^{1}

^{2}

^{3}

^{2}

^{1}

^{2}

^{3}

The mobile blockchain has been recognized as an emerging solution to address the security and privacy issues in a mobile application system. The mining process in mobile blockchain requires high computing resources which could overwhelm that which mobile devices can offer. In this case, mobile edge computing servers (MESs) can be involved to offer computing services to miners in mobile blockchain. Note that the resources of MESs are also limited; MESs could further request resources from the cloud computing server (CCS). Accordingly, the issue of hierarchical computing resource allocation arises. In this paper, we first consider a simple case with single-seller multiple buyers and a hierarchical single-seller multibuyer combinatorial auction model is proposed to solve this problem, based on which efficient and truthful frameworks are provided. We then extend the model to consider multiple CCSPs and propose a hierarchical multiple-seller multiple-buyer combinatorial auction model. For both models, the winner determination problems are formulated and computationally tractable algorithms are proposed. Also, pricing schemes are proposed to ensure the property of incentive compatibility and individual rationality. Finally, we evaluate the proposed schemes via simulations.

With the development of mobile applications (e.g., e-commerce), the privacy and security issues have been receiving more attentions [

Note that the computing resources of MESPs could also be limited; MESPs may not satisfy the large resource requirements of various miners. In this case, the Cloud Computing Service Provider (CCSP) who usually owns a large amount of computing resources can be further considered. In this case, miners can first request computing services from the MESPs. If the resources in MESPs are insufficient, MESPs could further rent services from the CCSP. Accordingly, the issue of hierarchical computing resource allocation involving miners, MESPs, and CCSPs arises. Specifically, the miners are regarded as service buyers and the MESP and CCSP are considered service sellers.

In this paper, we investigate the issue of hierarchical computing resource allocation in mobile blockchain. For simplicity, we first consider the simpler case of a single CCSP and a hierarchical single-seller multibuyer combinatorial auction model is proposed based on which an efficient and truthful framework is provided. The hierarchical combinatorial auction model can be divided into the lower level auction and the upper level auction. Specifically, miners act as buyers and MESPs act as sellers in the lower level auction, while in the upper level auction, the CCSP is both an auctioneer and seller and the MESPs act as the buyers. MESPs can be viewed as the middlemen who have no intrinsic valuations and demands, and their revenues are gained from resale. MESPs’ valuations depend on the demands from miners. We then extend the model to consider multiple CCSPs and propose a hierarchical multiple-seller multibuyer combinatorial auction model.

For both the proposed hierarchical combinatorial auctions, the following three issues need to be addressed: (1) how to design the hierarchical combinatorial auction mechanism for mobile blockchain, (2) how to formulate the winner determination problems (WDPs) in both lower level auction and upper level auction and how to solve them in an efficient way, and (3) how to design incentive compatible pricing schemes. In this work, we will solve all the issues above. The main innovations and contributions of this paper can be concluded as follows:

A single-seller multibuyer hierarchical combinatorial auction mechanism is proposed to address the two-level resource allocation problems for mobile blockchain. And we then extend the model to a more general multiple-seller multibuyer

WDPs are formulated for CCSP and MESPs, and corresponding solvable algorithms for WDPs are also proposed

Pricing schemes are proposed to ensure the property of incentive compatibility

Simulations show high resource utilization and efficiency of the proposed schemes

The rest of this paper is structured as follows. Related work are reviewed in Section

Recently, to address the problem on the blockchain mining, many studies have been done from the perspective of a game theory [

The auction mechanism has been widely used in resource allocation in different fields, e.g., radio resource allocation [

In addition, deep learning method is used in [

The mining process in blockchain networks is shown in Figure

The process of mining in blockchain networks.

Miner

During the mining process, the miners are competing with each other to become the first to solve the PoW problem and broadcast to reach an agreement. The generation of new blocks obeys the Poisson distribution which holds a constant rate

Judging from the above mining process, getting rewards successfully needs two steps. The first is successfully mining, and the second is timely propagation certification. The success rate of mining is directly proportional to the hash power. So, we use the hash power to represent the probability of successfully mining which is formulated as follows:

The time of propagation is related to the size of the transaction. Here, we use

In this case,,

After substituting

The blockchain mining protocol is maintained by the blockchain owner, which includes a fixed bonus

In this section, we firstly propose a single-seller multibuyer hierarchical combinatorial double auction model for the hierarchical resource allocation in mobile blockchain considering only one CCSP as the single seller in the upper layer auction. Then, we describe how to place the bid in this auction model. After receiving all the bids, the auctioneer needs to determine the winning bids, which is then formulated as the winner determination problem (WDP). Then, the algorithms for solving the WDP are provided. Finally, the price charged for the winning bidders are calculated.

We first consider the scenario where there is a single CCSP,

System model of hierarchical resource allocation for blockchain mining.

To jointly solve the issue of two-level resource allocation, a two-level hierarchical auction mechanism is designed as shown in Figure

The model of a single-seller multibuyer hierarchical auction.

Each mobile blockchain miner needs to place a bid for requested resources which reflects her valuation in the lower level auction. Since mobile blockchain users cannot know the winners’ number and the total resources’ number available until the auction is over, mobile user

After submitting (

After the auction result is released, user

After submitting (

For MESPs, bidding expressions differ from that of mobile users, because MESPs as the middlemen have no intrinsic demands and valuations. In the lower level auction, each MESP

After receiving all bids from bidders, which group of bids to be accepted needs to be decided by the auctioneer. For the proposed model in this paper, the WDPs for MESPs and CCSP need to be formulated. Specifically, the WDP for MESPs is shown as follows:

Specifically, the WDP for the CCSP is formulated as follows:

Backward induction is used to solve the winner determination problems in the hierarchical auction. Specifically, we first consider the lower level auction. Deliberately stated here, as a hierarchical optimization problem, how to jointly solve WDP for each layer is of key importance.

In the lower level, multiple miners and multiple MESPs exist. And the corresponding WDP is an integer programming problem which is NP-hard. For achieving a satisfying approximate optimal solution with low complexity, we propose a greedy algorithm motivated by [

1. Initialization: Set

2. For submitted bid

3. Match up one by one according to corresponding order until MESPs do not have enough resources to allocate and set relative

Similar to previous

There is a single CCSP as a seller and multiple MESPs as buyers in the upper level. The WDP problem in the upper level is also an integer programming problem. In this part, we propose two schemes to solve the WDP. Firstly, for a small-scale problem, a dynamic programming-based algorithm motivated by [

Accordingly, we can have

Considering the auctioneer’s computing power and the scale of the problem, a greedy algorithm which takes “bid density” into account is also proposed to obtain an approximate optimal solution as shown in Algorithm

1. Collect bids

2. Calculate the optimal value function

3. Output: using

1.Initialization: set

2. For submitted bid

3. Allocate computing resources to corresponding MESPs until the CCSP do not have enough resources and set relative

The design of pricing scheme is important for achieving incentive compatibility in the auction with which each bidder will always bid truthfully. The VCG [

In this case, we design a VCG-like pricing scheme which can be adapted to approximate algorithms, in which a base price is introduced. Specifically, each type of resources has a base price, and a user who is the winner will pay the larger one between the base price and the VCG price. The charged price can be formulated as follows:

For solving the WDP for CCSP, we propose two algorithms which are an exact algorithm based on dynamic programming algorithm and an approximate algorithm based on greedy algorithm, and the pricing scheme for them should be different. Specifically, for dynamic programming-based algorithm, we use VCG pricing. The VCG price of bidder

The pricing scheme in the greedy algorithm is similar to that in the lower level auction.

In this part, we will make an analysis on the properties of the proposed single-seller multibuyer hierarchical auction mechanism.

The hierarchical single-seller multibuyer auction mechanism proposed in this paper is individually rational for all truthful bidders in both two levels.

The VCG solution and the precise WDP solution algorithm proved to be individually rational [

The hierarchical single-seller multibuyer auction mechanism proposed in this paper is incentively compatible for all truthful bidders in both two levels.

To prove incentive compatibility, we have to prove monotonicity and critical payment property. Monotonicity can be proved immediately. Specifically, a bidder can increase its ranking order by increasing the bid value, as the bidder’s surplus at each level increases.

Then, we will prove critical payment property. If we can find the critical value that a bidder must bid to win the auction, the critical payment property can be proved. Donate user

The hierarchical single-seller multibuyer auction mechanism proposed in this paper achieves allocation efficiency with truthful bidding in both two levels.

This result can be easily proved from the winner determination problem formulation for which the sum of accepted bids is maximized.

We now extend the single-seller multibuyer hierarchical auction to a multiple-seller multibuyer one. Specifically, a set of

Proposed combinatorial double auction model.

Proposed combinatorial double auction model.

With submitted asks and bids, the winnerdetermination problem can be formulated to determine the winning bidders. In the lower level double auction, the WDP is formulated as follows:

For solving the multiple-seller multibuyer hierarchical double auction, backward induction is also used. For the lower level auction, similar schemes in the previous section can be applied. And for solving the WDP in the upper level, we propose a Lagrange multiplier algorithm.

In computer science, we often use “divide-and-conquer” to reduce computational complexity to solve the complex problems [

The algorithm solves the problem by the way of decomposing the problem into two or more subproblems, which are easily solved directly. The solution then combines these subproblems to construct a solution to the original problem. Lagrangian relaxation is used to provide a technique to decompose the problem into several subproblems. One way to reduce the burden of solving WDP calculations is to use Lagrangian relaxation to establish a fictitious market to determine allocation.

The basic idea of Lagrangian relaxation is to relax some of the constraints of the original problem on the objective function with a penalty term. That is, infeasible solutions to the original problem are allowed, but they are penalized in the objective function in proportion to the number of infeasibility. Under the condition of relaxed constraints selected, the constraint conditions are selected so that the optimization problem under the remaining constraints is easy in a certain sense. In the above WDP problem in the lower level, we observe that the coupling between different operations is caused by C1 and C2 constraints.

Let

Then, we depose the Lagrangian relaxation problem into two subproblems shown as follows:

where

Although the combinatorial double auctions can be decomposed into a number of SS and BS which can be easily solved for given Lagrangian multipliers

A key issue of the Lagrangian relaxation method is how to determine the value of the Lagrangian multipliers. We use a subgradient method to determine Lagrange multipliers by iteratively adjusting their values according to methods that violate the corresponding constraints. The value of the Lagrangian multipliers will increase and its value will decrease if the corresponding constraint is violated.

Our approach to finding solutions to dual problems is based on the iterative scheme by adjusting Lagrange multipliers based on the solution of SS and BS. Let

Lagrangian multipliers are adjusted according to the subgra- dients defined above. If subgradient

In our algorithm, donate

For pricing the winning bidders for ensuring the incentive compatibility, we also design a VCG-like scheme which can suit approximate algorithms, in which we introduce a base price. Each type of resources has a base price, and a user who is the winner will pay the larger one between the base price and the VCG price. The charged price can be formulated as follows:

For solving the WDP for CCSPs in the upper level auction, we propose two algorithms which are all approximate algorithms, the pricing scheme of which is similar to that in lower level auction. In this case, the description of the pricing scheme will be omitted here.

In this part, we will make an analysis on the properties of the proposed multiple-seller multibuyer hierarchical auction mechanism.

The hierarchical multiple-seller multibuyer auction mechanism proposed in this paper is individually rational for all truthful bidders in both two levels.

The VCG solution and the precise WDP solution algorithm proved to be individually rational [

The hierarchical multiple-seller multibuyer auction mechanism proposed in this paper is incentively compatible for all truthful bidders in both two levels. The proof process is similar to the previous section and is omitted here.

The hierarchical multiple-seller multibuyer auction mechanism proposed in this paper achieves allocation efficiency with truthful bidding in both two levels.

This result can be easily proved from the winner determination problem formulation for which the sum of accepted bids is maximized.

For numerical analysis, we consider a CCSP with 10000 MIPS computing capacity. There are two MESPs each of which reserves 3000 MIPS computing capacity, and the leftover 4000 MIPS computing capacity are available for the upper level. There are 20 miners who want to be miners to solve the PoW problem considered in this paper. Each mobile user requests

For comparison, a fixed sharing scheme is considered, where each MESP reserves 4000 MIPS computing capacity in the upper level. Besides, a rand allocation scheme is considered in the upper level auction in which each MESP reserves computing capacity ranging from 3000 to 4000 MIPS. For the hierarchical combination auction problem, the upper level auction adopts Algorithm

In order to investigate the results, Figure

The average social welfare achieved by different algorithms.

The average resource utilization achieved by different algorithms.

Average resource utilization achieved by different algorithms with different numbers of miners.

In addition, we compare the scheme which only considers edge computing with our proposed scheme which jointly considers edge computing and cloud computing in a satisfaction ratio with increasing number of mobile users as shown in Figure

The comparison in satisfaction ratio with increasing number of mobile users.

Also, we compare different algorithms using the multiple-seller multibuyer hierarchical auction. For comparison, we use the greedy algorithm termed as GA in the lower level auction and use both the greedy algorithm and the Lagrange multiplier algorithm termed as LA to resolve resource allocation problem in the upper level auction. We compare the average resource utilization achieved by different algorithms with increasing number of mobile users shown in Figure

The average resource utilization achieved by different algorithms with increasing number of mobile users.

The total utility achieved by different algorithms with increasing number of mobile users.

In this paper, a hierarchical combinatorial auction model has been proposed to solve the problem of resource allocation for mobile blockchain. Specifically, we have formulated winner determination problems (WDPs) for mobile edge computing providers and the cloud computing provider, and relevant computationally tractable algorithms to solve these problems have also been proposed. Besides, different pricing schemes have been designed to determine the final prices. The properties of the proposed schemes have also been proved theoretically. We then extend a single-seller multibuyer hierarchical auction to a multiple-seller multibuyer hierarchical auction, for which corresponding WDPs, tractable algorithms, and pricing scheme are also designed. Finally, the effectiveness of the schemes are verified by simulation and comparison.

The simulation data used to support the findings of this study may be released upon application to Yuanyuan Xu, who can be contacted at yuanyuan-xu@hhu.edu.cn.

This paper was presented in part at the IEEE GLOBECOM 2019 [

There is no conflicts of interest with others.

This work is partly supported by the Fundamental Research Funds for the Central Universities under Grant No. B200202189 and Grant No. NE2018107, the National Natural Science Foundation of China under Grant No. 61801167 and Grant No. 61701230, and the Natural Science Foundation of Jiangsu Province under Grant No. BK20170805.