A Bilateral Game Approach for Task Outsourcing in Multi-access Edge Computing
Zheng Xiao, Dan He, Yu Chen, Anthony Theodore Chronopoulos, Schahram Dustdar, Jiayi Du
aa r X i v : . [ c s . D C ] A ug A Bilateral Game Approach for Task Outsourcingin Multi-access Edge Computing
Zheng Xiao*,
Member , IEEE , Dan He, Yu Chen, Anthony Theodore Chronopoulos,
Senior member , IEEE ,Schahram Dustdar,
Fellow , IEEE , and Jiayi Du,
Member , IEEE
Abstract —Multi-access edge computing (MEC) is a promising architecture to provide low-latency applications for future Internet ofThings (IoT)-based network systems. Together with the increasing scholarly attention on task offloading, the problem of edge servers’resource allocation has been widely studied. Most of previous works focus on a single edge server (ES) serving multiple terminal entities(TEs), which restricts their access to sufficient resources. In this paper, we consider a MEC resource transaction market with multipleESs and multiple TEs, which are interdependent and mutually influence each other. However, this many-to-many interaction requiresresolving several problems, including task allocation, TEs’ selection on ESs and conflicting interests of both parties. Game theory canbe used as an effective tool to realize the interests of two or more conflicting individuals in the trading market. Therefore, we proposea bilateral game framework among multiple ESs and multiple TEs by modeling the task outsourcing problem as two noncooperativegames: the supplier and customer side games. In the first game, the supply function bidding mechanism is employed to model the ESs’profit maximization problem. The ESs submit their bids to the scheduler, where the computing service price is computed and sent tothe TEs. While in the second game, TEs determine the optimal demand profiles according to ESs’ bids to maximize their payoff. Theexistence and uniqueness of the Nash equilibrium in the aforementioned games are proved. A distributed task outsourcing algorithm(
DTOA ) is designed to determine the equilibrium. Simulation results have demonstrated the superior performance of
DTOA in increasingthe ESs’ profit and TEs’ payoff, as well as flattening the peak and off-peak load.
Index Terms —Multi-access edge computing (MEC), Internet of Things (IoT), Task outsourcing, Bidding mechanism, Noncooperativegame, Nash equilibrium. ✦ NTRODUCTION
The Internet of Things (IoT) is a system of interrelatedtens of billions of resource-hungry terminal entities (TEs),such as, sensors, wearable devices and unmanned aerialvehicles, which transfer data over a network with little orno human intervention. With the development of TEs andwireless networks, the demand for low-latency computingservices has been growing exponentially. This paves the wayfor the development of Multi-access edge computing (MEC).Multi-access edge computing (MEC) enables a power-ful cloud at the edge of the network. MEC decentralizesnetworks and allows any enterprise or mobile operator toplace a cloud at the edge, adjacent to the user. In the MECparadigm, plenty of machines are placed at the edge of thenetwork so that computing services can be deployed onthem for fast execution [1]. The MEC locates edge servers(ESs) with limited storage and computing resources at theedge of networks. Since the computation capabilities andbattery lives of TEs are limited, TEs offload computationallyintensive tasks (e.g., program execution) to ESs (e.g., 4G/5Gbase stations). The ESs execute these offloaded tasks and • *Corresponding Author • Zheng Xiao, Dan He, Yu Chen and Jiayi Du are with College of ComputerScience and Electronic Engineering, Hunan University, Hunan, China,410082. E-mail: { zxiao, danhe } @hnu.edu.cn, [email protected],[email protected],. • Anthony Theodore Chronopoulos is with Department of Computer Sci-ence, University of Texas, San Antonio, TX, USA 78249 and DeptComputer Engineering Informatics 26500 Rio, University of Patras,Greece. Email: [email protected]. • Schahram Dustdar is with Distributed Systems Group, Vienna Universityof Technology, Vienna, Austria. Email: [email protected]. return results to TEs. However, to take full advantage ofthese available computational resources of ESs, TEs’ tasksneed to be allocated appropriately [2] .Due to the resource constraints of ESs, some computa-tionally intensive tasks will be offloaded to cloud servers(CSs), which are normally distantly located. In that case,a higher transmission latency may be generated, whichseriously degrades the quality of service (QoS). Moreover,task offloading from ESs to CSs incurs extra latency andenergy consumption due to communication between TEsand CSs. Some important problems to be solved are howto satisfy the latency requirements of TEs and reduce theenergy consumption in MEC. In order to meet a strict QoS,Nafiseh et al. [3] proposed a two-sided matching mechanismfor edge services considering QoS requirements in terms ofservice response time.In order to achieve low-latency and energy-saving com-putations, several studies on task offloading in MEC havebeen proposed. Xinchen Lyu et al. [4] have attempted toencapsulate the latency requirements in offloading tasksand designed a selective offloading scheme. This schemeis achieved by enabling the devices to be self-denied or self-nominated for offloading. This can save energy consump-tion and minimize delays for task offloading. Authors in [5]developed a threshold-based strategy to improve the QoS,which combines the advantages of ESs’ with lower latencyand abundant computational resources of CSs. A priorityqueue is also applied to solve the delay problem, whereindelay-sensitive tasks are executed ahead of delay-toleranttasks.However, most previous studies examined a single ES serving multiple TEs. In fact, the proliferation of applica-tions puts a heavy load on the ESs. Since the ES has limitedcomputing capacity and can’t accommodate enough tasks,some task processes may have to wait longer to wake up.Thus, for a single ES is hard to cope with concurrent tasksof multiple TEs, which will hinder the development andpopularity of MEC. In the future MEC market, there will bemultiple different ESs offering optional computing serviceto TEs. Hence, TEs can choose different ESs according totheir real-time and cost requirements. In that case, multipleESs provide computing services in parallel, which can accel-erate the speed of task processing and alleviate offloadingdelays.Task outsourcing has been employed as an effectiveparadigm by accommodating as many on-demand tasks aspossible. Its principle is to distribute TEs’ tasks in differenttime slots according to ESs’ load of each time slot. Ourwork focuses on the problem of offloaded tasks outsourc-ing. It differs from most existing works which focus onthe necessity of offloading or on selecting which tasks tobe offloaded to ESs or CSs. The TEs’ offloaded tasks aremapped to ESs according to their resource capacities. Thus,task outsourcing can enhance scalability of MEC and satisfyTEs’ dynamic service demands.Fig. 1 describes a MEC communication network in IoT,where ESs are deployed densely near TEs. The MEC net-work is similar to a real competitive market, in which awide range of TEs can be grouped into virtual clusters andcompete for the ESs’ limited wireless resources. They areinterdependent and mutually influence each other. Severalbase stations with ESs also compete with each other towin more TEs. The ESs can communicate with the TEs viascheduler and inform them of their real-time service prices.Through the scheduler, TEs can participate in ESs selection,and make wise decisions regarding their daily computingresources consumption.
Terminal entities (TEs) Signal LinkBase station with edge server (ES)
Fig. 1: The MEC network scenario in IoT.However, one of the main challenges of task outsourcingis to consider the interests of both parties. On the one hand,ESs aim to get more profits, and strive for attracting moreTEs to use their computing resources. On the other hand,a rational TE will choose a task strategy that maximizes itsown payoff. Game theory can be used as an effective tool tomodel the interests of two or more conflicting individualsin trading markets and also load balancing in distributedsystems [6]. The game solution was proved to be a Nash equilibrium solution for the noncooperative game.In this paper, we propose a bilateral game frameworkamong multiple ESs (the suppliers) and multiple TEs (thecustomers) to model the task outsourcing problem as twononcooperative games. These two games are related to eachother and are played simultaneously. In the first game, thesupply function bidding mechanism is employed to modelthe noncooperative game among ESs. In the proposedscheme, each ES, with limited or idle resources, submitsa bid to reveal the available capacity “supplied” to themarket. Then the scheduler collects these bids and computesa service price to clear the market so that the supply ofthe resource to be traded equals the demand. In particular,all TEs are charged the same service price at one time slot.The scheme can maximize the profits of ESs. In the secondgame, in order to reduce costs, TEs determine the amount ofassigned tasks for each time slot based on the price of thattime slot. If the price of one time slot is high, there wouldbe fewer tasks assigned, and if the price is low, there wouldbe more tasks. This framework can encourage TEs to assignfewer tasks during peak times or shift some tasks to off-peaktimes, which flattens the demand curve by peak clipping orvalley filling.In summary, the contributions of this paper are: • A bilateral-game framework is developed to modelthe interactions among TEs and ESs. • A supply function bidding mechanism is proposed,where each ES submits a bid to reveal the availablecapacity “supplied” to the market. • A DTOA is designed to compute the Nash equilib-rium. • Simulations show that the proposed mechanismachieves the maximization of bilateral interests.The remainder of this paper is organized as follows. InSection 2, the related work about task outsourcing in MEC isintroduced. Section 3 models the task outsourcing problemin the ESs’ and TEs’ sides. In Section 4, a
DTOA is designedto compute the Nash equilibrium in both sides. Section 5presents simulations showing the performance of the newapproach using
DTOA . Finally, conclusions are presented inSection 6.
ELATED WORK
In recent years, significant attention has been devoted to theresource allocation in MEC networks [7]. Shi Yan et al. [8]studied the access selection for unmanned aerial vehicles(UAV) and bandwidth allocation of the base station (BS) ina UAV assisted IoT communication network. Wherein, theaccess competition among groups of UAVs is modeled as adynamical evolutionary game. The bandwidth allocation ofBSs is formulated as a noncooperative game. Authors in [9]examined resource allocation for a multi-TE MEC offload-ing system, which is formulated as a convex optimizationproblem for minimizing the weighted sum of mobile energyconsumption.Chunlin Li et al. [10] analysed a radio and computingresource allocation problem between an access point andmultiple devices in MEC system. They designed a time average computation rate maximization algorithm to de-termine the optimal transmit power, and time allocationfor the wireless devices. Junhui Zhao et al. [11] studied acloud-MEC collaborative computation offloading problemthat offloads tasks to automobiles in MEC vehicular net-works. They developed a tasks allocation optimization andcollaborative computation offloading scheme to decide theoptimal strategies. An offloading algorithm for shorteningthe computation time and increasing the system utility wasalso designed. Yunlong Gao et al. [12] studied the optimaltradeoff between resource consumption and user experiencein designing MEC systems. Cosmin Avasalcai et al. [13]introduced a decentralized resource management algorithmwith the purpose of deploying IoT applications at the edgeof the network such that end-to-end delay is minimized.Since the resources of an ES are limited, the ES can notundertake the tasks coming from multiple TEs. So multipleESs are needed. Nevertheless, the matching problem be-tween multiple ESs and multiple TEs becomes a key issue.Heli Zhang et al. [14] modeled the matching relationshipbetween ESs and TEs as a commodity trading by apply-ing a multi-round sealed sequential combinational auctionmechanism. In [15], the authors studied task offloading invehicular MEC environments and modelled the interactionsbetween edges and tasks as a matching game. They furtherdeveloped two standalone heuristic algorithms to minimizethe average delay while taking the energy consumption andvehicle mobility constraints into consideration. A three-tierIoT fog network was proposed in [16], in which all fognodes, data service operators and data service subscribersare jointly optimized to achieve the optimal resource alloca-tion in a distributed fashion.Furthermore, authors in [17, 18] adopted a price-basedmechanism to design efficient resource allocation in a MECnetwork. For example, [17] proposed a price-based dis-tributed method to manage the offloaded tasks from users.Wherein, edge cloud sets prices to maximize its revenue andeach user makes an optimal decision to minimize her/hisown cost. The work [18] proposed a price-based resourceallocation mechanism among the MEC server and multiplebase stations (BSs). The MEC server tries to provide pricesto BSs so as to maximize its own revenue while the BSsdetermine the computing space to improve the quality ofexperience.To summarize the related work above, we observe thatthe existing resource allocation and matching problem inMEC generally involves edge nodes and clients using re-source from an edge node. However, most of these studiesfocus either on the system performance or ESs’ benefits,while ignoring the TEs’ pursuit of maximizing payoff.Against this backdrop, our paper tries to balance the ob-jectives of both ESs and TEs. In this paper, we also adopta priced-based supply bidding mechanism to solve theresource allocation problem.
YSTEM M ODEL
As shown in Fig. 2, we consider a scheduler-based resourcestransaction market, which consists of M ESs and N TEs ina MEC network. ESs act as suppliers who sell computing
TE 1
Scheduler
TE 2
TE N ES 1ES 2ES M
Load price
Demand Profiles
Demand1
Demand2Demand3
TotaldemandTEs ESs (cid:23430)(cid:23460)(cid:23471)(cid:23462)(cid:23480)(cid:23471)(cid:23460)(cid:23479)(cid:23468)(cid:23473)(cid:23466)(cid:23395)(cid:23479)(cid:23467)(cid:23464)(cid:23395)(cid:23460)(cid:23466)(cid:23466)(cid:23477)(cid:23464)(cid:23466)(cid:23460)(cid:23479)(cid:23464)(cid:23395)(cid:23471)(cid:23474)(cid:23460)(cid:23463)(cid:23430)(cid:23460)(cid:23471)(cid:23462)(cid:23480)(cid:23471)(cid:23460)(cid:23479)(cid:23468)(cid:23473)(cid:23466)(cid:23395)(cid:23479)(cid:23467)(cid:23464)(cid:23395)(cid:23475)(cid:23477)(cid:23468)(cid:23462)(cid:23464)
Fig. 2: Diagram of a resources transaction market.resources to TEs and TEs act as customers who purchaseresources from ESs. The bidirectional interaction betweenESs and TEs is performed through a scheduler, which servesas a third-party agency outsourcing TEs’ tasks to ESs.On the one hand, TEs submit their demand profiles tothe scheduler via a communication network. On the otherhand, ESs compete with each other for acquiring more TEs,and submit bids based on strategies of their opponents andtheir own resource capacities. As a response, the schedulercalculates the service price and the aggregated load basedon ESs’ bids and TEs’ total demand. They are mutuallydependent upon each other as decisions on either side canhave a bearing on those of the other side. After receivingthe real-time price signal, the TEs will update their demandprofiles. Since the aggregate load depends on the TEs’demand profiles, the behavior of TEs will affect the ESs’bidding strategies. The aforementioned process is repeateduntil both customers and suppliers are satisfied.We divide one day into a set of T ( T = 24 ) time slots,denoted as T = { , · · · , T } . The set of ESs and TEs arerepresented as M = { , · · · , M } and N = { , . . . , N } . Howmany resources should ESs provide to the market and howthe ESs’ bids affect the TEs’ demand profiles are questionsworth investigation. We next present the model of both sidesin the MEC resources transaction market. For ES j ∈ M , let C j,t ( . ) denote the cost function of ES j at time slot t ( t ∈ T ) . Let R j,t ( . ) denote the revenuefunction of ES j at the t th time slot by providing thecomputational load. The profit of ES equals the revenueby providing computing service minus its cost of systemoverhead. Therefore, the profit P j,t of ES j at time slot t canexpressed as follows: P j,t = R j,t ( . ) − C j,t ( . ) . (1)We consider that each ES is selfish and tries to maximizeits own profit. Thus, the interaction among the profit max-imizer ESs can be modeled as a noncooperative game. TheESs are the players while the bid profiles are the strategies.Let λ j,t denote the bid of ES j at time slot t . The target ofeach ES j is to find the optimal bid λ j,t to maximize itsprofit, which can be defined as: maximize λ j,t P j,t j ∈ M , t ∈ T . (2) By substituting Equ. (1) into Equ. (2), we can get maximize λ j,t R j,t ( . ) − C j,t ( . ) j ∈ M , t ∈ T . (3)We denote by f j,t the task load that ES j willing togenerate in the time slot t . We assume that the service priceof different ESs in one time slot is the same and denoted as p e ( t ) at time slot t . The revenue of each ES is equal to theproduct of its load and the service price. Hence, the revenueof ES j at time slot t can be represented as R j,t = f j,t · p e ( t ) . (4)Similar to [19], the ES j ’s cost function is defined as aquadratic function C j,t ( f j,t ) = a j, f j,t + a j, f j,t + a j, , where a j, , a j, and a j, are positive coefficients and modelthe fact that different ESs incur different costs for servingthe tasks. We note that the cost function is increasing andconvex. Substituting Equ. (4) into Equ. (3), the optimizationproblem can be further rewritten as maximize λ j,t f j,t · p e ( t ) − C j,t ( f j,t ) subject to f j,t ≥ , j ∈ M , t ∈ T . (5) The demand of each TE consists of two parts: a base demandand a shiftable demand. On the one hand, a base demand isprimarily concerned with real-time tasks, which have highpriority. On the other hand, a shiftable demand has lowpriority real-time requirements and it can be assigned at anytime slot. The shiftable demand profile of TE i ( i ∈ N ) isdefined as χ i = ( χ i, , . . . , χ i,T ) and the base demand of TE i at time slot t is denoted as r i,t , which is known and fixed.The utility of TE i represents the profit that TE i receiveswhen it completes tasks and is denoted as U i ( . ) . Exactly, theutility function of TE i is the utility for the tasks rather thanthe service time or applications. Similar to [20], we employthe quadratic utility function because it is non-decreasingand its marginal benefit is non-decreasing, U i ( x ) = w i,t x − α i,t x , ≤ x ≤ w i,t α i,t w i,t α i,t , x > w i,t α i,t , (6)where x = ( χ i,t + r i,t ) , w i,t and α i,t , i ∈ N are coefficientsthat reflects the dynamic changes of TE i ’s demand.The payout function quantifies the payout that TE i needs to pay the ESs task completion. Without loss ofgenerality, we define the payout of TE i ’ as the product ofdemand and the service, i.e. P ayout i,t = ( χ i,t + r i,t ) · p e ( λ t , L t ) . (7)The payoff function quantifies the final benefits of TE i andrepresents the satisfaction of using the service. Thus, wedenote the payoff of TE i as its utility minus payout i.e. P ayof f i = U tility i − P ayout i . (8) Let u i denote the payoff of TE i . By substituting Equ. (6) andEqu. (7) into Equ. (8), we can obtain u i ( χ i , χ − i ) = X t ∈T (cid:16) U i ( χ i,t + r i,t ) − ( χ i,t + r i,t ) p e ( λ t , L t ) (cid:17) , (9)where χ − i denotes the vector of the demand profile of otherTEs and χ − i = ( χ , . . . , χ i − , χ i +1 , . . . , χ N ) . In Equ. (9),the utility is a function related to ( χ i,t + r i,t ) .Each TE tries to maximize its payoff by determiningits shiftable demand profile. Thus, the interaction betweenTEs can be modeled as a noncooperative game. The TEsare participants while the shiftable demand profiles are thestrategies of the noncooperative game.Let χ ∗ i denote the optimal demand profile of TE i in theNash equilibrium and Q totali denote the total daily shiftabledemand of TE i which is fixed and known. Let L t denotethe aggregate load demand of the ESs at time slot t and L t = P j ∈N ( χ j,t + r j,t ) . Considering TE i , the optimizationproblem can be formulated as follows when other TEs’profiles are fixed: maximize χ i u i ( χ i , χ − i ) subject to X t ∈T χ i,t = Q total i ,χ i,t ≥ , ∀ i ∈ N . (10) In this section, we employ a supply function bidding mecha-nism to model the relationship between market demand forservices and its price. We use a class of supply functionswith parameters. The bids submitted by ESs reveal theiravailable resource capacities “supplied” to the market.TABLE 1: Definitions of Mathematical Notations
Notation Definition L t TEs’ total load demand at time slot tf j,t The supply function of the ES j at time slot tp e ( t ) The computing service price at time slot tp , · · · , p K K break points of the price-wise linear function ofall ESs λ kj,t The slope of the function between the break points p k − and p k λ j,t The slope of the function between the origin andbreak point p The notations used in the supplier side model are pre-sented in Table 1. We assume that the supply function f j,t ischosen from the family of increasing and convex price-wiselinear functions of p e ( t ) [21]. Fig. 3(a) shows an increasingand convex piece-wise linear supply function. The abscissa p e ( t ) indicates the price and the ordinate f j,t denotes theload supplied by the TE j at time slot t . There exists K breakpoints on the abscissa of the Fig. 3(a). λ kj,t ≥ represents theslope of the function between the break points p k − and p k .Fig. 3(b) shows the affine supply function. (cid:28595) Fig. 3: (a) Piece-wise linear. (b) Affine supply functions.At time slot t ( t ∈ T ) , we use the vector λ j,t =( λ j,t , · · · , λ Kj,t ) to denote the bid profile of ES j ( j ∈ M ) .Thus, we obtain f j,t ( p e ( t ) , λ j,t ) = λ j,t p e ( t ) , ≤ p e ( t ) ≤ p λ kj,t p e ( t )+ λ k − j,t p k − , p k − < p e ( t ) ≤ p k . (11)It is assumed that each ES submits λ j,t as a bid profileto the scheduler at time slot t . For each ES j , the bidprofile describes the number of tasks that it is willing toadmit. We can use λ t to represent the bid profiles of allESs at time slot t and λ t = { λ ,t , · · · , λ M,t } . In responseto ESs, the scheduler sets the price p e ( t ) to clear market.In economics, market clearing means the supply of what istraded equals the demand, so that there is no leftover supplyor demand. In this case, the demand of all TEs is the sameas the load supplied by all ESs. Although the fluctuationin TEs’ demand will drive changes in ESs’ bid profiles,the demand and supply remains balanced. The equivalencefurther builds up the connection between the supplier gameand the customer game. Hence, it can expressed as X j ∈M f j,t ( p e ( t ) , λ j,t ) = L t , t ∈ T . (12)According to Equ. (11) and Equ. (12), we have L t = X j ∈M (cid:0) λ j,t p e ( t ) (cid:1) , ≤ p e ( t ) ≤ p X j ∈M (cid:16) λ kj,t p e ( t )+ λ k − j,t p k − (cid:17) , p k − < p e ( t ) ≤ p k . (13)According to Equ. 13, we can further calculate the serviceprice function as follows: p e ( t ) = L t P j ∈M λ j,t , ≤ p e ( t ) ≤ p L t − P j ∈M (cid:16) λ k − j,t p k − (cid:17)P j ∈M λ kj,t , p k − < p e ( t ) ≤ p k . . (14)At time slot t , the service price of different ESs is the same.In [22], the affine supply function f j,t ( p e ( t ) , λ j,t ) = λ j,t p e ( t ) is used as a special case of the aforementionedpiece-wise linear functions. Almost all the results of affine supply functions can be generalized to the piece-price affinesupply function [23]. As for Equ. (14), it can be concludedthat the affine function is equivalent to the piece-wise linearsupply function between two break points. Each piecewisefunction of Fig. 3(a) can be regarded as a linear function inFig. 3(b). As a matter of fact, the term λ k − j,t p k − is fixedwhen we are between break points p k − and p k . Therefore,without loss of generality, we generalize the results from theaffine functions to piece-wise linear functions. Therefore, thecomputing service price can be given as follows for an affinesupply function: p e ( t ) = L t P j ∈M λ j,t , t ∈ T . (15)For simplicity, we use the notation λ j,t instead of λ j,t to represent the affine supply function of ES j . Meanwhile,we use λ t = ( λ ,t , . . . , λ M,t ) to denote the bids profile forall ESs at time slot t . As Equ. (15) shows, the computingservice price is related to λ j,t ( j ∈ M ) and L t . Hence, theprice function can be denoted as p e ( λ t , L t ) . As suggestedby Equ. (11), supply function f j,t for ES j can be expressedas f j,t ( p e ( λ t , L t ) , λ j,t ) = λ j,t L t P r ∈M λ r,t , t ∈ T . (16)Similar to the computing service, the supply functioncan be represented by f j,t ( λ t , L t ) . Let λ − j,t denote thesubmitted bids of other ESs except for ES j . So it can be de-fined as λ − j,t = ( λ ,t , . . . , λ j − ,t , λ j +1 ,t , . . . , λ M,t ) . Hence,According to Equ. (5) and Equ. (16), the profit function ofES j is rewritten as P j,t ( λ j,t , λ − j,t ) = λ j,t L t (cid:0)P r ∈M λ r,t (cid:1) − C j (cid:18) λ j,t L t P r ∈M λ r,t (cid:19) . (17)When other ESs’ bids are fixed, the ES j tries to findthe optimal bid λ ∗ j,t by solving the following optimizationproblem: maximize λ j,t λ j,t L t (cid:0)P r ∈M λ r,t (cid:1) − C j (cid:18) λ j,t L t P r ∈M λ r,t (cid:19) subject to λ j,t ≥ , j ∈ M , t ∈ T . (18) The following section will explain that the ES’s game (Equ.(18)) has a unique Nash equilibrium, as shown by the lemmabelow.
Lemma 3.1.
Assume that the bids profile in Nash equilibriumat time slot t is denoted as λ ∗ t . When the Nash equilibrium isreached, it will satisfy λ ∗ j,t < P r ∈M ,r = j λ ∗ r,t for all ESs.Proof. The function Π j,t ( λ j,t , λ − j,t ) is expressed as follows: Π j,t ( λ j,t , λ − j,t ) = λ j,t L t (cid:0)P r ∈M λ r,t (cid:1) . (19) As the formula above suggests, Π j,t ( λ j,t , λ − j,t ) is thefirst term in P j,t ( λ j,t , λ − j,t ) . From Equ. (19), we can calcu-late the first derivative function as follows d Π j,t ( λ j,t , λ − j,t ) dλ j,t = L t · (cid:0)P r ∈M λ r,t (cid:1) − λ j,t L t (cid:0)P r ∈M λ r,t (cid:1)(cid:0)P r ∈M λ r,t (cid:1) (20)Let d Π j,t ( λ j,t , λ − j,t ) dλ j,t > , we can get X r ∈M λ r,t ! − λ j,t X r ∈M λ r,t > . (21)The Equ. (21) is equivalent to λ j,t + X r ∈M ,r = j λ r,t − λ j,t λ j,t + X r ∈M ,r = j λ r,t > . (22)From Equ. (22), we can derive that ≤ λ j,t < X r ∈M ,r = j λ r,t . In summary, we can conclude that P j,t ( λ j,t , λ − j,t ) is anincreasing function when ≤ λ j,t < P r ∈M ,r = j λ r,t . And itbecomes a decreasing function when λ j,t ≥ P r ∈M ,r = j λ r,t .Thus, in order to maximize profit, we should meet the con-straint ≤ λ j,t < P r ∈M ,r = j λ r,t . In the Nash equilibrium,the bid of ES j at time slot t is denoted as λ ∗ j,t . Therefore, wecan conclude that λ ∗ j,t < P r ∈M ,r = j λ ∗ r,t ( j ∈ M ) .Similar to [24], the proof for the following theorem givenas follows. Theorem 3.1.
The ES’s noncooperative game has a unique Nashequilibrium. Furthermore, the Nash equilibrium is the solution ofthe following convex optimization problem: maximize ≤ f j,t < Lt P j ∈M − Ψ j ( f j,t ) subject to P j ∈M f j,t = L t , (23) where Ψ j ( s j,t ) = (cid:18) L t − f j,t L t − f j,t (cid:19) C j ( f j,t ) − Z f j,t L t C j (Π j )( L t − j ) d Π j . (24) Proof.
According to lemma 3.1, we can infer that the loadsupplied by each ES at time slot t does not exceed L t / at the Nash equilibrium. The Lagrange function of theoptimization problem in Equ. (23) is denoted as F . Thus,we have F = X j ∈M − Ψ j ( f j,t ) + φ X j ∈M f j,t − L t , (25) where φ denotes the Lagrange multiplier. We can obtainthe following expression through the first-order optimalityfunction. ∂F∂f ∗ j,t ! (cid:0) f j,t − f ∗ j,t (cid:1) ≤ , ∀ j ∈ M , (26)where f ∗ j,t is defined as the supply function in equilibrium,while φ ∗ is the Lagrange multiplier in equilibrium.From Equ. (24), ( ∂F /∂f ∗ j,t ) can be expressed as follows: ∂F∂f ∗ j,t = φ ∗ − L t − f ∗ j,t L t − f ∗ j,t ! C ′ j (cid:0) f ∗ j,t (cid:1) . (27)We assume the first-order optimality condition for theoptimization problem in Equ. (18). Thus, we obtain (cid:18) ∂P j,t ∂λ j,t (cid:19) (cid:0) λ j,t − λ ∗ j,t (cid:1) ≤ , ∀ j ∈ M . (28)From Equ. (17), ∂P j,t /∂λ j,t is calculated as follows: ∂P j,t ∂λ j,t = p e ( λ t , L t ) − L t − f ∗ j,t L t − f ∗ j,t C ′ j (cid:0) f ∗ j,t (cid:1) . (29)By substituting Equ. (29) into Equ. (28), we can write theoptimality condition for Nash equilibrium as follows p e ( λ t , L t ) − L t − f ∗ j,t L t − f ∗ j,t C ′ j (cid:0) f ∗ j,t (cid:1)! (cid:0) λ j,t − λ ∗ j,t (cid:1) ≤ . (30)From Equ. (26) and Equ. (30), we can see that theLagrange multiplier is actually the price p e ( λ t , L t ) of thecomputing service. In addition, the optimality conditionEqu. (26) is equivalent to Equ.30. Therefore, the existenceand uniqueness of the Nash equilibrium is equivalent toproving the existence and uniqueness of the optimal pointof problem Equ. (23).In Theorem 3.1, it is proved that the ES’s game hasa unique Nash equilibrium solution, whose strategies aredetermined by the aggregate load L t . Besides, a ES canscale-up and scale-down its resource capacity according todifferent market demands. Thus, ESs will bid differently fordifferent levels of load.We next analyze the existence of Nash equilibrium forthe customer side game, which is proved by the theorembelow. Theorem 3.2.
The customers’ optimization problem is a convexprogramming problem. In fact, the customer side game Equ. (10) isan n-person game. It has a unique pure strategy Nash equilibrium.Proof.
From the above discussion it follows that the objec-tive function in Equ. (10) is equal to X t ∈T U i ( χ i,t + r i,t ) − X t ∈T ( χ i,t + r i,t ) + P j ∈N ,j = i ( χ j,t + r j,t ) P r ∈M λ r,t ! . (31)Let k = 1 / ( P r ∈M λ r,t ) , k > . For simplicity, we denote theright part of Equ. (31) as follows: h i ( χ i , χ − i ) = X t ∈T k (cid:18) ( χ i,t + r i,t ) + X j ∈N ,j = i (cid:18) χ j,t + r j,t (cid:19)(cid:19) . We have ∇ χ i h i ( χ i , χ − i ) = (cid:20) ∂h i ( χ i , χ − i ) ∂χ i,t (cid:21) Tt =1 = (cid:18) ∂h i ( χ i , χ − i ) ∂χ i, , · · · , ∂h i ( χ i , χ − i ) ∂χ i,T (cid:19) = 2 k ( χ i,t + r i,t ) + X j ∈N ,j = i ( χ j,t + r j,t ) Tt =1 (32)and the Hessian matrix is as follows: ∇ χ i h i ( χ i , χ − i ) = k k · · · k k k · · · k ... ... . . . ... k k · · · k N × T . (33)This further leads to X T ∇ χ i h i ( χ i , χ − i ) X = 2 k ( X + X + · · · + X N × T ) ≥ , ∀ X = ( X , X , · · · , X N × T ) T . Therefore, the Hessian matrix of h i ( χ i , χ − i ) is positivesemi-definite and h i ( χ i , χ − i ) is convex. Moreover, sincethe utility function U i ( . ) is continuous and strictly concavein the strategy space, the payoff function Equ. (9) of eachTE i ( ∀ i ∈ N ) is strictly concave. So the objective func-tion in Equ. (31) is concave. Hence, Equ. (10) is a convexoptimization problem. Meanwhile, since the constraints ofEqu. (10) are inequalities or linear equations, the feasibledomain is convex. Thus, the TEs’ optimization problemis a convex programming problem. Hence, the TE’s gameis a strictly concave N -person game. Since the demandprofile sets are closed, bounded and convex, the existenceof Nash equilibrium can be proved based on [25, Theorem1]. Analogously to [25, Theorem 3], for a concave N -persongame, there exists a unique equilibrium solution. Therefore,the theorem is proved.In the Nash equilibrium, for any given ESs’ bids, noTE can increase its payoff by a unilateral change on itsstrategy. In the next section, a task outsourcing algorithmis developed to determine the point for both ES and TE’sgames. ISTRIBUTED T ASK O UTSOURCING A LGORITHM
In this section, we propose a distributed task outsourcingalgorithm to demonstrate the interaction among TEs andESs. Our method is referred as
DTOA . Let g be the iterationnumber.Notations:Let χ gi,t denote the demand profiles of TE i in iteration g at time slot t and vector χ gi denote the demand profile of TE i for all time slots. The matrix χ = ( χ , . . . , χ t , . . . , χ T ) T denotes the demand profiles of all TEs in iteration g for alltime slots. Let matrix λ = ( λ , . . . , λ t , . . . , λ T ) T denote thebids of all ESs for all time slots. L gt denotes the aggregateloads in iteration g at time slot t . p ge ( λ gt , L gt ) denotes thecomputing service price in iteration g at time slot t .As shown in Fig. 4, the interaction between ESs and TEscan be modeled as a two-stage game. They interact with each other to determine optimal bids and demand profiles.The detailed process is depicted in Algorithm 1 and 2. • The ESs try to maximize their profits by determiningtheir own bids according to optimization functionEqu. (18). • The TEs will then adjust their demand profiles fol-lowing optimization function Equ. (10). (cid:23446)(cid:23462)(cid:23467)(cid:23464)(cid:23463)(cid:23480)(cid:23471)(cid:23464)(cid:23477) (cid:23430)(cid:23460)(cid:23471)(cid:23462)(cid:23480)(cid:23471)(cid:23460)(cid:23479)(cid:23468)(cid:23473)(cid:23466)(cid:23395)(cid:23479)(cid:23467)(cid:23464)(cid:23395)(cid:23460)(cid:23466)(cid:23466)(cid:23477)(cid:23464)(cid:23466)(cid:23460)(cid:23479)(cid:23464)(cid:23395)(cid:23471)(cid:23474)(cid:23460)(cid:23463)(cid:23430)(cid:23460)(cid:23471)(cid:23462)(cid:23480)(cid:23471)(cid:23460)(cid:23479)(cid:23468)(cid:23473)(cid:23466)(cid:23395)(cid:23479)(cid:23467)(cid:23464)(cid:23395)(cid:23475)(cid:23477)(cid:23468)(cid:23462)(cid:23464) (cid:23447)(cid:23432)(cid:23395)(cid:23468)(cid:23439)(cid:23474)(cid:23460)(cid:23463)(cid:23395)(cid:23480)(cid:23475)(cid:23463)(cid:23460)(cid:23479)(cid:23464)(cid:23432)(cid:23446)(cid:23395)(cid:23469) (cid:23429)(cid:23468)(cid:23463)(cid:23478)(cid:23395)(cid:23480)(cid:23475)(cid:23463)(cid:23460)(cid:23479)(cid:23464)
Fig. 4: Interactions between the ESs, TEs and scheduler.
Algorithm 1
TE’s game Initialization : g = 0 . Randomly initialize TEs’ demand profiles. repeat for (each time slot t ∈ T ) do Receive L gt from the scheduler. Update the bid λ gt by Algorithm 2. Receive the updated p ge ( λ gt , L gt ) from the scheduler. for (each TE i ∈ N ) do χ g +1 i,t = h χ gi,t + η ∂u i ( χ gt ) ∂χ gi,t i + . end for end for g := g + 1 . until (cid:13)(cid:13) χ g − χ g − (cid:13)(cid:13) < ǫ . Algorithm 2
ES’s game
Input:
Total load at time slot t : L t , t ∈ T and t . Output:
Bids of all ESs at time slot t: λ t . Initialization : Randomly initialize ESs’ bid profiles for thefirst time. Receive L t from the scheduler. for (each ES j ∈ M ) do λ g +1 j,t = h λ gj,t + η ∂P j,t ( λ gt ) ∂λ gj,t i + . end for return λ t .The DTOA can be described as follows. Firstly, the sched-uler randomly initializes the TEs’ demand profiles and ESs’bid profiles. Secondly, the TE i ( i ∈ N ) sends the shiftabledemand profile χ gi to the broker and receives L gt from it.Then, the ESs will receive a signal to update their bids basedon the following iterative equation: λ g +1 j,t = " λ gj,t + η ∂P j,t ( λ gt ) ∂λ gj,t + , ∀ t ∈ T . (34)where η is the step size. [ · ] + in Equ. (34) is the projectiononto the feasible set defined by the constraints λ j,t ≥ . It is noticed that the ES j ( j ∈ M ) does not know otherESs’ bids. In this aspect, the DTOA can also preserve theprivacy of participants. Thirdly, the computing service price p ge ( λ gt , L gt ) is updated by the scheduler according to Equ.(15). The TEs will further be informed to update theirshiftable demand profiles using a gradient boosting method: χ g +1 i,t = " χ gi,t + η ∂u i ( χ gt ) ∂χ gi,t + , ∀ t ∈ T . (35) η is the step size. [ · ] + in Equ. (35) is the projection ontothe feasible set defined by the constraints P t ∈T χ i,t = Q total i and χ i,t ≥ . It is worth remarking that Equ. (10)needs the updated price p ge ( λ gt , L gt ) and L gt to determine (cid:16) ∂u i ( χ gt ) /∂χ gi,t (cid:17) . Besides, since (cid:16) ∂u i ( χ gt ) /∂χ gi,t (cid:17) only de-pends on its own demand profile and the price and thereis no need to know the demand profile of other TEs. Thus,this fact protects the privacy of the TEs. Finally, the stoppingcriterion of the algorithm is checked by the scheduler. If therelative change of shiftable demand profiles during two con-secutive iterations is lower than the value ǫ , the iterationscan be stopped. Otherwise, the TEs will continue computingtheir demand profiles based on the newly updated price andbids.The optimization problems Equ. (18) and Equ. (10) willconverge to the optimal point by the projected gradientmethod. In the end, the algorithm will converge. In theequilibrium, the ESs are playing their equilibrium strategiesaccording to TEs’ tasks strategies, and the TEs also choosetheir equilibrium strategies based on ESs’ submitted bids.Thus when the Nash equilibrium is reached, none of theESs and TEs improve their profit. ERFORMANCE E VALUATION
In this section, we present a simulation experiment to val-idate our theoretical analysis. We assume a MEC resourceexchange market has 10 ESs and 1000 TEs, which are willingto participate in the
DTOA scheme. There are 24 time slots.The relevant parameters of the model are shown in Table2. The base demand r i,t of each TE at each time slot israndomly selected from [9660, 37065]. The shiftable demandrefers to real-time, non-shiftable tasks, which reflects thechanges in the total demand of all TEs at different timeslots. Since most loads are running in real-time pattern,it is plausible to assume relatively low shiftable loads forTEs. The shiftable demand χ i,t of each TE is assumed tobe chosen randomly from 10% to 12% of its base demand.And the total demand is the sum of the base demand andshiftable demand. Considering the generation cost function c ( f j,t ) = a j, f j,t + a j, f j,t + a j, for each ES j ( j ∈ M ) ,we assume that a j, is randomly generated in the interval[4.76e-6, 4.76e-5], a j, = 0 . and a j, = 0 . . The initialvalues of η and η are set as 0.05 and 0.01 respectively.In order to find the optimal solution, the step size of nextiteration will be a little less than the previous one, namely η = η *0.985 and η = η *0.98. The initial bids λ j,t of ESs areall set as 20000. The α i,t is set as 0.5 and ω i,t is randomlyselected from interval [0.8, 1.0]. Also, the ǫ is set equal to 0.3. TABLE 2: System Parameters System parameters Value(Fixed)-[Varied range]
Base demand r i,t [9660, 37065]Shiftable demand χ i,t [10%,12%]*Base demand a j, [4.76e-6, 4.76e-5] a j, (0.001) a j, (0.001)step size η (0.05), η = η *0.985step size η (0.001), η = η *0.98ES’s bid λ j,t (20000) α i,t (0.5) ω i,t [0.8,1.0] ǫ (0.3) Fig. 5: Convergence of ES 1-10’s bids at time slot 5.Fig. 6: Convergence of shiftable loads for TE 21-30 at timeslot 5.
The performance of our proposed
DTOA is evaluated interms of its convergence. Fig. 5 and Fig. 6 show the conver-gence of ESs’ bids and TEs’ shiftable loads at time slot 5.From Fig. 6, these ten TEs (TEs 21-30) are randomly selectedfrom 1000 TEs. The speed of convergence to the equilibriumpoint depends on the step sizes and the stopping criterion ǫ . As the number of iterations increases, the bids and theshiftable load demands start from the initial values and theygradually converge to stable values. In our experiment, thealgorithm converges after around 248 iterations. Hence, theproposed DTOA is efficient and verifies the theoretical proofpresented above.To demonstrate the computational complexity of thealgorithm, we evaluate the running time of the algorithmfor different number of TEs and ESs. As shown in Fig. 7,the running time of the algorithm increases linearly withthe number of TEs N and it is almost independent of M .This is because that by increasing the number of TEs andESs, the number of updates for TEs and ESs will increaseproportional to N and M , respectively. The update processfor TEs takes more time comparing with the updates forESs since the TEs need to consider load shifting during T time slots (the projected gradient), which make the updateprocess more complex. From Fig. 7, the running time of thealgorithm is acceptable even for large number of TEs. So itcan be concluded that the algorithm is efficient and can beimplemented in scenarios with large number of TEs.
100 300 500 800 1000 1500 20000306090120150
Running time of the algorithm (s)
Number of TEs 5 ESs 10 ESs 15 ESs 20 ESs
Fig. 7: Running time of algorithm for different number ofTEs and ESs
By participating in the
DTOA scheme, the payout (see Equ.(7)) represents TE’s expenditure on purchasing computingresources. Fig. 8 shows the daily total payout for TE 1 toTE 30 before algorithm and after algorithm. Compared withbefore algorithm, the total payout of each TE after algorithmis reduced. We can see that TEs can save around 5% of theypayout by participating in
DTOA scheme. The vertical axisof Fig. 8 shows the payouts of TEs are so huge, and evena 5% savings reduces a large expenditure. Furthermore, as shown in Fig. 9, the daily total payoff of each TE afterusing the algorithm has increased than before algorithm.The payoff (see Equ. (9)) is the utility of the calculation tasksminus the payout. Although the green bar is only a littlemore than the red bar chart, the payoff has also increased alot because its magnitude is large and arrives .Fig. 10 displays the total profit of ESs 1-10 before andafter algorithm. The total profit of ES is the sum of the profitof all time slots. From Fig. 10, we can see that the totalprofit of each ES increases after applying DTOA becausethe aggregate load profile becomes smoother; and hence,the ESs’ generation cost decreases. Besides, the suppliersaim to submit optimal bids that maximize their profits ineach time slot. The results of the algorithm are in line withexpectations, which shows the supplier side’s individualrationality of our proposed method.
For simulations, the initial state of TEs’ demand is assumedto be load profile before algorithm. The aggregate loadprofile becomes smoother after the
DTOA . The dashed circleshows the fluctuation of the demand including valley fillingand peak clipping. Normally, the peak load demand is26200, while the peak load demand decreases to 23800 inthe case of the
DTOA . Therefore, the peak load demandsare shifted from peak to off-peak time slots. Furthermore,the load demand for each TE is shifted to time slots withhigher w j,t , which brings a higher payoff to the TEs. Thisdemonstrates that the proposed DTOA performs satisfacto-rily in reducing the peak load demand.Fig. 12 shows the PAR (peak-to-average ratio) indexwith and without task scheduling in 24 time slots and fordifferent number of TEs. Before algorithm, since there arehigh peak load and low average load, PAR index is high.By applying
DTOA , the peak clipping and valley filling areachieved and the PAR index is low even for high number ofTEs. This demonstrates that the proposed task schedulingmethod can shift the shiftable loads from peak periods tooff-peak periods effectively.
In this section, we discuss the influence of some parameterson the convergence speed of the algorithm. The convergencespeed of the algorithm is reflected in the round of algo-rithm updates (iteration numbers). The smaller the iterationnumbers, the faster the algorithm converges. The bigger theiteration numbers, the slower the algorithm converges.Fig. 13 shows the influence of the parameter ǫ on itera-tion numbers. ǫ is the stopping criterion of the algorithm. Ascan be seen, the smaller the parameter ǫ , the more iterationsand the slower the algorithm convergence. This fact showsthat the stricter of the stopping criterion, the more times thealgorithm needs to be updated.In Fig. 14, the influence of the parameter η on iterationnumbers are shown. Since the parameter η will change inevery round, as shown in Table 2, we set different initialvalue of parameter η to show its impact on iterationnumbers. As can be seen, when other parameters are fixed,with the initial value of parameter η becomes larger, thenumber of iterations also increases. In summary, the speed Fig. 8: Daily total payout for TE 1 to TE 30.Fig. 9: Daily total payoff for TE 1 to TE 30. Fig. 10: Daily total profit for ES 1 to ES 10.
Peak clippingValley filling
Fig. 11: Base and total demand before and after algorithm.The peak shaving is achieved by using
DTOA (the dashedcircle).
100 300 500 800 1000 1500 20001.041.081.121.161.20
Peak-to-average ratio (PRA))
Number of TEs before algorithm after algorithm
Fig. 12: Peak-to-average ratio with and without taskscheduling of the algorithm convergence is related to the setting of someparameters.
Iteration numbers
The parameter e Fig. 13: The influence of the parameter ǫ on iteration num-bers. Iteration numbers
Initial value of the parameter h Fig. 14: The influence of the parameter η on iterationnumbers. ONCLUSION
In this paper, we analyze a practical resources transactionmarket in a MEC network, where multiple different ESsoffering the optional computing service to TEs. Since theresources of each ES are limited, the dynamic demand of itsTEs may not be met during spikes in demands. To overcomethe bottleneck of resource limitation, task outsourcing hasbeen regarded as an effective paradigm by accommodatingas many on-demand tasks as possible. We focus on the taskoutsourcing problem among multiple ESs and multiple TEs.A bidding mechanism is utilized to describe the servingrelationship between ESs and TEs, where the two partiesare assigned as sellers and buyers. The computing resourcesof ESs are regarded as commodities. Simulations results demonstrate that the algorithm in-creases the ESs’ profit and reduces the peak load by shiftingthe load demand to off-peak periods. Meanwhile, the TEs’payoff are also increased by participating in game process.As for future research, we will focus on the computingoffloading of ESs in a three-tier IoT MEC networks. A CKNOWLEDGMENTS
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Econometrica ,vol. 33, no. 3, pp. 520–534, 1965. Zheng Xiao received the Ph.D. in Computer Sci-ence from Fudan University, China, in 2009, andB.S. in Communication Engineering from HunanUniversity in 2003. He is currently an associateprofessor in College of Information Science andEngineering of Hunan University. His major re-search interests include distributed artificial in-telligence, high performance computing, paralleland distributed systems, intelligent informationprocessing and collaborative optimization. Hehas published over 30 journal articles and con-ference papers. He is currently an associate editor of IEEE Access. Heis a member of IEEE and CCF.
Dan He received the B.S. degree in computerscience and technology from Jiangxi NormalUniversity, Nanchang, China, in 2018. She iscurrently working toward the M.S. degree at theCollege of Information Science and Engineer-ing, Hunan University, Changsha, China. Herresearch interests focus on high performancecomputing, modeling and resource scheduling incloud computing systems, big data pricing andgame theory.
Yu Chen is currently working toward the M.S.degree at the Collage of Information Scienceand Engineering, Hunan University, China. Hisresearch interests focus on machine learningand natural language processing.
Anthony Theodore Chronopoulos obtained aPh.D. in Computer Science from the Universityof Illinois at Urbana-Champaign in 1987. He isa full Professor at the Department of ComputerScience, University of Texas, San Antonio, USAand a visiting professor, Department of Com-puter Engineering & Informatics, University ofPatras, Greece. He is the author of 83 journaland 73 peer-reviewed conference proceedingspublications in the areas of Distributed and Par-allel Computing, Grid and Cloud Computing, Sci-entific Computing, Computer Networks, Computational Intelligence. Heis a Fellow of the Institution of Engineering and Technology (FIET), ACMSenior member,
IEEE
Senior member.
Schahram Dustdar is Full Professor of Com-puter Science (Informatics) with a focus on In-ternet Technologies heading the Distributed Sys-tems Group at the TU Wien. He is Chairmanof the Informatics Section of the Academia Eu-ropaea (since December 9, 2016). He is ele-vated to IEEE Fellow (since January 2016). From2004-2010 he was Honorary Professor of In-formation Systems at the Department of Com-puting Science at the University of Groningen(RuG), The Netherlands. From December 2016until January 2017 he was a Visiting Professor at the University ofSevilla, Spain and from January until June 2017 he was a VisitingProfessor at UC Berkeley, USA. He is a member of the IEEE ConferenceActivities Committee (CAC) (since 2016), of the Section Committeeof Informatics of the Academia Europaea (since 2015), a member ofthe Academia Europaea: The Academy of Europe, Informatics Section(since 2013). He is recipient of the ACM Distinguished Scientist award(2009) and the IBM Faculty Award (2012). He is an Associate Editorof IEEE Transactions on Services Computing, ACM Transactions onthe Web, and ACM Transactions on Internet Technology and on theeditorial board of IEEE Internet Computing. He is the Editor-in-Chief ofComputing (an SCI-ranked journal of Springer).