I Introduction
Mobile edge computing (MEC) represents a promising technology to reduce the latency for 5G and beyond networks [1, 2]. It allows users to offload their computational intensive task to servers in close proximity, and thus, significantly enhancing their computation capacities and prolonging their lifespan. The whole process in general consists of the following three sequential phases [3]: 1) uplink data transmission; 2) task processing at the MEC server; and 3) results feedback through the downlink. The uplink and downlink transmission phases involve the allocation of wireless resources, while the task processing phase concerns the computational resources. As a result, a joint allocation of wireless and computational resources is required to deliver a decent system performance[4].
Initially, MEC was applied to singleuser systems, where the optimization variables include the transmission power, the offloading decision and ratio as well as the central processing unit (CPU) frequency for computing [5, 6]. The singleuser systems were then extended to the multiuser ones [7, 8, 9], where how to share the wireless and computational resources among the users directly affect the system performance. Recently, nonorthogonal multiple access is envisioned as a promising access technology [10], and its application to MEC has been considered, for example in [11]. Note that the above mentioned works only apply to singlecell systems. The general scenario of multicell systems is attracting great attention recently [12, 13, 14]. Compared with singlecell systems, a new problem that emerges in multicell systems is how to match the users to the access points (APs).
In this paper we consider a multiAP multiuser MEC system, where each user can access multiple APs and utilize their computation resource. This generalizes the previous works [12, 13, 14], where each user only offloads to one AP. Meanwhile, as in [14], we consider the general case with flexible bandwidth allocation across both the APs and users. The system objective is to minimize the sum energy consumption under response time constraints. The formulated problem is shown to be nonconvex. To handle it, we first investigate the complexity of optimization of a part of the system parameters. On this basis, we propose an iterative resource allocation procedure that converges to local optimum. To evaluate the proposed iterative solution, we compare it with the lower and upper bounds defined by less or more flexible multicell MEC architectures. Presented results validate the necessity of free selection of APs. Meanwhile, binary allocation, where all users select the AP with the highest share of their load provides a performance close to parallel processing. This facilitates its application to large systems, where the level of parallel processing is low.
Ii System Model
We consider a multicell MEC system that consists of users, and APs, each equipped with a MEC server. We denote the set of users by , and APs by . We consider that each user generates a computationally intensive and delay sensitive task, which is characterized by three parameters, the size of the input data, the number of CPU cycles required to perform the computation, and the completion time constraint . To save energy consumption at the user for processing the tasks, and satisfy the delay constraint, each user offloads its computing task to one or multiple APs for processing. That is, each user offloads part of its input data, i.e., to AP , satisfying and . For simplicity we assume that this parallel processing has no communication or computation overhead [15]. The objective of the considered MEC system is to minimize the energy consumption for data transmission under the delay constraint, by jointly allocating the data to be sent to the APs, the wireless and the computing resources.
Iia Wireless resource management
The overall system bandwidth is Hz, which should be appropriately shared among the users. We consider flat fading channel and orthogonal access with frequency division multiple access. Denote the corresponding channel gain for user to AP by . Then, the achievable data rate at user to AP is given by
(1) 
where is the corresponding transmission power, while denotes the allocated bandwidth, satisfying . Besides, is the noise power spectral density coefficient.
Accordingly, the transmission time and the resulting transmission energy consumption are respectively given by
(2) 
IiB Computing resource management
Let us denote the computational capacity of the MEC server for AP by . We denote the computing resource allocated to user as , satisfying . Then, the computational time of user ’s task is given by . We assume that there is a linear relation between and , i.e., , where is the coefficient [16]. Then, we have
(3) 
Iii Problem Formulation and General Results
We consider the problem of total transmission energy minimization, under the constraint on the completion time of the computational tasks. That is, for each user , the sum of the transmission and computational times should not violate the maximum delay , i.e., . We disregard the time needed for the downlink transmission of the results, since it concerns usually small amounts of data [17, 18, 19, 20]. Additionally, we do not consider the energy consumption of the computation at the MEC servers, since it is independent from the resource allocation (i.e., all computing needs to be performed at the MEC servers anyway, and consumes the same energy [19]).
The delay constraint then can be turned into the following rate requirement:
(4) 
The resource allocation concerns the allocation of bandwidth, power, the computing resource and the data for each user on each AP. The energy minimization problem can be formulated as
(5a)  
s.t.  (5b)  
(5c)  
(5d)  
(5e) 
where are the matrix of allocated powers , bandwidth , computational resource and data size , respectively. Inequality constraints (5b) reflect the minimum data rate requirement for each user on each AP. Constraints (5c) limit the bandwidth, while (5d) restrict the computing resource. Constraints (5e) limit the data size.
To solve P1, we need to jointly allocate the wireless and computing resources and data, which are coupled in a nonlinear way through the delay constraint. To progress, we first state the following theorem.
Theorem 1
Under any given bandwidth, computing resource and data size allocation , , the energy consumption is minimized when , holds and the transmission power is set as
(6) 
where is the minimum rate that still fulfills the delay requirement, i.e., .
Proof:
When , and are given, the energy consumption of the users is independent, and minimizing the total energy consumption is equivalent to minimizing that of each user on each AP. Without loss of generality, we look at , which can be reformulated as
(7) 
According to (7), it is clear that increases with . Therefore, is minimized when the minimum power is used. Meanwhile, to satisfy the delay constraint, we have , i.e., . At equality the achieved rate is , which in turn results a transmission time of . This concludes the proof.
Based on Theorem 1, Problem P1 can be simplified as
(8a)  
s.t.  (8b)  
(8c)  
(8d) 
This is the problem that we will evaluate in detail.
Theorem 2
Problem P2, that is, the optimal joint allocation of bandwidth , computing resource and data size is a nonconvex problem.
Proof:
P2 requires to jointly allocate bandwidth, computing resource and data size, and is difficult to handle. In the following, we show that P2 is actually nonconvex. Without loss of generality, we only consider . We first consider over and . Denote its Hessian matrix as , which is
(9)  
It can be verified that does not always hold. For example, by setting , we obtain . Since all leading principle minors of a convex function should be greater or equal than , this indicates P2 is nonconvex.
Iv Complexity of Subproblems and the Joint Resource Allocation Algorithm
Problem P2 considers three decision variables, and . In this section we provide a discussion about the complexity of the subproblems, where some of the variables are considered as given input parameters. This discussion helps us to find ways for decomposing the optimization problem into tractable subproblems. It also guides future system design, where the network, the computing resources and the application may be controlled by three different parties.
Iva Complexity of Subproblems
Let us first consider subproblems of P2 with a single free decision variable.
Theorem 3
Subproblems of P2, where two of the three variables of and are fixed, are convex problems.
Proof:
To prove the theorem, we need to consider the following three subcases:

[label=)]

Optimizing data size . According to , we have
Therefore, the considered problem is convex.

Optimizing bandwidth . According to , we have
Therefore, the considered problem is also convex.

Optimizing computing resource . We replace variables with to get
(10a) s.t. (10b) In (10), equality (10b) is clearly not affine, and thus, the feasible set is nonconvex. To address it, we relax the equality constraint and substitute (10b) with
(11) As a consequence of Theorem 1, the energy consumption decreases if is increased. Thus, for the optimal solution, equality is achieved in (11), which means substituting (10b) with (11) will not change the solution. Then, for inequality constraint (11), its second derivative is , and thus, it is convex. For the objective function, we have
Now let us consider the joint optimization problems, where one of the variables is fixed, while the other two are optimized jointly.
Theorem 4
Subproblems of P2, where is one of the decision variables, are nonconvex problems.
Proof:

[label=)]

Optimizing data size and transmission bandwidth under fixed processing power allocation . This has been shown in Theorem 2, namely, the proof of the nonconvexity of P2.

Optimizing data size and processing power under fixed transmission bandwidth allocation . In this case, the Hessian matrix is given by
where , and .
It can be verified that does not always hold, which indicates the problem is nonconvex. For example, by setting , we obtain .
However, the third subproblem of two decision variables, that is, the case when the size of the data blocks is fixed, is more tractable, according to the following theorem.
Theorem 5
The subproblem of P2, where is fixed and the transmission bandwidth and processing power allocation needs to be optimized, is a convex problem.
Proof:
We replace variables with to get
(12a)  
s.t.  (12b)  
(12c) 
First, equality constraint (12b) is affine. (12c) can be relaxed to (11), which is convex. Last, let us consider the objective function (12a). The Hessian matrix is given by
where , while . Besides, . After some algebraic manipulations, it can be verified that holds for all , which indicates (12a) is convex. This completes the proof.
Corollary 1
Consider the case, when each node can connect to a single AP only, and this AP is predefined (e.g., the one with best SNR). Then, the problem of transmission bandwidth and processing resource allocation is convex.
Proof:
This is a special case addressed by Theorem 5, where has only one nonzero element .
IvB Iterative Resource Allocation for MultiAP Processing
In this section, an Iterative Resource Allocation algorithm is proposed to solve the nonconvex joint resource allocation problem P2. As shown in Algorithm 1, the proposed algorithm follows two iterative steps: i) the Data Allocation Algorithm (DAA) updates to allocate the data, for given bandwidth and computing resource allocation and , and ii) the Bandwidth and Computing resource Allocation Algorithm (BCAA) updates and to allocate bandwidth and computing resource, for given . We denote by and the energy consumption of user after optimizing and , respectively, and is the stop condition. Additionally, denotes the initial value for
, and it can be obtained using a fixed allocation, e.g., equal allocation, or a random allocation, e.g., following a uniform distribution.
The Data Allocation Algorithm (DAA): The data allocation problem under given bandwidth and computing resource is shown to be convex in Theorem 3. Therefore, we can use the KarushKuhnTucker (KKT) condition to derive the optimal . The KKT condition for data is given by (13) at the top of next page.
(13) 
Note that in (13) is the Lagrange dual variable.
For given , the above equation can be used to obtain . Specifically, we have , which indicates that grows with , and thus a bisection search can be used to obtain by comparing with 0. Now the problem lies in how to obtain . When is increased, will decrease to ensure . Meanwhile, needs to hold. Consequently, can also be obtained with bisection search, by comparing with .
The resulting DAA consists of two loops: an outer loop to find the value of and an inner loop to determine the data allocation . The computational complexity is
The Bandwidth and Computing resource Allocation Algorithm (BCAA): Under given data allocation, the joint bandwidth and computing resource allocation problem is shown to be a convex one in Theorem 5, and thus, the optimal solution can be obtained using convex optimization tools, e.g., interiorpoint method. Nonetheless, considering that the bandwidth allocation is global for the entire network, while the computing resource allocation is local at each AP, we propose to apply an iterative algorithm to solve the joint bandwidth and computing resource allocation. The algorithm is presented in detail in our previous work [14], here we summarize it briefly. The proposed algorithm consists of two iterative steps: i) the Bandwidth Allocation Algorithm (BAA) updates to allocate bandwidth across and within the APs, for given (), and ii) the Computation resource Allocation Algorithm (CAA) updates to allocate the computing resource at each AP, for given bandwidth allocation .
Bandwidth Allocation Algorithm (BAA): Assume that the computing resource allocation is given. Then the KKT condition for is given by
(14)  
where is the introduced auxiliary variable, satisfying . As for in (13), the bisection method can be used to obtain and . Likewise, the resulting BAA consists of two loops: an outer loop to find the value of and an inner loop to determine the bandwidth allocation . The computational complexity is .
Computing resource Allocation Algorithm (CAA): Under given bandwidth allocation, the computing resource allocation is independent across the APs. Thus, the energy minimization for each AP is equivalent to that of the overall system. Let us consider AP , . The KKT condition for is given by
where is the introduced auxiliary variable, satisfying . As for and , the bisection method can be used to obtain and . Likewise, the resulting CAA consists of two loops: an outer loop to find the value of and an inner loop to determine the computing resource allocation . The computational complexity is .
BAA and CAA have to be repeated until convergence. Since the bandwidth and computing resource allocation problem is convex, convergence is guaranteed and the converged local optimum is also the global optimum.
Convergence and Complexity:
Theorem 6
The Iterative Resource Allocation algorithm converges in finite steps.
Proof:
In both lines 9 and 10 of Algorithm 1, the energy consumption decreases, or remains unchanged. Since there is a lower bound for the energy consumption, e.g., 0, the Iterative Resource Allocation algorithm always terminates, either by reaching the lower bound, or by achieving a decrease less than . Therefore, convergence is guaranteed.
Denote the number of iterations required for Algorithm 1 and BCAA to converge by and , respectively. The total computational complexity is .
V Numerical Results
In this section numerical results are presented to evaluate the effectiveness of the proposed energy minimization algorithm. Specifically, first we evaluate the convergence of the Iterative Resource Allocation algorithm. Then we evaluate what is the level of distributed processing under the optimized resource allocation, to see whether this possibility leads to significant performance gains.
We consider a small cell multiAP multiuser scenario, where image processing tasks are offloaded to the edge computing servers. The network parameters are set as for example in [21, 8, 22], while the task requirements follow [23, 8, 24]. More specifically, the number of APs and users are set to and . The users are placed uniformly randomly within the entire coverage region. The pathloss model follows , where is the distance in meter. The total bandwidth is 10 MHz, while the thermal noise density is 174 dBm/Hz. The data size of the computing task is 1.5 Mbits, while the delay constraint is 0.5 s. The computing coefficient is , and thus the computing need is 1.5 G CPU cycles. The computing capacity at each AP is 25 G CPU cycles/s.
Let us first investigate how the proposed iterative algorithm, described in Algorithm 1, converges. We consider three different initializations of the user data: Prop1 denotes the case when the data for each user are equally allocated among the APs; Prop2 is obtained following a uniformly random data size allocation; and finally Prop3 represents the case when of the data are allocated to the AP with the best channel condition, while the rest are equally allocated to the remaining APs. The stop condition is mJ. According to the results, the algorithm converges to the same solution, independently from the initial data allocation. This indicates that the proposed algorithm is robust to the initialization. Moreover, Prop3 requires the least number of iterations to approach the converged solution in all scenarios, which implies the proposed algorithm converges to a solution where users associate most of the data to their best APs.
We then plot the energy consumption versus the delay requirement in Fig. 2. As expected, the energy consumption decreases as the delay requirement becomes looser, for all considered scenarios. Lastly, we investigate how much the opportunity of parallel processing is utilized in the allocations. Fig. 3 considers only the users that connect to more than one AP, and shows what is the largest share of load sent to one of them, namely . The values are rather high for all cases, showing that most of the users have a preferred AP. This validates why an unbalanced initialization as Prop3 on Fig. 1 leads to fast convergence for the Iterative Resource Allocation.
Vi Conclusion
In this paper we investigated the join wireless and computing resource allocation for a multiAP multiuser MEC network. The general case with parallel processing and global bandwidth sharing was considered, and the system objective was to minimize the sum transmission energy under response time constraints. The formulated problem was shown to be nonconvex. To address it, we first investigated the complexity of optimizing a part of the system parameters, and based on the results proposed an Iterative Resource Allocation procedure with guaranteed convergence. Presented numerical results show that the proposed iterative algorithm converges rapidly to the local optimum. Moreover, by comparing the proposed iterative algorithm with the lower and upper bounds, it is clear that free selection of APs is crucial for obtaining decent system performance.
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