Computing Resource Allocation for Blockchain-Based Mobile Edge Computing

1  Introduction

Numerous mobile devices have become the indispensable instruments in people’s work, study, and living due to the fast development of mobile networks [1]. Intelligent bracelets, smart garbage disposals, smart signal light controls, smart home appliances, and other intelligent applications have quickly garnered popularity and public attention. It is widely acknowledged that these intelligent applications have significantly improved social efficiency, raised human productivity, and improved people’s quality of life [2,3]. Innumerable innovative applications call for considerable computing power, which imposes a significant barrier for the IoT devices with limited resources and energy [4,5]. Cloud computing offers the users on-demand services through a pool of computing resources. However, cloud data centers are typically located far from mobile devices (users), and hence mobile cloud computing faces the issues of high latency, slow channel transmission, etc. [6,7].

Mobile Edge Computing (MEC) [8–10] has been developing to address the issues mentioned above by offloading computing tasks to edge servers. Therefore, the quality of service for task processing, including lower energy consumption and reduced execution delay, can be considerably enhanced by offloading computing workloads from users to edge servers [11,12]. However, the users and edge servers are not fully mutually trusted [13]. Blockchain is considered as a viable strategy to solve the problem of trust [14,15]. Blockchain uses community verification to synchronize a decentralized ledger replicated over several nodes, unlike conventional digital ledger systems that depend on a reliable central authority [16–18]. Accordingly, both academia and business are paying attention to building MEC based on blockchain [19].

In the blockchain-based MEC system shown in Fig. 1, each edge server functions as a node in both MEC and blockchain. As an MEC node, each edge server processes users’ tasks and uploads the information about each task, e.g., user, time, task workload, results, etc., to the blockchain. As a blockchain node, the edge server treats the information of each task as a transaction and generates the blocks, each containing several transactions. Each transaction contains the encrypted file and the hash of the data and the result related to the task execution. The sources of the transactions in a block are all the users served by the edge server. That is, the transactions in a block may come from several users. After creating a block, the edge server propagates it to other edge servers which will verify the blocks. In this way, blockchain can provide trust between users and edge servers by transparent task-related information sharing. The transactions recording the information of the tasks are stored in blockchain, such that they cannot be modified. The anti-tampering characteristics of blockchain guarantee the authenticity and legitimacy of each transaction, thus allowing the users to check and track the details of the task related information. Each user can also evaluate the trustworthiness of other users based on their transaction history on the blockchain. If there is a dispute or controversy over a transaction, the transactions recorded in the blockchain can be used as a reliable evidence, which is used to resolve the dispute [20].

Figure 1: Blockchain-based MEC system

In blockchain-based MEC, each edge server runs both blockchain tasks and users’ offloaded tasks simultaneously. Note that there is a trade-off between the resource allocation for MEC and blockchain tasks. Therefore, the resource allocation for the blockchain and the MEC is crucial for the processing delay of blockchain-based MEC. Most of the existing research tackles the problem of resource allocation in either MEC [21–23] or blockchain [24–26], which leads to the unfavorable performance of blockchain-based MEC. In this paper, we tackle the problem of how to allocate the computing resources of edge servers to the MEC and blockchain tasks with the objective of minimizing the total system processing delay. The contributions of this paper are as follows:

(1) We study the problem of resource allocation for both MEC and blockchain tasks to reduce the total processing delay of blockchain-based MEC, different from the existing research that focuses on only one of MEC and blockchain.

(2) We propose a computing resource Allocation algorithm for Blockchain-based MEC (ABM), which consists of four steps. First, we prove that the problem is convex. Second, we prove that the problem satisfies the Slater’s condition, and hence the problem and its dual satisfy strong duality. Third, we use the Karush-Kuhn-Tucker (KKT) conditions and the partial derivatives of the Lagrangian function to obtain the solution with the dual variables. Finally, we decide the values of the dual variables via the subgradient projection method, which obtains the solution to the problem.

(3) Simulation results demonstrate that ABM converges and effectively reduces the processing delay of blockchain-based MEC.

The rest of the paper is organized as follows. The related work is introduced in Section 2. The problem is defined in Section 3. Section 4 proposes the algorithm. In Section 5, the experimental results are presented. The paper is summarized in Section 6.

2  Related Work

In blockchain-based MEC, each edge server executes both MEC and blockchain tasks. Both of these two kinds of tasks consume computational resources, and there is a trade-off between the resource allocation for MEC and blockchain tasks. Therefore, the resource allocation for the blockchain and the MEC is crucial for the processing delay of blockchain-based MEC. Nevertheless, most of the existing research tackles the problem of resource allocation in either blockchain or MEC.

Some research studied the problem of computing resource allocation for MEC. Ren et al. [21] proposed a hybrid communication and computing resource allocation approach for delay minimization in a multi-user time-division multiple-access MEC offloading system. Nosrati et al. [27] proposed a community-based solution for the management of data replication based on the model of communication latency between computing and storage nodes. Feng et al. [28] proposed a method to maximize the user utility while reducing the operator’s energy usage in a MEC system with multi-user radios, by jointly optimizing the radio power allocation at the base station and the power allocation for processing users’ offloaded data. Hong et al. [29] proposed an optimal resource allocation strategy that primarily addresses the issue of assigning MEC computing resources to a group of cars within a time window, so as to reduce the overall task processing delay. Liu et al. [30] proposed an edge framework that contains four groups under different locations between mobile edge nodes and users for the resource allocation problem to reduce the total cost of users and edge servers. Gao et al. [31] built a task offloading and resource allocation model which uses the overall computational cost as the objective function. Spallina et al. [22] proposed a deep Q-learning network for the resource allocation problem to reduce the task offloading energy usage while maintaining users’ quality of service. Guo et al. [32] developed an online greedy heuristic algorithm for the resource optimization problem with the aim to maximize the long-term overall performability. Zhang et al. [33] investigated the user association and offloading decision problems for satelliteaerial integrated computing networks, where users can offload their tasks to high-altitude platforms (HAPs) and the satellite with edge computing servers. Zhang et al. [34] proposed a blockchain based containerized edge computing platform for Internet of Vehicles. The platform was integrated with blockchain to improve the security of the network. A heuristic container scheduling algorithm was developed to schedule computing tasks to appropriate edge servers to reduce the computing latency. Gao et al. [35] proposed the resource control algorithm to improve the system performance under the unstable workload in vehicular networks. The proposed resource control algorithm adjusts the priority and proportion of the resource utilization according to the system status obtained from the resource monitor, such as transaction workload or CPU utilization status. Fan et al. [23] proposed a joint task offloading and resource allocation scheme for MEC. Aghapour et al. [36] proposed a deep reinforcement learning-based technique that divides the task offloading and resource allocation problem into two sub-problems, to reduce the delay and power consumption related to work offloading. Xiao et al. [37] proposed an algorithm to reduce users’ task-offloading overhead, by optimizing the computational offloading choice, the wireless resource allocation policy and the computational resource allocation strategy.

Some scholars conducted research on resource allocation for blockchain. Xia et al. [38] presented a three-phase auction method to allocate resources for mobile blockchain, which introduces a group buying mechanism to encourage edge servers to join the blockchain network. Kang et al. [13] proposed a reputation-based data-sharing scheme that uses contract technology and federation chains to accomplish secure data storage and avoid inappropriate data sharing. Toulouse et al. [39] investigated the problem of balancing workload of account-based blockchains such as Ethereum and proposed the method to predict transaction processing times. Jiao et al. [40] built an auction-based market model to determine the computing offloading strategy for miners and the allocation of computing resources to edge servers. Alfakeeh et al. [41] proposed the algorithms for the resource allocation of mining nodes, which utilizes the backpack algorithm to allocate the caching space of the mining nodes to resolve the problem of how to efficiently allocate resources so that data can be placed equitably in these fog mining nodes while maintaining the prioritization and sensitivity of patient data. Chang et al. [42] proposed an edge computing based blockchain incentive mechanism for miners to purchase edge server computing resources, and established a two-stage Stackelberg game between miners and edge servers. Fan et al. [43] constructed a computation offloading model composed of multiple miners, multiple ESPs, and a CSP for mobile blockchain and proposed a three-stage Stackelberg game for optimal pricing-based edge computing resource management. Zhang et al. [24] proposed a group-agent strategy with trust computing to ensure the reliability of edge devices during interactions and improve transmission efficiency. Wang et al. [25] proposed an incentive mechanism, in which the client chooses the amount of computational power allocated to each task via a two-stage Stackelberg game based on the rewards. Zhou et al. [44] proposed a blockchain-enabled spectrum trading system to maximize user benefits and designed a spectrum trading matching method. A Dynamic Credit Aggregate Signature Byzantine Fault Tolerant (DCABFT) algorithm based on Boneh-Lynn-Shacham (BLS) aggregated signatures was proposed, which aims to reduce the transaction latency and increase the user quality of service while addressing the latency and energy consumption of blockchain consensus algorithms. Zhang et al. [26] proposed two auction mechanisms for the blockchain network to maximize societal welfare and offer an efficient resource allocation method for edge computing service providers. Baranwal et al. [45] proposed a distributed auction-based resource allocation system in the context of edge computing-enabled industrial IoT. The technique eliminates the need of a trustworthy third party (the auctioneer) using consortium blockchain and smart contracts. Gao et al. [8] proposed a hierarchical resource scheduling scheme for the Vehicular Internetworking System (VINS) to allocate the computational resources, in order to increase the flexibility and efficiency of the system’s resource utilization and to boost the performance of the blockchain-based VINS.

Both MEC and blockchain tasks require computational resources for execution in blockchain-based MEC. Due to the limited computational capability of each edge server, an imbalance in the allocation of computational resources to blockchain and MEC will result in a large system processing delay. However, most of the current research tackles the problem of resource allocation for either blockchain or MEC, and little research has been conducted on the optimization of resource allocation for both blockchain and MEC simultaneously. Feng et al. [46] proposed a joint optimization framework for blockchain-enabled MEC by jointly optimizing user offloading decisions and computational resource allocation to achieve the trade-off between the energy consumption of the MEC system and the latency of the blockchain system. In the framework, some of the users’ tasks cannot be offloaded to the edge servers when they are making the choices of what tasks to be processed. However, this kind of offloading decisions are not reasonable, when the users have to but are unable to process some computation intensive tasks, e.g., real-time video stream processing, etc. As for the energy consumption, it is often difficult for the edge to obtain the factors of power, usage time, etc., of the large amount of heterogeneous users in the applications, which makes it hard to optimize the system energy consumption [47]. Furthermore, the energy consumption of users has no impact on the edge’s utility. Therefore, in this paper, we study the problem of how to allocate the computing resources of edge servers for the MEC and the blockchain tasks with the objective of minimizing the total system processing delay.

3  Problem Definition

3.1 System Model of Blockchain-Based MEC

There are multiple users (IoT devices, sensors, wearables, etc.) and edge servers in the blockchain-based MEC system. Each edge server with constrained computing resources serves an area, and the users in the area can offload the tasks to the edge server. At the same time, the edge servers comprise the blockchain. After running the users’ tasks, each edge server uploads the information related to the task execution, e.g., user, time, task workload, result, etc., to the blockchain. Each edge server, as a blockchain node, treats the task related information as a transaction and generates blocks, each containing several transactions. Each transaction contains the encrypted file and the hash of the data and the result related to the task execution. The sources of the transactions in a block are all the users served by the edge server. That is, the transactions in a block may come from several users. The blocks are broadcast to all blockchain nodes which verify the blocks. The block generation and verification are referred to as blockchain tasks. Accordingly, as shown in Fig. 1, each edge server runs two kinds of tasks: those offloaded by users and those in blockchain. Table 1 lists the symbols and notations used in this paper.

3.1.1 Latency of Task Execution in MEC

Denote the sets of users and edge servers as 𝒩={1,2,⋯,n,⋯,N} and ℳ={1,2,⋯,m,⋯, M}, respectively. The users offload their tasks to the edge servers using wireless communication. The system employs a shared frequency multiplexing method, wherein all edge servers can utilize identical wireless resources. The users adopt orthogonal spectrums for data transmission, with a collective bandwidth of B Hz. xn,m is a binary variable indicating if user n is in the service area of edge server m. The execution delay of the task is expressed as:

Tn,m=xn,m(Tn,mt+Tn,mc)(1)

where Tn,mt and Tn,mc represent the delay of transmitting the task related data from user n to edge server m and the latency of executing the task of user n on edge server m, respectively. Tn,mt and Tn,mc are calculated as follows:

Tn,mt=Dnrn,m(2)

Tn,mc=DnCmfn,m(3)

where Dn is the task size of user n, Cm specifies the number of CPU cycles that edge server m requires to process 1 bit input data, and fn,m represents the frequency of CPU cycles allocated to edge server m for processing the task of user n. Note that the variations in the bandwidth and power of the channel are small, when the user is within the coverage area of an edge server and has low mobility in the applications. In this case, the transmission rate is fixed [48]. The rate of data transmission from user n to edge server m can be calculated as:

rn,m=BUmlog2⁡(1+γn,m)(4)

where Um represents the number of users served by edge server m. γn,m is the channel interference plus noise ratio (CINR) between user n and edge server m:

γn,m=Pn,mgn,m∑i,j≠n,mPmaxgi,j+θ2(5)

where Pmax indicates the maximum transmission power of each user, and θ2 denotes the noise power. Pn,m and gn,m respectively represent the transmission power and channel gain from user n to edge server m.

3.1.2 Latency of Blockchain Task Execution

The total latency Tk of task execution in blockchain for block k comes from three sources: block generation delay, block transmission delay and block verification delay:

Tk=TI+Tt+Tv(6)

where TI represents the time required for the blockchain to generate a new block, and Tt denotes the transmission delay of the block during consensus. Tv denotes the block verification delay, which is a constant. Therefore, Tv has no impact on the computing resource allocation of edge servers. The time required for the blockchain to generate a new block is calculated as follows:

TI=1K∑m∈ℳ∑n∈𝒩xn,mInCmfb,m(7)

where fb,m represents the CPU cycle frequency allocated by edge server m to process the block generation tasks. K is the total number of blocks generated by the blockchain, which is determined by the consensus round time (block intervals) [49]. In denotes the size of the transaction related to the task from user n, which is calculated as:

In=Hash(Dn+Rn)+Qn(8)

where Hash(⋅) is a hash function that computes the hash value of the task related information. Rn and Qn denote the sizes of the execution result of the task from user n and the encrypted files related to the task from user n, respectively.

The block transmission delay Tt is calculated as:

Tt=SbE(9)

where Sb represents the block size, i.e., the number of bytes in the block, and E denotes the data transmission rate between the edge servers.

3.2 Problem Model

Each edge server executes the tasks in both MEC and blockchain. In this paper, we study how to allocate the computing resources of each edge server to MEC and blockchain, with the aim to minimize the total system processing latency. That is, our objective is to

minfn,m,fb,m(∑m∈ℳ∑n∈𝒩Tn,m+∑k∈KTk)s.t.C1: fb,m+∑n∈𝒩xn,mfn,m≤Jmaxm∀m∈ℳC2: fn,m>0,fb,m>0∀n∈𝒩,m∈ℳ(10)

Constraint C1 stipulates that the computational resources assigned by each edge server must be within the processing capacity of the edge server. Constraint C2 states that the edge server should allocate the computing resources to both MEC and blockchain.

4  Algorithm

The variables in the computing resource allocation problem in this paper are fn,m and fb,m, and there exists capacity constraint C1. The fractional planning problem, a well-known NP-hard problem, can be reduced to our problem. Therefore, our problem is also NP-hard.

In this section, we propose a computing resource Allocation algorithm for Blockchain-based MEC (ABM), which consists of four steps as shown in Fig. 2. First, we prove that the problem is convex. Second, we prove that the problem satisfies the Slater’s condition, and hence the problem and its dual satisfy strong duality. Third, we use the KKT conditions and the partial derivatives of the Lagrangian function to obtain the solution with the dual variables. Finally, we decide the values of the dual variables via the subgradient projection method, which obtains the solution to the problem.

Figure 2: Diagram of ABM

We denote f1=fn,m and f2=fb,m. The total system processing delay can be expressed as:

O=∑m∈ℳ∑n∈𝒩xn,m(Dnrn,m+DnCmf1)+∑k∈K(1K∑m∈ℳ∑n∈𝒩xn,mInCmf2)+SbE+Tv(11)

where SbE and Tv are constant. The original problem can be rewritten as:

minf1,f2Os.t.C1:f2+∑n∈𝒩xn,mf1≤Jmaxm∀m∈ℳC2:f1>0,f2>0∀n∈𝒩,m∈ℳ(12)

The following equations can be obtained by calculating the first-order derivative of objective function O:

∂O∂f1=∑m∈ℳ∑n∈𝒩xn,mDnCm(−f12)(13)

∂O∂f2=∑k∈K1K∑m∈ℳ∑n∈𝒩xn,mInCm(−f22)(14)

The second-order derivative of objective function O can be calculated as follows:

∂2O∂f12=∑m∈ℳ∑n∈𝒩xn,m2DnCmf13(15)

∂2O∂f22=∑k∈K1K∑m∈ℳ∑n∈𝒩2xn,mInCmf23(16)

The principal subequations of the Hessian matrix for each order have values greater than 0, and hence we can derive that the Hessian matrix is positive definite. Consequently, according to Theorem 4.1, objective function O is convex. Furthermore, the function in constraint C1 exhibits convexity, and hence the original problem is also convex.

Theorem 4.1. For a function f(x), if the Hessian matrix of f(x) is positive definite, then f(x) is a convex function.

Proof. The Hessian matrix H(x) of function f(x) is represented as follows:

H(x)=∇2f(x)=[∂2f∂x12⋯∂2f∂x1∂xn⋮⋱⋮∂2f∂xn∂x1⋯∂2f∂xn2](17)

where ∇2f(x) represents the second-order partial derivative of function f(x). Given that H(x) is positive definite, the primary subequations of every order in the upper left corner of H(x) are greater than 0. The expressions for the first-order and second-order principal subequations are as follows:

∂2f∂x12>0,|∂2f∂x12∂2f∂x1∂x2∂2f∂x2∂x1∂2f∂x22|>0(18)

Consider any two points, denoted as x and y, belonging to the definition domain. Given that H(x) is positive definite, we can derive the following equation according to Taylor’s equation

f(y)=f(x)+∇f(x)T(y−x)+12(y−x)TH(ξ)(y−x)>f(x)+∇f(x)T(y−x)

where ∇f(x)T represents the transpose of the first-order derivative of function f(x). The inequality above satisfies the definition of a convex function, and hence we can conclude that function f(x) is convex.

Lemma 4.1. The resource allocation problem in this paper satisfies the Slater’s condition.

Proof. The Slater’s condition holds for the definition domain of x, if there exists an interior point (not a point on the boundary)x∗ satisfying the inequality constraints in which the equal sign is taken. According to the analysis, there must be f1 and f2 in our solution, such that f2+∑n∈𝒩xn,mf1<Jmaxm. Therefore, the resource allocation problem in this paper satisfies the Slater’s condition.

According to Lemma 4.1, we can derive that the original problem and its dual problem meet strong duality, since they both satisfy the Slater’s condition and are mutually adequate to satisfy strong duality. The optimal solution using the KKT conditions can be found, if the original problem is convex and fulfills strong duality. We find the optimal solution by calculating the partial derivative of the Lagrangian dual function via the KKT conditions, using dual variables. The original problem’s Lagrangian function can be written as follows:

L(f1,f2,μn)=O+∑n∈𝒩μn(f2+∑n∈𝒩xn,mf1−Jmaxm)(19)

where μn is a Lagrange multiplier and satisfies μn≥0. The Lagrange dual function is expressed as:

D(μ)=minf1,f2∈C1,C2L(f1,f2,μn)(20)

where μ=(μ1,μ2,⋯,μn)≥0. We can relax the original problem as an unconstrained optimization problem:

minf1,f2∑m∈ℳ∑n∈𝒩xn,m(Dnrn,m+DnCmfn,m)+∑k∈K(1K∑m∈ℳ∑n∈𝒩xn,mInCmfb,m)+SbE+Tv+∑n∈𝒩μn(f2+∑n∈𝒩xn,mf1−Jmaxm)(21)

Following the definition of KKT conditions, we can derive

∂L(f1,f2,μn)∂f1=∑m∈ℳ∑n∈𝒩xn,mDnCm(−f12)++∑n∈𝒩μn∑n∈𝒩xn,m=0(22)

∂L(f1,f2,μn)∂f2=∑k∈K1K∑m∈ℳ∑n∈𝒩xn,mInCm(−f22)+∑n∈𝒩μn=0(23)

We can further obtain

f1∗=(∑m∈ℳ∑n∈𝒩xn,mDnCm∑n∈𝒩μn∑n∈𝒩xn,m)12(24)

f2∗=(∑k∈K1K∑m∈ℳ∑n∈𝒩xn,mInCm∑n∈𝒩μn)12(25)

where f2∗ and f1∗ denote the optimal values of the CPU cycle frequencies allocated by the edge server to process MEC tasks and blockchain tasks, respectively. Therefore, we need to determine the dual variables μn, so as to decide the values of f1∗ and f2∗. The dual problem of the original problem is as follows:

maxμD(μ) s.t. μ≥0(26)

According to Theorem 4.2, we can obtain the dual variables via the subgradient projection method, since the objective function and constraints in the original problem are convex.

Theorem 4.2. Δμn(t)=Jmaxm−(f2+∑n∈𝒩xn,mf1) and μn(t+1)=μn(t)−i(t)⋅Δμn(t), where t is the iteration index, i(t)=αt is the step size, and α(>0) is a pre-determined constant.

Proof. From D(μ), we have the following inequality:

D(μ ′)≥∑m∈ℳ∑n∈𝒩xn,m(Dnrn,m+DnCmf1∗)+∑k∈K(1K∑m∈ℳ∑n∈𝒩xn,mInCmf2∗)+SbE+Tv+∑n∈𝒩μn(f2∗+∑n∈𝒩xn,mf1∗−Jmaxm)(27)

After substitution, we have the following inequality:

D(μ ′)≥D(μ)+∑n∈𝒩(μn ′−μn)(f2+∑n∈𝒩xn,mf1−Jmaxm)(28)

Subgradient means that points x and y belonging to the definition domain of the function satisfy:

f(y)≥f(x)+gT(y−x)(29)

where g is the subgradient of f. For D(μ), its subgradient is

f2+∑n∈Nxn,mf1−Jmaxm(30)

Therefore, we can use the subgradient projection method to decide the values of the dual variables. That is

Δμn(t)=Jmaxm−(f2+∑n∈𝒩xn,mf1)(31)

and

μn(t+1)=μn(t)−i(t)⋅Δμn(t)(32)

where t is the iteration index and i(t) is the step size.

The users in the service area of edge server m are the inputs of Algorithm 1. The outputs are the CPU cycle frequencies fn,m∗ and fb,m∗ that are allocated by the edge server to process the MEC and blockchain tasks, respectively. Step 1 initializes μ, tmax, δ. Steps 2−4 use Eqs. (24) and (25) to determine the CPU cycle frequency f1∗ assigned to the MEC tasks and the CPU cycle frequency f2∗ allocated to the blockchain tasks, respectively. Step 5 derives Δμn(t) according to Theorem 4.2 and then uses the subgradient projection method to calculate binary variable μn(t+1) based on step size i(t) and binary variable μn(t). Steps 6−11 iteratively update the dual variables until the two-paradigm of the difference between μn(t+1) and μn(t) is less than the pre-defined threshold δ. In each iteration, the algorithm updates fn,m∗ and fb,m∗ as f1∗ and f2∗, respectively.

5  Simulation

5.1 Simulation Setup

In the simulations, there are 60 users and 6 edge servers in the blockchain-based MEC system. The parameters of the system are listed in Table 2. The performance of ABM is evaluated by utilizing three benchmark algorithms, namely IADA [46], JCCR [33] and DSRC [24].

(1) IADA [46]: The algorithm first adopts binary search to obtain the resource allocation for processing MEC tasks, and then solves the stationing point to obtain the resource allocation for processing blockchain tasks.

(2) JCCR [33]: Each edge server allocates a fixed amount of computing resources to process MEC tasks.

(3) DSRC [24]: Each edge server allocates a fixed amount of computing resources to process blockchain tasks.

5.2 Simulation Results

Fig. 3 shows the total latency performance of ABM with different step sizes. ABM converges after 4 iterations, when step size i(t) is 0.1t. ABM requires 7 iterations to converge, when step size i(t) is 0.12t. However, ABM cannot converge, when i(t)=0.2t. The results demonstrate that a large step size may result in oscillation or even failure to converge during the iteration process, while a too small step size may lead to slow convergence. Therefore, we set i(t)=0.1t in the following simulations.

Figure 3: Impact of the initial step size in ABM on the total latency

The convergence performance of the proposed algorithm is illustrated in Fig. 4. It is noteworthy that the latency of the algorithm exhibits a decreasing trend as the number of iterations increases and the algorithm converges to a stable value. For example, the total latency decreases by 40.78% from the first to the fourth iteration, when the user task size is 2 MB. After the fourth iteration, the total latency stabilizes and does not change with the increase in the number of iterations. Furthermore, the latency of our algorithm is positively correlated to the computing task size. The total latency rises as the computing task size increases. For example, the total latency with the task size of 2.5 MB is 62.45% higher than that with the task size of 1 MB, after 4 iterations.

Figure 4: Convergence performance of ABM

It can be observed from Fig. 5 that the total latency of IADA, JCCR, and DSRC is higher than that of ABM with a given number of users. The total latency of ABM is 4.85% and 16.45% lower than that of IADA, 15.21% and 29.11% lower than that of JCCR, and 17.02% and 15.21% lower than that of DSRC, when the numbers of users are 60 and 90, respectively. The total computing task size that the edge server needs to process increases with the increasing number of users. JCCR and DSRC use a fixed resource allocation strategy. In contrast, ABM utilizes the KKT conditions and the subgradient projection method to dynamically allocate the computing resources of the edge servers, such that the computing resources of the edge server can be dynamically allocated for MEC and blockchain tasks according to the number of users. The dynamic resource allocation is better able to adapt to the changes in the types and the number of tasks, which leads to improved resource utilization and system performance. IADA considers dynamic allocation. However, the resource allocation for MEC and blockchain tasks is performed individually, which results in the inferior performance to our algorithm.

Figure 5: Impact of the number of users on the total delay

Fig. 6 illustrates the impact of the number of servers on the overall delay, when the number of edge servers increases from 6 to 10. The total latency of all algorithms reduces with more edge servers. As the number of edge servers increases, the number of users served by each edge server decreases, which reduces the total size of the tasks to be processed by each edge server. Therefore, the overall latency is reduced. ABM achieves up to 8.15%, 17.39% and 18.64% lower system latency than IADA, JCCR and DSRC, respectively. ABM jointly optimizes the computing resource allocation for the MEC and the blockchain tasks, which leads to superior performance.

Figure 6: Impact of the number of edge servers on the total latency

Fig. 7 demonstrates the impact of the user’s task size on the total delay. We can observe that the total delay of the four algorithms exhibits an increasing trend with the increase in the user’s task size. The increase in the user’s task size requires more computing resources and time for processing and more time for data and block transmission. Consequently, the overall delay experiences an increase. Fig. 7 illustrates that ABM consistently outperforms IADA, JCCR and DSRC in terms of the total latency. When the user’s task sizes are 1, 2, and 3 MB, the total latency of ABM is respectively lower than that of IADA by 13.63%, 6.81% and 8.82%, less than that of JCCR by 20.52%, 25.33% and 25.30%, and lower than that of DSRC by 17.39%, 19.64% and 13.88%. ABM can dynamically allocate computing resources according to the change in task sizes by jointly optimizing the resource allocation for MEC and blockchain tasks. Therefore, ABM achieves a notable decrease in the overall system latency compared to IADA, JCCR and DSRC.

Figure 7: Impact of the user’s task size on the total latency

Fig. 8 shows the total latency of the four algorithms with various block sizes. It can be observed that the total delay increases with the increasing block size. The block with a large size contains more transactions, which means that the edge server needs more time to aggregate enough MEC task related information for block generation. A large block size also means that the blockchain needs more time to transmit and validate the block. ABM jointly optimize the resource allocation for MEC and blockchain tasks, such that the resource allocation can adapt to various block sizes to achieve a favorable total latency performance. Our algorithm outperforms IADA, JCCR and DSRC. Specifically, when the block sizes are 4, 6, and 8 MB, the total latency of ABM is 8.18%, 7.69% and 9.75% less than that of IADA, 25.65%, 28.57% and 28.84% lower than that of JCCR, and 26.08%, 25.64% and 26.88% less than that of DSRC, respectively.

Figure 8: Impact of the block size on the total latency

Fig. 9 shows the total latency with different clock cycles required for the edge server to process 1 bit data. The total latency increases with the increase of the number of clock cycles Cm required for the edge server to process 1 bit data. A large Cm means that each edge server takes long time to process the MEC and blockchain tasks. ABM uses the KKT conditions and the subgradient projection method to optimize the allocation of edge servers’ computing resources, which can flexibly adjust the resource allocation according to the task processing complexity. Fig. 9 indicates that ABM achieves lower total latency than IADA, JCCR and DSRC. When Cm is 400, 600 and 800, the total latency of ABM is 7.56%, 12.77% and 13.79% less than that of IADA, 17.39%, 18.96% and 18.84% lower than that of JCCR, and 22.64%, 26.15% and 26.31% less than that of DSRC, respectively.

Figure 9: Impact of the number of clock cycles to process 1 bit data in the edge server on the total latency

Fig. 10 depicts the total system delay with different processing capacities of the edge server. The total latency declines with the increasing processing capacity of the edge server. A higher processing capacity leads to short processing time of MEC and blockchain tasks. Accordingly, the total delay of each algorithm reduces, as the processing capacity of the edge server increases. Fig. 10 illustrates that ABM achieves better total latency performance than IADA, JCCR and DSRC. Specifically, the total latency of ABM is up to 7.04%, 7.69% and 10.25% lower than that of IADA, 18.51%, 22.58% and 25.53% less than that of JCCR, and 9.58%, 14.28% and 22.21% lower than that of DSRC, respectively.

Figure 10: Impact of the processing capacity of the edge server on the total latency

Fig. 11 illustrates the total latency vs. the transaction size. The total latency rises, as the transaction size increases. The edge server requires more time to generate the blocks with a large transaction size. ABM always incurs less total latency than IADA, JCCR and DSRC, due to the joint optimization of resource allocation for MEC and blockchain tasks. The total latency of ABM is less than that of IADA by 8.33%, 9.67% and 8.62%, and lower than that of JCCR by 12.38%, 10.34% and 14.52%, and less than that of DSRC by 14.56%, 13.46% and 15.62%, when the task sizes are 400, 700 and 1000 B, respectively.

Figure 11: Impact of the transaction size on the total latency

6  Conclusions and Future Work

In this paper, we studied how to allocate the computing resources of edge servers to the MEC and blockchain tasks with the aim to minimize the total processing delay of blockchain-based MEC. For the problem, we proposed a computing resource Allocation algorithm for the Blockchain-based MEC system (ABM), which consists of four steps. First, we prove that the problem is convex. Second, we prove that the problem satisfies the Slater’s condition, and hence the problem and its dual satisfy strong duality. Third, we use the KKT conditions and the partial derivatives of the Lagrangian function to obtain the solution with the dual variables. Finally, we decide the values of the dual variables via the subgradient projection method, which obtains the solution to the problem. We conducted experiments through simulations. ABM demonstrates superior performance in terms of the total system latency under various parameters, including the number of users, user task sizes, block sizes, number of clock cycles required for edge servers to process 1 bit data, number of edge servers, processing capacity of the edge servers, block sizes and step sizes.

The users’ tasks may have different priorities, which have a significant impact on the resource allocation. In our future work, we will work on how to allocate the computing resources of edge servers, to provide the QoS of different tasks.

Acknowledgement: The authors would like to express their gratitude to all who supported this research. We are thankful for the insightful comments from anonymous reviewers, which have greatly improved this manuscript. We also appreciate the Anhui Province Key Laboratory of Industry Safety and Emergency Technology for technical support.

Funding Statement: This work is partially supported by the Key Research and Development Project in Anhui Province of China (Grant No. 202304a05020059), the Fundamental Research Funds for the Central Universities of China (Grant No. PA2023GDSK0055) and the Project of Anhui Province Economic and Information Bureau (Grant No. JB20099).

Author Contributions: The authors confirm contribution to the paper as follows: study conception and design: Yuqi Fan, Wanbo Zhang; algorithm design: Wanbo Zhang, Yuqi Fan; analysis and interpretation of results: Wanbo Zhang, Jun Zhang, Xu Ding, Jung Yoon Kim; draft manuscript preparation: Wanbo Zhang, Yuqi Fan. All authors reviewed the results and approved the final version of the manuscript.

Availability of Data and Materials: All data is synthesized using MATLAB and the generation method is described in detail in the simulation section.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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