Towards Truthful Auction Mechanisms for Task Assignment in Mobile Device Clouds

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1 Towards Truthful Auction Mechanisms for Task Assignment in Mobile Device Clouds Xiumin Wang, Xiaoming Chen, Weiwei Wu School of Computer and Information, Hefei University of Technology, Hefei, China College of Information Science and Electronic Engineering, Zhejiang University, China School of Computer Science and Engineering, Southeast University, Nanjing, China chen Corresponding author: Weiwei Wu, Abstract Despite the increased capabilities of mobile devices, resource-demanded mobile applications still transc what can be accomplished on a single device. As such, mobile device cloud (MDC, an environment that enables computation-intensive tasks to be performed among a set of nearby mobile devices, offers a promising architecture to support real-time mobile applications. To stimulate mobile devices to execute tasks for others, it is essential to design an incentive mechanism that appropriately charges the owners of the tasks, acted as the buyers, and rewards the mobile devices, acted as the sellers. In this paper, we propose two truthful auction mechanisms for two different task models, heterogeneous and homogeneous task models, which assume the different and the same resource requirements of the tasks, respectively. Specifically, for heterogeneous task model, we propose an efficient heuristic winning bids determination algorithm to allocate the tasks, and decide the payment of each seller for its winning bids. For homogeneous task model, we design an optimal winning bid determination algorithm, and propose a Vickrey- Clarke-Groves (VCG based auction mechanism to determine the payment of each bid. Both theoretical analysis and simulations show that the proposed auction mechanisms achieve several desirable properties such as individual rationality, truthfulness and computational efficiency. I. INTRODUCTION In recent years, mobile devices including smart phones have become ubiquitous in people s daily life, whereas they are still seriously constrained by their limited battery capacities and computation capabilities [1] [3]. Various application tasks, such as face recognition and natural language processing, exceed the limits of mobile devices. To solve this issue, one may offload their complex or resource-demanded tasks to remote cloud [4] [6]. However, this may suffer large internet latency and energy consumption, due to long communication distance. Fortunately, recent work shows that computational offloading can also be performed among a set of mobile devices forming what we called Mobile Device Clouds (MDCs [7], [8]. Generally, utilizing the unused resources of the nearby mobile devices in MDCs can achieve better system performance, e.g., reducing the latency and network congestion with short-range communications [9] [13]. Nevertheless, it is non-trivial to ask mobile devices to share their resources or execute tasks for others, as these may also incur extra cost to themselves, such as battery outage. In the literature, several works have been done to motivate mobile devices to provide their resources in mobile cloud platform [12] [16]. For example, [12] imposes a bill backlog on each mobile device. Once a device s bill backlog exceeds a threshold, it will be unable to get services from others. Although the fairness of mobile devices can be well achieved, it might not appeal to many mobile devices, as the mobile devices are always forced to provide services. The work in [13] also discusses the possible incentive schemes that can be designed in MDCs. However, it only considers the cost of the waiting time and cellular payment during task allocation, while omitting the practical resource requirement of tasks. [14] constructs a Stackelberg game model to investigate the interaction between the tasks and the mobile devices that can participate in task execution. The Stackelberg equilibrium is also achieved by appropriately determining the price charged to each task and the amount of tasks that each mobile device can work on. Nevertheless, this approach does not capture the preference of task execution. It is noted that tasks executed on different mobile devices may get different performances including completion time, communication latency, etc. Under such a circumstance, different mobile devices should get heterogeneous payments for task executions. Auction [17], a popular trading form that can efficiently allocate resources of sellers to buyers at competitive prices, has been widely used as one of the most popular incentive schemes in many areas. By viewing the resource trading system as an ecosystem, the auction mechanism can appropriately address the conflicts between the buyer s and seller s interests, the internal competitions among themselves, and guarantee the truthfulness of the submitted information. Based on this, [15], [16] design two auction mechanisms for cloudlet resource sharing in mobile cloud computing. However, these works assume the homogeneous resource requirement of the tasks. Moreover, [15], [16] limit the auction mechanism in a one-toone matching manner, which omits the fact that the resourcerich devices can support the resource requirements of multiple buyers in a practical system. Task allocation in MDCs also involves a resource trading between the owners of the tasks and the mobile devices in the system. In this paper, we aim to design an efficient auction mechanism to solve the task

2 assignment problem in MDCs, by considering a more practical model for resource requirements of the tasks and the resource availability of mobile devices. Auction mechanisms in other areas have also been studied in the literature [18] [23]. Specifically, Vickrey-Clarke-Groves (VCG auction [18], [19] requires that the optimal allocation of the resources must be guaranteed, otherwise, the truthfulness cannot be achieved. [2] designs an auction mechanism for crowdsourcing. However, it assumes that each mobile device holds a set of predefined services, and if one of the bids submitted by a mobile device wins the auction, all its predefined services are traded, even if some of them are useless to the tasks. Moreover, it does not consider the resource requirements of task executions. In a similar way, [21] proposes a location-aware auction mechanism when assigning tasks in crowdsourcing. A dynamic auction mechanism is proposed in [22], which captures the mobile users mobility patterns through a sequence of correlated auctions. Nevertheless, most of the previous works omit the resource requirements of the tasks and the resource availabilities at mobile devices, which is quite important due to the resource scarcity of mobile devices in MDCs. Compared with the existing works, the auction problem considered in this paper mainly has the following three differences: (1 the mobile tasks may require different amount of resources for executions; (2 the number of tasks that can be executed at each mobile device is not a fixed value, deping on its resource availability and the resource requirements of the tasks allocated to it; (3 mobile device may have different preferences to the tasks allocated to it, because of different computation complexities etc. To capture the interaction between the owners of the tasks and the mobile devices participating in the system, we aim to design an efficient auction mechanism for task allocation in MDCs, where the tasks (or saying the owners of the tasks act as the buyers of mobile resources, and the mobile devices owning the resources act as the sellers. In the rest of the paper, we use the terminology of buyers and tasks, sellers and mobile devices, interchangeably. Specifically, we consider two task models, heterogeneous and homogeneous task models, respectively. In heterogeneous task model, each task is assumed to require different amounts of resources for execution, while homogeneous task model assumes the same amount of resource requirement for each task. The main contributions of this paper can be summarized as follows. For heterogeneous task model, we design an efficient auction mechanism to allocate the tasks and determine the payment. We prove that the proposed auction mechanism achieves certain properties, such as computational efficiency, individual rationality, and truthfulness. For homogeneous task model, we convert the task allocation problem into finding a minimum weight bipartite matching in an auxiliary bipartite graph. Based on that, we design an optimal auction mechanism based on VCG auction. Theoretical results also show the computational efficiency, individual rationality, and truthfulness of the scheme. We validate the desirable properties of the proposed auction mechanisms through simulations. The rest of the paper is organized as follows. Section II describes the system model and problem formulation. The auction mechanisms for heterogeneous and homogeneous task models are designed in Section III and Section IV, respectively. Section V presents the simulation results. Finally, we conclude the paper in Section VI. II. PROBLEM FORMULATION In this section, we first introduce the auction model. Then, we formulate the problem and present the desirable properties of the auction. A. Auction Model We consider a set of m indivisible tasks in T = {t 1, t 2,, t m } to be allocated to mobile devices in MDCs for executions. Without loss of generality, assume that running each task requires a certain amount of resources, e.g., CPU, memory, battery etc. Let r j be the amount of resources required for task t j. Suppose that D = {d 1, d 2,, d n } is the set of mobile devices in MDC that are willing to participate in task execution. Each mobile device d i submits its bids for the tasks, B i = {b i,1, b i,2,, b i,m }, where bid b i,j B i denotes the cost claimed by d i for executing task t j. To ease presentation, we use B = n i=1 B i to denote the set of all the bids submitted by the sellers in D. It is noted that, the bids to different tasks may not be the same, even from the same mobile device. This is reasonable, as different tasks may consume different batteries or completion time on mobile devices. Moreover, as each mobile device is also with limited resources, we use R i to denote the maximum amount of resources that can be provided by d i. As mentioned above, in our auction model, the tasks requiring the resources for executions act as the buyers, while the mobile devices owning the resources act as the sellers. To assist the matching between the buyers and sellers, a trusted third party auctioneer is necessary to administrate the trading. The detail of the auction process is described as follow. The auctioneer collects the tasks to be executed, including the information of the resource requirements of the tasks. Then, it initiates the auction to the sellers in D. Each seller in D privately submits its bids and the resource availability information to the auctioneer. Based on the bids and resource requirement/availability information, the auctioneer follows a predefined auction mechanism to determine the set of winning bids and the corresponding payments of the sellers. To ease presentation, we define the following variants: { 1, if di wins its bid for task t j ; x i,j =, otherwise. (1 p i,j : the final payment offered to d i for bid b i,j. (2

3 B. Problem Formulation Due to the distributed nature of mobile devices, the real execution cost of each task is generally private and unknown to others. In this case, some selfish mobile devices may manipulate the claimed cost to gain higher utility. To differentiate from the submitted bid, we use c i,j to denote the true cost of device d i for executing task t j. Then, for each device d i D, we can define its utility as follows: u i = (p i,j x i,j c i,j, d i D, (3 which is the net revenue earned at device d i by executing tasks for others. Firstly, the auctioneer needs to determine the winning bids and allocate the tasks to the corresponding mobile devices. Such winning bids determination problem can be formulated as follows. n min x i,j b i,j, (4 subject to i=1 x i,j r j R i, d i D, (5 n x i,j = 1, t j T. (6 i=1 In the above formulation, the objective is to minimize the sum of the claimed costs for executing tasks at mobile devices. The constraint in Eq. (5 limits that the overall amount of resources required by the tasks allocated to d i is no more than R i. The constraint in Eq. (6 denotes that each task must be allocated to one and exactly one mobile device. Secondly, based on the selected winning bids, the auctioneer further decides the final trading payment of the seller for each of its winning bid. It is noted that, seller d i is willing to make trade with t j for its bid b i,j, only if the payment is no less than its claimed cost b i,j. That is, p i,j b i,j. The goal of our work is to design a truthful auction mechanism that solves the above two problems. The designed mechanism should achieve the following properties. Individual rationality: it means the utility of each mobile device should be nonnegative, i.e., u i for d i D, which is a basic condition to stimulate the mobile devices participating in task execution. Computational efficiency: The algorithm should be solved in polynomial time, which is important to a practical system. Truthfulness: The bid submitted by each mobile device should be truthful, i.e., b i,j = c i,j where c i,j is the true cost of executing task t j at device d i. In the following sections, we aim to design truthful auction mechanisms by considering two different task models, heterogeneous and homogeneous task models, respectively. In heterogeneous task model, the tasks in T may require different amounts of resources. As such, the winning bids determination problem defined in Eq. (4 is NP-hard, by a polynomial-time reduction from Multiple Knapsack problem [24], a well-known NP-hard problem. For homogeneous task model, where the tasks require the same amount of resources, we will show in Section IV that the winning bids determination problem can be converted into finding a minimum weight matching in an auxiliary bipartite graph, which can be optimally solved in polynomial time. III. TRUTHFUL ACTION MECHANISM FOR HETEROGENEOUS TASK MODEL We now consider the heterogeneous task model, where the tasks require different amounts of resources. In this case, the winning bids determination problem is a NP-hard problem, and finding the optimal solution is with high complexity. Hence, we instead design a greedy algorithm which obtains a suboptimal solution with low computational complexity. Then, we propose a payment scheme to determine the payoff of each bid. Finally, we analyze the properties of the proposed mechanism. A. Greedy Winning Bids Determination Scheme For each bid submitted by the mobile devices, e.g., b i,j, we first define a parameter β i,j, named cost per unit resource, as follows: β i,j = b i,j r j. (7 In general, the bid with smaller cost per unit resource should have higher chance to win the bid. Hence, based on this parameter, we design a greedy algorithm to solve the winning bid determination problem. The greedy rule is to pick the next most unit-cost-efficient bid as the winning bid to allocate the task, until all the tasks are assigned. Specifically, we define T as the set of the tasks that have been assigned with mobile devices so far, and B as the set of the bids that can still be chosen as the winning bids so far. Initially, we have T =, B = m i=1 B i and x i,j =. Then, in each step, we conduct the following procedure. Select the bid, e.g., b i1,j 1, with the least cost per unit resource among the bids in B. That is, β i1,j 1 = min{β i,j b i,j B}. Add b i1,j 1 as one of the winning bids, i.e., x i1,j 1 = 1, and allocate the task t j1 to mobile device d i1, i.e., T = T {t i1 }. Correspondingly, remove bid β i1,j 1 from B. We then remove the bids that may violate the resource constraint of the device, or the bids that are useless for future task allocations. That is, remove the bid b i,j from T, if b i,j satisfies any of the following two conditions: Task t j has already been allocated with mobile device in the previous steps, i.e., n i=1 x i,j = 1. As such, the bids to task t j become useless, and should be removed.

4 Algorithm 1: Winning bids determination begin T =, B = B; x i,j =, for i, j; Calculate β i,j for each bid b i,j B; while T T do Choose the bid with the least cost per unit resource from B, denoted as b i1,j 1 ; Set b i1,j 1 as one of the winning bids, i.e., x i1,j 1 = 1; Remove b i1,j 1 from B; for b i,j B do Remove b i,j from B, if j = j 1 ; for d i D do Remove each bid b i,j from B, if R i m xi,jrj min {r j t j T T The resource left at device d i is not enough to support any other task in T T. That is, { R i x i,j r j min r j t j T T }. (8 Under such a circumstance, d i is unable to accept any new winning bid. Continue the above process, until all the tasks are allocated with mobile devices, i.e., T = T. The detail of the above process is presented in Algorithm 1. B. Payment Strategy Based on the winning bids determined in the above subsection, we then introduce how to decide the payment of each mobile device/seller for its winning bids. The payment should be such determined that each seller will honestly reports its real cost. It is noted that the bid that loses in the above subsection, will be paid by, i.e., p i,j = if x i,j =. The main idea of our payment scheme is to decide the payments of the winning bids one by one. Without loss of generality, we use B w as the set of winning bids selected in the above subsection, i.e., B w = {b i,j x i,j = 1}. The payment of winning bid b i,j B w is then decided as follows: Remove bid b i,j from the consideration bids in B. That is, B = m i=1 B i {b i,j }. Re-select the winning bids from set B as in Algorithm 1, until the task t j is allocated with a mobile device, e.g., d i. That is, the bid b i,j wins its bid for task t j without the presence of bid b i,j. We then set the payment for device d i s winning bid b i,j determined in Algorithm 1, as b i,j (i.e., the bid sent by device d i. The detail of deciding the payment for winning bid b i,j is presented in Algorithm 2. We continue the above process, until considering all the winning bids in B w. } ; Algorithm 2: The payment for winning bid b i,j begin T =, B = B\{b i,j }; x i,j =, for b i,j B; Calculate β i,j for b i,j B; while b i,j B x i,j < 1 do Choose the bid with the least cost per unit resource from B, denoted as b i1,j 1 ; Set b i1,j 1 as one of the alternative winning bids, i.e., x i 1,j 1 = 1; Remove b i1,j 1 from B; Remove all the conflicting bids as in Algorithm 1; Set p i,j = b i,j for winning bid b i,j, where x i,j = 1; Fig. 1. r 1 =2 r 2 =2 r 3 =3 r 4 =4 t d 1 t t 3 t 4 d 2 d 3 R 1 =6 R 2 =4 R 3 =5 An example to illustrate the winning bids determination process To illustrate the above process, we use Fig. 1 as an example. We assume that four tasks t 1, t 2, t 3, t 4 are to be allocated to mobile devices d 1, d 2, d 3. The amount of resources required by each task or available at each device is given near each vertex. The edge between task t j and device d i denotes the value of the bid b i,j. According to Algorithm 1, four bids b 3,1, b 3,3, b 1,4, b 2,2 will be selected as the winning bids. Furthermore, with Algorithm 2, the alternative winning bids for b 3,1, b 3,3, b 1,4, b 2,2 are b 1,1, b 2,3, b 2,4, b 1,2 respectively. Hence, tasks 1, 2, 3, 4 are allocated to devices 3, 2, 3, 1 with payments 2, 3, 3, 5 respectively. C. Theoretical Analysis We now analyze the properties of the proposed mechanism, including individual rationality, truthfulness, and computational efficiency. Lemma 1 The proposed auction mechanism achieves the individual rationality for each mobile device. Proof: Firstly, for a mobile device d i that has no winning bids, according to Algorithm 2, its payoff is zero, i.e., u i =. Secondly, for a mobile device d i that has winning bids, its utility can be calculated as u i = (p i,j x i,j c i,j = 3 5 b i,j B w (p i,j b i,j. (9 4

5 We then show that the utility obtained for each of its winning bids, e.g., b i,j B w, is non-negative, p i,j b i,j. When b i,j wins, its claimed cost must be smaller than its payment p i,j = b i,j, as otherwise, b i,j will win its bid instead of b i,j. Hence, for each winning bid that truthfully reports its bid, its utility must be nonnegative. We then analyze the computational efficiency as follows: Lemma 2 The proposed winning bids determination and payment schemes both have polynomial-time computational complexity. Proof: We first discuss the computation complexity of Algorithm 1. In Algorithm 1, we need to choose T winning bids. To select each winning bid, the bid with the least cost per unit resource in B must be found, and its complexity is O(mn. Furthermore, updating set B consumes at most O(mn complexity. Then, the complexity of selecting one winning bid is at most O(mn. Hence, the complexity of selecting all the winning bids for the tasks in T is O(m 2 n with Algorithm 1. We then analyze the complexity of Algorithm 2, which determines the payment for each winning bid. As presented, to obtain the payment of winning bid b i,j, Algorithm 2 needs to find an alternative bid for task t j, whose complexity is at most O(m 2 n. Then, the computation complexity of determining the payments of all the winning bids is O(m 3 n. To sum up, both Algorithm 1 and Algorithm 2 have polynomial-time complexity, which thus proves the above lemma. To demonstrate the truthfulness, we should prove that each mobile device will honestly submit all of its real costs. According to [25], the proposed mechanism is truthful if and only if it satisfies two conditions: 1 the wining bids determination algorithm is monotonic, and 2 each winning bid is paid the critical value. The definitions of monotonicity and critical value [25] are described as follows: Definition 1 Monotonicity: For each bid b i,j, if b i,j wins, then bid b i,j also wins, where b i,j = b i,j δ and δ >. Definition 2 Critical Value: For each bid b i,j, there is a critical value b i,j. Then, if the bid declares a cost that is not larger than b i,j, it must win, otherwise, it will not win. Based on above definitions, we can obtain that Lemma 3 The winning bids determination process in Algorithm 1 is monotonic. Proof: Suppose that b ik,j k is one of the winning bids determined in the k-th step with Algorithm 1. Then, there are k 1 bids won in the previous k 1 steps. Let (b i1,j 1, b i2,j 2,, b ik,j k be the list of the winning bids that have been determined in the first k steps. Assume that b ik,j k was replaced by another bid, e.g., b i k,j k, where b i,j = b i k,j k δ and δ >. According to Algorithm 1, it is easy to get that bid b i k,j k must also win in the k-th step or even earlier step. Thus, we prove that Algorithm 1 is monotonic. Lemma 4 Each winning bid is paid the critical value, with Algorithm 2. Proof: Suppose that b ik,j k wins its bid for task t jk, in the k-th step with Algorithm 1, and b i,j k is an alternative winning bid for task t jk without the presence of b ik,j k, determined with Algorithm 2. In this case, the payment of b ik,j k is set to b i,j k, i.e., p ik,j k = b i,j k. It is obvious that another bid b i,j k = p ik,j k δ, where δ >, would win, because its cost per unit resource must be lower than the cost per unit resource of bid b i,j k. On the contrary, the bid b i,j k = p ik,j k +δ, where δ >, will not win, as its cost per unit resource must be higher than that of b i,j k. Hence, we prove the above lemma. With the above analysis, we can analyze the truthfulness as follows. Theorem 1 The proposed auction mechanism is truthful. Proof: According to [25], we can easily get the theorem with Lemma 3 and Lemma 4. IV. AUCTION MECHANISM FOR HOMOGENEOUS TASK MODEL In this section, we consider the homogeneous task model, where each task is assumed to require the same amount of resources, i.e., r j = r. We first introduce an optimal winning bids determination algorithm. Then, we propose a VCG-based auction mechanism. Finally, we analyze the performance of the proposed mechanism. A. Optimal Winning Bids Determination Scheme Since the maximum amount of resources available at d i is R i, the number of tasks that d i can accept is constrained by Ri r and the overall number of tasks in T, m. In this case, the constraint in Eq. (5 becomes { x i,j min R } i r, m, d i D. (1 Then, the winning bids determination problem can be transformed into finding a minimum weight bipartite matching, where the bipartite graph G(L 1 L2, E is constructed as follows: For each task t j, we use a vertex l j to denoted it, and add it to L 1. In other words, L 1 = {l j t j T, 1 j m}. For each device d i, we add min { R i r, m} vertices to L 2, where we use v i,k to denote the k-th vertex of device d i. Hence, L 2 = {v i,k d i D, 1 k min { Ri r, m} }. For each pair of vertices (l j, v i,k, where l j L 1, v i,k L 2, we add an edge between them. That is, E = {(l j, v i,k l j L 1, v i,k L 2 }. For each edge (l j, v i,k E, we assign weight b i,j on it. With the above graph G(L 1 L2, E, we can get that

6 Lemma 5 The winning bid determination problem of minimizing the overall cost is equivalent to finding a minimum weight bipartite matching in graph G(L 1 L2, E, when r j = r for j. l 1 l 2 l 3 l 4 Proof: We first show that a feasible bipartite matching in graph G(L 1 L2, E is one of the feasible solutions of the problem defined in Section II-B. Without loss of generality, we use M = {(l j, v i,k } to denote a feasible matching in G(L 1 L2, E. Moreover, we set x i,j = 1 if there exists a matching (l i, v i,k M for k {1,, min { Ri r, m}, otherwise, we set it to. It is observed that for d i D, we have { x i,j min R } i r, m. (11 (l j,v i,k M This is reasonable, as for each device d i, there are only R i r vertices in L 2, and thus at most R i r matchings to it will be chosen. Thus, the matching M satisfies the constraint in Eq. (1. Moreover, the bipartite graph G(L 1 L2, E is a complete bipartite graph, which reveals that there exists a feasible matching with cardinality min{ L 1, L 2 }. Hence, the constraint in Eq. (6 can also be met. To sum up, a feasible matching in G(L 1 L2, E is one of the feasible solutions of the winning bids determination problem. Then, the matching with minimum weight in graph G(L 1 L2, E denotes a solution with minimum overall cost of task allocation. Hence, our problem can be converted into finding a minimum weight matching in G(L 1 L2, E. With the above Lemma, winning bids determination problem with the objective of minimizing the overall cost can be optimally solved by running an existing minimum weight matching algorithm in graph G(L 1 L2, E, which has polynomial-time computational complexity [25]. Let M = {(l j, v i,k } be the optimal matching with minimum weight in graph G(L 1 L2, E. Then, we can get the solution of winning bids determination as follows: set x i,j = 1 if there exists an edge (l j, v i,k M, where 1 k min { Ri r, m}, otherwise, we set x i,j =. Still take Fig. 1 as an example. We assume that all the tasks require the same amount of resources, e.g., r j = r = 2. Then, the auxiliary bipartite graph can be constructed in Fig. 2, where we omit the weights of the edges in the graph for simplicity. By running the existing minimum weight matching algorithm, we can get optimal winning bids {b 1,4, b 2,2, b 3,1, b 3,3 }. That is, the tasks t 1, t 2, t 3, t 4 will be assigned to devices 3, 2, 3, 1, respectively. B. Payment Strategy with VCG Mechanism In the above subsection, the winning bids can be optimally determined in polynomial time. To guarantee the truthfulness of the submitted bids, we now design a VCG auction mechanism to determine the payment to each mobile device that wins bids. VCG auction is one of the most popular auctions that ensure the truthfulness when the optimal allocation can be v 1,1 v 1,2 v 1,3 v 2,1 v 2,2 v 3,1 v 3,2 Fig. 2. The auxiliary bipartite graph for Fig. 1 with r j = r = 2 Algorithm 3: The payments of winning bids begin Construct G(L 1 L2, E by considering all the bids in B; Find a minimum weight matching M in graph G(L 1 L2, E; Set x i,j = 1 if there exists an edge (l j, v i,k belonging to the found matching M, for k; Calculate the overall cost C R B with {x i,j }; for x i,j = 1 do Construct new graph G (L 1 L2, E without the presence of b i,j ; Finding a minimum weight matching M in graph G ; Calculate the overall cost C R B\{b i,j } on the found matching M ; Set p i,j = C R B\{b i,j } ( C R B b i,j ; for x i,j = do Set p i,j = ; found. In VCG auction, the bid winning the game will be paid with the opportunity cost that its presence incurs to others. Without loss of generality, we use CB R and CR B\{b i,j} to denote the minimum overall cost required under the above optimal winning bids determination scheme, with and without the presence of the bid b i,j, by using the available resources R = {R i d i D}. It is noted that CB R x i,jb i,j denotes the total cost except for bid b i,j under the above optimal winning bids determination scheme. Then, bid b i,j s opportunity cost is defined as the difference between CB\{b R and i,j} CB R x i,jb i,j. In this way, bid b i,j s payment, denoted by p i,j, can be derived as follows: p i,j = C R B\{b i,j} ( C R B x i,j b i,j. (12 The detail of the designed VCG mechanism is presented in Algorithm 3. C. Theoretical Analysis We now theoretically analyze the performance of the above VCG-based auction mechanism on achieving the truthfulness, individual rationality, and computational efficiency. We first discuss the truthfulness. Lemma 6 The proposed VCG-based auction mechanism guarantees the truthfulness of the bids.

7 Proof: We assume that when bid b i,j is submitted truthfully, i.e., b i,j = c i,j, the decision of task allocation is denoted as {x i,j } with the optimal winning bids determination process shown in the above subsection. We also use p i,j to denote the payment offered for bid b i,j with truthful information. Suppose that bid b i,j is submitted untruthfully, i.e., b i,j c i,j. Then, there are two cases for bid b i,j : 1 x i,j = x i,j ; 2 x i,j x i,j. Case 1 (x i,j = x i,j : If x i,j = x i,j =, it is easy to get that their utilities are the same and both. Alternatively, if x i,j = x i,j = 1, we can obtain that C R B = C (R\{Ri} {R i r j} B\{b i,j } + b i,j (13 where (R\{R i } {R i r j } means that the left resources at device d i should exclude the resource used for task t j. Then, we have p i,j = C R B\{b i,j} ( C R B b i,j = C R B\{b i,j} ( C (R\{R i} {R i r j } B\{b i,j } + b i,j b i,j = C R B\{b i,j } C(R\{R i} {R i r j } B\{b i,j }. (14 It is noted that CB\{b R and i,j} C(R\{R i} {R i r j } B\{b i,j } are both indepent of the bid b i,j submitted by device d i. In this case, p i,j = p i,j. Hence, the utilities are the same. Case 2 (x i,j x i,j : We now derive the utility of bid b i,j in truthful case as follows: p i,j x i,jc i,j = C R B\{c i,j} ( C R B x i,jc i,j x i,j c i,j = C R B\{c i,j } CR B = C R B\{c i,j } ( C (R\{Ri} {R i x i,j rj} B\{c i,j } + x i,jc i,j ( CB\{c R i,j } C (R\{Ri} {R i x i,jr j} B\{c i,j } + x i,j c i,j = C R B\{c i,j } C(R\{Ri} {R i x i,jr j} B\{b i,j} x i,j c i,j = CB\{c R ( i,j} CB R x i,j b i,j xi,j c i,j = p i,j x i,j c i,j. (15 Hence, the utility with truthful information c i,j is no less than the utility with untruthful information b i,j. We then compare the utilities of the truthful and untruthful cases at device d i, denoted as u i and u i respectively, as follows: u i u i = ( p i,j x m i,jc i,j (p i,j x i,j c i,j = ( p i,j x i,jc i,j (p i,j x i,j c i,j. (16 To sum up, only if submitting the bids truthfully, the best utility can be achieved, which thus proves the above lemma. Based on the above result, we now analyze the individual rationality as follows. Lemma 7 With the above scheme, each mobile device achieves nonnegative utility. Proof: As analyzed in Lemma 6, each bid must be submitted truthfully, i.e., b i,j = c i,j. We now study the utility that can be achieved for each bid, by considering two cases: 1 x i,j =, and 2 x i,j = 1. Case 1 (x i,j = : In this case, the utility achieved for bid b i,j is. Case 2 (x i,j = 1: We can derive the utility of bid b i,j as follows: p i,j x i,jc i,j = C R B\{c i,j} CR B. (17 It is noted that CB\{c R i,j } must be no less than CR B. This is because, CB\{c R i,j } is only a feasible solution where b i,j fails its bid, and thus must be no less than the optimal solution CB R. Hence, the utility is no less than. To sum up, for each truthful bid, the utility achieved must be nonnegative. Hence, the sum of the utilities of all the bids submitted by each device must be also nonnegative, which proves the above lemma. Lemma 8 The proposed VCG-based auction mechanism is computationally efficient. Proof: As descried in Section IV-A, winning bids determination can be optimally solved with an existing minimum weight matching algorithm, whose complexity is at most O((max{ L 1, L 2 3. As L 1 = m and L 2 nm, the complexity of our winning bids determination process is at most O(n 3 m 3. Moreover, according to Algorithm 3, the complexity of determining the payments for all the winning bids is O(n 3 m 4. Thus, the proposed auction mechanism runs in polynomial time, and is computationally efficient. V. SIMULATION RESULTS In this section, we conduct simulations to evaluate the performance of the proposed mechanisms. Specifically, we study the properties such as individual rationality, truthfulness, computational efficiency, and system efficiency. Similar to [15], [16], we randomly generate the bids of the buyers according to a uniform distribution within (, 1]. The amount of resources required by each task is randomly selected from [1, 1], whereas the amount of resources available at each device is generated from [1, 3]. Moreover, these generated resource information must be able to guarantee that each task can finally be allocated to a mobile device successfully, as otherwise, we cannot find any feasible solution of the problem, i.e., violating the constraint in Eq. (6.

8 (a In heterogeneous case (b In homogeneous case.25.2 The final payment The submitted cost.2 The final payment The submitted cost.2 The bids/payments The selected winning bids The bids/payments The selected winning bids The utility of a single buyer with heterogeneous tasks with homogeneous tasks Fig. 3. Performance on individual rationality The ratio of the submitted bid to the truthful cost A. Performance on Individual Rationality We first investigate the performance of the proposed mechanisms on individual rationality. In this simulation, we study the proposed mechanisms for both heterogeneous and homogeneous task models. As shown in Fig. 3, we set m = 4, n = 2. Among all the winning buyers and sellers, we select 2 pairs of them, and present their submitted bids and the corresponding payments. From Fig. 3, we can see that in both models, the final payment earned at each seller must be no less than the bid it submitted. In other words, the individual rationality of the buyers can be achieved with the proposed auction mechanisms, which validates the theoretical analysis. B. Performance on the Truthfulness We now study the performance of the proposed mechanisms on guaranteeing the truthfulness of the bids. As shown in Fig. 4, the value in x-axis is defined as the b i,j c i,j. ratio of the submitted bid to the truthful valuation, i.e., For both heterogeneous and homogeneous task models, we randomly select one of the sellers, e.g., d i, and generate its bids b i,j according to the ratio shown in x-axis. Specifically, when the ratio is 1, the submitted bid b i,j is equal to the truthful information c i,j. According to Fig. 4, it is observed that the maximum utility is achieved when the seller ss b the truthful information, i.e., i,j c i,j = 1. Moreover, the larger the gap between the submitted bid and the truthful valuation, the lower the utility achieved at the seller. That is, only if the seller submits truthful information, the best utility can be obtained, which guarantees the truthfulness of the bids. C. Performance on System Efficiency and Running Time Finally, we study the performance of the proposed mechanism on the system efficiency and the running time. Here, system efficiency is defined as the overall execution cost of the tasks, which is the objective function defined in Eq. (4. It is noted that in homogeneous task model, the minimum execution cost can be achieved by running a minimum weight matching algorithm in polynomial time, which achieves the best system efficiency. Hence, we here omit homogeneous task model, and only investigate the system efficiency for heterogeneous task model. Fig. 5. The overal cost Fig. 4. Performance on the truthfulness Optimal task assignment scheme The proposed mechanism The number of tasks Performance on system efficiency for heterogeneous task model For fairly comparison, we introduce a baseline algorithm, named optimal task allocation scheme, by running an exhaustive searching algorithm in exponential time. After optimally allocating tasks to the corresponding mobile devices, the traditional VCG auction can then be used to decide the final payments of the sellers, so as to guarantee the truthfulness of the bids. As shown in Fig. 5, we set n = 2, and vary the number of tasks from 5 to 4. It is observed that our proposed mechanism performs worse than the optimal task allocation scheme in minimizing the overall execution cost. However, the gap between them is very small. Moreover, we present the running times of the proposed mechanism and the optimal task assignment scheme in Fig. 6. It shows that although the proposed mechanism achieves lower system efficiency, its running time is much less than the optimal task allocation scheme, which is particularly important for a large mobile cloud system. Furthermore, from Fig. 5 and Fig. 6, we can also see that both the overall cost and the running time increase with the number of tasks.

9 The running time (ms Fig x 1 4 Optimal task assignment scheme The proposed mechanism The number of tasks Performance on running time for heterogeneous task model VI. CONCLUSION In this paper, we consider mobile task assignment problem in MDCs. To capture the interactions between the owners of the tasks and the mobile devices, we construct an auction model, where the mobile devices act as the sellers of resources, and the tasks act as the buyers. We design two auction mechanisms for two task models, heterogeneous and homogeneous tasks, respectively. Specifically, for heterogeneous task model, we design an efficient sub-optimal winning bids determination algorithm and decide the payment for each winning bid. For homogeneous task model, we show that the optimal winning bids determination problem can be converted into finding a minimum weight matching in an auxiliary bipartite graph, based on which, a VCG-based auction mechanism is designed to decide the payments of the winning bids. 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