This paper studies a wireless powered mobile edge computing (MEC) system with fluctuating channels and dynamic task arrivals over time. We jointly optimize the transmission energy allocation at the energy transmitter (ET) for WPT and the task allocation at the user for local computing and offloading over a particular finite horizon, with the objective of minimizing the total transmission energy consumption at the ET while ensuring the user's successful task execution. First, in order to characterize the fundamental performance limit, we consider the offline optimization by assuming that the perfect knowledge of channel state information and task state information (i.e., task arrival timing and amounts) is known a-priori. In this case, we obtain the well-structured optimal solution in a closed form to the energy minimization problem via convex optimization techniques. Next, inspired by the structured offline solutions obtained above, we develop heuristic online designs for the joint energy and task allocation when the knowledge of CSI/TSI is only causally known. Finally, numerical results are provided to show that the proposed joint designs achieve significantly smaller energy consumption than benchmark schemes with only local computing or full offloading at the user, and the proposed heuristic online designs perform close to the optimal offline solutions.

Authors

• 36 publications
• 79 publications
• 43 publications
• Real-Time Resource Allocation for Wireless Powered Multiuser Mobile Edge Computing With Energy and Task Causality

This paper considers a wireless powered multiuser mobile edge computing ...
04/29/2020 ∙ by Feng Wang, et al. ∙ 0

• Optimal Resource Allocation for Wireless Powered Mobile Edge Computing with Dynamic Task Arrivals

This paper considers a wireless powered multiuser mobile edge computing ...
02/23/2019 ∙ by Feng Wang, et al. ∙ 0

12/16/2020 ∙ by Yizhen Xu, et al. ∙ 0

• Wireless Powered User Cooperative Computation in Mobile Edge Computing Systems

This paper studies a wireless powered mobile edge computing (MEC) system...
09/05/2018 ∙ by Dixiao Wu, et al. ∙ 0

• Multi-Agent Meta-Reinforcement Learning for Self-Powered and Sustainable Edge Computing Systems

The stringent requirements of mobile edge computing (MEC) applications a...
02/20/2020 ∙ by Md. Shirajum Munir, et al. ∙ 20

In this paper, we investigate the computation task with its sub-tasks su...
11/25/2020 ∙ by Xuming An, et al. ∙ 0

In this work, we consider a network of energy harvesting devices served ...
03/17/2018 ∙ by Mehdi Salehi Heydar Abad, et al. ∙ 0

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I Introduction

The integration of mobile edge computing (MEC) [3, 4, 1, 2, 6, 7, 5, 8] and wireless power transfer (WPT) [9, 10, 11, 12]

has recently emerged as a viable and promising solution to empower a large number of low-power wireless devices (such sensors and actuators) in Internet-of-things (IoT) networks, with enhanced and sustainable communication and computation. In such wireless powered MEC systems, energy transmitters (ETs) and MEC servers are deployed at the mobile network edge, either separated or co-located with access points (APs) or base stations (BSs) therein. Accordingly, wireless devices can harvest radio-frequency (RF) energy transferred from ETs over the air, and then rely on the harvested energy to execute their computation tasks via computing locally or offloading to MEC servers for remote computing. By exploiting both benefits of MEC and WPT, the wireless powered MEC is able to significantly prolong the network lifetime and even achieve sustainable network operation, with enhanced computation and communication capability at end devices. Therefore, this technique is envisioned to be of great importance to enable abundant IoT and artificial intelligence (AI) applications in the near future.

The wireless powered MEC systems face various new technical challenges due to the coupling of the wireless energy supply and the communication and computation demand at users. This thus calls for a new design framework to jointly optimize the WPT at ETs and the task execution via local computing and offloading at users, for maximizing the system performance. In the literature, the authors in [13]

first considered a single-user wireless powered MEC system with co-located ET and MEC server, with the objective of maximizing the probability of successfully computing given tasks at the user. Furthermore, the authors in

Despite such research progress, these prior works [13, 14, 15, 16, 17] focused on one-shot optimization under static wireless channels and given computation tasks at users, in which the time-dynamics in both WPT and task arrivals are overlooked. In practical wireless powered MEC systems, nonetheless, both wireless energy and computation task arrivals at users may fluctuate significantly over time, due to the randomness in wireless channels and the bursty nature of computation traffics, respectively. Therefore, both energy and task causality constraints are imposed at users, i.e., the energy (or task) amount cumulatively consumed (or executed) at any time instant cannot exceed that cumulatively harvested (or arrived) at that time. Under these new constraints, how to adaptively manage the ET’s wireless energy supply over time-varying channels to support users’ dynamic computation demands with random task arrivals is a fundamental but challenging problem that remains not well addressed yet. This thus motivates the current work.

• First, in order to characterize the fundamental performance limit, we consider the offline optimization by assuming that the perfect knowledge of channel state information (CSI) and task state information (TSI) (i.e., task arrival timing and amounts) is known a-priori. In this case, the energy minimization problem corresponds to a convex optimization problem. Then, we handle this problem by first considering the special scenario with static channels. In this scenario, we obtain a well-structured optimal solution by leveraging the Karush-Kuhn-Tucker (KKT) optimality conditions. The optimal solution shows that the ET should allocate the transmission energy uniformly over time, and the user should employ staircase task allocation for both local computing and offloading, with the number of executed task input-bits monotonically increasing over time.

• Next, we consider the general scenario with time-varying channels, in which the energy minimization problem becomes more challenging to solve. In this scenario, we show that this problem can be decomposed into two subproblems for the ET’s energy allocation and the user’s task allocation, respectively. Accordingly, we obtain the well-structured optimal solution via convex optimization techniques. It is shown that the ET should transmit energy sporadically at slots with causally dominating channel power gains, and the user should apply the staircase task allocation for local computing and the staircase water-filling for offloading with monotonically increasing computation levels over time.

• In addition, we also consider the online optimization when the knowledge of CSI and TSI is causally known, i.e., at each time slot, only the past and present CSI/TSI is available but the future CSI/TSI is unknown. Inspired by the structured optimal offline solutions obtained above, we develop heuristic online designs for the joint energy allocation (for WPT) at the ET and task allocation (for local computing and offloading) at the user, under both scenarios with static and time-varying channels.

• Finally, we provide numerical results to validate the performance of our proposed designs. It is shown that in both static and time-varying channel scenarios, the optimal offline solutions achieve significantly smaller energy consumption than benchmark schemes with only local computing or full offloading at the user, while the proposed heuristic online designs perform close to the offline solutions and considerably outperform the conventional myopic designs.

It is worth noting that the proposed joint energy and task allocation designs in wireless powered MEC systems are different from the task allocation in energy harvesting powered MEC systems [18, 19, 20] and the energy allocation in energy harvesting[21] or wireless powered communication systems [22]. First, unlike [18, 19, 20]

considering random and uncontrollable energy arrivals from ambient renewable sources (e.g., solar and wind energy), this paper considers the fully controllable energy supply from WPT at the ET, in which the energy allocation for WPT is an additional design degree of freedom for optimizing the system computation performance. Next, in contrast to

[21] and [22] with only communication energy consumption considered, this paper focuses on both communication (for offloading) and (local) computation energy consumptions at the user, thus making the demand side management (with task allocation) more challenging. Furthermore, it is also worth noticing that our prior work[24] addressed the system energy minimization problem in multiuser wireless powered MEC systems with co-located ET and MEC server at the AP subject to energy and task causality constraints at each user, in which joint energy and task allocation is optimized offline via standard convex optimization techniques. By contrast, in this paper we consider a different setup with the ET and MEC server separately located, under which the optimal offline solutions are obtained in well-structured forms to gain more design insights (instead of only numerical algorithms in [24]) in both static and time-varying channel scenarios, and new heuristic online designs are also proposed to facilitate practical implementation.

The remainder of the paper is organized as follows. Section II introduces the single-user wireless powered MEC system model and formulates the joint energy and task allocation problem of interest. Sections III and IV present the optimal offline solutions to the joint energy and task allocation problem in the scenarios with static and time-varying channels, respectively. Building upon the optimal offline designs, Section V presents heuristic online designs for the joint energy and task allocation. Section VI provides numerical results to demonstrate the effectiveness of the proposed designs, followed by the conclusion in Section VII.

Notation: For an arbitrary-size matrix , denotes the conjugate transpose. denotes the space of matrices with complex entries.

denotes the Euclidean norm of a complex vector

, denotes the absolute value of a complex scalar , and denotes the cardinality of a set . and

denote an identity matrix and an all-zeros vector/matrix, respectively, with appropriate dimensions;

denotes the distribution of a circular symmetric complex Gaussian (CSCG) random variable

with mean

and variance

, denotes the distribution of a uniform random variable within an interval , and stands for “distributed as”; denotes the statistical expectation. Furthermore, we define .

Ii System Model and Problem Formulation

 i∑j=1(ℓj+dj)≤i∑j=1Aj,  ∀i∈N. (1)

In addition, since the user needs to successfully accomplish the task execution before the end of the last slot , we have the task completion constraint as

 N∑j=1(ℓj+dj)=N∑j=1Aj. (2)

First, we consider the user’s local computing for executing the task input-bits at each slot . Let denote the number of central processing unit (CPU) cycles required for executing one task input-bit at the user, which generally depends on the types of applications and the user’s CPU architecture[25]. Accordingly, a total of CPU cycles are required for the user’s local computing. By applying the dynamic voltage and frequency scaling (DVFS) technique, in order to maximize the energy efficiency for local computing, the user should adopt a constant CPU frequency at each slot [14]. In this case, the user’s energy consumption for local computing at slot is expressed as [25]

 Eloc(ℓi)=Cℓiζ(Cℓiτ)2=ζC3ℓ3iτ2, (3)

where denotes the effective switched capacitance coefficient depending on the user’s CPU chip architecture.

Next, we consider the user’s computation offloading of the task input-bits at each slot . Let , , and denote the user’s transmission power, the channel power gain, and the system bandwidth for task offloading from the user to the AP, respectively. The transmission rate for offloading (in bits-per-second) from the user to the AP at slot is expressed as

 ri=Blog2(1+giqiΓσ2), (4)

where denotes the signal-to-noise ratio (SNR) gap due to the practical adaptive modulation and coding (AMC) scheme employed at the user[27] and denotes the power of the additive white Gaussian noise (AWGN) at the AP receiver. For notational convenience, we define in the sequel. In this case, we have in order for the user to offload the task input-bits to the AP. As a result, the user’s transmission energy consumption for task offloading at each slot is given by

 Eoffli(di)=τqi=τσ2gi(2diτB−1). (5)

Notice that both in (3) and in (5) are convex functions with respect to and , respectively.

Ii-B Energy Beamforming for WPT at ET

In this subsection, we consider the energy beamforming for WPT at the ET to wirelessly charge the user node. At each slot , let denote the energy-bearing signal at the ET, where is assumed without loss of generality. Also, let (with ) and denote the energy beamforming vector and transmission power at the ET, respectively. Then, the transmitted energy signal of the ET is . Let denote the channel vector from the ET to the user for downlink WPT. The harvested energy by the user in this slot is then given by , where denotes the RF-to-DC (direct current) energy conversion efficiency. Note that we consider a constant RF-to-DC energy conversion efficiency, by assuming that the ET can properly adjust its transmission power level such that the user’s received RF power is always within the linear regime for RF-to-DC conversion at the rectifier[9, 10]. More specifically, we assume that the ET employs the maximum ratio transmission (MRT) energy beamforming to maximize the transferred energy towards the user[11] by setting , . As a result, the energy harvested by the user at slot is given by

 EEHi(pi)=τηhipi, (6)

where denotes the channel power gain for WPT from the ET to the user at slot .

Note that the user’s local computing and task offloading are both powered by the wireless energy transferred from the ET, thereby achieving sustainable computation and communication. In practice, the harvested energy at each slot can only be utilized at the present and subsequent time slots. In this case, the user is subject to the so-called energy causality constraints[21, 22], i.e., at each slot , the cumulatively consumed energy amount (for local computing and task offloading) at the user (i.e., ) cannot exceed that cumulatively harvested from the ET at that slot (i.e., ). As a result, we have

 i∑j=1(Eloc(ℓj)+Eofflj(dj))≤i∑j=1EEHj(pj),  ∀i∈N. (7)

Ii-C Problem Formulation

 (P1): min{pi≥0,ℓi≥0,di≥0} N∑i=1τpi (8a) s.t.  i∑j=1(Eloc(ℓj)+Eofflj(dj))≤i∑j=1τηhjpj,  ∀i∈N (8b) i∑j=1(ℓj+dj)≤i∑j=1Aj,  ∀i∈N∖{N} (8c) N∑j=1(ℓj+dj)=N∑j=1Aj. (8d)

Notice that the solution of problem () critically depends on the availability of the knowledge of CSI (i.e., and ) and TSI (i.e., ). In this paper, we first focus on the offline optimization with non-causal CSI and TSI, i.e., the CSI of and and the TSI of are perfectly known a-priori. The offline optimization serves as the fundamental performance upper bound (i.e., the ET’s transmission energy consumption lower bound) for all the designs under imperfect and/or causally known CSI/TSI, which thus helps draw essential insights to motivate practical designs. In this case, since the computation energy consumption functions and are convex functions with respect to and , respectively, problem () is a convex optimization problem that can be efficiently solved by standard convex optimization techniques[26]. In Sections III and IV, we will obtain well-structured optimal solutions to problems (1) in the scenarios with static and time-varying channels, respectively. Next, inspired by the optimal offline solutions, in Section V we will consider the online optimization of problem () with causal CSI/TSI available, i.e., at each slot , only the CSI of and and the TSI of for the previous and present slots are perfectly known, but , , and for future slots are unknown.

Iii Optimal Energy and Task Allocation Under Static Channels

In this section, we consider the offline optimization of problem (1) under the special scenario with static channels, where and , . In this scenario, we define for notational convenience. Accordingly, the energy minimization problem () is reduced as

 (P2): min{pi≥0,ℓi≥0,di≥0}N∑i=1τpi (9a) s.t.  i∑j=1(Eloc(ℓj)+Eoffl(dj))≤i∑j=1τηhpj,  ∀i∈N (9b) (???) and (???).

In the following, we obtain the well-structured optimal solution to problem (). To start with, we notice that at the optimality of problem (), the -th constraint in (9b) must be tight. In other words, we have

 N∑i=11ηh(Eloc(ℓi)+Eoffl(di))=N∑i=1τpi, (10)

since otherwise one can always achieve a smaller objective value of problem () by decreasing the energy amount allocated at slot without violating constraints (8c) and (8d). Substituting (10) into the objective function, it is evident that the optimal task allocation solution of and to problem () can be obtained by equivalently solving the following total computation energy consumption minimization problem:

 (P2.1): min{ℓi≥0,di≥0}N∑i=11ηh(Eloc(ℓi)+Eoffl(di)) s.t.  (???) and (???).

Let denote the optimal solution to problem (2), where the task allocation corresponds to the optimal solution to problem (). Furthermore, under and , any energy allocation that satisfies the constraints in (9b) and (10) is actually the optimal solution of to problem (). Therefore, in the following we first obtain and by solving problem () and then find based on (9b) and (10).

Iii-a Obtaining Optimal {ℓ∗∗i} and {d∗∗i} by Solving Problem (P2.1)

Note that problem () is a convex optimization problem that satisfies the Slater’s condition. Therefore, strong duality holds between problem () and its Lagrange dual problem [26]. Let , , and , denote the Lagrange multipliers associated with the constraints in (8c) and (8d) in problem (), and and , , denote the Lagrange multipliers associated with and , respectively. The following KKT conditions are sufficient and necessary for and to be the primal and dual optimal solutions to problem ()[26]:

 ℓ∗∗i≥0, d∗∗i≥0, ¯θ∗∗i≥0, θ–∗∗i≥0, ∀i∈N, μ∗∗j≥0, ∀j∈N∖{N} (11a) i∑j=1(ℓ∗∗j+d∗∗j)−i∑j=1Aj≤0, ∀i∈N∖{N}, N∑j=1(ℓ∗∗j+d∗∗j)−N∑j=1Aj=0 (11b) ¯θ∗∗iℓ∗∗i=0, θ–∗∗id∗∗i=0, μ∗∗i[i∑j=1(ℓ∗∗j+d∗∗j)−i∑j=1Aj]=0,  ∀i∈N (11c) 3ζC3(ℓ∗∗i)2ηhτ2+N∑j=iμ∗∗j−¯θ∗∗i=0,  ∀i∈N (11d) σ2ln2Bηhg2d∗∗iτB+N∑j=iμ∗∗j−θ–∗∗i=0,  ∀i∈N, (11e)

where (11a-b) denote the primal and dual feasible conditions, (11c) denotes the complementary slackness conditions, and (11d) and (11e) mean that the gradients of the Lagrangian with respect to and vanish at and , , respectively. Based on the KKT conditions in (11) together with some algebraic manipulations, one can obtain the optimal solution in a closed form to problem () in the following theorem.

Theorem 1

For problem (), the optimal number of task input-bits for local computing and for offloading are expressed as

 ℓ∗∗i=τ√ηh[νi]+3ζC3,  ∀i∈N, (12a) d∗∗i=τBlog2(max[νi/(σ2ln2Bηhg), 1]),  ∀i∈N, (12b)

respectively, where , .

Proof:

Based on (11a–d), the optimal number of task input-bits for local computing is obtained as in (12a), where , . Similarly, based on (11a–c) and (11e), we obtain the optimal number of task input-bits for offloading as in (12b) for any .

For ease of description, we refer to in (12) as the computation level at slot . Based on the KKT optimality conditions in (11), it is verified that the computation level is always nonnegative, i.e., , . In addition, since and , , the computation level increases monotonically over slots, i.e., . Furthermore, we define and refer to slot as a transition slot if the computation level increases strictly after this slot, i.e., . It is clear that the last slot is always a transition slot. Let collect all the transition slots within the horizon such that for and . Based on Theorem 1 and the monotonically increasing nature of computation levels over time, we establish the following proposition.

Proposition 1

The optimal task allocation of and for problem () satisfies the so-called staircase property below.

• The number of task input-bits for local computing and for offloading both increase monotonically over slots, i.e., and .

• If slot is a transition slot, then it holds that , i.e., the task buffer at the user is completely cleared after this slot.

Proof:

Based on (12), since , it yields that and . Therefore, the first property of Proposition 1 is proved.

To prove the second property of Proposition 1, we consider the cases of and , respectively. First, since all the cumulative tasks should be successfully computed before the end of the horizon, it must hold that . Next, we consider one particular slot that is a transition slot, i.e., the computation level increases strictly after slot  with . Given and , we have . Based on the complementary slackness conditions in (11c), it follows that . The second property of Proposition 1 is thus verified.

Notice that the staircase task allocation in Proposition 1 is reminiscent of the staircase energy allocation for energy harvesting powered wireless communications[21]. Motivated by [21], we employ a forward-search procedure to find the optimal transition slot set, denoted by , and then obtain the optimal task allocation for problem () (or equivalently ()), as presented as Algorithm 1 in Table I and explained in detail as follows.

Algorithm 1 is implemented by induction, in which we start by searching the first optimal transition slot , followed by , , , until the last optimal transition slot . In particular, the search of the -th optimal transition slot is stated as follows. We define for convenience. First, let denote the set of candidate transition slots. Then, for each candidate transition slot , we compute as the unchanged number of task input-bits executed per-slot over slots . Next, we choose as the -th optimal transition slot, since this slot admits the smallest unchanged number of task input-bits per slot among all candidate slots in set . Given the optimal transition slot obtained, we have

 ℓ∗∗i+d∗∗i=1π∗∗k−π∗∗k−1π∗∗k∑j=π∗∗k−1+1Aj,  ∀i∈{π∗∗k−1+1,…,π∗∗k}, (13)

where and are given in (12a) and (12b), respectively. Accordingly, we can find via a bisection search based on (13) and consequently find and . Therefore, by performing the above procedures iteratively, the optimal task allocation of and is finally obtained. Note that the task allocation obtained in Algorithm 1 always satisfies the staircase property in Proposition 1 and equivalently the KKT conditions in (11). Therefore, Algorithm 1 is ensured to achieve the optimal solution to problem () and thus problem ().

Iii-B Obtaining Optimal Energy Allocation {p∗∗i} to Problem (P2)

Now, under the optimal task allocation obtained by Algorithm 1, it remains to find the optimal energy allocation to problem () based on (9b) and (10). Notice that based on Proposition 1, the allocated number of task input-bits and thus energy consumption at the user (for local computing and offloading) both monotonically increase over time. As a result, based on (9b) and (10), one optimal energy allocation solution to problem () is to uniformly allocate energy for WPT over time by setting

 p∗∗i=1τηhNN∑j=1(Eloc(ℓ∗∗j)+Eoffl(d∗∗j)),  ∀i∈N. (14)

By combining Algorithm 1 and (14), we finally obtain the optimal offline solution to problem ().

Example 1

For illustration, Fig. 2 shows the optimal offline solution to problem () with dynamic task arrivals , where the number of slots is set to be and other system parameters are set same as those in Section VI. As shown in Fig. 2(a), there are in total transition slots (i.e., , , and ), and the user’s task buffer becomes empty after each of these transition slots. It is also observed that both and increase monotonically over time, and they remain unchanged within the corresponding transition slot intervals (i.e., , , and ). The observations in Fig. 2(a) vividly corroborate the staircase task allocation structure (for local computing and offloading) as stated in Proposition 1. As shown in Fig. 2(b), the proposed uniform energy allocation for WPT at the ET is easy to implement in practice for meeting the user’s monotonically increasing computation energy demands over time.

Iv Optimal Energy and Task Allocation Under Time-Varying Channels

In this section, we present the optimal solution to problem () under the general scenario with time-varying channels, where the channel power gains for WPT and for offloading may change over slots. Let denote the optimal solution to problem ().

Iv-a Decomposition of Problem (P1)

In this subsection, we decouple problem () into two subproblems for optimizing energy allocation and task allocation , respectively. To this end, we first define the set of causality dominating slots (CDSs) for WPT from the ET to the user as [22]

 NCDS ≜{1}∪{i∈{2,…,N}∣∣hi>hj,∀1≤j

where . It is clear that in set , the channel power gain for WPT is strictly increasing over the CDS index , i.e., . Then, we have the following theorem.

Theorem 2

Under any given task allocation of and at the user, the optimal energy allocation to problem () is given by

 pi=⎧⎨⎩1τηhϕk∑ϕk+1−1j=ϕk(Eloc(ℓj)+Eofflj(dj)),if i=ϕk, k∈{1,…,|NCDS|},0,if i∈N∖NCDS, (16)

where is defined for convenience.

See Appendix -A.

Remark 1

Theorem 2 reveals the following two essential insights on the optimal energy allocation for WPT at the ET in wireless powered MEC systems over time-varying channels.

• First, in order to the meet the energy demand at the user within the horizon, the ET should transmit wireless energy to the user only at CDSs, i.e., , , and , . This is intuitively expected, since the user can always harvest a larger amount of energy when the ET allocates energy to an earlier CDS in rather than to other non-CDSs in .

• Second, the amount of energy harvested by the user at each CDS equals that consumed by the user at the CDS interval , i.e., , . This is because the channel power gains for WPT at CDSs are strictly increasing over time, and thus the ET only needs to allocate the exact amount of energy at the current CDS to meet the user’s energy demand during the corresponding CDS interval.

Based on Theorem 2 and to facilitate the description, we define as the effective channel power gain at slot for WPT from the ET to the user, where . By substituting the optimal ’s (16) back into the objective function of problem (), it yields that

 N∑i=1τpi =|NCDS|∑k=1ϕk+1−1∑i=ϕk1ηhϕk(Eloc(ℓi)+Eoffli(di))=N∑i=11ηh′i(Eloc(ℓi)+Eoffli(di)). (17)

Based on (17), we can obtain the optimal task allocation solution of and to problem () by solving the following weighted sum energy minimization problem:

 (P1.1): min{ℓi≥0,di≥0}N∑i=11ηh′i(Eloc(ℓi)+Eoffli(di)) s.t.(???) and (???).

In the following, we first derive the optimal task allocation solution of and at the user by solving problem () and then obtain the optimal energy allocation at the ET to problem () by using Theorem 2.

Iv-B Obtaining Optimal Task Allocation {ℓ∗i,d∗i} by Solving Problem (P1.1)

As problem () is a convex optimization problem that satisfies the Slater’s condition, strong duality holds between problem () and its Lagrange dual problem. Let , , , , and , , denote the Lagrange multipliers associated with the constraints in (8c) and (8d), , and , respectively. The following KKT conditions are necessary and sufficient for and to be the primal and dual optimal solutions to problem ()[26].

 ℓ∗i≥0,d∗i≥0,¯δ∗i≥0,δ–∗i≥0, ∀i∈N,λ∗j≥0, ∀j∈N∖{N} (18a) i∑j=1(ℓ∗j+d∗j)−i∑j=1Aj≤0, ∀i∈N∖{N}, N∑j=1(ℓ∗j+d∗j)−N∑j=1Aj=0 (18b) ¯δ∗iℓ∗i=0,δ–∗id∗i=0,λ∗i[i∑j=1(ℓ∗j+d∗j)−i∑j=1Aj]=0,  ∀i∈N (18c) 3ζC3(ℓ∗i)2ηh′iτ2+N∑j=iλ∗j−¯δ∗i=0,  ∀i∈N (18d) σ2ln2Bηh′igi2d∗iτB+N∑j=iλ∗j−δ–∗i=0,  ∀i∈N, (18e)

where (18a–b) denote the primal and dual feasible conditions, (18c) denotes the complementary slackness conditions, and (18d) and (18e) mean that the gradients of the associated Lagrangian with respect to and vanish at and