnu.edu.cn. • Bo Yang and Zhiyong Li are with the College of Computer Science andElectronic Engineering of Hunan University, National SupercomputingCenter in Changsha, Key Laboratory for Embedded and Network Com-puting of Hunan Province, Changsha, China, 410082.Manuscript received xx xx, 2017; revised July xx, xxxx. identical service from various ECC nodes in a physical field,whereas a ECC node can also provide service for multipleusers. Accordingly, it is inevitable to consider the loadbalancing among ECC nodes, so that the system can providebetter experience for the users. The existing researches forECC mainly focused on the task offloading between themobile device and the edge cloud node, such as user ap-plication partitioning and offloading [3], [4], access control[5], [6] and so on. Although these researches in these areasare very important, they did not taken into account bothof the resource competition among users and the queuingdelay of user tasks in ECC systems.Without loss of generality, most of the mobile users in theECC system are time-sensitive and independent from eachother. The objective of the user is to minimize the expectedresponse time of her tasks while she makes decisions fortask offloading. However, to capture this objective, thereare several significant points that need further considera-tion. For examples, the users of ECC systems are free toact independently in a selfish manner, such that achievingsatisfactory experience for users is difficult due to no centralauthority in the system; the response time of the user tasknot only refers to its running time and transmission delay,but also includes its queuing delay on the ECC node. No-tably, the processing capability of the mobile terminal is farlower than that of the edge cloud node in fact. Nevertheless,the mobile users can also execute their own part of taskslocally when external computing resources provided by theedge cloud node cannot meet their requirements [3], [5].However, currently task offloading strategies completelyconsidering above scenarios have not been well studied.Therefore, it is still a challenge to design an efficient, sta-ble and distributed method of task offloading in the ECCsystem.
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Game theory can provide a natural paradigm to designdecentralized mechanisms [7], which can help obtain anin-depth analytical understanding of the task allocationproblem of ECCs. Accordingly, in this study, we introducethe game theoretic approach to address the aforementionedissues. In our model, ECC users, who act as players, com-pete with one another in sharing resources distributed inmultiple ECC nodes in a physical field. To minimize theaverage response time of tasks, they can decide whichECC nodes will process their requirements by observingthe running scenario, such as the available computing re-sources, the transmitting delay and the service levels of theirrequirements. Our main contributions are summarized asfollows.Our schemes consider the expected response time thatincludes both of the queueing time and the transmitting de-lay for tasks. We establish a queueing model to characterizethe computing node in the ECC system. The task offloadingproblem of each user is viewed as a centralized problem inour schemes, because each mobile user independently actswith a selfish manner. Furthermore, to ease the difficultyof solving the problem, Nash bargaining solution (NBS)method is adopted to calculate the optimal strategy for themobile user.Additionally, we propose a non-cooperative game frame-work for the mobile edge cloud system, in which eachmobile user can selfishly minimize her payoff by making op-timal strategy for her task offloading with the NBS method.We prove the existence of the Nash equilibrium of theproposed game. Moreover, corresponding algorithms aregiven to find the best response of the mobile user, and canconverge to an efficient equilibrium after several iterations.The experimental results demonstrate that our approachesare effective and efficient.The rest of this paper is organized as follows. We firstdiscuss related works in Section 2, and then introduce thesystem model in Section 3. We provide the formulation ofthe resources allocation problem and present the detaileddescription of the proposed algorithms in Sections 4 and 5,respectively. We discuss the simulation results that demon-strate the effectiveness of our approach in Section 6. Finally,the conclusions are drawn in Section 7.
ELATED W ORK
In recent years, due to the improvement of communica-tion technology and mobile terminals, more resources ofcloud computing have been transferred to the network edgeclosed to customers. The existing researches on ECCs aremainly focus on the problem model and resource manage-ment of the system [8]. In this section, we will discuss theassociated work from following aspects: the task partition,and the task offloading among multiple users in ECCs.Generally, a mobile computing task is consist of multipleprocedures, such as the augment reality (AR) task, whichhas five critical procedures: the video collection, the tracker,the mapper, the object recognizer, and the renderer. Amongthese procedures, the computation-intensive procedures,i.e., the tracker, mapper and object recognizer can be of-floaded for cloud execution, others can be performed locally [8]. In this way, the mobile user can enjoy various benefitsfrom ECCs using distinct partition algorithms. Hence, a se-ries of task offloading approaches are proposed to optimizethe system performance [3], [9]–[11].Ref [3] focused on the upper bound of the system per-formance, and investigated the problem of the task partitionand placement, in which the user’s task and the physicalcomputing system are mapped into two types of nodes thatare labelled as the requirement and the available resourcein a graph respectively, and then an approximate onlinealgorithm was proposed to solve the optimal matching forthe two types of nodes in this graph. Considering the sce-nario where the available resources are constrained in ECCs,authors of Ref [9] investigated the task partition methodsthat minimize the expected time of user tasks. Xiang et al. proposed a consolidated method to coordinate the taskpartition while conducting the task offloading strategy. Intheir work, saving energy is realized by reducing the timeof the net interface working at high power [10]. Chang et al. studied the task placement approach for the application ofInternet of Thing in the edge cloud, where the time-sensitivecomponents of the task are executed on the edge cloud nodeby splitting the task seamlessly [11].Above investigations primarily focused on minimizingenergy consumption and time delay, or considering thetradeoff of them. In fact, the multi-step task partitioningand placement in ECCs still encounter numerous challengeseven if it is a simple task partitioning and placement. More-over, these investigations mostly used heuristic algorithmsto solve such problems [12]. It is well known that thesemethods generally have high computing complexity, andhence can not guarantee the performance. Furthermore,above studies neglected the task congestion or queuingdelay, i.e., the user task arriving on an edge computingnode may have to wait for a certain time before it can beexecuted. It would significantly harm the user’s experienceif the waiting time is too long.For the scenario of multiple users, the popular schemesemployed by existing studies for the resource managementin ECCs can be classified into two types: centralization anddecentralization. In these investigations [13]–[15] with cen-tralized strategies, the edge cloud node has the completedinformation of all users, such as the resource request ofthe user, makes decisions for resource allocation and thensends them to users. Ref [14] studied resource allocation fora multiuser mobile edge cloud system, where the optimalresource allocation is formulated as a convex optimizationproblem for minimizing the weighted sum mobile energyconsumption under the constraint on computation latency.Chen et al. aimed to minimize the overall cost of energy,computation and delay for all users, and formulated theoptimization of the offloading decision as a non-convexquadratically constrained quadratic program [13]. Theirwork is further extended in the literature [15], e.g., thecomputation resource allocation and processing cost weretaken into account.For the investigations with decentralization approaches,Sardellitti et al. formulated the offloading problem as a jointoptimization of the radio resources and the computationalresources, and provided a distributed resource schedulingalgorithm using successive convex approximation technique
OURNAL OF L A TEX CLASS FILES, VOL. , NO. , SEPTEMBER 3 to minimize the overall users’ energy consumption, whilemeeting latency constraints [16]. Similarly, authors of lit-erature [17] proposed a distributed resource schedulingalgorithm to reduce energy consumption and shorten ap-plication completion time in mobile edge clouds. However,the above studies did not considered the queuing delay andthe user selfishness in the scenario of multiple users.Moreover, the non-cooperative game approach, as a typeof decentralized method, has obtained extensive attentionsin current mobile edge computing fields. Ref [5], [18] as-sumed that the user tasks can run on the local terminalor the edge cloud, modelled the competing traffic channelamong multiple users as a non-cooperative game, and pro-posed a task offloading approach being able to reach Nashequilibrium (NE). Cardellini et.al. investigated a scenarioin which multiple non-cooperative users share the limitedcomputing resources of close-by cloudlets in a fixed fieldand can selfishly determine to assign their computations toany of the three tiers, e.g., a local tier of mobile terminals, amiddle tier of ECCs, and a remote tier of cloud servers [19].Nevertheless, their work is for a single mobile edge cloud,and ignored a fact that there are multiple mobile edge nodesthat can simultaneously offer services for users in a physi-cal area. Considering a scenario that exists multiple userscompeting sharing computation resources provisioned bymultiple ECCs in a field, Li et al. employed the queuingmodel to characterize the mobile edge cloud server, andproposed a non-cooperative game algorithm to find the op-timal computation offloading strategy for the mobile usersand multiple ECCs [20]. Although similar technologies areused in our work, the difference from their work is that wecombine the Nash Bargain Solution (NBS) and the conceptof non-cooperative game to calculate the optimal allocationscheme of the user, such that the analytic solution of the taskallocation for each user can be quickly obtained.
YSTEM M ODEL
In this section, we introduce the models for the mobileedge cloud and user respectively. For the convenience ofthe readers, the major notations used in this paper are listedin Table 1.
Similar to WiFi APs, in a given physical area, we as-sume that there exists a mobile edge cloud system M , which has m edge cloud nodes (ECN), M = { ECN , ECN , · · · , ECN m } . An edge cloud node can beenvisioned as a small data center such as Cloudlet, which isable to provision service for the users closed to it, avoidingthe unacceptable delay yielded by the uncertain networkinfluence while the user task is transmitted to remote cloudsthrough multiple network nodes. Moreover, there are anumber of mobile users who can access certain edge cloudnodes in the area. Let N = { M U , M U , · · · , M U n } repre-sent the set of mobile users. Fig. 1 describes the model of theedge cloud system.Assume that each mobile user generates tasks in terms ofa Poisson process and independently of other mobile users.The task execution requirements (measured by the number TABLE 1Notations
Symbol Meaning i Subscript of the mobile user or terminal MU i Mobile user or terminal ij Subscript of the computing node(e.g., edge cloud node,local terminal) M Set of the edge cloud node m The number of edge cloud nodes N Set of the mobile user
ECN j Edge cloud node j ˆ µ i Available processing rate of MU i ˆ λ i Task arrival rate of MU i T li Average response time of the tasks deployed on localcomputing for MU i T ei Average response time of the tasks deployed on ECNsfor MU i T i Overall average response time of the tasks for MU i ρ i Task allocation probability vector for MU i ρ ij Probability assigning the task belonged to MU i to com-puting node jµ ij Processing rate that computing node j can provision to MU i H i Set of available computing nodes for MU i ρ − i Task allocation probability vector of the users excepteduser i L ij Mean delay of transmitting a task of MU i to computingnode jµ ij Available processing rate for MU i on computing node j Q Joined strategy set of all players T maxi Acceptable maximum value of the expected task re-sponse time for MU i µ ij Initial processing rate specified by MU i for computingnode j Local Edge Cloud Edge CloudLocal LocalLocalEdge Cloud
Fig. 1. Mobile edge cloud model of instructions to be executed) are i.i.d. exponential randomvariables r with mean ¯ r . An edge cloud node j performsthe user task with speed s j (measured by the number ofinstructions executed in one unit of time), and the executiontimes on the node are i.i.d. exponential random variables x j = r/s j with mean x j = r / s j . Hence, the averageprocessing rate of the edge cloud node in one unit of timecan be denoted by µ j = 1/ x j . Moreover, an edge cloudnode maintains a queue with infinite capacity for waitingtasks, where the first-come-first-served queuing disciplineis adopted. Such an edge cloud node can be modeled as anM/M/1 queuing system.Notably, considering the fact that the service time ofuser tasks obeys general distributions, the M/G/1 queuingmodel should be more appropriate for them. However, there OURNAL OF L A TEX CLASS FILES, VOL. , NO. , SEPTEMBER 4 is still not a simple close-form solution for this queuingmodel, which can help us to understand the effect of thetask size variability on the mean response time [21].Let λ j denote the task arrival rate on edge cloud node ECN j with an average processing rate µ j . In terms ofLittle law [21], the average response time of ECN j can becalculated by T j = 1 µ j − λ j . (1)For the system stability, the task arrival rate must belower than the processing rate in an ECN. Hence, thefollowing inequality should be hold. λ j < µ j , ∀ j ∈ M . (2)Otherwise, the queue at the ECN will build up to infinityand the expected response time for the user task will beinfinite. Generally, the edge cloud user has a certain computingcapability, which is far lower than that of the ECN, andcan also offload her tasks to closed ECNs or remote cloudservers by the wireless network, while having to tackle thetasks with realtime requirements. Without loss of generality,in order to maximize her own profit, an edge cloud usermay distribute strategically her tasks to suitable points (suchas closed ECNs and her own intelligent terminal). Fig. 2characterizes the model of the edge cloud user.
Local Edge CloudTasks (cid:79)
Fig. 2. Edge cloud user model
Minimizing the expected response time of the task hasbecome an inevitable problem in the mobile edge computingdue to the time-sensitive requirement of the task. Notably,saving energy is also extremely significant in the scenarioswith constrained energy provisioning. Nevertheless, it isno longer the primary goal in numerous edge computingoccasions, e.g., pilotless automobile, speech recognition, ar-gument reality and so on, even it may be negligible in thesecases. Accordingly, for simplification, we do not take energysaving into account in this study.Moreover, a user terminal with a certain computingcapability can be regarded as a small server. Hence, we alsoemploy the queuing model to characterize it. Let ˆ λ i denotethe task arrival rate of M U i in a time interval, and ˆ µ i is heraverage processing rate for the task. Given that the quantityof tasks assigned to ECNs by M U i is λ ei , then the number oftasks remaining at her own terminal is denoted by λ li , whichis written by λ li = ˆ λ i − λ ei , (3) where λ ei ∈ [0 , ˆ λ i ] . (4)According to Eq. (1), the average response time of tasksdeployed on local terminal for M U i is given as T li = 1ˆ µ i − λ li . (5)While M U i ’s tasks are deployed on ENCs for execution,the total time overhead for these tasks mainly consists ofthe processing time T pi and the transmitting time T ti , and isgiven as T ei = T pi + T ti . (6)Here, the returned time of computing results is omitted.In our models, the mobile user, as an intelligent and selfishindividual, is always greedy to search the computing nodes,which can promise the maximization of her own interests,and then allocates her tasks to them such that her interestsare maximized. That is, minimizing the average responsetime of the task means the maximization of user interests inour work.Furthermore, Each mobile user should be aware of thefact that the ECNs are serving other mobile users, whilemaking decisions of task offloading. Hence, the interestsof each mobile user rely on the strategies of others. It isreasonable that using the game theory to model the compe-tition among the mobile users, which can help obtain an in-depth analytical understanding of the service provisioningproblem of ENCs. ON - COOPERATIVE GAME FOR TASK OFFLOAD - ING
From aforementioned discussions, the mobile user can beviewed as the player in the non-cooperative game, andselfishly makes strategy profiles to maximize her interest.Following the routine, the following problem is how to for-mulate an non-cooperative game, which can quickly reacha stable situation where the task offloading strategy of eachuser is optimal and no one wants to change it.
In the edge computing situation, a mobile user can dis-tribute her tasks to any closed computing nodes includingECNs and her own terminal. For representation conve-nience, throughout the paper, the edge cloud node and userlocal terminal are collectively called computing nodes. Theset of available computing nodes for
M U i is represented by H i = M ∪ {
M U i } .In our models, the mobile users are rational, and actindependently in a selfish manner that maximizes their owninterests. Let ρ i denote a probability profile determining M U i ’s task allocation. To minimize the task response time, M U i will do her best to achieve an optimal ρ i . ρ i = { ρ i , · · · , ρ ij , · · · , ρ iH } , where ρ ij denotes the probability by that M U i assigns hertasks to computing node j , and P Hj =1 ρ ij = 1 , H = m + 1 .The strategy profile set of M U i is given as OURNAL OF L A TEX CLASS FILES, VOL. , NO. , SEPTEMBER 5 Q i = ρ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H X j =1 ρ ij = 1 and ρ ij ≥ . (7)While having determined the task allocation probabilityvector ρ i , M U i will assign her tasks to computing nodes interms of it. In addition, the available computing capabilityof each computing node for M U i is represented as follows: S i = { µ i , · · · , µ ij , · · · , µ iH } , where µ ij represents the processing rate that computingnode j can provision to M U i .Let the strategy profile of other mobile users is ρ − i = { ρ , · · · , ρ i − , ρ i +1 , · · · , ρ n } , and the time of transmitting atask from M U i to computing node i is denoted by L ij (if thedestination is herself, i.e., i = j , then L ij = 0 ). CombiningEqs (1) and (6), the utility function of M U i , i.e., her overallexcepted response time of the task, can be written as T i ( ρ i , ρ − i ) = H X j =1 ρ ij µ ij − ρ ij ˆ λ i + ρ ij ˆ λ i L ij ! (8)Obviously, the mobile user, as a selfish individual, isbound to seek an optimal strategy to maximize her interests.Here, the optimal strategy of M U i is defined as follows Definition 4.1. ( Optimal Strategy ) Given the strategy profilesof other mobile users ρ − i , M U i ’s optimal strategy is ρ ∗ i ∈ Q i , ifshe prefers ρ ∗ i to any other strategy ρ i ∈ Q i . That is T i ( ρ ∗ i , ρ − i ) ≤ T i ( ρ i , ρ − i ) , i = 1 , · · · , n. (9)Generally, a non-cooperative game comprises of the setof players, the strategy of the player and the set of players’strategies [7]. In our models, the mobile user is the player inthe game, the player collection is denoted by N , the strategyset of each player M U i is given by Q i , and the joined strat-egy set of all players is represented by Q = Q × · · · × Q n .Formally, we characterize the above game as a 2-tuple G = hQ , T i , where T = ( T , · · · , T n ) . Given the joinedstrategies of other participants ρ − i , the objective of M U i is to achieve the optimal strategy ρ ∗ i ∈ Q i , which minimizesthe excepted response time of her tasks, T i ( ρ i , ρ − i ) . That is ρ ∗ i ∈ arg min ρ i ∈Q i T i ( ρ i , ρ − i ) , ( ρ i , ρ − i ) ∈ Q . (10)The Nash equilibrium has a beneficial self-stability prop-erty, such that all selfish players at equilibrium can achieve amutually satisfactory solution and no one has the incentiveto change anymore. Therefore, given that the jointed opti-mal strategies of other participants ρ ∗− i , we have followinginequality for the mobile user M U i at the Nash equilibrium. T i ( ρ ∗ i , ρ ∗− i ) ≤ T i ( ρ i , ρ ∗− i ) , i = 1 , · · · , n. (11)Obviously, the existence of the Nash equilibrium in anon-cooperative game is an essential condition ensuring thesystem stability. Hence, more attentions should be placed onit while designing a distributed task offloading mechanismwith the game theory. Following aforementioned discussions, the existence of theNash equilibrium implies that the approach proposed byus is feasible. Hence, we will discuss and analyse the Nashequilibrium existence for the non-cooperative game G = h Q, T i in this subsection. Lemma 4.1.
For each
M U i , given a convex and compact strategyset Q i , and the function of the task expected response time T i ( ρ i , ρ − i ) , which is continuously differentiable on ρ i ∈ Q i , thefunction T i ( ρ i , ρ − i ) is convex on ρ i ∈ Q i for any strategy profile ρ − i .Proof. Obviously, the strategy set of each users Q i is con-vex and compact in our game model, and the function ofthe task expected response time T i ( ρ i , ρ − i ) is continuouslydifferentiable on ρ i . In terms of literatures [22], [23], ifthe Hessian matrix of the function T i ( ρ i , ρ − i ) is positivesemidefinite,then the above lemma would hold.According to Eq. (8), the first derivative of the function T i ( ρ i , ρ − i ) for M U i ’s strategy vector ρ i is given as ∇ ρ i T i ( ρ i , ρ − i ) = (cid:20) ∂T i ( ρ i , ρ − i ) ∂ρ ij (cid:21) Hj =1 = (cid:18) ∂T i ( ρ i , ρ − i ) ∂ρ i , . . . , ∂T i ( ρ i , ρ − i ) ∂ρ iH (cid:19) = µ i ( µ i − ρ i ˆ λ i ) + 2 ρ i ˆ λ i L i ,. . . , µ iH ( µ i − ρ iH ˆ λ i ) + 2 ρ iH ˆ λ i L iH ! , and its Hessian matrix is expressed as ∇ ρ i T i ( ρ i , ρ − i ) = diag ((cid:20) ∂ T i ( ρ i , ρ − i ) ∂ ( ρ ij ) (cid:21) Hj =1 ) = diag " µ ij ˆ λ i ( µ ij − ρ ij ˆ λ i ) + 2ˆ λ i L ij Hj =1 . (12)We learn that inequality µ ij > ρ ij ˆ λ i must hold in terms ofExpression (2); otherwise, the queue will build up to infinityand the expected response time for the task will be infinite.Accordingly, the diagonal matrix in Eq. (12) has all diagonalelements being positive. Hence, the Hessian matrix of thefunction T i ( ρ i , ρ − i ) is positive semidefinite and the resultfollows.This remark completes the proof. Theorem 4.1.
Non-cooperative game G = h Q, T i exists a Nashequilibrium at least.Proof. In terms of the results of literatures [24], [25], an non-cooperative game existing a Nash equilibrium must satisfytwo conditions: first, the strategy set Q i for each user M U i isan non-empty convex closed and upper-bounded subset onEuclidean space; second, given the joined strategies of otherusers ρ − i , the utility function of M U i , i.e., the function of thetask expected response time T i ( ρ i , ρ − i ) is continuously dif-ferentiable and convex for any strategy ρ i ∈ Q i . Obviously,the two conditions are satisfied in the game G = h Q, T i according to lemma 4.1 and the result follows. OURNAL OF L A TEX CLASS FILES, VOL. , NO. , SEPTEMBER 6 this remark completes the proof.
LGORITHM DESIGN
In this section, we focus on the implement of the distributedtask offloading approach for the non-cooperative game G formulated above. Here, a decentralized task offloadingframe promising the maximization of the user utility isproposed. Before the system arrives a Nash equilibrium, every stepof improvement update carried by the participant in thegame G is to minimize the task expected response time,such that her utility reaches maximization. That is, the bestresponse of each participant is the solution for the followingoptimization problem (labeled P1).P1 minimize T i ( ρ i , ρ − i ) , i = 1 , · · · , n. (13)However, it is significantly difficult to directly addressthe above problem. In fact, while making the optimal de-cision for the task offloading at each step of improvementupdate, every participant considers only the computing re-sources being available for herself, but the affection yieldedby others’ strategies on her.Hence, the optimization problem for each participantmay be viewed as a centralized decision problem. Takingadvantage of Nash bargaining solution (NBS) addressingsuch problems [26]–[28], we will adopt the NBS approach tosolve it in our work.Generally, each user has a maximum toleration for theaverage response time of the task. Let T maxi denote it for M U i . Therefore, the following constraint for any user mustbe met when she conducts task allocation on a computingnode. µ ij − λ maxij + λ maxij L ij ≤ T maxi , (14)where λ maxij is the maximum rate with that M U i can assignher tasks to computing node j under given constraints, suchas T maxi , µ ij and L ij . The available processing rate µ ij can beobtained by observing the current state of computing node j by M U i . By several algebraic calculation, We have thefollowing results in terms of the above inequality. λ maxij = T maxi + L ij µ ij ± ω L ij , (15)where ω = q ( T maxi + L ij µ ij ) − L ij ( T maxi µ ij − . Notably, the following constraints must be satisfied. µ ij > λ maxij ≥ , (16) T maxi − λ maxij L ij > . (17)Otherwise, the system will be impractical. Constraint(16) guarantees the non-negative of the performance pro-vided to M U i by computing node j . Expression (17) impliesthat the transmitting delay can not exceed to the maximum value being acceptable for the user. Therefore, Equation (15)can be rewritten as λ maxij = T maxi + L ij µ ij − ω L ij . (18)If either of expressions (16) and (17) cannot be satisfiedon computing node j , then the maximum rate λ maxij withthat M U i can assign tasks to this computing node shouldbe set to 0.For applying the NBS method in our game model, theinitial processing rate on computing node j for M U i isdenoted by µ ij , which is the maximum processing rate thatcomputing node j can provision to M U i without violatingconstraints (16) and (17). Obviously, there is µ ij = λ maxij ,those computing nodes with initial processing rate 0 will beremoved. Moreover, because the initial processing rate hasconsidered the maximum transmitting delay of tasks, thetraffic affect on the task response time is omitted to reducethe difficulty of our problem in the following NBS method.According to the above propositions and the NBS defi-nition [29], formally, problem P1 can be converted into thefollowing optimization problem (labeled P2).P2 minimize − H X j =1 ln( µ ij − ρ ij ˆ λ i ) , i = 1 , · · · , n, (19)subject to X Hj =1 ρ ij = 1 , (20) ρ ij ≥ , j = 1 , · · · , H, (21) ρ ij ˆ λ i ≤ µ ij , j = 1 , · · · , H. (22)As the utility function of the mobile user is reset byExpression (19), let T ′ denote their new utility functioncollection, the non-cooperative game G can be rewritten as G ′ = hQ , T ′ i . Theorem 5.1.
Non-cooperative game G ′ = h Q, T ′ i exists aNash equilibrium at least.Proof. Through observing the formulation of P2, we canlearn that Q i , as the strategy set of M U i , is a non-emptyconvex closed and upper-bounded subset on Euclideanspace, and the objective function Eq. (19) is also continu-ously differentiable and convex for any strategy ρ i ∈ Q i .Therefore, similar to the proof of Theorem 4.1, we can alsoassert that the game G ′ has at least a Nash equilibrium.This remark completes the proof.Given µ ij = λ maxij , ∀ j ∈ { , · · · , H } and sorting allavailable computing nodes for M U i in descending order oftheir initial performance µ i ≥ µ i ≥ · · · ≥ µ iH , we havethe following conclusion. Theorem 5.2. the solution ρ i = { ρ i , · · · , ρ ij , · · · , ρ iH } ofoptimization problem P2 is given by ρ ij = µ ij ˆ λ i − P kj =1 µ ij − ˆ λ i k ˆ λ i , j = 1 , · · · , k ;0 , j = k + 1 , · · · , H. (23) OURNAL OF L A TEX CLASS FILES, VOL. , NO. , SEPTEMBER 7 where k ∈ { , · · · , H } is the largest number exactly satisfyingthe following expression. µ ik > P kj =1 µ ij − ˆ λ i k . (24) Proof.
In our models, assume that the total arrival rate oftasks generally does not exceed the total processing rateof the systems. Hence, the constraint (22) is omitted in ourproblem.Let θ ≥ , η j ≥ , j ∈ { , , · · · , H } denote the La-grange multipliers. The Lagrangian of problem P2 is givenas follows: L ( ρ i , · · · , ρ iH , θ, η , · · · , η H )= − H X j =1 ln( µ ij − ρ ij ˆ λ i ) − θ ( H X j =1 ρ ij − − H X j =1 η j ρ ij . (25)The first-order Kuhn-Tucker conditions and constraintsare given as follows: ∂L∂ρ ij = ˆ λ i µ ij − ρ ij ˆ λ i − θ − η j = 0 , (26)subject to the constraints ∂L∂θ = X Hj =1 ρ ij − , (27) η j ρ ij = 0 , η j ≥ ρ ij ≥ , j = 1 , · · · , H. (28)In terms of Eq. (28), if ρ ij = 0 , we have η j ≥ ; otherwise η j = 0 . Therefore, combining the Equations (26) and (28),we have ˆ λ i µ ij − ρ ij ˆ λ i (cid:26) = θ, if ρ ij > ≥ θ, if ρ ij = 0 . (29)This implies that those computing nodes with lowerperformance can be out of the service for M U i , because notasks are assigned to them. Hence, only ρ ij > needs tobe considered under this case. Let k is an integer promising ρ ij > for all j ∈ { , · · · , k } . The Kuhn-Tucker conditionsis rewritten as: ˆ λ i µ ij − ρ ij ˆ λ i = θ, j = 1 , · · · , k, (30)subject to k X j =1 ρ ij = 1 . (31)By adding the equation given by Eq. (30) for all j ∈{ , · · · , k } and some algebraic calculation, we have θ = k ˆ λ i P kj =1 µ ij − ˆ λ i . (32)Using the above result in Eq. (30), we can obtain ρ ij = µ ij ˆ λ i − P kj =1 µ ij − ˆ λ i k ˆ λ i , j = 1 , · · · , k. (33)If we search k from H to 1, it is easy to determine thelargest number k satisfying the condition µ ik > P kj =1 µ ij − ˆ λ i k , which guarantees ρ ij > for all j ∈ { , · · · , k } .This remark completes the proof.Following the result of Theorem 5.2, the Optimal task of-floading method for M U i is proposed in Algorithm 1 (called OTOM ). In this algorithm, firstly the initial processing per-formance of each computing node is determined, and thensorted in descending order (lines 2-3). Appropriate comput-ing nodes for
M U i are picked to deploy tasks such that herutility can achieve maximization (lines 4-8). It is obviousthat those computing nodes with lower performance willbe removed from the service for M U i , once the algorithmis finished. Final, each selected computing node is set to atask allocation probability, others are set to 0 (lines 9-11). Asfor the complexity of Algorithm 1, its overhead primarilyfocuses on the computation of sorting for computing nodesand determining the computing nodes participating in theservice with dual loops. Hence, the computation complexityof Algorithm 1 is O ( m log( m )) . Algorithm 1
OTOM( S i , L i , T maxi , ˆ λ i ): Optimal task offload-ing method for M U i Input:
The available processing rate for
M U i on each comput-ing node: S i = { µ i , · · · , µ iH } .The mean delay of M U i transmitting tasks to eachcomputing node: L i = {L i , · · · , L i H } .The acceptable maximum response time for M U i : T maxi .The mean task arrival rate on M U i : ˆ λ i . Output:
M U i ’ task allocation vector: ρ i = { ρ i , · · · , ρ iH } . ρ i ← { , · · · , } , initialize M U i ’ task allocation vector Calculate the initial performance of each computingnode according to Eq. (18), S ← { µ i , · · · , µ iH } Sort S in descending order υ ← P | S | j =1 µ ij − ˆ λ i | S | while υ > µ i | S | do S ← S \{ µ i | S | } υ ← P | S | j =1 µ ij − ˆ λ i | S | end while for j ← to | S | do ρ ij ← µ ij − υ ˆ λ i end for return ρ i In this subsection, we proposed a distributed task offload-ing algorithm (called
DITOA ) for non-cooperative game G ′ , which are elaborated in Algorithm 2. The players (themobile users) in the game act asynchronously. First, theexpected task response time and allocation probability forall players are preset to 0 (lines 1-2). Each player employs OTOM algorithm to update her own optimal strategy inround-robin manner (lines 5-10). Specially, to recalculatethe task offloading strategy, we reduce the number of tasks
OURNAL OF L A TEX CLASS FILES, VOL. , NO. , SEPTEMBER 8 that were assigned to each computing node in the previousround (line 7). Moreover, a deviation ξ , as the conver-gence condition, is built to measure whether the systemhas reached a Nash equilibrium. That is, the algorithm getsthrough when the cumulative deviation difference betweentwo adjacent iterations is less than ξ (line 16). While havingcompleted the strategy update, the mobile user computesthe cumulative deviation and then broadcasts it to otherplayers (lines 11-13). In just doing so, other players candecide whether to continue updating their task offloadingstrategies based on the advertised information in practice. Algorithm 2
DITOA( ξ ): Distributed task offloading algo-rithm Input:
Acceptable deviation: ξ T i ← , ∀ i ∈ { , · · · , n } ρ i ← { , · · · , } , ∀ i ∈ { , · · · , n } repeat sum ← for each M U i do Obtain free service rate { µ i , · · · , µ iH } and transmit-ting delay L i ← {L i , · · · , L i H } by inspecting theavailable computing nodes for M U i µ ij ← µ ij + ρ ij ˆ λ i , ∀ j ∈ { , · · · , H } S i ← { µ i , · · · , µ iH } ρ i ← OTOMU ( S i , L i , T maxi , ˆ λ i ) Update the task offloading strategy by ρ i Calculate response time T i ’ using Eq. (8) sum ← sum + | T ′ i − T i | Advertise cumulative deviation sum to all players T i ← T ′ i end for until sum ≤ ξ The above algorithm can be started periodically or whensystem parameters change. For example, when the taskarrival rate on a mobile user changes, she runs immedi-ately the algorithm to update her task offloading strategy,and then broadcast the deviation to others. Their strategieswill continue to be updated until the Nash equilibrium isreached. Besides, the processing rate on the edge cloudnodes also need to be reported to mobile users in ourmethods. The measurement of the processing rate can referto the literatures [30]–[32]. In addition, given that the mobileusers employ
OTOM to calculate their optimal strategiesin Algorithm 2, the computation complexity of
DITOA is O ( nm log( m )) . After several iterations, the algorithm willconverge to an efficient equilibrium, and then terminates. XPERIMENTAL R ESULTS
In the section, we conduct extensive simulation experimentsto measure the performance of the approaches proposedby us, which include the expected task response time, theconvergence speed of the proposed algorithms and so on.For comparison purposes, we also implemented twoalternative task allocation methods: proportional-schemealgorithm (called PS) proposed by the literature [33]; globaloptimal scheme (denoted as GOS) employed by the litera-tures [30], [34], [35]. Our method is labeled DITOA. The PS algorithm runs in such a manner that assignstasks to each computing node in proportion to its processingrate. Hence, the faster the processing rate of the computingnode is, the greater the probability of assigning tasks to itis. Allocating tasks to computing node j with PS algorithm, M U i can employ following expression to calculate the taskallocation probability. ρ ij = µ ij P Hj =1 µ ij . (34)With respect to the global optimal scheme, it is a central-ized method, and its objective is to minimize the expectedresponse time for tasks in our experiments. However, takingthe traffic delay of tasks into account, it is significantlydifficult to achieve the optimal solution with GOS in ourproblem. For simplification, the transmitting delay is omit-ted when we perform our experiments using GOS. In our simulation experiments, assume that there are sixedge cloud nodes in a fixed physical field. The average pro-cessing rate of each edge cloud node follows an exponentialdistribution, which is elaborated in Table 2.
TABLE 2The average processing rate of edge cloud nodes
Edge cloud nodes 1 2 3 4 5 6Average processingrate (tasks/s) 80 60 100 160 90 70
For the task transmitting delay L ij , we configure it asa random number distributed evenly in the interval [5ms,100ms]. The maximum acceptable response time T maxi for M U i , i ∈ , · · · , n , is randomly generated in the interval[200ms, 280ms]. Additionally, the number of mobile usersin our experiments is set to 20, the task arrival rate of eachuser equals to the product of her scaling factor f i and theaggregated task arrival rate λ . For example, the task arrivalrate of M U i can be written as λ i = λf i . The scaling factor f i , i = 1 , · · · , n , and the local processingrate for each mobile user are given in Table 3. TABLE 3Scaling factors f i and the local processing rate for each user Mobile users 1-3 4-7 8-12 13-17 18-20 f i Since the game method proposed by us needs multipleiterations to reach the Nash equilibrium, the convergencerate of the algorithm becomes a significant metric estimatingits performance. In this subsection, we will estimate theconvergence rate of our methods on two aspects: acceptable
OURNAL OF L A TEX CLASS FILES, VOL. , NO. , SEPTEMBER 9 deviate ξ and the load rate of the system. The experimentalparameters consist to the configuration provided above.For comparison, two types of system initial scenariosare considered in our simulation experiments. The first isthe initial 0 in which the computing nodes are not assignedany task; the second is the initial P in which the tasks havebeen assigned to each computing node in proportion to theirprocessing rates.Firstly, given an acceptable deviate ξ = 0 . , whichmeans the stop condition of our method and the arrival ofthe Nash equilibrium, we estimate the convergence rates ofour algorithm on different system utilization rates, whichare varied in the interval [0.1,0.7] with step 0.1. More-over, the transmitting delay and the maximum acceptableresponse time for users are generated randomly in ourexperiments, hence, each experiment on different systemutilization rates is carried out 100 times and its averageresults is described in the Fig. 3. E x pe c t ed nu m be r o f i t e r a t i on s System utilization Initial_0 Initial_P
Fig. 3. System utilization rate and convergence rate
Fig. 3 shows that given the convergence condition ξ =0 . , the average number of iterations of DITOA increaseswith the increase of the system utilization rate. The conver-gence rates varies evenly after the system utilization ratereaches 0.2. Furthermore, we can observe that the averagenumber of iterations in the case of initial P is lower than thatof initial 0 in Fig. 3. Generally, the task distribution obeyingthe case of initial P is consist with the real scenario wherethe initial state of task allocation on computing nodes isproportional to their processing rate, so the experimentalresults demonstrate the effectiveness of our schemes.Next, given the system utilization rate is set to 0.5, weestimate the convergence rate of our methods on distinctacceptable deviates. Similar to the above experiments, werun the experiments 100 times and then calculate the av-erage number of iterations as experimental results. Fig. 4shows that the convergence rate of initial P is prior to thatof initial 0 . It implies that the system in the case of initial P is more close to the Nash equilibrium. Additionally, whilethe acceptable deviate is set to 0.01, the average number ofthe iteration of our algorithm is 18. The above experimentresults demonstrate that the algorithm proposed by us canconverge quickly in an acceptable deviate range. E x pe c t ed nu m be r o f i t e r a t i on s Deviation Initial_P Initial_0
Fig. 4. Acceptable deviate and convergence rate
In this subsection, we will estimate the average responsetime of user tasks by varying the utilization rate of thesystem. The parameters of simulation experiments are givenin terms of previous experiment configurations. Consider-ing that the transmitting delay and the acceptable deviateare set randomly, similar to the above experiments, eachsimulation experiment is also carried out 100 times. Theconvergence condition (acceptable deviate) ξ is set to 0.001in the experiments. E x pe c t ed r e s pon s e t i m e ( s e c ) The users(0.1X system utilization) (a) DITOA E x pe c t ed r e s pon s e t i m e ( s e c ) The users(0.1X system utilization) (b) PS
Fig. 5. 0.1X system utilization rate and the average response time ofuser tasks
OURNAL OF L A TEX CLASS FILES, VOL. , NO. , SEPTEMBER 10 E x pe c t ed r e s pon s e t i m e ( s e c ) The users(0.4X system utilization) (a) DITOA
A B C D E F G H I J K L M N O P Q R S T0.000.020.040.060.080.100.120.14 E x pe c t ed r e s pon s e t i m e ( s e c ) The users(0.4X system utilization) (b) PS
Fig. 6. 0.4X system utilization rate and the average response time ofuser tasks
Figs. 5 , 6 and 7 present the average task response timewith standard deviate for PS and DITOA methods, whenthe system utilization rates are at 0.1X, 0.4X and 0.7X respec-tively. The experimental results show that the expected taskresponse time of our method is lower than that of PS methodat distinct system utilization rates. The main reason is thatthe strategy employing by users in our method is optimalfor individuals, rather than PS method is not optimal forindividuals due the task allocation in proportion to theprocessing rates of computing nodes. Accordingly, it makesthe performance of our method is superior to that of PS.According to the previous configurations, GOS methoddoes not need to be executed repeatedly because its trans-mitting delay is omitted. Furthermore, the result yielded byGOS is the optimal average response time of the whole sys-tem. Hence, GOS is not included in the above experiments.It is obvious that the task response time of each user in PSand DITOA is different. To compare PS, DITOA and GOS,we adopt the following expression to normalize their overallaverage response time of the tasks. P ni =1 ˆ λ i n X i =1 ˆ λ i T i ( ρ i , ρ − i ) . (35)Fig. 8 describes the overall average response time ofthe task for the three methods at distinct system utilizationrates. It can been seen that the task response time of GOSis the lowest, and PS is the highest. The primary reason E x pe c t ed r e s pon s e t i m e ( s e c ) The users(0.7X system utilization) (a) DITOA E x pe c t ed r e s pon s e t i m e ( s e c ) The users(0.7X system utilization) (b) PS
Fig. 7. 0.7X system utilization rate and the average response time ofuser tasks is that the transmitting delay is ignored in GOS method.Specially, while the system utilization rate is low, the taskprocessing time on the computing node is far less than thetask transmitting delay in DITOA and PS. Obviously, GOS issuperior to the other two methods. Moreover, as the increaseof the system utilization rate, the difference between theprocessing time and the transmitting delay decreases, andthe performance of DITOA approaches that of GOS. Insome senses, GOS can be regarded as the optimizationbenchmark in our experiments. Therefore, the experimentalresults validate the effectiveness of our methods.
ONCLUSION
In this paper, we investigated the task offloading problemin the mobile edge cloud computing. We employ NBS ap-proach to compute the mobile user utility, and then present anon-cooperative game framework and associated algorithmto address the problem, such that each mobile user canminimize the expected response time of tasks in a com-promise scenario (approaching a Nash equilibrium point).The presented algorithm has relatively low complexity anddistribution execution characteristic, and thus, can be easilyimplemented to improve the reliability and robustness ofthe system. The effectiveness of our approach was assessedby performing simulated experiments. The experimentalresults demonstrated that our approach could outperform
OURNAL OF L A TEX CLASS FILES, VOL. , NO. , SEPTEMBER 11 A gg r ega t ed e x pe c t ed r e s pon s e ( s e c ) System utilization DITOA PS GOS
Fig. 8. System utilization rate and the overall average response time ofthe task alternative popular methods achieving better performance.Besides, our proposed approach can be applied to otherresource allocation models.In the future, we plan to explore the VM or containermigration among multiple mobile edge clouds as an exten-sion of our work. We are also interested in implementing aprototype allocation system in an experimental edge cloudplatform to further study the performance of our proposedapproach. A CKNOWLEDGMENTS
This work was partially supported by the NationalNatural Science Foundation of China (No. 61173107,91320103, 61672215, U1613209), National High-tech R&DProgram of China (863 Program) (No. 2012AA01A301-01),the Special Project on the Integration of Industry,Education and Research of Guangdong Province, China(No.2012A090300003), Science research project of Depart-ment Education of Hunan Province, China (No.2016C0270),Social Science Foundation of Hunan Province, China (No.16YBA050) and the Science and Technology Planning Projectof Guangdong Province, China (No.2013B090700003). Thecorrespondence author is Zhiyong Li. R EFERENCES [1] F. Liu, P. Shu, H. Jin, and L. Ding, “Gearing resource-poor mobiledevices with powerful clouds: architectures, challenges, and appli-cations,”
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