The Fisher Market Equilibrium Price Based Multi-Access Edge Computing Coalition Formation

33 Abstract The emergence of multi-access edge computing (MEC) aims at extending cloud computing capabilities to the edge of the radio access network. As the large-scale IoT services are rapidly growing, a single edge infrastructure provider (EIP) may not be suﬃcient to handle the data traﬃc generated by these services. The coalition method has been used in MEC for resource optimization, latency, energy consumption reduction, computation oﬄoading, etc. However, the majority of research does not consider the price of computing resources corresponded to a container. Moreover, each SP does not choose EIP with the highest cost-performance to sign a medium/long-term computing resource purchase or lease contract. In this work, we consider a scenario with a collection of SPs with diﬀerent budgets and several EIPs distributed in geographical locations. During the ﬁrst phase, we get the market equilibrium price and select the optimal EIPs to make a deal by solving the Eisenberg-Gale convex program. In the second stage, using a mathematical model, we maximize EIP’s proﬁts and form stable coalitions between EIPs by a distributed coalition formation algorithm. Numerical results demonstrate that the eﬀectiveness of our method is signiﬁcantly better than the existing model.

services [1,2,3]. In MEC, the computing resource is deployed on the edge side, closing to users. In contrast to the service centralization of mobile cloud computing, the computation requests from users can be offloaded to edge nodes (ENs) closing to users. 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. [4]. However, the power supply limitation of a single edge node raises the question of how an edge node provides high computing resources. As the large-scale IoT services, such as healthcare [5], smart cities [6], agriculture monitoring [7], and many others [8] are rapidly growing, a single EN structure may not be sufficient to handle these data traffic. In addition, cloud data centers (DCs) are often geographically distant from the end-user, which may cause high backhaul traffic. It may induce that the latency constraints of offloaded computation tasks cannot be satisfied due to the propagation delay from the ENs to the DCs. The above reasons drive physical resources and workload sharing between ENs in a peer-to-peer manner by forming coalitions [9].
A coalition is a collection consists of participants in a game to complete a specific task, generating with the arrival of new requests and breaking up with the completion of request processing [10,11]. However, affected by the number of edge nodes, computing power, optimization targets, and constraint conditions, the coalition formation is more complicated in natural edge scenarios. Hence, it is one of the significant challenges in the field of edge computing [12,13]. We currently consider a situation where service requests are hosted into containers executed through a virtualization platform running on ENs. The objective of each service is to single out the edge infrastructure providers (EIPs) with maximized cost-effectiveness and find its price in line with market laws under the premise of guaranteeing its QoS.
Furthermore, the EIPs' goal is to maximize their profit by amortizing their costs, which requires that the aggregated request load submitted to containers must be intense enough to amortize its costs.
The objective of services induces a novel market-based solution framework that aims to get the EIPs maximized cost-effectiveness and obtain a price in line with the market laws. The basic idea of the method used in this work is to assign different prices to a computing resource block (CRB) corresponding to a container of different EIPs. According to the market rules, highly in-demand CRBs are priced high while under-demanded CRBs' prices are low. Moreover, we assume that each SP has a specific budget for CRB procurement. And the budget is used to capture service priority. Given the CRBs prices, each SP buys CRBs from the EIPs with the maximized cost-effectiveness. When each SP spends the total budgets and CRBs of EIPs are entirely sell out, the resulting prices form a market equilibrium.
Aims at the second goal, if each EIP agrees to cooperate by sharing their requests and computing resources, they can increase their profits. In particular, an EIP can increase its net profit by turning off some ENs and transferring its workloads to other EIPs, or running workloads originated from other EIPs to improve its revenues. Here, depending on the market price and optimal EIPs for SPs, we use a suitable algorithm to decide what conditions the EIPs willing to cooperate. The main contributions of this work are summarized as follows: • We consider a scenario with a collection of SPs with different budgets and several EIPs distributed in geographical locations. For the scenario, we introduce a system architecture of MEC in an area.
• By solving the Eisenberg-Gale convex program, each SP chooses EIP with the highest cost-performance to sign a medium/long-term computing resource purchase or lease contract and gets the market equilibrium price of the computing resource block.
• Depending on the results of the above phase, we use a mathematical model to maximize EIP's profit and a distributed algorithm to form a stable coalition.
• We conduct the simulation experiments for the above question and demonstrate that the effectiveness of our method is significantly better than the existing model.
The rest of paper is organized as follows. We first discuss related work in Section , and introduce the system model in Section . In Section , we provide a formulation of the market equilibrium price generating and present the detailed description of the cooperative game. We discuss the simulation results that demonstrate the effectiveness of our approach in Section , followed by the conclusion in Section .

Related work
The coalition method has already been investigated in recent years to maximize the profit of EIP. Notably, the methods based on game theory have also been used in the field of mobile/multi-access edge computing for resource optimization [14,15], latency and energy consumption reduction [16,17], computation offloading [18,19], etc. GlebKoshevoy [20] et al. introduce Shapley value, nucleolus, kernel, and bargaining set in the cooperative game theory. Using time, utility, cost-performance [21] stability principle or the reasonableness principle [22], it continuously removes the relatively unstable coalition structure in the space of strategy and finally retains the approximately optimal coalition structure. T. Ling [23]  We contemplate that EIPs are distributed in different geographic locations in an area with distinct configurations and limited computing capacities. Each region hosts a set of edge nodes to run the services which are in charge of processing the data generated by user devices located in that area. We assume that each service has a definite budget for CRBs procurement and offloads its requests/data to MEC as many as possible. The service budget may be related to its net profits from users.
However, we do not focus on it to simplify the problem in this research. Through the maximum revenue generated by using the resource bundle of EIPs, we can evaluate the merit of an EIP to a service provider. An EIP may have different worth for different service providers because of the varied distance between an SP and an EIP. Intuitively, each SP would choose its favorite resource bundle with optimal cost performance.
There exists a platform between SPs and EIPs for the sake of collecting the information of EIPs (e.g., computing capacities) and SPs (e.g., budget, etc.) and computing an ME price of unit resource bundle, which maximize the SPs' revenue and also fully allocates the EIPs' resources. The ME price can act as an incentive mechanism to charge SPs and to reward MECs. On one side, if the amount of EIPs' resource bundles is fixed, the market-clearing price is increases with the SPs' budget increase. On the other side, anchored the SPs' budget, the price is inversely proportional to the number of EIPs' resources. To format a rational coalition among EIPs, we focus on computing a market-clearing equilibrium price that SPs spent all budgets and EIPs sold out all resources.

Problem formulation
APs have the same objective as EIPs, which is to prompt their net profit as much as possible. Without loss of generality, we assume that it is independent among areas to allocate containers/CRBs for services. We only consider the single geographic district, and extend it to multiple zones is straightforward. In general, we deem the EIPs' total computing capacity is steady, which is determined by predicting the whole region's resource requirements. That is not the point on which we focus. Nevertheless, the requests from users are dynamic variance along with the users' changing. Therefore, precisely predicting the demands originated from users is challenging. Moreover, to improve Qos and decrease EIPs' expenditure as much as possible, the MEC coalition is indispensable. Namely, sharing workloads and resource bundles to serve SPs can reduce EIPs' energy consumption costs and increase their revenues by hosting services on others' EIP. It is noteworthy that each EIP prefers to cooperate with some EIP to gain more profit than others. The coalition formed is unstable without the above incentive mechanism. That is to say, a member in an alliance can always find a more profitable confederation that leaves the current league.

EIP Selection and CRB pricing model
We denote M = {1, 2, · · · , M } and S = {1, 2, · · · , S} as the set of EIPs and SPs, where |M| = M , |S| = S, respectively. Denote i and j as the SP and EIP index, respectively. Assuming that each EIP j has c j homogeneous computing unit [28] (e.g., VMs, servers, etc.). Let a ij is the portion of the resources on EIP j allocated to SP i, 0 ≤ a ij ≤ 1. Then, c j a ij is the number of computing units allocated to SP i from EIP j. Denote a i = (a i,1 , a i,1 , · · · , a i,M ) as the vector of resources allocated to SP i from all the EIPs. We assume that the computing resource is dividable to support hosting container (VM/Docker). Therefore, c j a ij can be non-integer; and we define the budget of SP i as B i . The ultimate goal is choosing an EIP with the best cost performance for an SP.
Computing ME solutions include an equilibrium price vector p = (p 1 , p 2 , · · · , p M ), p j is the price of EIP j, and a resource allocation matrix A, the element in which at row ith and column jth is a ij . Note that p j is the price of all the resources of EIP j. Then one unit resource price equals pj cj , ∀j. We define u i (a i , p) as the SP i's utility function of the number of resources a i received from EIPs under the resource price p. Depend on the capacity constraints of EIPs, then we have Each SP is a player in the Fisher market game, in which the player aims to maximize its utility subject to the Definition 1 A market outcome that maximizes the utility of each buyer subject to its budget constraint and clears the market is called a market equilibrium [29].
In-state, (p * , A * ) is a market equilibrium if and only if satisfy the following conditions: • Condition 1 : For all i ∈ S, a i maximizes buyers i's utility given prices p * and • Condition 2 : Each item (e.g., computing resource block) j either is completely sold or has price 0, i.e., S i=1 a ij − 1 p j = 0, ∀j ∈ M.
• Condition 3 : All budgets get spent, i.e., M j=1 p j a ij = B i , ∀i ∈ S. A market equilibrium is guaranteed to exist if each item is desired by at least one buyer and each buyer desires at least one item [30]. Here, condition one guarantees that the equilibrium allocation a * i maximizes the utility of SP i at the equilibrium prices p * and SPs' budgets constraints B i . Condition two maximizes the EIPs' resource utilization, i.e., the resources of EIPs are entirely sold in the market. Condition three represents that each SP runs out of its budget to purchase resources from EIPs. Moreover, condition two and three is called the market clearing condition [31].

Service provider utility
For the sake of representation simplicity, we use u i (a i ) to denote the utility of SP i, i.e., u i (a i , p) = u i (a i ). The model considers linear functions for ease of investigation. We view the delay-sensitive applications and assume that the transmission bandwidth is large enough and the data size need transmission is relatively small.
Hence, we can neglect the transmission delay and only consider propagation latency and processing delay [32,33]. The sum total latency of a request from users consists of three parts: the round-trip latency between a user and a PoA, such as base station (BS) or access point (AP), the return journey delay between the PoA and an EIP's node which hosting the service application, and the processing latency of request. A detailed depiction is shown in Figure . Note that we only investigate the model from the aggregation layer to the MEC layer. For the sake of simplicity, we assume that each service is located at one PoA in a region (If a service's requests from several PoAs, we can take sum over all the PoAs' requests). When there is more than one PoAs for a service, we take the sum over all PoAs requests processed by the MEC. Let T max i as the maximum latency tolerance of service i , then Denote λ max ij as the maximum number of requests that EIP j's nodes can process.
We use M/M/1 queues to model the processing latency and assume the workload is assigned among computing units evenly [32,33,34,35]. The processing latency l p ij can be represented as , where µ ij is the processing rate of a computing unit of EIP j to deal with service i's requests, λ ij is the arrival rate of request from service i to EIP j. In addition, to assure the stability of the queue, the inequality λij aij < µ ij must hold. Otherwise, the delay of the queue will be infinite. Combining formulas (1) and (2), we can derive If l n ij < T max i , the maximum number of requests from service i processed by the EIP j is Denote r i as the profit of successfully serving a request from service i [34]. Hence, the revenue of service i u ij (a ij ) = r i g ij c j a ij . Let v ij = r i g ij c j , and then we have Thus, Obviously, r i g ij can be a valuation index which is the gain of service i from one unit resource of EIP j. Accordingly, v ij is a valuation index of all resources of EIP j.
It is simple to verify that u i (a i ) is a linear function with degree d = 1.

Solution
Define the ratio vij pj is the bang per buck [37] of EIP j to SP i, which is the utility gained by SP i per unit amount of money spent on EIP j. Thus, we can define pj } as the maximum bang per buck (MBB) [37]. Only SP i buys resources from EIP of α (i), will each SP spend entire budget and maximize its utility. Derive from definition 2, the buyers' utility is linear. Hence, by solving the Eisenberg-Gale (EG) convex program, we can find the market equilibrium. The EG convex program can be described as a ij ≥ 0, ∀i, j.
By setting a ij > 0 and its value is small enough such that the constraints (8)-(10) are stringent inequality. Thus, Slater's condition holds, and then Karush-Kuhn-Tucker (KKT) conditions are necessary and adequate for optimality [38]. Using ϕ i , p j , and ζ ij to represent the dual variables affiliated with restrictions (8), (9), and (10), severally. According to the Lagrangian, we have The KKT conditions are From the KKT conditions, we can derive Note that variable p j can be seen in the EG convex program as the pang per buck of EIP j to SP i. The subsequent theorem reveals the connection between the EG convex program and the market equilibrium solution. Besides, some properties of the ME are also represented.

Theorem 1
The optimal solution of the EG convex program is a market equilibrium. Particularly, the Lagrangian dual variables of the EIPs' capacity restrictions (9) are the equilibrium prices. In addition, the allocation at the equilibrium is Pareto optimal and envy free, which also satisfies the sharing-incentive and proportionality properties [39].
At the equilibrium, the resource allocation maximizes the utility and exhausts each service's budget. Moreover, each SP i only purchases resources from ENs in α i . Additionally, SPs' optimal utilities and equilibrium prices are exclusive.

Optimal Coalition Allocation Utility
An EIP j aims at increasing its net margin as much as possible. We assume that each service application is encapsulated in a container with a computing resource block, and each EIP's edge nodes need a virtualization platform to run the container.
EN (j) represents the set of edge nodes of EIP j. Let w z (η) be the expected power consumption of edge node z.
, where η ∈ [0, 1] is the utilization of resource (CPU) on EIP edge node z, W max z and W min z represent the power consumption with idle and fully utilized CPU states, respectively. The net margin rate of EIP α (i) denoted by M α(i) .
, where W e (x e , η e ) = x e W min e + W max e − W min e η e ; x e is a binary variable that depicts the edge node state; when x e = 1, the state is switched on; conversely x e = 0, the state is switched off. n i denotes the number of containers/CRBs allocated for service i, and N i is the minimum number of containers/CRBs that satisfies the delay requirement of service i, which can derive from the inequality (4) and the stability condition λij aij < µ ij of the queue. η e represents the ratio of resources allocated to all the containers running on edge node e to its total resource capacity. E j is the price (per unit of time) charged EIP j for electricity. b i is binary variable. When n i < N i , b i = 1, which represents that CRBs allocated for service i cannot satisfy the Qos requirement, conversely, b i = 0. When SP i's Qos is not met, it will charge EIP j for a monetary penalty P i,j . R α(i),i represents the revenue that EIP α (i) earns for running a container for processing its requests from SP i. In equation (24), the first term on the right means the sum of EIP α (i)'s income to run n i containers, where the second term on the right denotes the overheads of EIP α (i).
The coalition formation algorithm (in Subsection ) needs to compute the coalition net margin Ψ (C) to get the payoffs that EIPs gain in any coalition C. By introducing the Mixed Integer Linear Program (MILP) model to solve a maximization problem, we can find the best allocation of the containers V onto the edge nodes N c = ∪ j∈C EN (j) to run the services S c = {i : α (i) ∈ C }. Denote m (i) as a mapping V → S c , which represents the service application run by container i. For the sake of simplicity, we define vector R = R α(i),i i∈Sc , n = (n i ) i∈Sc , b = (b i ) i∈Sc , P = P i,α(i) i∈Sc , W = (W j (x j , η j )) j∈Nc , E = E α(i) α(i)∈C,j∈EN (α(i)) (here, j is the edge node). The MILP model can be described as In the above MILP model, the binary variable x i = 1 if edge node i is turned on, otherwise x i = 0. The binary variable y ij = 1 if container j is allocated on edge node i, else y ij = 0. The non-negative real variable η i denotes the ratio of resources allocated to all the containers running on edge node i to its total resource capacity, and L i represents the maximum number of containers that can host on edge node i.
The non-negative integer variable n k denotes the number of containers allocated for service k. The objective function Ψ (C) of the maximization problem is the retained profits rate gained by coalition C composed of EIPs. It is an extension of the net profit M α(i) of EIP α (i) and has the following constraints.
Constraint condition (26) ensures that each container is hosted on one edge node at most. Limitation condition (27) assures that containers will not be allocated on edge nodes turned off. Equation

Coalition preference function
We assume that each EIP is a rational participant in a cooperative game to make optimal strategic decisions to increase their profits. Thus, every EIP may form a coalition to pursue its maximum yields, in which each EIP (coalition member) must go along with sharing its resources with others. An alliance formation is a dynamic process, where an EIP moves from one coalition with a lower utility to another with a higher. We model the process as a coalition formation cooperative game. Foremost, an EIP must ensure that joining a league gained is no worse than working alone.
In addition, to ensure that the benefits are not short-lived, the coalition, an EIP joined, needs to satisfy the following properties [40,41].
• Coalition stability property: A coalition is stable if no participant can deviate from the current federation to reach a better one subjectively.
• Fairness property: Any member in the current coalition expects that the resulting profits divide fairly among all players.
Hence, the EIPs desiderate finding a method to decide whether to join a coalition.
By modeling the coalition formation process as a cooperative game with transferable utility [42], each EIP can cooperate with others to maximize its net profit by using a coalition formation algorithm. Here, we use a hedonic game, where any player's gain is decided by the coalition members to which the player belongs; and the players' Given that M s is the EIPs set selected, then a coalition C ⊆ M s acts as an entity, representing an agreement that they must agree to share their resources and users among EIPs. At any time, we divide the participants set into a coalition partition Π = {C 1 , C 2 , · · · , C l }. Each element C k in partition Π is disjoint for k = 1, 2, · · · , l, i.e., ∪ l k=1 C k = M s and C i ∩ C j = ∅, i = j. Denote C Π (i) as a coalition containing player i in partition Π, ∀ i ∈ M s . Define Ψ (C | Π) as the utility value of a coalition C in partition Π of M s . For a hedonic game, the utility value of alliances is independent of each other, i.e., Ψ (C | Π) = Ψ (C), which can be calculated by formula (25) in subsection . Let φ i (C, Ψ) be a fraction coalition utility value received from coalition C by i, where i ∈ C. That is to say, φ i (C, Ψ) is the payoff of i in coalition C. Each player in cooperative games aims to get a stable coalition and gains rewards as high as possible. In parallel, it is critical to maintaining a stable coalition to distribute the federation value fairly among the members. Furthermore, every participant cannot improve its rewards by leaving the current coalition to join a new one. In a nutshell, the coalition value distribution is fair and can be reached using a profit allocation schedule with the following properties [43].
• Symmetry property: The efficiency property guarantees that the total coalition value is assigned. The symmetry property denotes that if two participants who make the same contribute to every subset composed by other players, they should obtain the same profit.
The dummy property represents that the members, nothing marginal contribution provided to any other coalitions, get zero profit. In addition, the fairness property means that any two participants who make the same contribution to the alliance will get the same payoff. Shapely value [44], a payoff allocation, satisfies the additivity, strong monotonicity and the above four properties. Here, we use the Shapely value defined in the literature [45], described as , where B is all possible subsets that do not contain i of coalition C, including the empty set. Define i as a preference relation that EIP i uses to compare all the possible coalition that it may join. The binary relationship needs to satisfy complete, reflexive, and transitive, etc., properties [41]. In other terms, ∀ C 1 , C 2 ⊆ M s , where i ∈ C 1 and i ∈ C 2 , C 1 i C 2 represents that participant i prefers being a member of C 1 than C 2 , or at least i prefers both equally. Hence, we use the following preference relation , where C 1 and C 2 are any two coalitions that contain member i. Φ i (·) is the preference function, described as , where h (i) is a set that contains the coalitions having been evaluated [46,47].
For member i, if C 1 is strictly better than C 2 , we use notation ≻ i to denote the preference relationship, i.e., C 1 ≻ i C 2 .

Coalition Formation Algorithm
After deriving the price of the computing resource block, coalition utility, and preference function from subsection -, in this subsection, we use a distributed algorithm named DCFA (Distributed Coalition Formation Algorithm) based on the hedonic shift rule [48] to get a stable coalition. By introducing the suitable distributed state management algorithms [49,50], coping with the distribution property of DCFA.
Describing simply the shift rule as, given a coalition partition Π = {C 1 , C 2 , · · · , C l } on set M s and a preference ≻ i , if and only if C k ∪ {i} ≻ i C Π (i), player i leaves its current coalition C Π (i) to join coalition C k ∈ Π ∪ ∅. The shift rule exploits the rational participants' selfishness to move from low profit to higher, ignoring the influence of this behavior on other individuals in the same coalition. The objective of DCFA is to let each member find the possible alliances it may join and check whether it is preferred to join. If a member decides to leave the current coalition to join a new one, we put the current coalition into set h (i) to avoid repeatedly visiting it. Given that each EIP can operate asynchronously and independently from others, we can implement DCFA using suitable distributed mechanisms, which The function DCF accepts the variable eip i and the global variable partition used to store the current coalition partition. The initial coalition partition is that every EIP is a coalition. In DCFA, A \ B represents that set A removes element B. The algorithm has two important properties, which are convergence and Nash-stability [41,52].
Theorem 2 Starting from any initial coalition structure, the DCFA algorithm always converges to a final partition.
Theorem 3 Any obtained final partition with algorithm DCFA is Nash-stable. Similar proof of theorem 2 and 3 can refer to the literature [53]. The property of Nash stability is a fundamental guarantee of convergence property. Moreover, introducing each member's visited history set makes the number of alternatives decrease monotonically to end with a stable coalition configuration.

Simulation Setup
In this subsection, we consider a square region with 10km×10km located in Anning District, Lanzhou (China), as shown in Figure . The locations of EIPs and SPs are generated randomly in the region. We randomly generated 100 EIPs and 1000 SPs locations in total, and assumed that each SP is located at one location. We randomly sampled with 8 EIPs, and 4 SPs from EIPs and SPs generated previously for the sake of clarity and analysis, i.e., M = 8, S = 4. We assume that the delay between an SP and an EIP is proportional to their distance. Suppose that the price for an SP to serve a request successfully is 2 to 3 per 100000. The physical infrastructure of each EIP has four identical edge nodes, and everyone has 40 CRBs, whose maximum  Table 1.

Experimental Results
According to the parameters set before, we get the valuation index v ij that is each service i for all resources of EIP j, which is shown in Figure . Obviously, if we normalize v ij corresponded to one CRB, it will have a generality. In this experiment, we assume that each EIP has an equal number of resources, such that the performance before and after normalization is equivalent. In Figure , (Figure ), we inferred that the result obtained above is reasonable. For instance, we only consider one influencing factor -distance,  By solving the EG program, we get the Fisher market equilibrium prices of all CRBs and unit CRB of EIPs under service providers' budgets, which described in Figure . We set three budgets of SPs to analyze the impact of which on the market price; they are the benchmark budgets (10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20), double of the benchmark, and a half of the benchmark. The effects show in Figure . From Figure , all the prices increase twice as the budget of each SP is double. By the same rule, all the prices reduce one-half, with each SP's budget reduced by half.
Hence, the value of the budget only affects the equilibrium prices by a scaling factor.
When fixing the budget, the prices act as a primary means to allocate resources only.
In the coalition formation algorithm, we vary the request intensity (arrival rate) of services, and the corresponding required containers are changing with it. The In Table 2 As the requests increase, we find that the benefits of cooperation increase first and then decrease. For one thing, the increase in requests made them obtain more benefits, as more requests take up more resources. For another, as the requests increasing, the number of edge nodes switched off are decrease. We depict the utilities of different methods (fixed one-one contract-FOOC, coalition) in Figure , capturing that the EIPs' utilities obtained by cooperation are markedly better than FOOC. And the arithmetic means of utilities of the coalition method are also not less than FOOC. The experiments point out that cooperation brings more significant benefits, and as the requests increase beyond a certain amount, the advantages decrease gradually.
To analyze the effect of parameter β on the coalition formed and the average utility of EIPs, we choose the arrival rate 2 as a case. The detailed results of the experiment are shown in Table 3 and Figure ,

Conclusion and future works
In this work, we consider a scenario with a collection of SPs with different budgets and several EIPs distributed in geographical locations. We aim to develop a price strategy to select EIPs with the most cost-effectiveness for each SP and get a stable coalition among EIPs selected with maximum EIP profits for different workloads.
For the scenario, we introduce the system architecture of MEC in an area. There are three layers: the user layer, the aggregation layer, and the MEC layer. Towards the first goal, we use the famous concept of General Equilibrium in Economics as an effective solution. It produces a Market Equilibrium with Pareto-efficient and fairness, etc., properties. We get the market equilibrium price and select the optimal EIPs for each SP by solving the Eisenberg-Gale convex program. Aim at the second purpose, depending on the price and EIPs selected, obtained by the first goal, we use a mathematical model to maximize EIP's profit and an algorithm to form a stable coalition. Numerical results demonstrate the effectiveness of the method.
In future work, we will consider the net profit of the service provider and analyze the EIPs profit and the coalition under its market equilibrium price.           The evaluation indicator of Service to EIP  The utilities on different EIPs The maximum arrival rate for different EIPs Figure 9 Market equilibrium prices of CRB on different EIP Figure 10 The impact of different budgets on the market equilibrium price The impact of β on average Ψ{i} and E[φi]