I Introduction
Mobile edge computing (MEC) enabling lowlatency, highbandwidth, and agile mobile services has attracted much attention in both academia and industry [1][3]. MEC is a paradigm that provides cloud services within the vicinity of mobile device via the radio access network. In contrast to the service centralization of mobile cloud computing (MCC), MEC aims to empower the network edge. Since the proximity, MEC has the virtue of low latency, mobile energy saving, privacy and security enhancement, contentawareness. Especially, one advantage is preprocessing of large data at the MEC server before sending it to the cloud. As fusion of wireless communications and mobile computing, MEC is viewed as a key technology of next generation networks, e.g., 5G, Internet of Things (IoT), Internet of Me, Tactile Internet, Social Networks, etc.
As a novel and hot research area, there are some works on MEC. In [4], the offloading from a single mobile device (MD) to multiple edge devices has been investigated under a proposed optimization framework. And the authors focus on the task allocation and computational frequency scaling. In [5], computation offloading together with interference management have been jointly studied under an integrated framework. In [6], performance guaranteed computation offloading has been investigated for mobileedge cloud computing. An energy minimizing optimization problem is proposed and solved by apply KarushKuhnTucker (KKT) conditions. In [7]
, resource allocation and tasks scheduling in heterogeneous cloud have been studied for delaybounded MEC. Maximization for the probability of meeting the delayrequirements is proved to be concave and an optimal algorithm is proposed. In
[8], the authors have investigated matching problem between the MEC service providers (SPs) and the user equipments (UEs) under a multiMEC and multiUE scenario. In [9], joint offloading and computing optimization have been investigated in wireless powered MEC systems. Energy cost minimization of access point is formulated in convex framework and optimal solution in semiclosed form is derived. In [10], the tradeoff between service response time and QualityofResult (QoR) has been identified in MEC. A new optimization framework that minimizes app energy consumption and service response time is proposed by jointly optimizing the QoR and the offloading strategy. In [11], an auctionbased profit maximization method has been proposed towards the hierarchical MEC. In [12], energy harvesting aided MEC has been investigated, and an online learning algorithm for offloading and autoscaling is proposed.Coalitional game theory has been widely applied in design and analysis of wireless communications system especially when cooperation occurs
[13]. In [14], a securityaware resource allocation approach of delivering mobile social big data has been proposed by utilizing joint matchcoalitional game. In [15], a trustbased hedonic coalitional game has been formulated towards trustworthy multicloud services communities, and a threefold solution is derived thereafter. In [16], A layered coalitional game algorithm is given for hierarchical cooperation in operatorcontrolled devicetodevice (D2D) communications. In [17], by using coalitional game theory, we have studied cooperative transmission in vehicular networks.Traditional data collection of edge MDs are transferred to the core network directly with high bandwidth cost and latency. Instead, data collection is performed at the edge of the network in MEC. Bandwidth consumption is greatly reduced and latency performance improves. We investigate the data offloading of MEC in the paper. Specifically, data of MDs are transmitted to neighboring MEC servers (offloading from MDs to MEC servers), where preprocessing (e.g., data analytic) is performed, over fading channels. After data preprocessing, the results are sent to the cloud server (over wire channels, e.g., fiber channels). As multiple MDs and multiple MEC servers are considered, the data offloading from MDs to MEC server becomes a challenge problem. How to schedule multiple MDs’ offloading?How to choose the MEC servers for offloading? How to encourage offloading among MDs and MEC servers so as to attain the virtue of MEC?
Utilizing coalitional game theory and pricing mechanism, we propose a joint coalitionpricing based data offloading approach. First, MDs and MEC servers form different coalitions. In each coalition, MDs are scheduled for data offloading to selected MEC server within the same coalition. Selection of MEC servers is closely related to our proposed notion “coverage probability ” which shows the availability of MEC servers to MDs. Furthermore, pricing mechanism is utilized to promote cooperation between MDs and MEC servers. In one hand, MDs could gain throughput increase when utilizing MEC server (compared with no MEC). On the other hand, MEC server charges the MDs that choose it as MEC service provider in each data offloading.
Ii System model
As illustrated in Fig. 1, consider an MEC network consisting of a cloud server, MEC servers, and MDs. The cloud server is represented as . and denote the sets of MEC servers and MDs, respectively. Denote the transmit distance of MD as . That is to say, within the sphere with radius around MD (referred to as transmit range), an MEC server could correctly receive the transmitted data. When MEC server is in the transmit range of MD , we call “ covers ”. Mobile device is active with probability , independently of other MDs, in each slot. When two or more mobile devices transmit simultaneously, it is referred to as a collision. We assume that the transmissions fail once a collision occurs.
MEC servers and MDs form coalitions to facilitate data transmission. Mobile devices in the same coalition can cooperate to schedule the data transmission, and only one MD can transmit at a slot to avoid interference. MEC servers that are in the same coalition with MD and cover MD constitute the feasible MEC server set of MD . In each coalition, a scheduler chooses an active mobile that can transmit while other active MDs remain silent.
When MD is scheduled to transmit at a timeslot, it selects one MEC server from its feasible MEC server set, and sends its data to the the selected MEC server for preprocessing and forwarding. If the feasible MEC server set is empty, i.e., no MEC server is available, the MD transmits to the cloud server directly (with a lower rate given fixed power). Then one MD only transmits to at most one MEC server for each transmission. Furthermore, as no more than one MD is scheduled to transmit at a given timeslot, one MEC server communicates with at the most one MD during a given timeslot. MEC server acts as a helper in data transmission from the MD to the cloud server. When MD utilizes MEC server for its transmission, MEC server charges MD with price per transmission.
When some MDs and MEC servers form a coalition, the MDs would share the channel with each other in time division multiple access (TDMA) mode, and they need to pay for the MEC servers. However, the MDs can avoid collision and improve bandwidth efficiency and latency. On the other hand, although there are costs in receiving, processing and forwarding the MDs signals, the MEC servers could achieve revenues by charging the MDs. In a word, both MDs and MEC servers have incentives to form coalitions.
Iii Coalitional game formulation
A coalitional game is uniquely defined by the pair , where is the set of players, and is the coalition value quantifying the worth of a coalition in a game. Any nonempty subset of is referred to as a coalition. In the paper, players are the MDs and the MEC server. That is to say, . is a coalition. Denote and . Consider a slot, let denote MEC server covers MD during the whole slot and .
The per bandwidth data rate increase of MD with the edge computing of the MEC server is denoted as .^{1}^{1}1Since the bandwidth efficiency and latency improves with the edge computing, then the equivalent per bandwidth data rate from the MD to the cloud server increases. Then the per bandwidth rate of MD when utilizing MEC server can be expressed as with being direct transmission rate to the cloud server under the same power.^{2}^{2}2Specially, when direct transmission is impossible,
Formally, the scheduler in is a map such that for all and iff .
The average effective throughput for MD can be expressed as
where denotes the ratio of timeslots that MD is chosen to transmit. For example, we can assume chooses the minimal element from the set of feasible transmitting MD . Denote with , then
(1) 
and
(2)  
In the paper, we assume that selection of the MEC server is performed according to a uniform probability distribution for simplicity. The average data rate for MD
with mobile edge computing, , is given bywhere . If , i.e., , . The average direct rate
where
is the probability that at least one MEC server in the coalition covers MD . Suppose , i.e., the data increase for MD is irrelevant to the selection of MEC servers, then and the average throughput for MD can be simplified as
Remark: If there are active MDs outside at a given timeslot, at least one MDs outside transmits simultaneously with the scheduled MD in no matter how the MDs outside form coalitions. Thus, there is no collision if and only if (iff) all MDs outside is inactive.
For MD , the average payment made to the MEC servers can be given by
with
If , . Assume , i.e., all the MEC servers set the same price for MD , then the average payment can be simplified as
Remark: The MEC server charges the MD once the MDs employs the MEC server for a transmission, it does not take the collisions into account. That is to say, the actions in other coalitions do not affect the charging.
The payoff of MD is determined by
For MEC server in , the revenue charged from the MDs can be given by
(3) 
where is the probability that MD employs MEC server for transmission, and it is given by
(4)  
Assume that MEC server receives the signal of MD at cost and the cost of preprocessing signal and forwarding to the cloud server as . The average cost of MEC server can be expressed as
(5) 
Remark: MEC server receives the message of MD once it covers MD (with probability ), and it preprocesses the message and forwards only when it is selected as the helper by MD (with probability ).
The payoff of MEC server is determined by
Define
be the set of feasible payoff vectors for
, we formulate the considered data collection problem as a coalitional NTUgame .Iv Coalitional game analysis
In this section, we carry out analytical analysis on the formulated game. First, we present two analytical results. Next, we investigate the stability of the game and propose a sufficient condition for the existence of the core.
In the beginning, we have the following lemma.
Lemma 1.
Let
denote the sum payoff of . When and , we have
That is to say, the pricing has no effect on the sum payoff in this case.
Proof:
Remark: Lemma 1 reveals the fact that the total revenues obtained by the MEC servers equal to the payments of all the MDs.
In addition, we obtain the second lemma.
Lemma 2.
A coalition should have at least one MD. Otherwise, and . That is to say, when there are only the MEC servers, the MEC servers have no stimuli to form coalitions and each MEC server will act alone.
Proof:
Remark: The function of the MEC server is preprocessing and forwarding the MD’s signal. So when there is no MD, it is meaningless to group only the MEC servers together.
On the other hand, when there is no MEC server in a coalition , i.e., ,
for and . Specially when , we derive
Hence, when satisfying , the MDs will form coalitions to improve the utility. Specifically, let with , based on (1) and (2), we can derive that if
(9) 
forming coalition is profitable.^{3}^{3}3Although forming may be not optimal, it is at least better than acting alone. Specially, when , i.e., all MDs have the same active probability, we can derive that (9) holds, then forming coalitions is always profitable in the case. The reason is as follows: Since the formulation of coalitions can avoid collision, then it can improve the effective throughput.
The definition for the core of our coalitional game is given as follows.
Definition 1.
The core of is defined as
The following lemma gives a sufficient condition for the existence of the core.
Lemma 3.
The core of is nonempty once the following conditions hold ():

, , , and .

or .

, and .
Proof:
When 1) holds, we can find , , , and to satisfy 2). If 2) does not holds, we have
for and for . Then, each MD and MEC server will act alone. In this case, the core is empty. When 3) holds, we can prove that is balanced [18]. Thus, the core is nonempty according to the BondarevaShapley theorem. ∎
V Numerical results
In the simulations, we consider 2 MDs (MD 1 and MD 2) and 2 MEC servers (MECs 3 and MECs 4). Two dimensional scenario is considered. MDs are uniformly distributed in an 2dimensional area with
, where is its center, MECs 3 locates at , MECs 4 locates at . topology implementations are averaged to get the mean performance.Fig. 2 shows the probability of MD 1 and MD 2 being covered by MECs 3 and MECs 4, respectively. From the figure, we can find that the coverage probabilities of MECs 3 over MD 1 and MD 2 are same, and the coverage probabilities of MECs 4 over MD 1 and MD 2 are same also. In addition, the coverage probability of MECs 4 is higher than that of MECs 3 and both increase with the increase of . We can explain by theoretical analysis. Since MD 1 and MD 2 are uniformly distributed, the coverage probability can be computed using Fig. 3 in twodimensional case. For any MD, when it is in the overlap of a circle centering at with radius , it can be coverage by MECs 3. Then, the coverage probability is the ratio between the area of the overlap and . With the increase of , the ratio increases (until it reaches 1). For the MECs 4, the center becomes and we can obtain similar results. Furthermore, as (7,6) is nearer to (i.e., area center) with the same , its overlap area is bigger than that for . Then the coverage probability of MECs 4 is higher than that of MECs 3.
There are 15 possible coalitional structures for 2MD and 2MECs, as listed in Table I. Applying Lemma 2, is similar as and is similar as . Meanwhile, as MD 1 and MD 2 as well as MECs 3 and MECs 4 are roleexchangeable in the scheme, we can see that is similar as ; is similar as ; , and are similar as ; and is similar as . In conclusion, we need to consider . In the following simulations, , , , , , , , and .
: {1,2,3,4}  : {1,3},{2,4}  : {1,2},{3,4} 
: {1,3,4},{2}  : {1,2,3},{4}  : {1},{2},{3,4} 
: {1,2},{3},{4}  : {1},{2,3,4}  : {1,2,4},{3} 
: {1},{2},{3},{4}  : {1,4},{2,3}  : {1,4},{2},{3} 
: {1},{3},{2,4}  : {1},{4},{2,3}  : {2},{4},{1,3} 
Fig. 4 plots the mean utility performance under , which is also referred to as the grand coalition, i.e., all nodes form one coalition. We can observe that the utility performance improves when the transmit distance becomes longer. It is because that with the expand of the transmit distance, the coverage probability increases. Then the MEC severs could take part in the data collection with higher probability, and the rate improves. The performance of MD 1 is better than that of MD 2 since MD 1 has priority in transmitting node selection in the scheme when MD 1 and MD 2 in the same coalition. MECs 4 has better coverage probability which leads to more revenues charged from MDs. Hence MECs 4 has better performance than MECs 3.
Fig. 5 demonstrates the mean utility performance under , where each node forms a coalition and transmit solely. We can find that the utility performance of MECs 3 and MECs 4 remains as zero. This can be explained by using Lemma 2. The utility performance of MD 1 and MD 2 are the same and remain constant. The can be viewed as a baseline.
Fig. 6 shows the mean utility performance under . MD 1, MECs 3, and MECs 4 are in a coalition. The performance improves with the increase of transmit distance. As MD 2 transmits solely, the utility remains constant with respect to transmit distance. The performance of MD 1 is better than that of MD 2. It verifies that the MEC could improve utility performance (especially by comparison with Fig. 5).
Fig. 7 illustrates the utility performance with . MECs 3 and MECs 4 respectively form a coalition. And the utility performance remain zero, which complies with Lemma 2. MD 1 and MD 2 form one coalition. Since no MECs available, the utility performance are constant. The performance of MD 1 is better than that of MD 2 because of its priority in transmitting selection within the coalition. In addition, compared to Fig. 5, we can see the utility increase of MD 1. That is to say, even without MEC server, forming coalition is profitable for MD. This can be explained as follows: When forming coalition, collisions can be avoided. Then the effective throughput increases and utility improves accordingly.
Fig. 8 depicts the average utility performance under setting. MD 1 and MECs 3 respectively constitute a coalition. Then the utility performance remains constant. MD 2 and MECs 4 form one coalition, and the utility increase with transmit distance. Furthermore, MD 2 has better performance than MD 1 and MEC4 has better performance than MECs 3. This verifies the advances of constituting coalition.
Fig. 9 draws the mean utility performance with . In the setting, MD 1 and MECs 3 form one coalition; MD 2 and MECs 4 constitute another coalition. Compared with Fig. 5, utility performance of each node has improvement. Additionally, MD 2 has better performance than MD 1 and MEC4 has better performance than MECs 3. This is because that MECs 4 has better coverage probability than MECs 3, then the MECs 4 & MD 2 coalition has better performance.
Fig. 10 plots the average utility performance under . MD 1, MD 2, and MECs 3 constitute one coalition. MECs 4 forms another coalition solely. Since MECs 4 act alone, its utility remains zero. MD 1 and MD 2 could utilize MECs 3 for MEC data transmission to the cloud server. Utility performance increases with increment of since bigger value of incurs higher coverage probability and transmission rate performance thereafter. MD 1 has better performance than MD 2 as MD 1 has selection priority in scheduling.
Comparing utility performance of the considered coalition structure, we find that the grand coalition, , has the best score. In addition, all utilities in the grand coalition are positive and , . Then the conditions of Lemma 3 hold, we claim that the core is not empty and is in the core.
Vi Conclusion
Data offloading in mutiMD and multiMECserver scenario is studied in the paper. We propose a coalition based pricing scheme. The proposed scheme incorporates MD scheduling, the MEC server selection and offloading encouragement. Theoretically, coalitional game is first formulated, and the game is analyzed thereafter. Especially, we give sufficient conditions for nonempty of the core. In the end, numerical results verify the effectiveness of the proposed scheme by showing the utility performance.
References
 [1] M. Patel et al., “Mobileedge computing,” Eur. Telecommun. Standards Inst., Sophia Antipolis, France, White Paper, 2014.
 [2] Y. Mao, C. You, J. Zhang, K. Huang, and K. B. Letaief, “A survey on mobile edge computing: The communication perspective,” IEEE Commun. Surv. Tutor., doi:10.1109/COMST.2017.2745201, to appear
 [3] N. Abbas, Y. Zhang, A. Taherkordi, and T. Skeie, “Mobile edge computing: A survey,” IEEE Internet of Things, doi:10.1109/JIOT.2017.2750180, to appear
 [4] T. Q. Dinh, J. Tang, Q. D. La, and T. Q. S. Quek, “Offloading in mobile edge computing: Task allocation and computational frequency scaling,” IEEE Trans. Commun., vol. 65, no. 8, pp. 3571  3584, Aug. 2017.
 [5] C. Wang, F. R. Yu, C. Liang, Q. Chen, and L. Tang, “Joint computation offloading and interference management in wireless cellular networks with mobile edge computing,” IEEE Trans. Veh. Tech., vol. 66, no. 8, pp. 7432 7445, Aug. 2017.
 [6] X. Tao, K. Ota, M. Dong, H. Qi, and K. Li, “Performance guaranteed computation offloading for mobileedge cloud computing,” IEEE Wireless Commun. Lett., doi:10.1109/LWC.2017.2740927, to appear.
 [7] T. Zhao, S. Zhou, X. Guo, and Z. Niu, “Tasks scheduling and resource allocation in heterogeneous cloud for delaybounded mobile edge computing,” in Proc. IEEE ICC, 2125 May 2017, Paris, France.
 [8] H. Zhang, F. Guo, H. Ji, and C. Zhu, “Combinational auctionbased service provider selection in mobile edge computing networks,” IEEE ACCESS, doi:10.1109/ACCESS.2017.2721957, to appear.
 [9] F. Wang, J. Xu, X. Wang, and S. Cui, “Joint offloading and computing optimization in wireless powered mobileedge computing systems,” in Proc. IEEE ICC, 2125 May 2017, Paris, France.
 [10] Y. Li, Y. Chen, T. Lan, and G. Venkataramani, “MobiQoR: Pushing the envelope of mobile edge computing via qualityofresult optimization” in Proc. IEEE ICDCS, 58 June, 2017, Atlanta, GA, USA
 [11] A. Kiani and N. Ansari, “Towards hierarchical mobile edge computing: An auctionbased profit maximization approach,” IEEE Internet of Things, doi:10.1109/JIOT.2017.2750030, to appear
 [12] J. Xu, L. Chen, and S. Ren, “Online learning for offloading and autoscaling in energy harvesting mobile edge computing,” IEEE Trans. Cogn. Commun. Netw., doi:10.1109/TCCN.2017.2725277, to appear
 [13] W. Saad, Z. Han, M. Debbah, A. Hjorungnes, and T. Basar, “Coalitional game theory for communication networks,” IEEE Signal Process. Mag., vol. 26, no. 5, pp. 77 97, Aug. 2009.
 [14] Z. Su and Q. Xu, “Securityaware resource allocation for mobile social big data: A matchingcoalitional game solution,” IEEE Trans. Big Data, doi:10.1109/TBDATA.2017.2700318, to appear.
 [15] O. A. Wahab, J. Bentahar, H. Otrok, and A. Mourad, “Towards trustworthy multicloud services communities: A trustbased hedonic coalitional game,” IEEE Trans. Serv. Comput., doi:10.1109/TSC.2016.2549019, to appear.
 [16] X. Lu, P. Wang, and D. Niyato, “Hierarchical cooperation for operatorcontrolled devicetodevice communications: A layered coalitional game approach,” IEEE WCNC, Mar. 912, 2015, New Orleans, LA, USA
 [17] T. Zhang, W. Chen, Z. Han, and Z. Cao, “Coalitional game theoretic approach for cooperative transmission in vehicular networks”, in Proc. IEEE ICC, Jun. 913, 2013, Budapest, Hungary.
 [18] R. B. Myerson, Game Theory, Analysis of Conflict. Cambridge, MA: Harvard Univ. Press, Sep. 1991.
Comments
There are no comments yet.