Performance Analysis for RF Energy Harvesting Mobile Edge Computing Networks with SIMO/MISO-NOMA Schemes

In this paper, we study an RF energy harvesting mobile edge computing network based on a SIMO/MISO system and NOMA schemes over Nakagami-m fading. Speciﬁcally, a multi-antenna user harvests RF energy from a power station by using a selection combining/maximal ratio combining scheme and oﬄoad its tasks to two MEC servers through downlink NOMA by employing transmit antenna selection/maximal ratio transmission scheme. Accordingly, we investigate the performance of six schemes, namely SC-TAS1, SC-TAS1, MRC-TAS1, MRC-TAS2, SC-MRT, and MRC-MRT, for this considered system. To evaluate the performance, exact closed-form expressions of successful computation probability are derived. We further propose the optimal algorithm to ﬁnd the best parameter sets to achieve the best performance. Moreover, the impacts of the network parameters on the system performance for these schemes are investigated. Finally, the simulation results are also provided to verify the accuracy of our analysis.


Introduction
Indeed, the next-generation wireless communication networks (NGWCN) (e.g., beyond 5G or 6G) will accommodate a massive number of user devices (e.g., Internet of Things (IoT)) and fulfill their capacity and computation demands [1]. However, it is beyond the possibility for existing wireless communication technologies with limited radio resources (e.g., spectrum and transmit power) and limited computation ability (i.e., computing and storing) to cope with such ever increasing capacity and computation demands.
Therefore, multiple candidate technologies (e.g., 3D MIMO, massive MIMO, mmWave, cognitive radio, cooperative communications, NOMA, intelligent of multiple antennas selection or combination schemes to NOMA systems have also been exploited to improve system performance [11][12][13]. The technique of transmit antenna selection (TAS) or maximal ratio transmission (MRT) can be deployed at the transmitter to assist the data transmission. The adaptive transmission mode switching between minimum mean square error beamforming and NOMA-based MRT for a system with one multi-antenna base station and two singleantenna users is investigated in [11]. In the work of [12], the system performance of a two-user cooperative multiple-input single-output (MISO) NOMA network with simultaneous wireless information and power transfer was investigated, where multiple antennas at source are exploited by utilizing TAS. The authors in [13] examined a TAS scheme and a user selection method to enhance the secrecy performance of the downlink MISO NOMA system.
Mobile edge computing (MEC) has been seen as a new evolution of cloud computing, in which the function of servers or access points moves towards the network edges to support the intensive computation needs of wireless devices in NGWCNs [14][15][16][17][18]. In this kind of computing approach, the edge servers serve as the computational access points to help accomplish the computation tasks of mobile computation-constrained devices through wireless links. There are two modes of computation offloading for mobile users: binary computation offloading and partial computation offloading. In binary computation offloading mode, the computation task that is highly integrated or relatively simple cannot be divided and has to be executed locally by itself or offloaded to the MEC servers. Meanwhile, in partial computation offloading mode, the computation task can be divided into two parts, one executed at the mobile device and the other offloaded to edge servers or access points. Some studies have investigated the combination of MEC with the NOMA technique to improve system performance in NGWCNs [19][20][21][22].
An uplink NOMA MEC system consisting of one multi-antenna base station (BS) and multiple singleantenna users is studied in [19]. In [20], the authors proposed a model that multiple users simultaneously offload their workload to one multi-antenna BS. The Lagrange method is used to solve the problem of optimizing the user's energy consumption. The offloading scheme in three different modes, namely the partial computation offloading, the complete local computation, and the complete offloading, was proposed in [21] for a NOMA MEC network, in which two users may partially offload their respective tasks to a single antenna MEC server through the uplink NOMA. The optimal solutions for an optimization problem to maximize the successful computation probability were obtained by jointly optimizing the parameters of this proposed scheme. The work of [22] studied NOMA MEC networks for both uplink and downlink transmissions in a general MEC communication scenario with multiple single-antenna users and multiple single-antenna access points. The studied results have shown that the use of NOMA can efficiently reduce the latency and energy consumption of MEC offloading compared to their conventional orthogonal multiple access (OMA) schemes.
Meantime, due to the energy-constrained wireless devices, the radio frequency energy harvesting (RF EH) technique is proposed and deployed in applications with quality-of-service requirements to prolong mobile users lifetime and maintain the coverage of wireless networks [23][24][25]. The prior research results have shown that the user computation performance can be improved by integrating RF EH and NOMA techniques into MEC networks [26][27][28][29]. Specifically, the IoT sensor nodes in [26] and the smart wearable devices in [27] can harvest RF energy and offload the heavy computation workload to the MEC server to satisfy the allowed delay under limited energy conditions. A similar model is proposed in [28], in which each user device can execute its task either locally at the mobile or by offloading to MEC using the energy harvesting from a single antenna base station. The joint computation offloading and resource allocation scheme is developed for the MEC system supporting multiple EH mobiles. A computation efficiency maximization framework was proposed for wireless-powered MEC networks based on uplink NOMA according to both partial and binary computation offloading modes [29]. The iterative algorithm and alternative optimization algorithm were proposed to solve the computation efficiency nonconvex problem. The NOMA-MEC model is particularly suitable for applications with demanding time and energy requirements such as autonomous vehicles [30], Industrial Internet [31], Wearable Virtual Reality device [32].
Besides, receiver can employ selection combining (SC) or maximal ratio combining (MRC) technique for combining signals to improve the system performance [33,34]. In work [34], the authors proposed a MEC network in which two multi-antenna computational access points support computation for a user under Nakagami-m fading. The operating protocol for the system is a combination of employ receiver antenna selection (RAS) or maximal ratio combining (MRC) at the receiver and implement selection combining (SC) or switch-and-stay combining (SSC). The paper concludes that the increased number of antennas helps improve system performance, i.e., reduce the latency and energy consumption effectively.
To the best of our knowledge, there is no prior work to study the integration of RF EH, multi-antenna technique, and downlink NOMA into MEC system. In 2 EAI Endorsed Transactions on Industrial Networks and Intelligent Systems 04 2021 -06 2021 | Volume 8 | Issue 27 | e2 this work, we consider the scenario that a single multiantenna user harvests the energy from the power station by using SC/MRC schemes and partially offloads its tasks to two MEC access points by applying TAS/MRT downlink NOMA schemes. We compared the system performance under protocols in term of successful computation probability (SCP). The main contributions of our paper are as follows.
• Six quadra-phase protocols for RF EH downlink NOMA mobile edge computing system based on SC/MRC and TAS/MRT schemes are proposed.
• Exact closed-form expressions of successful computation probability for these protocols of the considered system are derived.
• Two algorithms are proposed to find the optimal parameter set to achieve the best performance for this proposed system.
• The impact of the network parameters, e.g., transmit power, time switching ratio, power allocation ratio, and task bit allocation, on the system performance is examined by numerical results to verify the efficiency and effectiveness of deployment of RF EH, multi-antenna, and NOMA in MEC network.
The remainder 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 define the notations used in the next part of this work in Table 1. Fig. 1 depicts an RF EH NOMA MEC system in which an energy-constrained K-antenna user (S) harvests RF energy from a single antenna power station (P) to offload its task to two MEC servers located at single antenna access points (APs). Specifically, S has tasks with L bits/task to be executed. Due to the constraint of latency requirement and its limited computation ability, it may not execute its tasks locally. Therefore, it offloads its tasks to MEC servers, which have more strong computation ability. However, due to the limited battery problem, S has to harvest RF energy by exploiting SC/MRC schemes and uses it to offload its tasks. After RF energy harvesting, it transmits its task data using all harvested energy based on TAS/MRT and downlink NOMA schemes. We assume that the task follows datapartition model and can be arbitrarily divided into different subtasks [35][36][37] to apply NOMA. For instance, L 1 = L ( 0 ≤ ≤ 1) bits are offloaded to AP 1 , and Meaning m 0 ,m 1 ,m 2 Fading severity factor of Nakagami-m g 0 ,g 1 ,g 2 Channel power gain of the wireless links of P-S, S-AP 1 , S-AP 2 α Time switching ratio a Power allocation coefficient K Number of antennas of S P 0 Transmit power of power station P γ 0 Average transmit SNR η Energy conversion efficiency f i CPU-cycle frequency of AP i Task bit allocation coefficient c i The number of required CPU cycles for each bit of AP i B Channel bandwidth T The threshold of latency L The length of task P s Successful computation probability δ The algorithm accuracy factor the remained L 2 = (1 − )L bits are offloaded to AP 2 . Finally, AP s return the computing results to the user by the uplink NOMA scheme. The time flowchart of this considered RF EH NOMA MEC system is shown as Fig.  2. The entire protocol can divide into four phases as Algorithm 1. Specifically, 3 EAI Endorsed Transactions on Industrial Networks and Intelligent Systems 04 2021 -06 2021 | Volume 8 | Issue 27 | e2 • In the first phase (energy harvesting phase), S harvests energy from P during the time of τ 0 = αT , where α denotes the time switching ratio, i.e., 0 < α < 1, and T stands for transmission block time.
• In the second phase (offloading phase), S offloads its tasks to APs in duration τ 1 .
• In the third phase (computing phase), after successful transmission, the offloaded tasks are computed at the corresponding MEC APs during duration τ 2 .
• In the last phase (result downloading phase): After successful computation, the MEC APs feedback the computed results to S within τ 3 . Notice that due to the computed results is small data, we assume that τ 3 is very small compared to τ 0 , τ 1 as well as τ 2 and thus is neglected [21,29].
Assuming that all the channels have block Nakagamim fading, i.e., the channel power gain is constant over each block but vary independently between between different block and follows Nakagami-m distribution with parameter m. We also assume that P, S, APs operate in half-duplex mode and all harvested energy is used for offloading transmission.
Given the energy harvesting phase, the harvested energy of S during the duration of αT is given by where 0 < η ≤ 1 stands for the energy conversion efficiency of the energy receiver [24], P 0 denotes the transmit power of power station, g 0 is the channel power gain of link P -S. For the SC technique, an antenna at S is selected for RF EH to maximize the channel gain of the link P -S. Thus, the channel power gain g 0 is written as For the MRC technique, all of K antennas of S are used for RF EH to maximize the power harvesting. Therefore, the channel power gain g 0 , in this case, is calculated as Without loss of generality, we assume that the transmit power allocated to AP 1 is greater than that allocated to AP 2 , thus a is selected to satisfy the condition: 0.5 < a ≤ 1 to apply the NOMA scheme. S broadcasts the signal to APs by exploiting TAS/MRT schemes with transmit power P T calculated as follows During the offloading phase, S uses harvested energy to transmit a superimposed message signal to APs, where s 1 and s 2 are the messages for AP 1 and AP 2 , respectively; a stands for the power allocation coefficient.
Thus, the received signals at AP i corresponding to the k th antenna of S is written as where n ik is the additive white Gaussian noise (AWGN) with mean 0 and variance For applying the TAS scheme, an antenna at S, denoted as k * , is selected for transmission to maximize the channel gain of the link S -AP 1 , namely TAS1 case: Thus, in this TAS1 case, the channel power gains for the selected transmit antenna (k * ) of links S -AP 1 and S -AP 2 are given by Similarly, an antenna at S, denoted as k * * , can be selected to maximize the channel gain of the link S -AP 2 , namely TAS2 case, in which k * * is obtained by In this TAS2 case, the channel power gains corresponding to the selected transmit antenna of links S -AP 1 and S -AP 2 are written as For applying the MRT scheme, all of K antennas of S are used for communication. Therefore, the channel power gains, in this case, are expressed as The instantaneous signal-to-interference-noise ratio (SINR) at AP 1 to detect s 1 is given by where γ 0 ∆ = P0 σ 2 is the average transmit signal-to-noise ratio (SNR).
By applying successive interference cancellation (SIC) technique, AP 2 detects message s 1 and subtracts this component from the received signal to obtain its message s 2 . Therefore, the instantaneous SINR at AP 2 to detect 4 EAI Endorsed Transactions on Industrial Networks and Intelligent Systems 04 2021 -06 2021 | Volume 8 | Issue 27 | e2 s 1 is obtained by and the instantaneous SNR at AP 2 to detect s 2 is written as The instantaneous channel capacity of link S -AP 1 and link S -AP 2 are respectively obtained by where B is the channel bandwidth.
The transmission latencies of offloading to AP 1 and AP 2 are respectively expressed as where denotes the task bit allocation coefficient. The execution time τ of this system is calculated as follows where c i denotes the number of required CPU cycles for each bit of AP i , and f i stands for the CPU-cycle frequency at the AP i , i ∈ {1, 2}. Phase-1: S harvests energy from P by exploiting SC/MRC scheme.

4:
Phase-2: S broadcasts its tasks to APs by applying TAS1/TAS2/MRT and NOMA schemes. AP 2 uses SIC technique to detect s 2 .

5:
Phase-3: S waits for APs execute the received tasks.

6:
Phase-4: S downloads the computed results from APs. 7: end procedure Notice that the wireless links of P -S, S -AP 1 and S -AP 2 undergo the Nakagami-m fading which is a generalized fading model for practical communication scenarios. Therefore, the cumulative distribution function (CDF) and probability density function (PDF) of channel power gains, i.e., |h ik | 2 , (i ∈ {0, 1, 2} and 1 ≤ k ≤ K) are respectively given by (21) where λ i = E(|h ik | 2 ), m i ≥ 1/2 is the fading severity factor, in which m i = 1 corresponds to Rayleigh fading and m i = (V + 1) 2 /(2V + 1) approximates Rician fading with parameter V , E(.) stands for expectation operator.
For SC scheme, the CDF and PDF of g 0 can be respectively written as Similarly, for TAS1 case, the CDF and PDF of g 1 can be respectively written as lδ l . It follows that the CDF and PDF of g 2 can be respectively given by Similarly, for TAS2 case, the CDF and PDF of g 1 can be respectively expressed as It follows that the CDF and PDF of g 2 in this case can be respectively written as For MRT/MRC scheme, the CDF and PDF of g i can be respectively expressed as

Performance Analysis and Optimization
In this section, we present the performance analysis of this proposed system in terms of the successful computation probability (P s) and the optimization of the parameter set to achieve the optimal performance.

Performance analysis
In order to characterize the performance of a MEC system, P s is defined as the probability that all tasks are successfully executed within a given time T th > 0 [21]. Therefore, P s of this proposed system is written as where τ is calculated as (19). In order to evaluate the performance of six schemes, i.e., SC-TAS1, SC-TAS2, MRC-TAS1, MRC-TAS2, SC-MRT, MRC-MRT, for this considered RF EH NOMA MEC system, we obtain the following theorem. , λ1 + m2Ψ2 λ2 , µ 6 = m0 λ0 , and K v (.) is the modified Bessel function of the second kind and v th order.
Proof. See in Appendix A.
is constraint of three key parameters: data offloading ratio ( ), time switching ratio (α), and power allocation ratio (a). This constraint can be rewritten as follows:

Optimization: Problem formulation and solution
According to the above analysis, we formulate the optimization problem to maximize P s as follows: Note: SCPMP stands for successful computation probability maximization problem. Constraint (36b) describes the data offloading ratio for APs. Constraint (36c) describes the time switching ratio for energy harvesting. Constraint (36d) means that the power allocation ratio is chosen to apply NOMA and maximize P s. To solve (36), we propose two optimal algorithms: Algorithm 2 -SCPMS and Algorithm 3 -SCPMG, which is used to find the optimal set (P s max , * , α * , a * ) in the entire solution space. The Algorithm 2 -SCPMS is based on search method and is divided into two simple steps. We use 6 EAI three loops to cover the entire solution space ( , α, a). First, depending on which one of the six schemes the system operates, P s is calculated with the right formula. Next, for each set of parameters ( , α, a), we evaluate P s and update P s max . The algorithm terminates when ( , α, a) approach their upper limit, i.e., = 1, α = 1 and a = 1. We define δ is an accuracy factor, that is, the step where each parameter in ( , α, a) is updated in each loop. The smaller δ, the higher the accuracy. Thus, the total number of comparison operations that the SCPMS algorithm needs to perform is given by: As such, the complexity complexity of algorithm We propose the second algorithm for the SCPMP based on the Golden section search algorithm, i.e., the SCPMG. Algorithm 3 -SCPMG consists of four main steps: • Step 1: Similar to SCPMS algorithm, SCPMG chooses the P s formula according to the scheme that the system is operating.
• Step 2 (Initialization): Determine lower bound x l = ( l , α l , a l ) and upper bound x u = ( u , α u , a u ). We define δ is an accuracy factor. Normalize the variable a by using the equation Thus, the interval of each parameter ( , α,â) is ( 1 , α 1 , a 1 ) and x 2 = ( 2 , α 2 , a 2 ) such that: is the golden ratio.
• Step 3 (Evaluation): Next, we evaluate P s at x 1 and x 2 . In case P s(x 1 ) > P s(x 2 ), we update x l ← x 2 , x 2 ← x 1 , and x 1 ← x l + (x u − x l )/G. In the opposite case, we update x u ← x 1 , x 1 ← x 2 , and x 2 ← x u − (x u − x l )/G.

•
Step 4: We run step continuously in the loop until x u − x l < δ. Then, the algorithm terminates and the P s max occurs at xu+x l 2 .
So, SCPMG use three loops that cut the interval in 1 G each time they run; hence, its time complexity is O(log 3 ( 1 δ )).

Numerical Results and Discussion
In this section, we provide numerical results in terms of successful computation probability P s for SC-TAS1, SC-TAS2, MRC-TAS1, MRC-TAS2, SC-MRT, MRC-MRT schemes to reveal the impacts of key system parameters (data offloading ratio, time switching ratio, and power allocation coefficient) to the system performance. Furthermore, the simulation results are also provided to verify analytical results. The parameters used in this work are provided in  switch (Scheme) 8: case SC-TAS1: Calculate P s using SCP (a).

Impacts of average transmit SNR and the number of antennas
Figs. 3-8 depict the impacts of average transmit SNR γ 0 and the number of antennas K on P s with = 0.6, α = 0.3, and a = 0.8. Obviously, from these figures we can observe that γ 0 or/and K increase leading P s increases. It means that the computing performance of this considered system can be improved by increasing the average transmit SNR or/and the number of antennas. However, when γ 0 gets too large, P s will tend to be saturated. So in the proposed model, it is not necessary to increase the transmit power of the user too large.
As expected, in above figures, we observe that the P s of the NOMA scheme significantly outperforms that of the OMA scheme.

Impacts of the data offloading ratio
In Fig. 9, we examine P s as a function of the data offloading ratio with γ 0 = 10dB, K = 2, α = 0.3, and a = 0.8 for different schemes. From this figure, we can see that when increases from 0 to * , P s upgrades. However, if continuously increases from * to 1, P s degrades. This can be easily explained that applying NOMA AP 1 is allocated more transmit power than AP 2 , therefore when increases, P s upgrades. However, when continuously increases the load allocated for AP 1 increases. This leads the offloading and computing time of AP 1 longer and makes P s degrades, meanwhile the AP 2 's time is wasted. 8 EAI

Impacts of the time switching ratio
The impacts of the time switching ratio α on P s are depicted as Fig. 10 with γ 0 = 10dB, K = 2, = 0.6, and a = 0.8 for different schemes. From this figure, we can observe that when α increases from 0 to α * , P s upgrades, when α continuously increases from α * to 0.5, P s degrades. When α ≥ 0.5, P s is approximately 0 due to the condition (35). It can be explained that when α increases from 0 to α * , the more harvested energy the better SNR and the better P s. However, if α continuously increases the remained time for transmission and computation is less, this makes P s degrades.

Impacts of the power allocation coefficient
In Fig. 11, we investigate P s as a function of the power allocation coefficient a with γ 0 = 10dB, K = 2, = 0.6, and α = 0.3 for different schemes. From this figure, we can see that when a increases from max{0.5, ρ} to a * , P s upgrades and when a continuously increases from a * to 1, P s degrades. This can be easily explained that due to the offloading data for AP 1 is larger the AP 2 , thus when a increasing leads AP 1 allocated more transmit power than AP 2 , P s upgrades. However, when a continuously increases the power allocated for AP 2 decreases and makes P s degrades.
Remark 2. From Figs. 3-11, we can see that the MRC-MRT scheme is the best one, meantime the SC-TAS1 scheme is worst one. This is reasonable because the MRC-MRT scheme employs all antennas to harvest RF Figure 3. Impacts of the number of antennas K on P s of SC-TAS1 scheme Figure 4. Impacts of the number of antennas K on P s of SC-TAS2 scheme energy and offload tasks to APs, meanwhile in SC-TAS1 scheme only the best antenna is selected to harvest RF energy and offload task to APs, but it is simplest one.
Remark 3. From Figs. 3-11, we can observe that the analysis and simulation results are matching very well. It means that the correctness of our analysis has been verified.
4.5. Impacts of the length of task and the bandwidth Figure 12 presents the P s by the length of the task under different bandwidth. We fixed the power allocation to 0.8. We observe that P s decreases when L increases. It can be explained as the time that spent on the offloading phase increases when L increases (follow a formula (17), 9 EAI Endorsed Transactions on Industrial Networks and Intelligent Systems 04 2021 -06 2021 | Volume 8 | Issue 27 | e2 Figure 5. Impacts of the number of antennas K on P s of MRC-TAS1 scheme Figure 6. Impacts of the number of antennas K on P s of MRC-TAS2 scheme (18)), which reduces the time remaining for the data computing phase. Thus, both user and APs will likely not have enough time to handle all of their tasks, so the P s decreases. Figure 12 shows the apparent effect of bandwidth on system performance. The case where the bandwidth is sufficiently large (B = 100 MHz) offers much better performance than the lower bandwidth case (B = 50 MHz). In case the length of task is short, the effect of bandwidth is not significant, but the longer the task, the impact of bandwidth is very pronounced.

Optimization for successful computation probability
In Fig. 13, we verify the two optimization algorithms, i.e., SCPMS and SCPMG, to achieve the optimal performance in terms of P s for different schemes. We can easily see that both algorithms achieve the same optimum effect for the proposed model. In order to comparison, we also plot the P s with = 0.4, α = 0.3, and a = 0.6 in the non-optimal case. From this figure, we can see that when the optimization algorithm is deployed the values of P s is higher than non-optimal ones. In other words, the computing performance of this considered system can achieve the optimal value by using the optimal set (P s max , * , α * , a * ) for corresponding schemes. 10 EAI Endorsed Transactions on Industrial Networks and Intelligent Systems 04 2021 -06 2021 | Volume 8 | Issue 27 | e2 Figure 9. Impacts of the data offloading ratio on P s of different schemes Figure 10. Impacts of the time switching ratio α on P s of different schemes

Conclusion
In this paper, we have studied the RF energy harvesting NOMA mobile edge computing network. Six schemes, namely SC-TAS1, SC-TAS2, MRC-TAS1, MRC-TAS2, SC-MRT, and MRC-MRT are proposed for this system based on multi-antenna selection of user. We have also derived the exact closed-form expressions of successful computation probability for these corresponding schemes. Moreover, the optimization algorithm has been proposed to obtain the optimal performance. Finally, the numerical results have been provided to reveal the impacts of system parameters on performance. From these results, we observe that the performance of this considered system can be improved by increasing the transmit power and/or the number Figure 11. Impacts of the power allocation coefficient a on P s of different schemes Figure 12. Impacts of the length of task and the bandwidth of antennas and by selecting the optimal set of key parameters: data offloading ratio, time switching ratio, and power allocation coefficient.
In our future work, we will study the case of multiple input multiple output RF EH NOMA MEC system with imperfect channel state information. We also consider the system equality and scalability to expand the scope for the study. P s SC T AS1 = Pr max t 1 + , if a > ρ