Intelligent Reflecting Surface assisted RF Energy Harvesting Mobile Edge Computing NOMA Networks: Performance Analysis and Optimization

In this paper, we focus on the performance analysis and optimization of an RF energy harvesting (EH) mobile edge computing (MEC) network by the assistance of the intelligent reflecting surface (IRS) and non-orthogonal multiple access (NOMA) schemes. Specifically, a pair of users harvest RF energy from a hybrid access point (HAP) and offloads their tasks to the MEC server at HAP through wireless links by employing an IRS-aided and uplink NOMA scheme. To evaluate the performance of this proposed system, the closed-form expressions of successful computation and energy transfer efficiency probabilities are derived. We further formulate a multi-objective optimization problem and propose an algorithm to find the optimal energy harvesting time switching ratio value to achieve the best performance, namely SENSGA-II. Moreover, the impacts of the network parameters are provided to draw helpful insight into the system performance. Finally, the Monte-Carlo simulation results are shown to confirm the correctness of our analysis. The results have shown that the deployment of IRS can improve the performance of this considered RF EH NOMA system by increasing the number of reflecting elements.


Introduction
In recent years, the Internet of Things (IoT) has been proliferating and applied in many different areas, e.g., industry, agriculture, transportation, military, and so on. Many latency-sensitive applications exist, such as virtual reality, autonomous driving, and interactive online gaming [1]. The mobile edge computing (MEC) approach has been proposed to support new applications and services that have significant data with latency constraints [2][3][4]. In the MEC paradigm, the servers with be placed between transmitters and receivers, e.g., on a building wall, to create an intelligent communication environment. It is a cost-effective solution to enhance the performance of B5G and 6G wireless communication networks; thus, it has attracted increasing interest as a hot research topic [20][21][22][23][24]. For instance, the authors in [24] investigate the RIS-assisted communication downlink model for MIMO networks. Specifically, the base station communicates with users via RIS panels deployed in unmanned aerial vehicles (UAVs). The system is proven to deliver outstanding performance in terms of total network throughput and worst-case throughput of the users.
To the best of our knowledge, a few works focused on the combination of IRS and/or wireless power transfer and NOMA in MEC networks [25][26][27]. In [25], the NOMA and TDMA schemes of an IRS-aided two-NOMA-user MEC system are investigated to clarify the integration of IRS and NOMA in MEC networks. The results showed that NOMA is better than TDMA for the sum delay minimization when the users' cloudcomputing time is sufficiently short. The integration of RF EH, IRS, and NOMA into MEC networks was studied in [26], where a hybrid access point provides RF energy and computing services to multiple NOMA users. Under the dynamic IRS beamforming framework for NOMA and TDMA schemes, computation rate maximization problems were formulated and solved by jointly optimizing the IRS passive BF and the resource allocation. A system model of a pair of near and far NOMA users and an IRS to assist the far user in offloading tasks is investigated in [27]. The performance analysis of this IRS-aided MEC network was studied by deriving the closed-form expression of successful computation probability.
In this work, we extend the work of [27], where the integration of RF EH, IRS, and NOMA is combined into the MEC system. Specifically, we consider the scenario where two users harvest energy from the hybrid access point (HAP) and offload their tasks to the server located at HAP with the assistance of an IRS. The main contributions of our paper are as follows: • A quadra-phase protocol for RF EH IRS-aided mobile edge computing system based on the NOMA scheme is proposed.
• The considered system's closed-form expressions of successful computation and energy transfer efficiency probabilities are derived.
• The multi-objective optimization problem with two objective functions, i.e., successful computation probability and energy transfer efficiency probability, is formulated under the constraint of the time switching ratio. Accordingly, an algorithm 2 EAI Endorsed Transactions on Industrial Networks and Intelligent Systems 06 2022 -08 2022 | Volume 9 | Issue 32 | e5 based on a genetic algorithm, namely SENSGA-II, is proposed to find the optimal front to achieve the best performance for this proposed system.
• The impact of network parameters, e.g., transmit power, time switching ratio, bandwidth, and task bit allocation, on the system performance is examined by numerical results to verify the efficiency and effectiveness of RF EH, IRS, and NOMA deployment in the MEC network.
The rest of this paper is organized as follows. Section II presents the proposed system model. The performance analysis and optimization of this considered system are provided in Section III. The numerical results and discussion are shown in Section IV. Finally, we draw a conclusion for our work in Section V.

System and Channel Model
We describe the notations used in the next part of this paper in Table 1.
The small-scale fading vector of The small-scale fading vector of The small-scale fading vector of HAP − IRS links N Number of reflecting elements of IRS P 0 Transmit power of HAP γ 0 Average transmit SNR η Energy conversion efficiency f CPU-cycle frequency of HAP ξ The number of required CPU cycles for each bit of HAP W Channel bandwidth L The length of the task We consider an RF EH IRS NOMA MEC system depicted in Fig. 1 and the time flowchart of this considered system is shown in Fig. 2.
Due to the real-time requirement and limited computation resources, a pair of energy-constrained users U i , i ∈ {1, 2}, cannot implement their tasks locally. Hence, they harvest RF energy from a hybrid access point (HAP) to offload the tasks to a MEC server located at HAP, which has outperformed the computation resource. There are blocking objects between users and HAP in many practical scenarios. The signal of users cannot reach HAP and vice versa.  In this work, we assume that U 1 is near to HAP without blocking objects. Meantime, U 2 is far from HAP and there are blocking objects between U 2 and U 1 as well as U 2 and HAP. Thus, an IRS consisting of N reflecting elements is deployed in the cell to assist U 2 's transmission. Assuming that the element spacing of the IRS is high enough so that the small-scale fading associated with two different reflecting elements is independent. Moreover, all devices are assumed to have a single antenna and operate in the half-duplex mode. For the sake of simplicity, U i is assumed to have a number of tasks with the same length of L i bits for offloading.
We The transmission protocol of this considered system, namely an EH-IRS-MEC scheme, can be divided into four phases.
-Phase 1 (energy harvesting): HAP transfers the RF energy to U i during the time of τ 1 = αT , where α denotes the time switching ratio, i.e., 0 < α < 1, and 3 EAI Endorsed Transactions on Industrial Networks and Intelligent Systems 06 2022 -08 2022 | Volume 9 | Issue 32 | e5 T stands for transmission block time. Note that, IRS assists HAP in transferring RF energy to U 2 .
The harvested energy of U 1 and U 2 during the duration of αT are respectively written as where 0 < η ≤ 1 stands for the energy conversion efficiency of the receiver [28], and P 0 denotes the transmit power of HAP. -Phase 2 (task offloading): U 1 and U 2 an apply uplink NOMA scheme to offload their tasks to MEC server with the help of IRS during the time of (τ 2 ). The received signal of offloading tasks at HAP is expressed as where P Ui represents the transmit power of the i th user, i ∈ {1, 2}; s i denotes the offloading task of the i th user, i ∈ {1, 2}; w ∼ CN (0, σ 2 ) represents the AWGN at HAP. Φ ∆ = diag β 1 e jθ1 , β 2 e jθ2 , ..., β N e jθ N , where j = √ −1, β n ∈ [0, 1] and θ n ∈ [0, 2π), n ∈ {1, ..., N }, are the amplitude-reflection and phase-shift variable of the n th element of IRS. The duration of Phase 2 is calculated as: where L i and W i stand for the task length and channel bandwidth with respect to U i , respectively. γ i denotes the signal-to-interference-plus-noise ratio (SINR) and the signal-to-noise ratio (SNR) at HAP to decode s i . -Phase 3 (task executing): In this phase, the successive interference cancellation (SIC) is applied at HAP to mitigate the inter-user interference. At this time, HAP detects the signal s i separately and decodes and computes the tasks of P Ui during the time of τ 3 . The duration of Phase 3 is given by: -Phase 4 (result downloading): At last, during the time of τ 4 , HAP returns the computing results to users by applying the downlink NOMA scheme. In this work, we assume that τ 4 is very small compared to τ 1 , τ 2 as well as τ 3 and thus is omitted (τ 4 =0) [11,15].
The transmit power of U 1 and U 2 are respectively given by where a By applying SIC, according to (3), the SINR and SNR at HAP to decode s i are respectively obtained as follows: where γ 0 ∆ = P0 σ 2 is the transmit SNR. Note that, to improve the transmit environment, the parameters of IRS are reconfigured to obtain the best channel quality, i.e., to maximize Z = |HΦh 2 | = | N n=1 β n g 1n h 2n e jθn |. It means that the phase-shift variables can be adjusted to obtain Z max = N n=1 β n |g 1n ||h 2n |. For the sake of simplicity, we assume β n = 1, ∀n. Thus, according to [Lemma.2, [23]], the probability density function (PDF) and the cumulative density function (CDF) of Z max = N n=1 |g 1n ||h 2n | are expressed as PDF: CDF: , m H and m 2 denote the Nakagami-m fading parameters; γ(., .) is the lower incomplete gamma function.

Performance Analysis and Optimization
This section describes the performance analysis of this proposed system in terms of the successful computation and energy consumption probabilities. Furthermore, we formulate the optimization problem and propose the solution to solve this.

Performance analysis
We describe the performance analysis of this proposed IRS-aided MEC system in terms of successful computation probability and energy consumption probability.
The successful computation probability (Φ s ) is an important measure that is useful for system designers to evaluate the latency performance of the MEC networks [15]. Φ s is defined as the probability that all tasks are successfully executed within a required time T th . For this proposed system, Φ s is expressed as where T = T th − ξL f . To evaluate the latency performance of this proposed RF EH IRS-assisted MEC system with NOMA scheme, the Lemma 1 is derived as follows: Lemma 1. In this considered RF EH IRS-aided MEC NOMA system, the closed-form expression of the successful computation probability Φ s is given by where Moreover, the energy transfer efficiency is also an important performance metric that is used to characterize how much energy is actually transferred to offload in RF EH MEC networks. Definition 1. The energy transfer efficiency (EFE) is defined as the ratio of instantaneous transferred energy to the corresponding instantaneous total offloading rate. For this proposed system, EFE is given by where Remark 1. In contrast to common energy efficiency defined in existing works [29,30], we define the EFE from the instantaneous perspective to show the transferred energy and instantaneous data rate simultaneously. Obviously, µ e is a random value. Thus we consider its statistical property.
Definition 2. The energy transfer efficiency probability (EEP), denoted by Ψ e , is defined as the probability that the energy transfer efficiency falls below a predetermined threshold ψ th . For this proposed system, Ψ e is expressed as To evaluate the energy performance of this proposed RF EH IRS-aided MEC system, we derive the following Lemma 2 as follows: Lemma 2. In this proposed RF EH IRS-aided MEC NOMA system, the closed-form expression of the energy transfer efficiency probability Ψ e is expressed as where γ th = 2 , Q 2 is the complexityvs-accuracy trade-off coefficient.
Proof. See in Appendix B.

Optimization: Problem formulation and solution
We are interested in the design optimization problem to jointly optimize the SCP and EEP for the RF EH IRS-aided NOMA MEC system as follows: where (P1) belongs to the class of multi-objective optimization problems, here specifically, maximizing SCP and EEP. Constraint (18b) describes the condition of the time allocation factor of the users. From the performance definitions of the proposed system, it is easy to see that SCP and EEP are two correlated objective functions; i.e., maximizing SCP could negatively result in EEP maximization; and vice versa. Therefore, the optimal outcome (P1) is not unique but a set of optimal results, representing the best correlation between SCP and EEP. For single-objective optimization, the fitness values (FV) of objective function always have complete order because when considering any two FVs; namely Ω 1 and Ω 2 , there always exist two cases are Ω 1 < Ω 2 or Ω 1 ≥ Ω 2 . However, for the problem (P1), we need to use the dominant concept to compare the FVs, as follow Definition 3. A feasible solution α 1 dominates over the solution α 2 with α 1 ̸ = α 2 , denoted by α 1 ⪯ α 2 , if and only if, In this study, we use the Pareto curve to represent the optimal group of results (P1) [31]. A solution is the Pareto optimal if any others do not outperform it in the objective space. The set of possible solutions that are not dominated is called the Pareto optimal set, and their corresponding FVs are called Pareto Front (PF). Accordingly, we transform the objective functions so that (P1) can be expressed by PF. Specifically, the compensation functions of SCP and EEP, i.e., Φ s = 1 − Φ s , Ψ e = 1 − Ψ e , will be used as the objective function, and the problem is transformed into determining the PF of the minimum functions. Thus, the proposed multiobjective optimization problem is rewritten as follows: We propose to use the Non-dominated Sorting Genetic Algorithm II (NSGA-II) to determine the PF [32] for (P2), namely SENSGA-II. The SENSGA-II is a stochastic evolutionary multi-objective algorithm constructed based on combining two algorithms: genetic algorithm (GA) [33] to select and develop optimal individuals and crowding distance algorithm to get a reasonable distribution of optimal points on the PF [34]. Basic GA genetics concepts and operations, including crossover, mutation, and natural selection, are also used. The difference between the multi-objective and single-objective GA algorithms is in the fitness assignment, and how to maintain the elite population. The detailed presentation of SENSGA-II algorithm is given in Algorithm 1, specifically as follows: • Step 1: Randomly initialize n P op individuals p i , (i = 1, ..., n P op ) for the population P, where each element contains a gene that is also a solution of (P2) and satisfies (18b), i.e., α. Thus, in the t th generation, the population is described by the formula: • Step 2: Calculate the fitness value of the objective functions that need to be optimized simultaneously, i.e., SCP and EEP, for P(t).
• Step 3: Call the Non-dominated Sorting (NdS) procedure, which uses (19) to determine the ranking of the input P(t). NdS assigns a first rank, namely F(1), to individuals that are not dominant in the population and excludes them from consideration. Then find a new set of nondominated individuals to assign a second rank and so on until all individuals p i are checked, p i ∈ P(t).
Thus, the output of NdS is a set of individuals sorted by ranking, namely F(k), where k is the index of ranking. Details of NdS steps are given in Algorithm 2.
• Step 4: Call the Crowding distance calculator (CDC) procedure with the input F(k). This step aims to find the sum of Euclidean distances from each individual to two neighboring solutions in each front based on two objective functions. Thereby, CDC estimates the density of solutions surrounding the solution under consideration, as follows: where CD pi is the crowding distance of p i , Ω max and Ω min are the maximum and minimum of the fitness values, respectively, when using the objective function Ω, Ω ∈ {Φ s , Ψ e }. Details of CDC steps are given in Algorithm 3. • Step 5: The parent is selected based on their rank and crowding distance as follows: randomly select two solutions; if they have the same nondomination rank F(k), the solution which has CD pi higher will be chosen; otherwise, the solution with the lower rank will be selected. All selected parents are added to the matching pool to prepare for the crossover process. • Step 6: Arithmetical crossovers [35] were used in this study to crossbreed the parent p i and p j . Let ζ c be the crossover percentage. Two offspring individuals are given by the formula: where δ c is an uniform arithmetical crossover factor, that is randomly chosen from 0 to 1.

•
Step 7: In this step, non-uniform mutation, as (24), is applied such that the probability approaches zero as the number of evolution rounds increases. It makes the mutation operator efficient in the early stages of the algorithm, having a more negligible effect on the population as it evolves into the late stages.
, r is randomly chosen from 0 to 1, δ b is selected following the level of dependency on the maximum number of iterations, τ is the random number that may have zero or one. Let ζ m be the crossover percentage. Let the set of offspring based on the GA process be Q(t). 6 EAI Endorsed Transactions on Industrial Networks and Intelligent Systems 06 2022 -08 2022 | Volume 9 | Issue 32 | e5 Intelligent Reflecting Surface assisted RF Energy Harvesting Mobile Edge Computing NOMA Networks • Step 8: The NdS and CDC procedures are called to evaluate the elite-preservation population, denoted O, which is the combination set of P(t − 1) and Q(t), i.e., O(t) = P(t − 1) ∪ Q(t).
• Step 9: Select n P op elite individuals based on F(k) and CD pi which hold in Step 8. If the number of individuals belong to F(1) is less than n P op , the entire F(1) will be selected into P(t + 1). The rest will be taken from the F(k), until enough n P op individuals. The set of p i belong to F(1) is stored in P F . This process continues until the end of evolution.

7:
Choose Q(t) following ranking F(k) and crowding distance CD pi of p i ∈ P(t − 1)

12:
Call: CDC with O(t) 13: Updated P(t) 14: Store P F 15: end while 16: Return P F We further investigate the algorithm complexity of SENSGA-II as follows: In Algorithm 2, we perform rankings for 2n P op instances belonging to O(t) with two nested loops. In each loop, we evaluate two objective functions Φ s and Ψ e with algorithmic complexity of m 1 Q 1 and m 1 Q 2 , respectively. Thus, the total number of operations required for this procedure is: For Algorithm 3, the sorting and comparison algorithm is applied to 2 nP op instances with two objective functions, so the total number of operations required for this procedure is as follows:

4:
Init nonD p = ∅ is the set containing all the individuals that are not dominant than p i

5:
Init D p = 0 is the number of individuals that dominated p i 6: if p i ⪯ p j then Init CD pi = 0

5:
Calculate CD pi using (22) 6: end for 7: Return CD pi Finally, the elite individual n P op natural selection operation has a complexity of: The overall complexity of the SENSGA-II algorithm is: To evaluate the convergence of SENSGA-II, we use the diversity metric [32], denoted by D, which measures the extent of spread achieved among the obtained solutions. The lower D, the better convergence of the algorithm.
where D e1 and D e2 are the Euclidean distances from the extreme solutions to the neighboring solutions, respectively, for the two objective functions, D is the average Euclidean distance of PF. 7 EAI Endorsed Transactions on Industrial Networks and Intelligent Systems 06 2022 -08 2022 | Volume 9 | Issue 32 | e5

Numerical Results and Discussion
This section analyzes the system behaviors in terms of SCP and EEP in the uplink RF EH IRS-aided NOMA MEC system. Furthermore, the simulation results are also provided to verify analytical results. Unless otherwise defined, the investigation parameters configuration is depicted in TABLE 2 [36]. The maximum number of iterations MaxIt 300 The population size n P op 100 The crossover percentage ζ c 0.9 The mutation percentage ζ m 0.002

Impacts of average transmit SNR and the number of IRS elements
The impact of average transmit SNR (γ 0 ) and the number of IRS elements (N ) on the performance of the plotted system is shown in Fig. 3. The solid lines are curves describing the SCP, while the dashed lines represent the EEP of the system. We find that the analyzes of SCP and EEP are accurate as all simulation results coincide with the corresponding analytical results derived from (16) and (17). The first observation that can be drawn is that all three curves describing the system's SCP have the same form as increasing as γ 0 increases. In other words, system performance improves when γ 0 has an excellent value. It is evident that because γ 0 increases, the user is supported more in terms of energy and improves the efficiency of the offloading process. Nevertheless, when γ 0 gets too high, i.e., γ 0 exceeds 30 dB, the SCP tends to saturate.
Meanwhile, EEP tends to be the opposite of SCP, i.e., as γ 0 increases, EEP decreases. The reason is that EEP represents energy efficiency used, smaller EEP means high energy consumption, and conversely, large EEP means good energy efficiency. It is explained as follows: as γ 0 increases, that is, the transmit power at the HAP increases, and the energy used during offloading also increases, resulting in reduced energy efficiency gradually. Therefore, the trade-off between SCP and EEP is a concern when designing the proposed system. The transmit power parameters from the HAP need to be considered to ensure optimal SCP while keeping the EEP at an appropriate value. It also highlights the necessity of deploying multi-objective optimization algorithms for the proposed RF EH IRS-assisted NOMA MEC system.
Another observation mentioned is the significant dependence of SCP and EEP on N . Furthermore, as expected, as N increases, both SCP and EEP improve. The reason arises from two aspects. First, when increasing N , the different signal paths from HAP towards U 2 are added to the system, thus improving channel rank and spatial multiplexing gain. Second, the larger N is, the more energy can be collected by U 2 in the EH phase, resulting in enough power to use during the offloading phase, contributing to the elevation of the system's SCP. Furthermore, IRS reflectors can support U 2 energy collection without any transmit RF chains, so EEP will be significantly improved.

Impacts of the time switching ratio
The effect of the time switching ratio (α) on SCP and EEP is depicted in Fig. 4. We focused on observing the lines describing the SCP with the values γ 0 being 8 EAI Endorsed Transactions on Industrial Networks and Intelligent Systems 06 2022 -08 2022 | Volume 9 | Issue 32 | e5 Figure 4. Impacts of the time switching ratio 20 dB, 30 dB, and 40 dB, respectively. The obtained results show that SCP is a unimodal function according to the parameter α, i.e., SCP has only one maximum in the range α in [0, 1]. Thus, it is necessary to use the optimal algorithm to find α * such that SCP reaches its maximum value. The dependence on α is evident because the SCP difference when α changes are substantial: when α is slight, i.e., 0 < α < 0.2 or when α approaches 1, the system's SCP is very low. It means that the system has a high probability of not completing tasks within the maximum allowable delay time. Equations (1) and (2) explain this: when α is too low, the energy collected by the user is not enough for them to perform offloading, resulting in low SCP. Meanwhile, when α is high, the time to offload and compute decreases, and the system does not have enough time to reach the offloading target, i.e., low SCP. In contrast, the system obtains high SCP with an adequately designed α coefficient. Another observation, the trend in EEP in Fig.4 shows low energy efficiency as α increases. According to (6) and (7), an increase in α leads to a rise in the user's transmit power, resulting in a decrease in EEP. Fig. 5 describes the effect of task length (L 1 ) on SCP and EEP in three different bandwidth scenarios, W = 1 MHz, W = 2MHz, and W = 3 MHz. The system's SCP decreases as the task sequence length increases. It is consistent with formula (4) because the task sequence length is proportional to the offloading time tau 2 . Furthermore, as L 1 increases, the total number of bits that the HAP needs to process also increases, resulting in an increase in the computation time of tau 3 in phase 3. In short, the larger the offload bit sequence, the more budget time is required for offloading and processing, Figure 5. Impacts of the task length and the bandwidth resulting in SCP degradation. The impact is enormous when the system's bandwidth is low. Observing the curve describing SCP when W = 1 MHz, we see its slope is steep: L increases from 2 kB to 7 kB, then SCP decreases up to 10 times. Continuing to compare with the SCP signal in the other two cases, we conclude that the larger the bandwidth, the better the SCP of the system. Furthermore, increasing bandwidth also positively affects EEP: i.e., as W increases, so does EEP, meaning increased energy efficiency. Therefore, bandwidth is a parameter that needs special attention when designing the RF EH IRS NOMA MEC system in practice.

Impacts of the length of task and the bandwidth
More interestingly, the system EEP descriptor curve does not change with the change of task sequence length, and this trend is consistent for all three cases of bandwidth change. It can be explained by formula (17) when EEP is an independent function of the variable L 1 . In terms of physical significance, we only consider EFE in terms of energy dissipated during transmission and investigate its statistical characteristics. Accordingly, EEP is expressed independently of the constraint in terms of maximum system latency, which depends on task sequence length as observed in SCP lines. Thus, in terms of the EEP perspective, the system is only required to meet the given energy threshold, regardless of the transmission rate. In simpler terms, the system can operate in a way that transmits bits at a slow data rate while holding the EFE condition, thereby allowing the desired EEP to be achieved. EAI Endorsed Transactions on Industrial Networks and Intelligent Systems 06 2022 -08 2022 | Volume 9 | Issue 32 | e5 Figure 6. The Pareto front shows the optimal results for the two goals of (P2). 0.323) in PF, to compare with the random one D(0.96, 0.4) taken in the feasible solution space. Obviously, solution A has minimum Φ s and maximum Ψ e , while solution C has maximum Φ s and minimum Ψ e . It means that there is no superior solution between these two solutions. Continuing, the solution pair B and also belongs to PF because the compensation SCP is better than C's, while the compensation EEP of C is better. Next, we compare solution D with C, and we observe that the second goal of solution C is better than D, but conversely, the first goal of solution D is better than C. Thus, if we do not use the PF, D would belong to the same ranking as solution C. However, our solution demonstrates the fact that solutions C and B are not dominant over each other, but solution D is dominated by B. Therefore, D is not optimal and is a dominant solution. The concept of the multi-objective optimal solution is confirmed as shown in the formula (19).

Optimal algorithm experiments
Another comment is that compensation SCP increases to 0.9984, then compensation EEP will decrease and reach a limit (value is 0.323) and cannot be reduced further. Meanwhile, when the compensation EEP increases to 0.8813, the compensation SCP will decrease and reach the limit value of 0.521. The results show that if the design option favors SCP optimization, then solution A should be chosen. If the option favors EEP optimality, solution C should be selected, or if the balance is between SCP and EEP, solution B should be chosen.
Finally, we investigate the convergence of SENSGA-II using the diversity metric in Fig. 7. Initially, a high value of D indicates that the solutions are not asymptotic to PF. Then the value D decreases if we continue increasing the maximum number of iterations. We see that the convergence trend slows down, showing that the solutions are distributed more and more uniformly. The subgraph depicting the diversity metric at loops 250 to 300 shows that over 50 evolutions, D only improved by approximately 0.03. In the last loops, D converged to 0.0621 and could not improve further. Although the D value has dropped to a deficient and acceptable level after the M axIt loop, the optimal value of 0 has not yet been reached. This also demonstrates in Fig. 6, when multiple spaces on the PF are not covered. It shows that the approaches to solving multi-objective optimization problems based on meta-heuristic such as SENSGA-II have high efficiency, but the obtained result is always at a near-optimal level. Furthermore, the dependence on initialization parameters based on a GA-based algorithm must be done in the next studies.
Remark 2. From Figs. 3-6, we can observe that the analysis and simulation results are matching very well. It means that the correctness of our analysis has been verified.

Conclusion
This paper has studied the system performance and optimization of IRS-assisted RF EH MEC NOMA networks under Nakagami-m fading. Specifically, we investigate the NOMA-MEC model for two edge-users who need to offload tasks to proximity HAP. An IRS element is deployed to the system to support the far-user operations. Accordingly, we propose a protocol for the system consisting of 4 phases, namely the energy harvesting phase, offloading phase, data processing phase, and result downloading phase. We evaluate system performance based on two parameters, SCP and EEP, and detect their trade-offs. Therefore, we state a multi-objective maximization problem for SCP and EEP with the constraint of time switching 10 EAI Endorsed Transactions on Industrial Networks and Intelligent Systems 06 2022 -08 2022 | Volume 9 | Issue 32 | e5 ratio. We propose a low complexity algorithm, namely SENSGA-II, to determine the PF of the optimization problem. Next, we use the Monte-Carlo simulation to confirm the correctness of the theoretical analysis. The simulation results show that the system performance is highly dependent on parameters such as transmit power, time switching coefficient, task sequence length, and bandwidth. Furthermore, the PF obtained SENSGA-II indicates that extreme solutions and other optimal solutions are better than the solution of randomly choosing the α parameter.
In our future work, we will investigate the RF EH MEC NOMA system with multiple IRS assisting two users in energy harvesting and task offloading. We will apply other multi-objective optimization algorithms such as NSGA-III or MOPSO to reference the results obtained in this study.

Appendix A. Proof of Lemma 1
This appendix provides the detailed proof for Lemma 1. According to (14), we have: is the CDF of X = |h 1 | 2 follows the Nakagami-m fading model with fading parameter m 1 . Thus, it is given by where λ 1 = E |h 1 | 2 , E [.] stands for the expectation operator. Note that it is NLoS for m 1 = 1 and is LoS for m 1 > 1.
Step (3) is derived by applying the Gaussian-Chebyshev quadrature method with Q 2 is the complexity-vsaccuracy trade-off coefficient, x q = cos 2q−1 2Q2 π , u q =