# Joint Task Partition and Resource Allocation for Multiuser Cooperative Mobile Edge Computing

Authors: Gang Xie; Zhenzhen Wang; Yuanan Liu
Journal: Wireless Communications and Mobile Computing
 (2022)

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## Abstract

Exploiting the idle computation resources distributed at wireless devices (WDs) can enhance the mobile edge computing (MEC) computation performance. This paper studies a multiuser cooperative computing system consisting of one local user and multiple helpers, in which the user solicits multiple nearby WDs acting as helpers for cooperative computing. We design an efficient orthogonal frequency-division multiple access- (OFDMA-) aided three-phase transmission protocol, under which the user’s computation-intensive tasks can be executed in parallel by local computing and offloading. Under this setup, we study the energy consumption minimization problem by optimizing the user’s task partition, jointly with the communication and computation resources allocation for task offloading and results downloading, subject to the user’s computation latency constraint. For the nonconvex problem, we first transform the original problem into a convex one and then use the Lagrange duality method to obtain the globally optimal solution. Compared with other benchmark schemes, numerical results validate the effectiveness of the proposed joint task partition and resource allocation (JTPRA) scheme.


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## Body

## 1. Introduction

The real-time communication and computation of massive wireless devices (WDs) (e.g., smart wearable devices and laptops) promote the rapid growth of emerging applications (e.g., face recognition, smart grid, and autonomous driving) [1]. In fact, these applications or tasks may be computation-intensive and latency-critical, but WDs are generally of small size and only have the finite battery power. Hence, how to enhance their computation capabilities and reduce the computation latency is one crucial but challenging task to be handled. To deal with such limitations, mobile edge computing (MEC) has been proposed as a promising technology by providing cloud-like computing at the network edge (e.g., base stations (BSs) and access points (APs)) [2].Various efforts have been devoted to handling technical challenges against different computation task models. Two extensively adopted task models in the current research works are partial and binary offloading, respectively. Note that in partial offloading, the mutual dependency of the computing tasks significantly affects the computation offloading process [3]. That means partial offloading can be classified as the task-call graph and the data-partition model. Also, there exists different types of MEC system architectures, such as single user single-server [4–7], multiuser single-server [8–14], and single/multiuser multiserver [15–18].Due to the increasing number of WDs, resource contention may occur on MEC servers. Under this circumstance, cooperative computing provides a viable solution by utilizing abundant idle computation resources distributed at WDs [19, 20]. In a basic two-user device-to-device (D2D) cooperative computing system, [19] jointly optimized both users’ local computing and task offloading decisions over time, in order to minimize their weighted sum-energy consumption. Under a single-user single-helper single-server setup, [21, 22] jointly optimized the communication and computation resources allocation at both the user and helper based on time-division multiple access (TDMA) and nonorthogonal multiple access (NOMA), respectively. In a cellular D2D MEC system, [23] proposed a joint task management architecture to achieve efficient information interaction and task management. Also, by integrating D2D into the MEC system, [24] jointly optimized D2D pairing, task split, and the communication and computation resource allocation, in order to improve the system computation capacity.In the above research works, the user is mostly assumed to cooperate with single helper at the same time. In practical design, multiple helpers can simultaneously share their own computation and communication resources to help the user [20, 25–27]. In the D2D MEC system, [25] jointly optimized helpers’ selection and the communication and computation resource allocation for minimizing the energy consumption. In [26], a multihelper MEC with NOMA-based cooperative edge computing has been presented to maximize the total offloading data subject to the latency constraints. However, the system model in [25, 26] ignores results downloading, which may be not applicable for practical design. Thus, [20, 27] focus on the joint task offloading and results downloading. In the D2D-enabled multihelper MEC system, [27] jointly optimized the time and rate for task offloading and results downloading, as well as the computation frequency for task execution, in order to minimize the computation latency. Unlike the binary offloading model in [20, 27], it investigated a multiuser computational offloading scheme, in which the controlling user partially distributes its computing tasks to multiple trusted helpers. Also, [20] ignores computation resource of the local user and dynamic management of computation frequency.Despite the recent research progress, cooperative computing still faces some technical challenges. First, the previous research works mostly consider cooperating with the MEC server or single helper. When multiple helpers share unused resources to help the user, how to effectively coordinate the cooperation between the user and multiple helpers for achieving computing diversity remains challenging, especially when the helper number becomes large. Second, the previous research works generally ignore potential performance improvement brought by dynamic management of computation frequency as well as results downloading. When the MEC system considers these options, how to solve such a complex problem is also challenging.Motivated by this, we consider a multiuser cooperative MEC system consisting of one user and multiple nearby WDs serving as helpers. The user has individual computation tasks to be executed within a given time block. To implement the cooperation between the user and the helpers, the time block is divided into three phases. In the first phase, the user simultaneously offloads the computing tasks to multiple nearby helpers. In the second phase, the helpers execute their assigned computation bits. In the three phase, the helpers send the computation results back to the user. Under this setup, this paper develops an energy-efficient multiuser cooperative MEC design by optimizing the user’s task partition, jointly with the communication and computation resource allocation for task offloading and results downloading. The main contributions of this paper are summarized as follows.(1)
We propose an MEC framework for multiuser cooperative computing, in which the user can simultaneously offload the computing tasks to multiple nearby helpers(2)
We design an OFDMA-aided three-phase transmission protocol involving results downloading, which efficiently coordinates the cooperation between the user and multiple nearby helpers(3)
For the energy consumption minimization problem, we optimize the user’s task partition, jointly with the communication and computation resource allocation for task offloading and results downloading. Due to nonconvexity of this problem, we first transform it into a convex one and then use the Lagrange duality method to obtain the globally optimal solutionThe rest of this paper is organized as follows. Section2 introduces the system model. The proposed joint task partition and resource allocation problem is formulated in Section 3. The joint task partition and resource allocation algorithm is presented in Sections 4. Section 5 provides numerical results, followed by the conclusion in Section 6.Notation is as follows: we employ uppercase boldface letters and lowercase boldface ones for matrices and vectors, respectively. ∆ is represented by “denoted by” xab. And [x]+ is denoted by b,mina,x and max0,x, respectively. A continuous random variable z uniformly distributed over [a, b] is denoted by z ∼ U [a, b]. |A| denotes the determinant of a matrix A. Moreover, lu+∑k=1Klk=Lu and R+ stand for the sets of nonnegative real vectors of dimension K and positive real numbers, respectively.


## 2. System Model

As shown in Figure1, we consider a multiuser cooperative MEC system, which consists of one user and a set.Figure 1
System model of multiuser cooperative MEC.K=∆1,⋯,K of nearby helpers all equipped with single antenna. We focus on a time block with length T, where the user should execute the computing tasks with data-size Lu (in bits) within this block. Here, T is no larger than the channel coherence time [21]. Suppose that there is a central controller that is responsible for collecting the network information, such as the global channel state information (CSI), accordingly, the central controller can send the optimized strategies to the user and helpers to take actions [21, 22]. For easy implementation, it is further assumed that the task offloading and result downloading channel reciprocity are leveraged in this paper [20].Specifically, theLu bits generally can be divided into K+1 independent parts for local computing and offloading to the helpers, respectively. Let lu≥0 and l1≥0,⋯,lK≥0 denote the numbers of bits for local computing at the user and offloading to K helpers, respectively. Then, we have
(1)lu+∑k=1Klk=Lu.


### 2.1. Local Computing

Thelu bits are executed locally with the optimal central process unit (CPU) frequency given as [21]
(2)fu=culuT,wherecu denotes the number of CPU cycles for computing 1-bit input-data at the user. Note that fu is subject to the maximum frequency constraint, that is,
(3)fu≤fumax.Accordingly, the user’s energy consumption for local computing is given by(4)Eucomp=γuculufu2.whereγu denotes a constant related to the user’s hardware architecture [21]. Replacing fu in (4) with (2), Eucomp can thus be reexpressed as
(5)Eucomp=γuculu3T2.


### 2.2. Remote Computing at Helpers

The OFDMA-aided three-phase transmission protocol is shown in Figure2. At first, the user first offloads lk bits to the k-th helper with duration toffk via OFDMA in the task offloading phase, k∈K. Then, the k-th helper executes its assigned bits with duration tcompk in the task execution phase. At last, in the result downloading phase, the k-th helper sends the computation results back to the user with duration tdlk via OFDMA. Note that the cooperation between the user and K helpers does not affect each other. To meet the user’s latency requirement, we have the following time constraint:
(6)tkoff+tkcomp+tkdl≤T,∀k∈K.Figure 2
An illustration of the OFDMA-aided three-phase transmission protocol.In the following, we describe the OFDMA-aided three-phase transmission protocol in detail.


#### 2.2.1. Phase I (Task Offloading)

Lethoffk denote the channel power gain from the user to the k-th helper, k∈K. The achievable offloading rate at the k-th helper is given by
(7)lk=tkoffrkoffpkoff,whereW in Hz denotes one frequency resource block, poffk is the transmit power for offloading data to the k-th helper, and σk2 is the power of additive white Gaussian noise (AWGN) at the k-th helper. Hence, we have the offloaded bits lk from the user to the k-th helper as
(8)lk=tkoffrkoffpkoff.Accordingly, the total energy consumption for task offloading consumed by the user is expressed as(9)Euoff=∑k=1Ktkoffpkoff.


#### 2.2.2. Phase II (Task Execution)

After receivinglk bits, the k-th helper executes with the optimal CPU frequency given as
(10)fk=lkckT−tkoff−tkdl,whereck denotes the number of CPU cycles for computing 1-bit input-data at the k-th helper. Similarly as in (3), fk is also subject to the maximum frequency constraint, that is,
(11)fk≤fkmax,∀k∈K.Consequently, the energy consumption for cooperative computation at thek-th helper is expressed as
(12)Ekcomp=γklkck3T−tkoff−tkdl2,whereγk denotes a constant related to the k-th helper’s hardware architecture [21].


#### 2.2.3. Phase III (Result Downloading)

After executing the user’s assigned bits, thek-th helper begins sending the computation results back to the user via OFDMA. Let hkdl denote the channel power gain from the k-th helper to the user. The achievable downloading rate from the k-th helper is given by
(13)rkdlpkdl=Wlog21+pkdlhkdlσ02,wherepkdl is the transmit power of the k-th helper, and σ02 is the power of AWGN at the user. The corresponding computation results are thus given by
(14)qlk=tkdlrkdlpkdl,whereq∈R+ denotes the normalized ratio between the size of computation results and the size of computing tasks [20]. The energy consumption for results downloading consumed by the k-th helper is expressed as
(15)Ekdl=tkdlpkdl.


## 2.1. Local Computing

Thelu bits are executed locally with the optimal central process unit (CPU) frequency given as [21]
(2)fu=culuT,wherecu denotes the number of CPU cycles for computing 1-bit input-data at the user. Note that fu is subject to the maximum frequency constraint, that is,
(3)fu≤fumax.Accordingly, the user’s energy consumption for local computing is given by(4)Eucomp=γuculufu2.whereγu denotes a constant related to the user’s hardware architecture [21]. Replacing fu in (4) with (2), Eucomp can thus be reexpressed as
(5)Eucomp=γuculu3T2.


## 2.2. Remote Computing at Helpers

The OFDMA-aided three-phase transmission protocol is shown in Figure2. At first, the user first offloads lk bits to the k-th helper with duration toffk via OFDMA in the task offloading phase, k∈K. Then, the k-th helper executes its assigned bits with duration tcompk in the task execution phase. At last, in the result downloading phase, the k-th helper sends the computation results back to the user with duration tdlk via OFDMA. Note that the cooperation between the user and K helpers does not affect each other. To meet the user’s latency requirement, we have the following time constraint:
(6)tkoff+tkcomp+tkdl≤T,∀k∈K.Figure 2
An illustration of the OFDMA-aided three-phase transmission protocol.In the following, we describe the OFDMA-aided three-phase transmission protocol in detail.


### 2.2.1. Phase I (Task Offloading)

Lethoffk denote the channel power gain from the user to the k-th helper, k∈K. The achievable offloading rate at the k-th helper is given by
(7)lk=tkoffrkoffpkoff,whereW in Hz denotes one frequency resource block, poffk is the transmit power for offloading data to the k-th helper, and σk2 is the power of additive white Gaussian noise (AWGN) at the k-th helper. Hence, we have the offloaded bits lk from the user to the k-th helper as
(8)lk=tkoffrkoffpkoff.Accordingly, the total energy consumption for task offloading consumed by the user is expressed as(9)Euoff=∑k=1Ktkoffpkoff.


### 2.2.2. Phase II (Task Execution)

After receivinglk bits, the k-th helper executes with the optimal CPU frequency given as
(10)fk=lkckT−tkoff−tkdl,whereck denotes the number of CPU cycles for computing 1-bit input-data at the k-th helper. Similarly as in (3), fk is also subject to the maximum frequency constraint, that is,
(11)fk≤fkmax,∀k∈K.Consequently, the energy consumption for cooperative computation at thek-th helper is expressed as
(12)Ekcomp=γklkck3T−tkoff−tkdl2,whereγk denotes a constant related to the k-th helper’s hardware architecture [21].


### 2.2.3. Phase III (Result Downloading)

After executing the user’s assigned bits, thek-th helper begins sending the computation results back to the user via OFDMA. Let hkdl denote the channel power gain from the k-th helper to the user. The achievable downloading rate from the k-th helper is given by
(13)rkdlpkdl=Wlog21+pkdlhkdlσ02,wherepkdl is the transmit power of the k-th helper, and σ02 is the power of AWGN at the user. The corresponding computation results are thus given by
(14)qlk=tkdlrkdlpkdl,whereq∈R+ denotes the normalized ratio between the size of computation results and the size of computing tasks [20]. The energy consumption for results downloading consumed by the k-th helper is expressed as
(15)Ekdl=tkdlpkdl.


## 2.2.1. Phase I (Task Offloading)

Lethoffk denote the channel power gain from the user to the k-th helper, k∈K. The achievable offloading rate at the k-th helper is given by
(7)lk=tkoffrkoffpkoff,whereW in Hz denotes one frequency resource block, poffk is the transmit power for offloading data to the k-th helper, and σk2 is the power of additive white Gaussian noise (AWGN) at the k-th helper. Hence, we have the offloaded bits lk from the user to the k-th helper as
(8)lk=tkoffrkoffpkoff.Accordingly, the total energy consumption for task offloading consumed by the user is expressed as(9)Euoff=∑k=1Ktkoffpkoff.


## 2.2.2. Phase II (Task Execution)

After receivinglk bits, the k-th helper executes with the optimal CPU frequency given as
(10)fk=lkckT−tkoff−tkdl,whereck denotes the number of CPU cycles for computing 1-bit input-data at the k-th helper. Similarly as in (3), fk is also subject to the maximum frequency constraint, that is,
(11)fk≤fkmax,∀k∈K.Consequently, the energy consumption for cooperative computation at thek-th helper is expressed as
(12)Ekcomp=γklkck3T−tkoff−tkdl2,whereγk denotes a constant related to the k-th helper’s hardware architecture [21].


## 2.2.3. Phase III (Result Downloading)

After executing the user’s assigned bits, thek-th helper begins sending the computation results back to the user via OFDMA. Let hkdl denote the channel power gain from the k-th helper to the user. The achievable downloading rate from the k-th helper is given by
(13)rkdlpkdl=Wlog21+pkdlhkdlσ02,wherepkdl is the transmit power of the k-th helper, and σ02 is the power of AWGN at the user. The corresponding computation results are thus given by
(14)qlk=tkdlrkdlpkdl,whereq∈R+ denotes the normalized ratio between the size of computation results and the size of computing tasks [20]. The energy consumption for results downloading consumed by the k-th helper is expressed as
(15)Ekdl=tkdlpkdl.


## 3. Problem Formulation

In this paper, we aim to minimize the total energy consumption of the multiuser cooperative MEC system (i.e.,Euoff+Eucomp+∑k=1KEkdl+Ekcomp) by jointly optimizing the user’s task partition, the task offloading time, the result downloading time, and the transmit power of the user and helpers, subject to the user’s computation latency constraint T. Specifically, the energy consumption minimization problem is formulated as
(16a)P1:minl,toff,tdl,poff,pdlEuoff+∑k=1KEkdl+Ekcomp+Eucomps.t.lk≤tkoffrkoffpkoff,∀k∈K,(16b)qlk≤tkdlrkdlpkdl,∀k∈K,(16c)∑k=1Ktkoffpkoff≤Emaxoff,(16d)tkdlpkdl≤Emaxdl,∀k∈K,(16e)tkdl≤T,tkoff≤T,∀k∈K,(16f)l∈ℝ≥0K+1,toff∈ℝ≥0K,tdl∈ℝ≥0K,poff∈ℝ≥0K,pdl∈ℝ≥0K1,3,11,wherel≙lu,l1,⋯,lK∈ℝ≥0K+1,toff≜t1off,⋯,tKoff∈ℝ≥0K,tdl≜t1dl,⋯,tKdl∈ℝ≥0K,poff≜p1off,⋯,pKoff∈ℝ≥0K, and pdl≜p1dl,⋯,pKdl∈ℝ≥0K. (1) denotes the user’s task partition constraint, (3) and (11) denote the maximum CPU frequency constraints at the user and helpers, respectively, (16a) and (16b) denote the constraints for data transmission between the user and the helpers, and (16c) and (16d) denote the transmission energy consumption constraints at the user and helpers, respectively. Note that in problem (P1), we replace the equality in (8) and (14) as the inequality constraints (16a) and (16b), respectively. (16a) and (16b) should be met with strict equality at optimality of problem (P1). This is consistent with intuition. Because of the coupling of tkoff and pkoff and tdlk and pdlk in the objective function and the constraints (16a) and (16b), problem (P1) is nonconvex.


### 3.1. Feasibility of Problem (P1)

Before solving problem (P1), we need to guarantee its feasibility so that the multiuser cooperative MEC system can support the latency-constrained task execution. LetLmax denote the maximum data size in bits supported by the proposed MEC system within duration T. There is no doubt that only when Lmax≥Lu, problem (P1) is feasible, or problem (P1) is infeasible. Hence, we check the feasibility of problem (P1) by determining Lmax. Intuitively, Lmax is obtained when the user and helpers make full use of the communication and computation resources in the proposed MEC system. This corresponds to letting the constraints (3) and (11) be met with strict equality in problem (P1). Then, the data maximization problem is formulated as
(17a)P2:Lmax≜maxtoff,tdl,poff,pdl,lTfumaxcu+∑k=1KT−tkoff−tkdlfkmaxcks.t.T−tkoff−tkdlfkmaxck≤tkoffrkoffpkoff,∀k∈K,qT−tkoff−tkdlfkmaxck≤tkdlrkdlpkdl,∀k∈K,∑k=1Ktkoffpkoff≤Emaxoff,tkdlpkdl≤Emaxdl,∀k∈K16e,16f.Due to the similarity of between problems (P1) and (P2), problem (P2) can be solved like problem (P1). By comparingLmax and Lu, we finally check the feasibility of problem (P1).


## 3.1. Feasibility of Problem (P1)

Before solving problem (P1), we need to guarantee its feasibility so that the multiuser cooperative MEC system can support the latency-constrained task execution. LetLmax denote the maximum data size in bits supported by the proposed MEC system within duration T. There is no doubt that only when Lmax≥Lu, problem (P1) is feasible, or problem (P1) is infeasible. Hence, we check the feasibility of problem (P1) by determining Lmax. Intuitively, Lmax is obtained when the user and helpers make full use of the communication and computation resources in the proposed MEC system. This corresponds to letting the constraints (3) and (11) be met with strict equality in problem (P1). Then, the data maximization problem is formulated as
(17a)P2:Lmax≜maxtoff,tdl,poff,pdl,lTfumaxcu+∑k=1KT−tkoff−tkdlfkmaxcks.t.T−tkoff−tkdlfkmaxck≤tkoffrkoffpkoff,∀k∈K,qT−tkoff−tkdlfkmaxck≤tkdlrkdlpkdl,∀k∈K,∑k=1Ktkoffpkoff≤Emaxoff,tkdlpkdl≤Emaxdl,∀k∈K16e,16f.Due to the similarity of between problems (P1) and (P2), problem (P2) can be solved like problem (P1). By comparingLmax and Lu, we finally check the feasibility of problem (P1).


## 4. Optimal Solution

In this section, we first transform problem (P1) into a convex one and then present an efficient algorithm to obtain the globally optimal solution.To accomplish this target, we introduce two auxiliary variable vectorsyoff≜y1off,⋯,yKoff and ydl≜y1dl,⋯,yKdl with ykoff=tkoffpkoff and ykdl=tkdlpkdl, ∀k∈K. Then, it holds that poffk=ykoff/toffk if toffk>0, and pkoff=0 if either ykoff=0 or tkoff=0. Similarly, this also applies to pdlk=ykdl/tdlk. By substituting pkoff=ykoff/tkoff and pkdl=ykdl/tkdl, problem (P1) can be reformulated as
(18a)P1.1:minl,toff,tdl,yoff,ydl∑k=1Kykoff+ykdl+γklkck3T−tkoff−tkdl2+γulucu3T2,s.tlk≤tkoffrkoffykofftkoff,∀k∈K,(18b)qlk≤tkdlrkdlykdltkdl,∀k∈K,(18c)∑k=1Kykoff≤Emaxoff,(18d)ykdl≤Emaxdl,∀k∈K,(18e)tkdl≤T,tkoff≤T,∀k∈K,(18f)l∈ℝ≥0K+1,tdl∈ℝ≥0K,toff∈ℝ≥0K,ydl∈ℝ≥0K,yoff∈ℝ≥0K.1,3,11Lemma 1.
Problem (P1.1) is a convex problem.Proof.
It is obvious that the functionrkoffx is a concave function with respect to x≥0. As the perspective operation maintains convexity, xrkoffy/x is jointly concave with respect to x≥0 and y≥0 [28]. Similarly, this also applies to xrkdly/x. Therefore, the set defined by the constraints (18a)–(18d) is convex. The Hessian of lk3/T−tkoff−tkdl2 is
(19)H=6lkT−tkoff−tkdl26lk2T−tkoff−tkdl36lk2T−tkoff−tkdl36lk2T−tkoff−tkdl36lk3T−tkoff−tkdl46lk3T−tkoff−tkdl46lk2T−tkoff−tkdl36lk3T−tkoff−tkdl46lk3T−tkoff−tkdl4.
The leading principal mirrors ofH are given by
(20)Δ1=6lkT−tkoff−tkdl2≥0,Δ2=6lkT−tkoff−tkdl26lk2T−tkoff−tkdl36lk2T−tkoff−tkdl36lk3T−tkoff−tkdl4=0.From the above analysis, we can validate thatlk3/T−tkoff−tkdl2 is convex and so is lu3/T2. Hence, problem (P1.1) is convex.In view of Lemma1, to gain engineering insights, we next leverage the Lagrange duality method to solve problem (P1.1).[28]Letλ1∈RK≥0 and λ2∈RK≥0 indicate the dual variables related to the constraints in (18a) and (18b), respectively, and let μ1∈R≥0, μ2∈R, and μ3∈RK≥0 be the dual variables related to the constraints in (18c), (1), and (18d), respectively.Defineλ≜λ1,λ2, μ≜μ1,μ2,μ3, λ1≜λ1,1,⋯,λ1,K, λ2≜λ2,1,⋯,λ2,K, and μ3≜μ3,1,⋯,μ3,K. The partial Lagrangian of problem (P1.1) is given by
(21)Ltoff,tdl,yoff,ydl,l,λ,μ=∑k=1K1+μ1ykoff+μ3,k+1ykdl−μ3,kEmaxdl+γklkck3T−tkoff−tkdl2+λ1,k−μ2+λ2,kqlk−λ1,ktkoffrkoffykofftkoff−λ2,ktkdlrkdlykdltkdl+γulucu3T2−μ2lu+μ2Lu−μ1Emaxoff.The dual function of problem (P1.1) is expressed as(22)gλ,μ=mintoff,tdl,yoff,ydl,lLtoff,tdl,yoff,ydl,l,λ,μs.t.3,11,18e,18f.As a result, the dual problem of problem (P1.1) is given by(23)P1.1‐dual:maxλ,μgλ,μs.t.λ∈ℝ≥02K,μ1≥0,μ2∈ℝ,μ3∈ℝ≥0K.DenoteΨ and λopt and μopt as the feasible set and the optimal dual variables for problem (P1.1-dual), respectively.Since problem (P1.1) is convex and satisfies Slater’s condition, there is zero duality gap between problems (P1.1) and (P1.1-dual) [28]. Next, we first find the dual function g (λ, μ) by solving problem (22) under any given (λ, μ) ∈ Ψ and then obtain λopt and μopt to maximize g (λ, μ).s.


### 4.1. Derivation of Dual Function g (λ, μ)

Denotetoff∗,tdl∗,yoff∗,ydl∗,l∗ as the optimal solution for problem (22) under any given (λ,μ) ∈ Ψ, toffopt,tdlopt,yoffopt,ydlopt,lopt as the optimal primal solution for problem (P1.1), respectively. In the following, we find the dual function g (λ,μ) by solving problem (22) under any given (λ,μ) ∈ Ψ. Equivalently, we decompose (22) into K+1 subproblems as follows:
(24)P1.1‐sub1:mint1off,t1dl,y1off,y1dl,l11+μ1y1off+μ3,1+1y1dl−μ3,1Emaxdl+γ1l1c13T−t1off−t1dl2+λ1,1−μ2+λ2,1ql1−λ1,1t1offr1offy1offt1off−λ2,1t1dlr1dly1dlt1dls.t.11,0≤t1off≤T,0≤t1dl≤T,0≤l1,0≤y1off,0≤y1dl,⋮,P1.1‐subK:mintKoff,tKdl,yKoff,yKdl,lK1+μ1yKoff+μ3,K+1yKdl−μ3,KEmaxdl+γKlKcK3T−tKoff−tKdl2+λ1,K−μ2+λ2,KqlK−λ1,KtKoffrKoffyKofftKoff−λ2,KtKdlrKdlyKdltKdl,s.t.11,0≤tKoff≤T,0≤tKdl≤T,0≤lK,0≤yKoff,0≤yKdl,P1.1‐subK+1:minluγulucu3T2−μ2lus.t.3.As these subproblems are independent of each other, they can be parallelly solved. Also, the optimal solutions for problems (P1.1-sub1)-(P1.1-subK+1) are presented in Lemmas 21,···, 2K, and 3, respectively. Note that we only show the proof of Lemma 2 since Lemmas 21,···, 2k−1, 2k+1,···, 2K, and 3 can be similarly proved via Karush-Kuhn-Tucker (KKT) conditions.Lemma 2.
Under given (λ,μ) ∈ Ψ, the optimal solution ykoff∗,ykdl∗,tkoff∗,tkdl∗,lk∗ to problem (P1.1subk) satisfies
(25)ykoff∗=tkoff∗pkoff∗,(26)ykdl∗=tkdl∗pkdl∗,(27)lk∗=Mk∗T−tkoff∗−tkdl∗,(28)tkoff∗=T,ρk,1>0,0,T,ρk,1=0,0,ρk,1<0,(29)tkdl∗=T,ρk,2>0,0,T,ρk,2=0,0,ρk,2<0,wherepkoff∗=λ1,kW/ln21+μ1−σk2/hkoff+,pkdl∗=λ2,kW/ln21+μ3,k−σ02/hkdl+, and
(30)Mk∗=μ2−λ1,k−λ2,kq3γkck30fkmax/ck,μ2−λ1,k−λ2,kq≥0,0,μ2−λ1,k−λ2,kq<0,(31)ρk,1=λ1,krkoffpkoff∗−α1fkmaxck−2γkckMk∗3−λ1,kWhkoff/σk2pkoff∗ln21+hkoff/σk2pkoff∗,(32)ρk,2=λ2,krkdlpkdl∗−α1fkmaxck−2γkckMk∗3−λ2,kWhkdl/σ02pkdl∗ln21+hkdl/σ02pkdl∗,(33)α1=0,Mk∗<fkmaxck,μ2−λ1,k−λ2,kq−3γkck3Mk∗2,Mk∗=fkmaxck.Proof.
Since problem (P1.1-subk) is convex and satisfies Slater’s condition, there is zero duality gap between problems (P1.1-subk) and its dual problem [28]. Hence, we use the KKT conditions to solve problem (P1.1-subk). The Lagrangian function of problem (P1.1-subk) is given by
(34)Lk=1+μ1ykoff+μ3,k+1ykdl+γklkck3T−tkoff−tkdl2−μ2lk+λ1,klk−tkoffrkoffykofftkoff+λ2,kqlk−tkdlrkdlykdltkdl+α1lk−T−tkoff−tkdlfkmaxck+a1tkoff−T−a2tkoff−b1lk−β1ykoff−η1ykdl+d1tkdl−T−d2tkdl,
wherea1, a2, b1, α1, β1, η1, d1, and d2 are the nonnegative Lagrange multipliers associated with tkoff≤T0≤tkdl, 0≤lk, lk≤T−tkoff−tkdlfkmax/ck,0≤ykoff,0≤ykdl,tkdl≤T, and 0≤tdlk, respectively.
According to the KKT conditions, it follows that(35)a1tkoff−T=0,(36)a2tkoff=0,(37)b1lk=0,(38)β1ykoff=0,(39)η1ykdl=0,(40)d1tkdl−T=0,(41)d2tkdl=0,(42)a1lk−T−tkoff−tkdlfkmaxck=0,(43)∂Lk∂tkoff=2γklkck3T−ykoff−ykdl3−λ1,kWlog21+ykofftkoffhkoffσk2+λ1,kWykoff/tkoffhkoff/σk2ln21+ykoff/tkoffhkoff/σk2+α1fkmaxck+a1−a2=0,(44)∂Lk∂tkdl=2γklkck3T−tkoff−tkdl3−λ2,kWlog21+ykdltkdlhkdlσ02+λ2,kWykdl/tkdlhkdl/σ02ln21+ykdl/tkdlhkdl/σ02+α1fkmaxck+d1−d2=0,(45)∂Lk∂ykoff=1+μ1−λ1,kWhkoff/σk2ln21+ykoff/tkoffhkoff/σk2−β1=0,(46)∂Lk∂ykdl=1+μ3,k−λ2,kWhkdl/σ02ln21+ykdl/tkdlhkdl/σ02−η1=0,(47)∂Lk∂lk=3γkck3lk2T−tkoff−tkdl2−μ2+λ1,k+λ2,kq+α1−b1=0,where (35)–(42) denote the complementary slackness condition, (43)–(47) are the first-order derivative conditions of Lk with respect to toffk, tdlk, ykoff, ykdl, and lk, respectively. Hence, we have (27) and (28) based on (45) and (46), respectively, and (29) holds due to (47). In addition, we have (29) based on (47) and some manipulations.Also, by substituting (25) and (27) into (43) and (26) and (27) into (44) and assuming ρk,1=a1−a2 and ρk,2=d1−d2, we have ρk,1 and ρk,2 in (31) and (28), respectively. Consequently, the optimal tkoff∗ and tkdl∗ are given in (28) and (29), respectively.Lemma 3.
Under given (λ, μ) ∈ Ψ, the optimal solution lu∗ to problem (P1.1-subK+1) is
(48)lu∗=Tμ23γucu30Tfumax/cu.Remark 4.
Note that in (28) and (29), if ρk,i=0 (for any i∈1,2), the optimal solution tkoff∗ or tkdl∗ is generally nonunique. In this case, we choose tkoff∗=0 or tkdl∗=0 for evaluating g (λ, μ). Since such choice may not be feasible or optimal for problem (P1.1), we add an additional step to find the primal optimal tkoffopt and tkdlopt, as will be shown in Section 4.3.By combining Lemmas 21,···, 2K with 3, g (λ,μ) is evaluated for any given (λ, μ) ∈ Ψ.


### 4.2. Obtainingλopt and μopt to Maximize g (λ, μ)

Withtoff∗,tdl∗,yoff∗,ydl∗,l∗ obtained, we then solve problem (P1.1-dual) to maximize g (λ, μ). Due to the property of g (λ, μ), the ellipsoid method is utilized to obtain λopt,μopt [29]. For the objective function in (22), one subgradient is given by [29]
(49)e=l1∗−t1off∗r1offy1off∗t1off∗,⋯,lK∗−tKoff∗rKoffyKoff∗tKoff∗,ql1∗−t1dl∗r1dly1dl∗t1dl∗,⋯,qlK∗−tKdl∗rKdlyKdl∗tKdl∗,∑k=1Kykoff∗−Emaxoff,Lu−lu∗−∑k=1Klk∗y1dl∗−Emaxdl,⋯,yKdl∗−Emaxdl.For the constraintsλ∈ℝ≥02K,μ1≥0,μ2∈ℝ and μ3∈RK≥0, the subgradients are e1,···,e2K, e2K+1, e2K+2, and e2K+3,···,e3K+2, respectively. Note that ei is of all zero entries except for the i-th entry being one.


### 4.3. Finding the Optimal Primal Solution to (P1)

Having obtainedλopt,μopt, we still need to further solve problem (P1.1). By replacing (λ,μ) with λopt,μopt in Lemmas 21,···, 2K, and 3, we obtain the corresponding poffopt=p1offopt,L,pKoffopt, pdlopt=p1dlopt,⋯,pKdlopt, Mopt=M1opt,⋯,MKopt, and luopt, respectively. However, due to the nonuniqueness of tkoff∗ and tkdl∗,k∈K, we implement an extra procedure to obtain the optimal solution of other variables for problem (P1). With poffopt, pdlopt, Mopt, and luopt, the optimal solution must satisfy lk=MkoptT−tkoff−tkdl, ykoff=tkoffpkoffopt, and ykdl=tkdlpkdlopt. By substituting them in (P1.1), we have the following linear program (LP) to obtain toffopt and tdlopt:
(50)mintoff,tdl∑k=1Ktkoffpkoffopt+tkdlpkdlopt+γkMkoptcu3T−tkoff−tkdl,s.t.MkoptT−tkoff−tkdl≤tkoffrkoffpkoffopt,∀k∈K,qMkoptT−tkoff−tkdl≤tkdlrkdlpkdlopt,∀k∈K,∑k=1Ktkoffpkoffopt≤Emaxoff,tkdlpkdlopt≤Emaxdl,∀k∈K,∑k=1KMkoptT−tkoff−tkdl+luopt=Lu,tkoff+tkdl≤T,∀k∈K,0≤tkoff,0≤tkdl,∀k∈K.Since problem (50) is an instance of LP, it can be solved by the interior-point method [28]. Finally, we obtain the globally optimal solution for problem (P1). The proposed joint task partition and resource allocation (JTPRA) scheme is thus summarized in Algorithm 1.Algorithm 1
1 Initialization: Given an ellipsoidε((λ,μ),A) contain-ing(λopt,μopt), where (λ,μ) is the center point of εandA≻0 characterizes the volume of ε.2repeat3 Obtainpoff∗, pdl∗, yoff∗, ydl∗, toff∗, and tdl∗ by Lemmas21,⋯,2K and 3 under given (λ,μ)∈Ψ, respectively;4 Compute the subgradients ofg (λ,μ), then update(λ,μ) using the ellipsoid method [29];5until (λ,μ) converge to a specified accuracy.6Setλopt,μopt⟵λ,μ.7Output: Obtain poffopt,pdlopt, and luopt based on 21,···,2Kand 3 by replacing(λ,μ) with λopt,μopt, and thencomputetoffopt, tdlopt, and lopt by solving the LP in (50).Remark 5.
With Lemmas 21,···, 2K, and 3, the following insights can be obtained as follows:
(1)
As for local computing, it is observed from Lemma3 that luopt generally increases as T becomes large. This indicates that the user prefers executing more tasks when the user’s computation latency constraint becomes loose(2)
As for cooperative computing, it is evident that, based on Lemma2, the offloading power pkoffopt increases as the channel power gain hoffk becomes stronger. That is, the user prefers offloading more tasks to the closer helper, in order to reduce the marginal energy consumption for offloading. Similarly, this also applies to pkdlopt


### 4.4. Complexity

The complexity of the ellipsoid method isON2, where N is the number of dual variables and N=3K+2 in (23) [29]. Moreover, the complexity of the interior-point method is OM3.5log1/ε where M is the number of optimal variables, log (1/ε) is the iteration complexity order, and M=2K is in (50) [28]. Hence, the total complexity of Algorithm 1 is OK3.5log1/1ε−ε.


## 4.1. Derivation of Dual Function g (λ, μ)

Denotetoff∗,tdl∗,yoff∗,ydl∗,l∗ as the optimal solution for problem (22) under any given (λ,μ) ∈ Ψ, toffopt,tdlopt,yoffopt,ydlopt,lopt as the optimal primal solution for problem (P1.1), respectively. In the following, we find the dual function g (λ,μ) by solving problem (22) under any given (λ,μ) ∈ Ψ. Equivalently, we decompose (22) into K+1 subproblems as follows:
(24)P1.1‐sub1:mint1off,t1dl,y1off,y1dl,l11+μ1y1off+μ3,1+1y1dl−μ3,1Emaxdl+γ1l1c13T−t1off−t1dl2+λ1,1−μ2+λ2,1ql1−λ1,1t1offr1offy1offt1off−λ2,1t1dlr1dly1dlt1dls.t.11,0≤t1off≤T,0≤t1dl≤T,0≤l1,0≤y1off,0≤y1dl,⋮,P1.1‐subK:mintKoff,tKdl,yKoff,yKdl,lK1+μ1yKoff+μ3,K+1yKdl−μ3,KEmaxdl+γKlKcK3T−tKoff−tKdl2+λ1,K−μ2+λ2,KqlK−λ1,KtKoffrKoffyKofftKoff−λ2,KtKdlrKdlyKdltKdl,s.t.11,0≤tKoff≤T,0≤tKdl≤T,0≤lK,0≤yKoff,0≤yKdl,P1.1‐subK+1:minluγulucu3T2−μ2lus.t.3.As these subproblems are independent of each other, they can be parallelly solved. Also, the optimal solutions for problems (P1.1-sub1)-(P1.1-subK+1) are presented in Lemmas 21,···, 2K, and 3, respectively. Note that we only show the proof of Lemma 2 since Lemmas 21,···, 2k−1, 2k+1,···, 2K, and 3 can be similarly proved via Karush-Kuhn-Tucker (KKT) conditions.Lemma 2.
Under given (λ,μ) ∈ Ψ, the optimal solution ykoff∗,ykdl∗,tkoff∗,tkdl∗,lk∗ to problem (P1.1subk) satisfies
(25)ykoff∗=tkoff∗pkoff∗,(26)ykdl∗=tkdl∗pkdl∗,(27)lk∗=Mk∗T−tkoff∗−tkdl∗,(28)tkoff∗=T,ρk,1>0,0,T,ρk,1=0,0,ρk,1<0,(29)tkdl∗=T,ρk,2>0,0,T,ρk,2=0,0,ρk,2<0,wherepkoff∗=λ1,kW/ln21+μ1−σk2/hkoff+,pkdl∗=λ2,kW/ln21+μ3,k−σ02/hkdl+, and
(30)Mk∗=μ2−λ1,k−λ2,kq3γkck30fkmax/ck,μ2−λ1,k−λ2,kq≥0,0,μ2−λ1,k−λ2,kq<0,(31)ρk,1=λ1,krkoffpkoff∗−α1fkmaxck−2γkckMk∗3−λ1,kWhkoff/σk2pkoff∗ln21+hkoff/σk2pkoff∗,(32)ρk,2=λ2,krkdlpkdl∗−α1fkmaxck−2γkckMk∗3−λ2,kWhkdl/σ02pkdl∗ln21+hkdl/σ02pkdl∗,(33)α1=0,Mk∗<fkmaxck,μ2−λ1,k−λ2,kq−3γkck3Mk∗2,Mk∗=fkmaxck.Proof.
Since problem (P1.1-subk) is convex and satisfies Slater’s condition, there is zero duality gap between problems (P1.1-subk) and its dual problem [28]. Hence, we use the KKT conditions to solve problem (P1.1-subk). The Lagrangian function of problem (P1.1-subk) is given by
(34)Lk=1+μ1ykoff+μ3,k+1ykdl+γklkck3T−tkoff−tkdl2−μ2lk+λ1,klk−tkoffrkoffykofftkoff+λ2,kqlk−tkdlrkdlykdltkdl+α1lk−T−tkoff−tkdlfkmaxck+a1tkoff−T−a2tkoff−b1lk−β1ykoff−η1ykdl+d1tkdl−T−d2tkdl,
wherea1, a2, b1, α1, β1, η1, d1, and d2 are the nonnegative Lagrange multipliers associated with tkoff≤T0≤tkdl, 0≤lk, lk≤T−tkoff−tkdlfkmax/ck,0≤ykoff,0≤ykdl,tkdl≤T, and 0≤tdlk, respectively.
According to the KKT conditions, it follows that(35)a1tkoff−T=0,(36)a2tkoff=0,(37)b1lk=0,(38)β1ykoff=0,(39)η1ykdl=0,(40)d1tkdl−T=0,(41)d2tkdl=0,(42)a1lk−T−tkoff−tkdlfkmaxck=0,(43)∂Lk∂tkoff=2γklkck3T−ykoff−ykdl3−λ1,kWlog21+ykofftkoffhkoffσk2+λ1,kWykoff/tkoffhkoff/σk2ln21+ykoff/tkoffhkoff/σk2+α1fkmaxck+a1−a2=0,(44)∂Lk∂tkdl=2γklkck3T−tkoff−tkdl3−λ2,kWlog21+ykdltkdlhkdlσ02+λ2,kWykdl/tkdlhkdl/σ02ln21+ykdl/tkdlhkdl/σ02+α1fkmaxck+d1−d2=0,(45)∂Lk∂ykoff=1+μ1−λ1,kWhkoff/σk2ln21+ykoff/tkoffhkoff/σk2−β1=0,(46)∂Lk∂ykdl=1+μ3,k−λ2,kWhkdl/σ02ln21+ykdl/tkdlhkdl/σ02−η1=0,(47)∂Lk∂lk=3γkck3lk2T−tkoff−tkdl2−μ2+λ1,k+λ2,kq+α1−b1=0,where (35)–(42) denote the complementary slackness condition, (43)–(47) are the first-order derivative conditions of Lk with respect to toffk, tdlk, ykoff, ykdl, and lk, respectively. Hence, we have (27) and (28) based on (45) and (46), respectively, and (29) holds due to (47). In addition, we have (29) based on (47) and some manipulations.Also, by substituting (25) and (27) into (43) and (26) and (27) into (44) and assuming ρk,1=a1−a2 and ρk,2=d1−d2, we have ρk,1 and ρk,2 in (31) and (28), respectively. Consequently, the optimal tkoff∗ and tkdl∗ are given in (28) and (29), respectively.Lemma 3.
Under given (λ, μ) ∈ Ψ, the optimal solution lu∗ to problem (P1.1-subK+1) is
(48)lu∗=Tμ23γucu30Tfumax/cu.Remark 4.
Note that in (28) and (29), if ρk,i=0 (for any i∈1,2), the optimal solution tkoff∗ or tkdl∗ is generally nonunique. In this case, we choose tkoff∗=0 or tkdl∗=0 for evaluating g (λ, μ). Since such choice may not be feasible or optimal for problem (P1.1), we add an additional step to find the primal optimal tkoffopt and tkdlopt, as will be shown in Section 4.3.By combining Lemmas 21,···, 2K with 3, g (λ,μ) is evaluated for any given (λ, μ) ∈ Ψ.


## 4.2. Obtainingλopt and μopt to Maximize g (λ, μ)

Withtoff∗,tdl∗,yoff∗,ydl∗,l∗ obtained, we then solve problem (P1.1-dual) to maximize g (λ, μ). Due to the property of g (λ, μ), the ellipsoid method is utilized to obtain λopt,μopt [29]. For the objective function in (22), one subgradient is given by [29]
(49)e=l1∗−t1off∗r1offy1off∗t1off∗,⋯,lK∗−tKoff∗rKoffyKoff∗tKoff∗,ql1∗−t1dl∗r1dly1dl∗t1dl∗,⋯,qlK∗−tKdl∗rKdlyKdl∗tKdl∗,∑k=1Kykoff∗−Emaxoff,Lu−lu∗−∑k=1Klk∗y1dl∗−Emaxdl,⋯,yKdl∗−Emaxdl.For the constraintsλ∈ℝ≥02K,μ1≥0,μ2∈ℝ and μ3∈RK≥0, the subgradients are e1,···,e2K, e2K+1, e2K+2, and e2K+3,···,e3K+2, respectively. Note that ei is of all zero entries except for the i-th entry being one.


## 4.3. Finding the Optimal Primal Solution to (P1)

Having obtainedλopt,μopt, we still need to further solve problem (P1.1). By replacing (λ,μ) with λopt,μopt in Lemmas 21,···, 2K, and 3, we obtain the corresponding poffopt=p1offopt,L,pKoffopt, pdlopt=p1dlopt,⋯,pKdlopt, Mopt=M1opt,⋯,MKopt, and luopt, respectively. However, due to the nonuniqueness of tkoff∗ and tkdl∗,k∈K, we implement an extra procedure to obtain the optimal solution of other variables for problem (P1). With poffopt, pdlopt, Mopt, and luopt, the optimal solution must satisfy lk=MkoptT−tkoff−tkdl, ykoff=tkoffpkoffopt, and ykdl=tkdlpkdlopt. By substituting them in (P1.1), we have the following linear program (LP) to obtain toffopt and tdlopt:
(50)mintoff,tdl∑k=1Ktkoffpkoffopt+tkdlpkdlopt+γkMkoptcu3T−tkoff−tkdl,s.t.MkoptT−tkoff−tkdl≤tkoffrkoffpkoffopt,∀k∈K,qMkoptT−tkoff−tkdl≤tkdlrkdlpkdlopt,∀k∈K,∑k=1Ktkoffpkoffopt≤Emaxoff,tkdlpkdlopt≤Emaxdl,∀k∈K,∑k=1KMkoptT−tkoff−tkdl+luopt=Lu,tkoff+tkdl≤T,∀k∈K,0≤tkoff,0≤tkdl,∀k∈K.Since problem (50) is an instance of LP, it can be solved by the interior-point method [28]. Finally, we obtain the globally optimal solution for problem (P1). The proposed joint task partition and resource allocation (JTPRA) scheme is thus summarized in Algorithm 1.Algorithm 1
1 Initialization: Given an ellipsoidε((λ,μ),A) contain-ing(λopt,μopt), where (λ,μ) is the center point of εandA≻0 characterizes the volume of ε.2repeat3 Obtainpoff∗, pdl∗, yoff∗, ydl∗, toff∗, and tdl∗ by Lemmas21,⋯,2K and 3 under given (λ,μ)∈Ψ, respectively;4 Compute the subgradients ofg (λ,μ), then update(λ,μ) using the ellipsoid method [29];5until (λ,μ) converge to a specified accuracy.6Setλopt,μopt⟵λ,μ.7Output: Obtain poffopt,pdlopt, and luopt based on 21,···,2Kand 3 by replacing(λ,μ) with λopt,μopt, and thencomputetoffopt, tdlopt, and lopt by solving the LP in (50).Remark 5.
With Lemmas 21,···, 2K, and 3, the following insights can be obtained as follows:
(1)
As for local computing, it is observed from Lemma3 that luopt generally increases as T becomes large. This indicates that the user prefers executing more tasks when the user’s computation latency constraint becomes loose(2)
As for cooperative computing, it is evident that, based on Lemma2, the offloading power pkoffopt increases as the channel power gain hoffk becomes stronger. That is, the user prefers offloading more tasks to the closer helper, in order to reduce the marginal energy consumption for offloading. Similarly, this also applies to pkdlopt


## 4.4. Complexity

The complexity of the ellipsoid method isON2, where N is the number of dual variables and N=3K+2 in (23) [29]. Moreover, the complexity of the interior-point method is OM3.5log1/ε where M is the number of optimal variables, log (1/ε) is the iteration complexity order, and M=2K is in (50) [28]. Hence, the total complexity of Algorithm 1 is OK3.5log1/1ε−ε.


## 5. Simulation Results

We provide simulation results for verifying the effectiveness of the proposed joint task partition and resource allocation (JTPRA) scheme in Section4, as compared against the following five benchmark schemes:
(1)
Local Computing with Optimal Frequency (LCOF): the computation tasks are executed locally with the optimal CPU frequency, and thus the optimal energy consumption for local computing isElocalopt=γuLucu3/T2(2)
Local Computing with Fixed Frequency (LCFF): the computation tasks are executed locally with the maximum CPU frequency, and thus the energy consumption for local computing isElocalfixed=γuLucufumax2(3)
Full Offloading (FO): the computation tasks are partitioned intoK parts for offloading to nearby helpers, which corresponds to solving problem (P1) by setting lu=0(4)
Joint Offloading Ratio and CPU Frequency (JORCF) [7]: in this scheme, the user adjusts both the offloading ratio and CPU frequency to cooperate with the MEC server(5)
Fixed Frequency (FF) [20]: let the constraints (3) and (11) be met with strict equality. This corresponds to solving problem (P1) by setting fu=fumax and fk=fkmax, k∈KIn the simulation, the distance between the user and helpers isd ∼ U [dmin, dmax] meters, where dmax=30meters and dmin=1meters. The path loss between any two nodes is modeled as bd−φ, where b=10−3 corresponds to the path loss at a reference distance of 1 meter, d denotes the distance from the user to a helper, and φ=3 is the path loss exponent [21]. Also, the helpers’ maximum CPU frequencies are assumed to be uniformly chosen from the set {1.6, 2.4, 3}GHz. The other parameters are set as shown in Table 1 unless otherwise specified.Table 1
Simulation parameters.
Frequency resource blockW1 MHzEffective capacitance coefficientγu=γ1=⋯=γK3×10−27Noise powerσ02=σ12=⋯=σK2-120 dBm [22]Computation intensitiescu=c1=⋯=cK103 cycle/bit [21]User’s maximum CPU frequencyfumax2 GHz [21]Available energy for data transmissionEmaxoff=Emaxdl0.5 jouleNormalized ratio between the size of computation results and the size of computing tasksq0.2Number of channel realizations500Figure3 shows the maximum data-size versus the block length T where Lu=0.2Mb and K=3. In the following simulation, under given T, we set Lu smaller than Lmax to guarantee the feasibility of problem (P1).Figure 3
The maximum data-size versus the block lengthT.Figure4 shows the average energy consumption versus data size Lu where T=0.15sec and K=3. It is observed that our proposed JTPRA achieves the minimum energy consumption than other schemes. Moreover, we have some observations as follows.
(1)
The average energy consumption by all the schemes increases asLu increases. JTPRA achieves significant performance gain over FF. This indicates the benefit of computation frequency optimization in energy saving for MEC(2)
For schemes with unchanged CPU frequency, FF achieves a lower energy consumption than LCFF. This is because the user prefers offloading the computing tasks to the helpers whose maximum CPU frequencies are below that of the user, compared with local computing(3)
For schemes involving optimizing CPU frequency, FO achieves significant energy reduction than LCOF, which is because the helpers’ optimal CPU frequencies are far below than that of the user. Also, by comparison with JORCF, JTPRA has about 69.9% energy consumption reduction on average. This indicates the performance gain brought by proximityFigure 4
The average energy consumption versus data sizeLu.Figure5 shows the average energy consumption versus the block length T where Lu=0.2Mb and K=3. We have generally similar observations in Figure 5 as in Figure 4. Specifically, it is observed that our proposed JTPRA has about 52.4% energy consumption reduction on average, compared with JORCF. Moreover, we have some observations as follows. (1) For schemes with unchanged CPU frequency, LCFF remains unchanged as T increases, while FF keeps almost unchanged. This indicates there is no need for the user and helpers to increase the transmission rate when the latency requirement is loose.Figure 5
The average energy consumption versus the block lengthT.(2) For schemes involving optimizing CPU frequency, the average energy consumption by all the schemes decreases asT increases, which is because as T increases, the user and helpers can lower down their optimal CPU frequency for consuming less energy. However, JORCF decreases with T very slowly. This indicates fixing the transmission rate is not conducive to improve the MEC performance.Figure6 shows the average energy consumption versus the frequency resource block W where T=0.15sec, Lu=0.2Mb, and K=3. We have generally similar observations in Figure 6 as in Figure 5. Specifically, it is observed that our proposed JTPRA has about 67.5% energy consumption reduction on average, compared with JORCF. Moreover, for schemes involving optimizing CPU frequency other than LCOF and JORCF, the average energy consumption first steadily decreases and then keeps almost unchanged as W increases. This is because a large W not only signifies a high transmission rate but incurs decreased transmission energy consumption between the user and the helpers.Figure 6
The average energy consumption versus the frequency resource blockW.Figure7 shows the average energy consumption versus the helper number K where Lu=0.2Mb and T=0.15sec.Figure 7
The average energy consumption versus the helper numberK.We have generally similar observations in Figure7 as in Figure 6. Specifically, it is observed that our proposed JTPRA has about 65.1% energy consumption reduction on average, compared with JORCF. Obviously, for schemes with unchanged CPU frequency, FF decreases as K becomes large. In addition, for schemes involving optimizing CPU frequency, JTPRA achieves a lower energy consumption than both LCOF and FO, while when K<2, JORCF achieves a lower energy consumption than JTPRA, while the reverse is true when Kbecomes large. This is because the more helpers whose optimal CPU frequencies are below that of the user are helpful for achieving more significant energy reduction.Figure8 shows the average energy consumption versus the maximum distance dmax from the user to the helpers where Lu=0.2Mb, T=0.15sec, and K=3. We have generally similar observations in Figure 8 as in Figure 7. Specifically, it is observed that our proposed JTPRA has about 15.1% energy consumption reduction on average, compared with JORCF. For schemes involving optimizing CPU frequency, FO first grows rapidly as dmax increases and then becomes even worse than both JORCF and LCOF, while JTPRA steadily increases as dmax increases. This is because as dmax increases, the channel gain between the user and the helpers becomes smaller, which leads to increased transmission energy consumption between the user and the helpers, and thus local computing is more beneficial than computation offloading at large dmax values.Figure 8
The average energy consumption versus the maximum distancedmax from the user to the helpers.


## 6. Conclusion

In this paper, we have proposed a novel joint task partition and resource allocation (JTPRA) scheme, in which nearby helpers share their own communication and computation resources to help the user. By considering an efficient OFDMA-aided three-phase transmission protocol, we proposed an energy-efficient design framework by jointly optimizing the user’s task partition, and the communication and computation resources allocation for task offloading and results downloading, subject to the user’s computation latency constraint. Based on convex optimization methods, we presented an efficient algorithm to obtain the globally optimal solution. Extensive numerical results demonstrated the merits of the proposed JTPRA scheme over alternative benchmark schemes.Due to space limitation, there are some other challenging problems to be handled in this paper, which are investigated as follows to inspire future work.(1)
Although this paper considered single-user multihelper model, our results are extendable to more general ones with multiuser multihelper. In this case, we can design helper selection policy to pair each user with one or multiple helpers, such that the helpers can use the proposed JTPRA scheme to help the computation of the paired user. However, how to efficiently handle the joint optimization problem of helper selection and resource allocation is a quite challenging problem worthy of further study(2)
Due to easy implementation of OFDMA, we designed the proposed protocol based on it in this paper. To further improve the system performance, we can next exploit other orthogonal multiple access schemes, e.g., NOMA schemes [22] and sparse code multiple access (SCMA) [30](3)
In terms of energy saving, we achieved the expected goal. But for MEC standardization, how to improve the propose scheme’s implementation like D2D, e.g., symbol synchronization and signaling interaction, is a difficult problem worth pursuing in the future [31]



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*Source: 5143640-2022-07-29.xml*