An Online Scheduling Algorithm for Energy Minimization in Wireless Powered Mobile Edge Computing Networks

Computation scheduling in conventional MEC networks has been extensively studied and was systematically summarized in [2] and [3]. Recently, the advancements in WPT technology bring in the possibility of building wireless powered MEC networks. To the best of our knowledge, the works in [4] and [5] are the first that simultaneously study energy harvesting and computation offloading in MEC networks, but they only consider simple networks with only one WD and one access point (AP). Their model was extended by subsequent researches to incorporate more WDs. The works in [6] and [7] consider the WP-MEC networks with two near-far WDs and try to resolve the so-called “double-near-far” effect, which occurs because a farther device harvests less energy from the AP but spends more power to communicate in longer distances. The authors in [8] and [9] aim to maximize the overall computation rate of all WDs in the networks, where an unmanned aerial vehicle was utilized to transmit energy in [9]. Due to the existence of binary offloading variables, the considered problems are generally formulated as mixed integer programming (MIP) problems and require a prohibitively long time to solve. In order to make real-time control decisions in fast fading environments, a deep reinforcement learning-based algorithm is proposed in [10] to obtain near-optimal solutions in large-scale WP-MEC networks. To the best of our knowledge, [11] is the only work that considers WP-MEC networks with multiple APs, where an approximation algorithm is derived to maximize the ratio of computation tasks completed before their deadlines.

The computation offloading considered in [4, 5, 6, 7, 8, 9, 10, 11] operate in binary mode, i.e. the tasks are non-splittable and are either processed by WDs or fully offloaded to APs. In addition to binary offloading, the scheduling problem is also investigated under partial offloading, where tasks can be divided into smaller parts and executed on WDs and APs concurrently. The work in [12] aims to optimize the energy consumption of multiple WDs and one AP, where the AP is assumed to have multiple antennas. The authors in [13] examined the computation efficiency maximization problem under both binary offloading and partial offloading modes. They also studied the problem under non-orthogonal multiple access (NOMA) in addition to the widely used time division multiple access (TDMA) scheme.

The researches described above generally focus on the one-shot optimization where the scheduling problem is only considered in a specific time interval. However, in practical settings, the WP-MEC networks are operated under sustainable manners and the control decisions at different times are mutually dependent. For example, the energy harvested in the current time frame, if not fully consumed, may be stored in the battery for later use. Inspired by this fact, many recent works try to solve the scheduling problem in online settings. A one-WD-one-AP model is considered in [14] where the energy consumption is minimized by optimizing the power of the energy transmitter and the offloading decisions of WDs. The model is extended to include multiple WDs in [15] where the long-term system throughput is maximized. In [16], the authors designed an online algorithm based on the Lyapunov optimization theory and presented a theoretical tradeoff between energy efficiency and delay.

In this paper, we study the computation scheduling problem in WP-MEC networks with multiple WDs and multiple APs. Different from [11], we formulate our problem under an online setting where both channel states and computation data arrivals fluctuate over time. This brings new challenges in algorithm design as decision variables are coupled along the timeline. To facilitate the process of tasks, the partial offloading mode is adopted in our model. Our objective is to minimize the long-term energy consumption by jointly optimizing the resource allocation in WPT and wireless communication stages. To avoid the mutual interference of concurrently emitted energy waves [17], only one AP is allowed to broadcast RF energy at the same time. By choosing different APs for WPT in turn, we also alleviate the “double near-far” effect because each WD has a chance to harvest energy from a closer AP. Our main contributions are summarized as follows.

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  We formulate the energy minimization problem for WP-MEC networks with partial offloading. To the best of our knowledge, this is the first work that considers multiple WDs and multiple APs under online settings.

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  We design an online algorithm by transforming the original problem into a series of deterministic optimization problems in each time slot based on the Lyapunov optimization theory. To reduce the time complexity of our algorithm, we propose a relax-then-adjust technique where the optimization problem is first relaxed and decomposed into several independent subproblems. After solving each subproblem, we adjust the computed values of variables to obtain a feasible solution.

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  To solve the non-convex computation offloading subproblem obtained in the previous step, we propose an iterative algorithm based on the Alternating Minimization method. We demonstrate that each iterative step can be solved in polynomial time and the algorithm converges to local optima.

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  Extensive simulations are conducted to validate the performance of our algorithm. Numerical results show that the total energy consumption is significantly reduced under various settings.

The rest of the paper is organized as follows. In Section II, we introduce the system model and the problem formulation. An online algorithm that jointly optimizes the WPT and computation scheduling is proposed in Section III. The simulations and related numerical results are presented in Section IV and we conclude our paper in Section V.

According to [17], when multiple APs transfer RF power simultaneously, their energy waves may interfere with each other and lead to a possible energy cancellation. To improve the efficiency of RF energy transfer, we select only one AP to broadcast RF energy during each time slot. By choosing different APs in turn, we also alleviate the “double near-far” effect because each WD has a chance to harvest energy from a close AP. We use $a^{T}_{j}(t)\in\{0,1\}$ to indicate whether AP $j$ broadcasts RF energy in slot $t$ and let $P^{T}_{j}(t)$ and $\tau^{T}_{j}(t)$ be the corresponding transmission power and transmission time. Then $a^{T}_{j}(t)$ should satisfy

and the energy consumption of AP $j$ during the WPT is

As in [16, 19], we assume the energy harvested from noise is negligible and adopt a linear model to characterize the energy harvesting circuit of WDs. In particular, the energy harvested by WD $i$ during the $t$-th time slot is

where $\mu_{j}\in(0,1)$ is the energy conversion efficiency of WD $i$. The harvested energy is stored in the battery of WDs. Let $B^{max}_{i}$ and $B_{i}(t)$ be the capacity and remaining power of the battery in WD $i$, then the update rule of $B_{i}(t)$ is

where $E^{L}_{i}(t)$ and $E^{O}_{i}(t)$ are the energy consumption for local computation and wireless communication, as explained in the next subsection. Due to the energy causality constraint, the energy consumption cannot exceed the available energy in the battery, so we must ensure

In each time slot $t$, let $A_{i}(t)$ be the amount of computation data arrived at WD $i$. Without loss of generality, we assume $A_{i}(t)$ is i.i.d. with average rate $\mathbb{E}[A_{i}(t)]=\lambda_{i}$. The arrived data can be either processed locally or offloaded to the APs. The length of queueing data $Q_{i}(t)$ at WD $i$ evolves according to the following equation

where $D^{L}_{i}(t)$ and $D^{O}_{i}(t)$ are the amount of locally processed data and offloaded data, respectively. Note that the new data may arrive at the end of slot $t$ and cannot be processed until the beginning of next slot, so $D^{L}_{i}(t)$ and $D^{O}_{i}(t)$ must satisfy

To reduce the carbon footprint, in this paper, we aim to minimize the energy consumption for processing computation data arrived at WDs by jointly optimizing the control decisions for WPT, local computation, and data offloading. A similar objective is also considered in [7, 16, 5]. Based on our model, the considered problem can be formulated as follows.

where $P^{max}_{i}$ and $f^{max}_{i}$ is the maximum offloading power and computation capacity of WD $i$. Constraint (9) ensures the time allocation is feasible under the TDMA protocol. The objective reflects APs’ long-term energy consumption. Since WDs harvest energy from APs, this is also the total energy consumption in the system.

For WDs without SWIPT capabilities, we only need to replace constraint (9) with

because the wireless communication is prohibited during WPT and thus the available communication time of all APs (not just the one that transfers energy) must exclude the WPT time. It should be noted that the algorithm proposed in the next section can also be conveniently adapted to this situation.

For convenience of description, we first define the battery shortage $B^{-}_{i}(t)$ as

According to the constraint (4), the energy consumed by WD $i$ on slot $t$ cannot exceed the remaining energy in its battery. From a long-term perspective, this implies the time-average harvested energy by WD $i$ is equal to or greater than its time-average energy consumption, so $B^{-}_{i}(t)$ is also stable in the long run. Let $\bm{\Theta}(t)=[\bm{Q}(t),\bm{B}^{-}(t)]$ be the combined queue vector, where $\bm{Q}(t)=(Q_{1}(t),Q_{2}(t),\dots,Q_{N}(t))$ and $\bm{B}^{-}(t)=(B^{-}_{1}(t),B^{-}_{2}(t),\dots,B^{-}_{N}(t))$. According to the Lyapunov optimization theory, we start by defining the quadratic Lyapunov function

and the conditional Lyapunov drift

Next, we combine $\Delta L(t)$ with the objective function and form the following drift-plus-penalty term

where $V$ is a tunable parameter that controls the trade-off between the energy consumption and the queueing delay of computation data. The following lemma provides an upper bound for $\Delta_{V}L(t)$.

The proof follows a standard procedure [21] and is omitted for brevity. Based on the Lyapunov optimization theory, we can obtain an approximately optimal algorithm of our problem by minimizing the right-hand side of (13) in every slot $t$. The resulting time-average energy consumption decreases at the rate of $O(1/V)$ and the time-average queueing delay increases at the rate of $O(V)$, presenting a $O(1/V)$-$O(V)$ tradeoff between the two metrics. As a result, we can approach the optimal energy consumption arbitrarily close by increasing the value of $V$. However, in our problem, directly solving the minimization problem is difficult because the decision variables are coupled in constraints such as (4) and (5). To reduce the complexity of our algorithm, we propose a relax-then-adjust technique to decompose the original minimization problem into independent subproblems. The details are presented in the next subsection.

In our problem, the objective is to minimize the system energy consumption used to process the computation data of WDs. When the amount of arrived workload is fixed, this is equivalent to maximizing the energy efficiency, which is defined as the ratio of total energy consumption to the corresponding aggregate accomplished computation data. Inspired by this interpretation, we define the marginal energy efficiency of local computation and computation offloading, denoted by $\epsilon^{L}_{i}(t)$ and $\epsilon^{O}_{i}(t)$ respectively, as follows

where we have assumed the time allocation $\tau_{i}(t)$ is given in deriving (14). According to the optimality conditions, the two marginal energy efficiency should be equal when the overall energy efficiency is maximized. Based on this fact, we can first relax constraint (4) and (5) and then adjust the values of $f_{i}(t)$ and $P_{i}(t)$ to obtain a feasible solution that makes $\epsilon^{L}_{i}(t)=\epsilon^{O}_{i}(t)$. A similar technique is also used to decouple the WPT and computation offloading variables in constraint (9). After relaxing constraints (4), (5), and (9), the optimization problem of minimizing the right-hand side of (13) can be decomposed into three subproblems by classifying variables into independent groups. In our algorithm, we first compute solutions for these subproblems and then adjust their values according to the optimality conditions. The details of our algorithm are described as follows.

In this subsection, we propose a heuristic algorithm for the optimization problem of computation offloading based on the Alternating Minimization (AM) method. The main idea underlying AM is to replace the difficult joint optimization with a sequence of easier optimization involving grouped subsets of the variables. In our problem, we partition the decision variables into two groups: the time allocation variables $a_{ij}(t)\tau_{i}(t)$ and the transmission power variables $P_{i}(t)$. According to the AM method, the original problem can be solved iteratively by solving the following subproblems in each step $k$

where $g(t)$ is the objective function in (17). Theoretical analysis guarantees that the computed solution converges to local minima [22].