Joint User Scheduling and Computing Resource Allocation Optimization in Asynchronous Mobile Edge Computing Networks

Mobile edge computing (MEC) provides powerful computing ability to edge devices [1, 2]. Numerous works have investigated MEC systems from the perspective of resource allocation. In [3], computing offloading and service caching are jointly optimized in MEC-enabled smart grid to minimize system cost. Work [4] proposed a reverse auction-based offloading and resource allocation scheme in MEC. With the aid of machine learning, a multi-agent deep deterministic policy gradient (MADDPG) algorithm is designed to maximize energy efficiency in [5]. However, deploying computing resources at edge servers of a wireless network faces several challenges. First, due to limited energy of edge servers, they may not be able to provide sufficient computation resource according to devices’ requirements [6, 7]. Second, executing all offloading tasks synchronously requires edge servers to wait the arrival of the task with maximum transmission delay which may not be efficient. Meanwhile, task scheduling sequence is nonnegligible in synchronous task offloading, which will also impact the network loads and task completion [8].

To address the first challenge, wireless power transfer (WPT) technology that exploits energy carried by radio frequency (RF) signals emerges [9]. Instead of using solar and wind sources, ambient RF signals can be a viable new source for energy scavenging. Harvesting energy from the environment provides perpetual energy supplies to wireless devices for tasks offloading [10]. Thus, WPT has been regarded as a promising paradigm for MEC scenarios. Combining WPT with MEC, the authors in [11] proposed a multi-user wireless-powered MEC framework aiming at minimizing the total energy consumption under latency constraints. In [12], considering binary computation offloading, the weighted sum computation rate of all wireless devices was maximized by optimizing computation mode selection and transmission time allocation. The work in [13] proposed a multiple intelligent reflecting surfaces (IRSs) assisted wireless powered MEC system, where the IRSs are deployed to assist both the downlink WPT from the access point (AP) to the wireless devices and the uplink computation offloading. However, the above works [14, 15, 16, 17, 11, 12, 13] assumed that all computational tasks offloaded by users will arrive at the server at the same time and then the server starts to process all tasks simultaneously, which is not efficient and even impractical due to users’ dynamic computational task processing requests [18].

Currently, only a few existing works [18, 19, 20, 21, 22, 23] optimized MEC networks under dynamic computation requests. The work in [18] designed a Whittle index based offloading algorithm to maximize the long-term reward for asynchronous MEC networks where computational tasks arrive randomly. In [19], the authors studied the co-channel interference caused by asynchronous task uploading in NOMA MEC systems. The work in [20] investigated the energy efficient asynchronous task offloading for a MEC system where computational tasks with various latency requirements arrive at different time slots. Task scheduling problem for MEC systems with task interruptions and insertions was studied in [21]. However, the above works [18, 19, 20, 21] that focused on the asynchronous task offloading neglected how the asynchronous task arrival affects the computation at the MEC server. The authors in [22] used a sequential computation method to solve the energy consumption minimization problem under asynchronous task arrivals. The work in [23] designed a computation strategy that only allows a task to be executed after the completion of the previous tasks. Yet, works in [22] and [23] are still constrained by their limited usage of the server computation capacity, and cannot act as resource efficient asynchronous task offloading solutions.

The sequential computation strategy [24] has shown to have the potential to improve the computation resource efficiency and task execution punctuality in an asynchronous MEC network. However, since the computation resource allocation at the server depends on the arrival of the offloaded tasks, the sequential scheduling of the tasks will inevitably affect system performance, which is a fact that has been wildly ignored [22, 25, 26].

The main contribution of this paper is a novel asynchronous MEC framework that jointly schedules tasks and allocates computation resource with optimized system energy efficiency. In brief, our key contributions include:

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  We develop a novel framework to manage computation resource for the sequential computation in asynchronous MEC networks. In particular, we consider a MEC network in which the edge devices sequentially harvest energy for transmission, offload their computational tasks to a MEC server, and then compete for computation source at the server to get their tasks accomplished. To achieve the high energy efficient task execution, a policy needs to be designed for determining the optimal task scheduling sequence, time and computational resource allocation. We pose this joint scheduling and resource allocation problem in an optimization framework and seek to find the strategy which minimizes the energy consumption of the tasks.

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  Then, a general benders decomposition (GBD) based algorithm is proposed to solve the formulated mixed-integer non-linear programming (MINLP) problem which is decomposed into a primal problem that allocates computation resource and time, and a master problem that schedules user tasks. By iteratively solving the primal problem and master problem, the optimal scheduling and resource allocation scheme is obtained.

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  To show the effectiveness of the proposed algorithm, we prove that the optimal energy efficient scheduling and resource allocation scheme also optimizes the task punctuality. Our analytical results also show that the optimal allocation scheme for a given offloading task follows a specific pattern: the computation frequency allocated to each task remains constant initially, then gradually decreases before eventually reaching zero. Notably, all tasks experience a simultaneous decrease, the time of which is given in a closed form, in terms of their required central processing unit (CPU) cycles. Leveraging these identified properties, we introduce a computation resource allocation algorithm that offers a low-complexity solution.

Simulation results demonstrate that the proposed asynchronous computing framework reduces $87.87\%$ energy consumption compared with conventional synchronous computing counterpart. Moreover, computational complexity of the proposed computation resource allocation algorithm is reduced by $100$ times compared with conventional interior point method.

The rest of this paper is structured as follows. Section II elaborates system model and problem formulation. In Section III, we investigate the properties of asynchronous frequency allocation with given time allocation and user scheduling. The joint optimization of user scheduling, time allocation, and computation resource allocation is rendered in Section IV. Simulation results are presented in Section V. Finally, Section VI draws the conclusions.

Consider a MEC network consisting of one MEC server, and a set $\mathcal{K}=\{1,2,\cdots,K\}$ of energy harvesting enabled edge devices. Within this network, each device $k\in\mathcal{K}$ needs to execute an $A_{k}$ bits computational task, and will offload its computational task to the MEC server. As shown in Fig. 1, the devices need to first harvest energy from the server to enable such offloading. Then, using the time division multi-access (TDMA) technique, the devices need to schedule their offloading toward the MEC server. In other words, the computational tasks offloaded by devices will arrive at the MEC server asynchronously. To this end, the MEC server will process each device’s computational task in an asynchronous manner. In particular, the server will process devices’ computational tasks according to the time that it receives each of these computational tasks.

The server and edge devices must complete their computational tasks within a time period $T$ which is divided into $\left(K+2\right)$ time slots. The duration of each time slot $n\in\{0,\cdots,N+1\}$ is represented by $\Delta t_{n}$, with $N=K$. Each device uses one time slot to offload its computational task. Let $a_{k,n}$ be the index to indicate whether device $k$ offloads its task to the server at time slot $t_{n}$. In particular, if device $k$ uses time slot $t_{n}$ to offload its computational task, we have $a_{k,n}=1$; otherwise, $a_{k,n}=0$. Since each device uses only one time slot and each time slot can only be allocated to one device, we have $\sum_{k=1}^{K}a_{k,n}=1,\ \forall n\in\{1,\cdots,N\}$, and $\sum_{n=1}^{N}a_{k,n}=1,\ \forall k\in\mathcal{K}$. Meanwhile, when $a_{k,n}=1$, device $k$ will harvest energy from time slot $t_{0}$ to $t_{n-1}$. Once task $k$ arrives at the MEC server, i.e., at time slot slot $t_{n+1}$, the server will process this computational task.

The task computation process of the server and a device jointly completing a computational task $k$ consists of three stages: 1) energy harvesting, 2) task offloading, and 3) remote computing. Next, we first introduce the process of the energy harvesting, task offloading, and remote computing stages. Then, the problem formulation is given.

In this section, we first analyze the properties of the optimal computation resource allocation, and then a low-complexity computation resource allocation algorithm is accordingly proposed. For ease of notation, we use $F_{n}$ $(\forall n\in\{1,\cdots,N\})$ to represent the computation cycles to complete the task that arrives at the server with order $n$. With given time slot allocation vector $\Delta\boldsymbol{t}$ and user scheduling matrix $\boldsymbol{A}$, problem (2) is simplified as follows:

Before solving problem (3), we provide the feasibility condition as follows.

Denote $\{\alpha_{m}\}$, $\{\beta_{n}\}$, and $\{\gamma_{n,m}\}$ as the non-negative Lagrangian multipliers associated with the maximum frequency constraints (3a), task computation completion constraints (3b) and non-negative frequency constraints (3c), respectively. The optimal computation resource allocation is given by the following proposition.

According to Propostion 2, we can use the sub-gradient method to obtain the optimal $\{\alpha^{*}_{m}\}$ and $\{\beta^{*}_{n}\}$ so as to acquire the optimal computation resource allocation. To further reduce the computational complexity and provide some design insights, the properties of the optimal solution of problem (3) are summarized in the following theorem.

According to property 1) in Theorem 3, the optimal frequency allocation scheme for a certain offloading task always follows a specific pattern: the frequency allocated to each device remains constant initially, then gradually decreases and eventually reaches zero. This property motivates us to deduce the condition $f^{*}_{n,i}>f^{*}_{n,i+1}$. The property 2) in Theorem 3 implies that the computation resource of the server is redundant at time slots from $t_{2}$ to $t_{i}$, while the maximum computation resource $F_{\max}$ is utilized at slots from $t_{i+1}$ to $t_{K+1}$.

According to property 1) in Theorem 3, unless $f^{*}_{n,n+1}=\cdots=f^{*}_{n,K+1}$ $(\forall n\in\mathcal{K})$, there always exists a special time slot $t_{\varkappa}$ we called “transition point” such that $f^{*}_{n,n+1}=\cdots=f^{*}_{n,\varkappa-1}>f^{*}_{n,\varkappa}\geq\cdots\geq f^{*}_{n,K+1}$ $(3\leq\varkappa\leq K+1)$. The transition point indicates the number of time slots that the computation resource remains the same. The computation resource decreases for all tasks at the transition point. The method to find out the transition point when it exists is given by property 3) in Theorem 3.

Property 3) in Theorem 3 also shows that the transition point is impacted by the computation ability of the server. We can directly determine the transition point $t_{\varkappa}$ utilizing property 3) in Theorem 3 without the need of solving problem (3). After determining the transition point, we have $\alpha^{*}_{m}=0$ $(2\leq m\leq\varkappa-1)$ according to property 2) in Theorem 3.

Fig. 3 depicts an illustration of properties in the optimal computation resource allocation. As can be seen, before the transition point $t_{\varkappa}$, the optimal $\alpha^{*}_{m}=0$ and $f^{*}_{n,m}$ keeps unchanged as $m=2,\cdots,\varkappa-1$. Based on Theorem 3, a low-complexity algorithm is proposed in Algorithm 1. First, we check the feasibility of problem (3) according to Proposition 1. Then, we determine the transition point $t_{\varkappa}$ based on property 3) in Theorem 3. If there is no transition points, which means that the computation resource of server is abundant, we can directly obtain the optimal solution $f^{*}_{n,n+1}=\cdots=f^{*}_{n,K+1}=\frac{F_{n}}{\sum_{m=n+1}^{K+1}\Delta t_{m}}$ for $n=1,\cdots,K$; otherwise, we obtain the transition point $t_{\varkappa}$ and have $\alpha^{*}_{m}=0$ for $m=2,\cdots,\varkappa-1$. Hence, we only need to find out the optimal $\alpha^{*}_{m}$ $(m=\varkappa,\cdots,K+1)$ and $\beta^{*}_{n}$ $(n=1,\cdots,K)$. Note that with given $\beta^{*}_{n}$ $(n=1,\cdots,K)$, we can obtain the optimal $\alpha^{*}_{m}$ by solving the following $(K-\varkappa+2)$ equalities

since the maximum frequency is utilized at time slots from $t_{\varkappa}$ to $t_{K+1}$. Since $G(m,\alpha_{m})$ decreases with respect to $\alpha_{m}$, the bisection method is adopted. It should be noticed that $G(m,\alpha_{m})$ achieves the maximum value of $\sum_{n=1}^{m-1}\sqrt{\frac{\beta_{n}}{3\kappa}}$ when $\alpha_{m}=0$ and the minimum value of $0$ when $\frac{\alpha_{m}}{\Delta t_{m}}\geq\max_{n=1,\cdots,m-1}\beta_{n}$. Therefore, the upper bound of $\frac{\alpha_{m}}{\Delta t_{m}}$ is set as $\frac{\alpha_{m}^{ub}}{\Delta t_{m}}=\max_{n=1,\cdots,m-1}\beta_{n}$. For the lower bound, we set $\frac{\alpha_{m}^{lb}}{\Delta t_{m}}=\frac{\alpha_{m-1}}{\Delta t_{m-1}}$ according to property 2) in Theorem 3. After obtaining $\alpha_{m}$ for $m=\varkappa,\cdots,K+1$, $\beta_{n}$ is updated by a sub-gradient method [31], where $\phi_{n}$ is the dynamically chosen step-size. Through repeating Steps 5 to 13 until the objective of (3) converges, we can obtain the optimal $\alpha^{*}_{m}$ for $m=2,\cdots,K+1$ and $\beta^{*}_{n}$ for $n=1,\cdots,K$.

The complexity of Algorithm 1 is $\mathcal{O}\left(\frac{(K+2-\varkappa)}{\sqrt{\epsilon_{1}}}\log_{2}(\frac{1}{\epsilon_{0}})\right)$, where $\epsilon_{0}$ denotes the accuracy of the bisection method and $\epsilon_{1}$ is the accuracy of the objective of problem (3). Compared with the complexity of $\mathcal{O}\left(\left(K^{2}+K\right)^{3.5}\right)$ by the interior point method, the complexity of the proposed algorithm is significantly reduced. Moreover, when $\varkappa$ is large, the complexity can be further reduced since more numbers of $\alpha^{*}_{m}$ are zeros.

In this section, we employ the GBD method to solve problem (2). The core idea of GBD method is decomposing the original MINLP problem into a primal problem related to continuous variables and a master problem associated with integer variables, which are iteratively solved²²2Interested readers may refer to [29, 32, 33, 34] for details. . Specifically, for problem (2), the primal problem is a joint communication and computation resource optimization problem with fixed user scheduling. The master problem optimizes user scheduling by utilizing the optimal solutions and dual variables of the primal problem. Next, we describe the detailed procedures.