1. Introduction

The advance of 6G communication would accelerate the comprehensive digital transformation of vertical industries. In the future, the Machine-Type Communication (MTC) within the framework of the Industrial Internet of Things (IIoT) 4.0 would cover different types of equipment in manufacturing, public services, and home management, serving as a key driver of the Fourth Industrial Revolution [1]. IIoT equipment has strict requirements for high-reliability, low-latency, and secure communication [2,3]. Visible Light Communication (VLC), which possesses inherent advantages including abundant spectrum resources, strong immunity to electromagnetic interference, and high security [4–6], could effectively support IIoT equipment connections and improves network throughput.

As a potential technology, Non-Orthogonal Multiple Access (NOMA) could provide the VLC network with higher flexibility and enhance spectrum utilization. The authors of [7] presented a theoretical framework for analyzing the performance of NOMA in a downlink VLC system. Results show that, NOMA can offer a high performance gain in a multi-user VLC system. In addition, the authors of [8] proposed a NOMA-based Light Fidelity (Li-Fi) system, where VLC was used for downlink communication and infrared light was applied to uplink communication to support high-energy efficiency and high-speed IIoT equipment. Furthermore, the authors of [9] considered a downlink multi-user VLC system utilizing NOMA technology, where the NOMA technology was proposed based on individuals and groups, therefore improving user fairness of large-scale equipment accessing the NOMA VLC network. The aforementioned studies demonstrate that applying NOMA technology to the VLC network could improve network performance. However, the VLC network is highly sensitive to blockages, and link interruptions could easily lead to performance degradation. In contrast, Radio Frequency (RF) links have a wider coverage area and stronger diffraction capability, and could serve as an effective supplementary mechanism for the VLC network to improve overall system capacity [10–12].

Applying NOMA technology to VLC/RF hybrid networks could extend the coverage of VLC network while supporting multi-user communication [13,14]. In NOMA VLC network, the authors of [15] employed an RF network as a relay to assist remote users, proposing a weighted energy efficiency indicator to capture the characteristics of the power grid while improving the system’s energy efficiency. Added to that, the authors of [16] constructed a NOMA VLC/RF system based on user mobility. Under incomplete Channel State Information (CSI), the developed system proved that the average energy efficiency of hybrid NOMA VLC/RF systems is better than that of the NOMA VLC system. Furthermore, the authors of [17] exhibited a service mechanism for NOMA VLC/RF networks where strong users could forward messages to weaker ones through VLC/RF links, and designed an adaptive service-oriented user collaboration and link selection strategy to improve system throughput and user reception performance. References [15–17] confirmed that the cooperative mechanism could effectively improve the energy efficiency of hybrid NOMA VLC/RF networks. However, when supporting Ultra-Reliable Low-Latency Communications (URLLC) equipment requirements, traditional long packet transmission technology increases end-to-end latency due to the large data processing and encapsulation overhead. In contrast, Short Packet Communication (SPC) technology is a promising URLLC technology that could enhance system reliability by transmitting smaller data units.

To more efficiently support URLLC requirements, SPC has been proposed for applications within wireless communication network [18]. For instance, the authors of [19] studied the SPC of the multiple-input multiple-output-NOMA two-hop relay systems and derived a closed form expression for the asymptotic block error rate. Simulation results showed that the systems met the requirements of URLLC. The authors of [20] used the Full-Duplex (FD)-cooperative NOMA SPC systems to forward information toward remote users. Compared to long-packet communication, FD C-NOMA-SPC yielded a narrower power allocation range and could achieve highly reliable transmission. Furthermore, the authors of [21] designed a NOMA system with SPC enhancement by allocating different powers to User Equipment (UE), where all users could get reliable communication at the same time. References [18–21] demonstrated that SPC technology could effectively support URLLC services, providing theoretical support for its application in the constructed RF-aided NOMA-VLC network. In wireless communication networks, the achievable instantaneous rate varies randomly. Therefore, it is challenging to ensure delay performance for delay-sensitive equipment while enhancing Service Reliability (SR).

Motivated by the above challenges, we propose a new resource optimization scheme that jointly controls blocklength and power allocation for the NOMA VLC/RF scenario. The proposed solution could efficiently guarantee the differentiated Quality of Service (QoS) requirements of multi-UE while enhancing the Service Capacity (SC) and SR. The main contributions of this work are as follows:

• An RF-aided NOMA VLC transmission model is established. Utilizing the SPC technology, the successful transmission probability of multi-UE is calculated, and we obtain an instantaneous service expression. Furthermore, the SC is quantified based on the effective capacity theory.
• Under the constraints of statistical delay QoS, SR, and transmission power, the optimization problem of maximizing SC is devised. To solve this problem efficiently, we propose a resource optimization algorithm combining blocklength control and power allocation.

2. NOMA VLC/RF networks model

In this paper, we analyze the NOMA VLC/RF downlink networks containing K MTC equipment. The specific scenario is illustrated in Fig 1. The VLC network enables data transmission through intensity modulation of Light Emitting Diode (LED) light sources. According to the different QoS requirements of UE, the UE in VLC/RF networks is classified into two categories: the number of MTC equipment with high-reliability and common demands is X and Y, respectively. There is an RF Access Point (AP) at the same height as the UE, and the VLC AP is placed in the center of the roof. The UE equipped with Photo Detectors (PD) communicates with the VLC network preferentially. If and only if the UE fails to decode the VLC signal, the RF network could continue to communicate with the UE as a compensation scheme. RF could serve multi-UE simultaneously. It is assumed that the channel is static or quasi-static. Therefore, the UE does not change the state of the channel during each transmission, and the CSI is completely known. As the diffusion non-line-of-sight link component is basically ignored, only the line-of-sight link component is considered in this work [22]. To achieve low-latency communication, each UE employs SPC technology to transmit the signal.

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Fig 1. NOMA VLC/RF networks model.

https://doi.org/10.1371/journal.pone.0353349.g001

Each UE in the VLC/RF networks has differentiated QoS requirements. The QoS level of each category considers the highest QoS requirements of the UE within the group. From a statistical perspective, the i-th MTC equipment must satisfy the probability , where d_i and denote the steady-state and maximum tolerable delay, respectively. Moreover, θ_i is the QoS index, and μ_i represents the average arrival rate [23]. For a particular , a larger QoS index θ_i corresponds to a stricter delay QoS constraint. To more clearly explain the signal transmission process of the NOMA VLC/RF networks, the following part selects X = 1 and Y = 1 as an example for research without loss of generality. Consequently, the equipment with high-reliability requirements and the one with common requirements are represented by UE₁ and UE₂, respectively.

2.1. Signal transmission process with NOMA VLC/RF networks

In the NOMA VLC/RF networks model constructed in this paper, UE_i (i∈{1, 2}) will send signal to both the VLC AP and RF AP at the same time. It is worth noting that UE_i can only receive the VLC or RF signal, but not both signals simultaneously. When the VLC signal is successfully decoded, the VLC, as the primary communication network, provides service to UE_i through NOMA technology. Conversely, if the VLC signal is incorrectly decoded, the RF network could be regarded as a compensation solution.

2.1.1. VLC signal transmission process based on NOMA technology.

In the adopted intensity modulation and direct detection scheme, the VLC network requires that the transmitted signal is real and non-negative. To generate real-valued signal, Hermitian symmetry is typically applied. Additionally, a DC bias is introduced at the LED to ensure the non-negativity of the transmitted signal. The superimposed signal received at the VLC AP is expressed as follows:

(1)

where P_t represents the VLC transmission power, I_DC denotes the DC bias of the signal, S_j (j∈{1,2}) is the information signal sent by VLC to UE_i (i.e., S₁ → UE₁, S₂ → UE₂), and α₁ and α₂ are the power coefficients of the signal assigned to UE_i.

After the optical-to-electrical power conversion and the elimination of the DC bias at the receiver, the received signal at the UE_i is expressed as follows:

(2)

where represents the VLC network noise, which is considered to be Additive White Gaussian Noise (AWGN) with zero values for both mean and variance. It is calculated using [24]. Moreover, B_V denotes the VLC network bandwidth, and N_V0 is the noise power spectral density of the VLC network. Finally, is the channel gain between VLC AP and UE_i, formulated as follows:

(3)

where is the Lambert coefficient, and represents the half-power angle of the LED. Moreover, A_c and R₀ are the receiving area and optical-to-electrical power conversion coefficient of PD, respectively. In addition, and represent the emission angle of the LED and the incident angle of the optical detector PD at the receiving end, respectively, whereas expresses the field of view of PD. In addition, λ_i is the straight-line distance between VLC AP and PD. Finally, and represent the gain of the optical filter and optical concentrator at the receiver, respectively.

At UE₂, S₁ is considered as interference, and S₂ is decoded directly. The Signal to Interference Noise Ratio (SINR) corresponding to the UE₂ decoded signal S₂ is expressed as follows:

(4)

where represents the normalized channel gain of UE_i. At UE₁, the receiver first performs the Successive Interference Cancellation (SIC) and decodes S₂. After successful decoding, S₂ is subtracted from the total signal, then S₁ is detected and decoded. Thus, the SINR expressions corresponding to the UE₁ decoded signals S₂ and S₁ are defined as follows:

(5)(6)

2.1.2. RF signal transmission process.

The transmitted signal x_Ri of UE_i at the RF AP can be written as follows:

(7)

is the information signal sent by RF to UE_i. Moreover, P_R represents the RF transmission power.

The signal y_Ri received by UE_i satisfies the following relation:

(8)

represents AWGN where its mean and variance are set to zero and , respectively. Moreover, is the channel gain between RF and UE_i, and it is expressed as follows:

(9)

where λ_Ri represents the straight-line distance from RF to UE_i, h_r expresses the fading channel, being an exponential random variable with a mean of 2.46 dB, and L(λ_Ri) is the path propagation loss expressed as follows:

(10)

where v is the path loss factor. In addition, λ_R0 = 1/m is the reference distance. The shadow component is denoted by , which is a Gaussian random variable with a zero mean and variance equal to 1.8 dB.

The SINR expression corresponding to the UE decode RF signal is expressed as follows:

(11)

where N_R0 expresses the noise power spectral density of the RF [17]. B_R denotes the RF channel bandwidth.

2.2. VLC signal transmission process based on OMA technology

To demonstrate the advantages of NOMA technology, this paper selects OMA as the comparison scheme of NOMA to implement the solution proposed. Through OMA, it is assumed that both UEs are distinguished by the time division multiple access technique to prevent interference between UEs. At UE_i, the transmitted signal can be written as follows:

(12)

The received signal of UE_i in the OMA solution is expressed as follows:

(13)

Finally, the SINR expression of the UE_i decoded signal is denoted as follows:

(14)

3. Design of resource optimization algorithm

3.1. Service Capability evaluation of NOMA VLC/RF networks

According to reference [25], short packet transmission technology is used in NOMA VLC/RF networks. Thus, the instantaneous bit per channel use (BPCU) of UE_i can be written as follows:

(15)

where and are the selection variables of VLC network and RF network, respectively. Moreover, and denote the blocklength of VLC network and RF network, respectively, and k represents the number of transmitted bits of UE_i.

For VLC/RF networks, at a given instantaneous transmission rate and blocklength , the decoding error probability of signal S_j at UE_i can be approximated as follows:

(16)

where the Gaussian function Q is . , i, j∈{1,2}.

The function can be obtained using the following:

(17)

The NOMA technology transmits signal to two UE simultaneously with different power allocations, so NOMA can serve UE₁ and UE₂ (i.e., ) at the same time. In the NOMA VLC network, according to Eq. (16), the decoding error probability of S₂ at UE₂ is approximately:

(18)

For UE₂, only S₂ should be decoded. There is only one decoding situation at UE₂. As a result, the probability of successful access of UE₂ is expressed as follows:

(19)

For UE₁, it is necessary to decode S₂ first and then decode S₁. The decoding error probability of S₂ and S₁ at UE₁ is defined as follows:

(20)

UE₁ is successfully accessed if and only if both signals S₁ and S₂ are decoded successfully. Then, the probability of successful access of UE₁ is expressed as follows:

(21)

When the UE fails to decode the VLC signal, the RF network provides service to the UE. Currently, the decoding error probability of (i∈{1,2}) at UE_i is defined as follows:

(22)

Moreover, the probability of successful access of UE_i is expressed as follows:

(23)

In the OMA VLC network, according to Eq. (16), since both UEs are assigned to different orthogonal resources, the probability of successfully access UE_i is defined as follows:

(24)

Due to the decoding error probability, the rate matched by the network for the UE is different. The VLC signal is decoded successfully, , the UE_i will get the service rate provided by the VLC network. However, the VLC signal decoding error occurs and the RF signal decoding is successful, , the RF network allocates the service with a rate of to the UE. Finally, when both the VLC signal and the RF signal are decoded incorrectly, , the actual data rate transmitted by the network is set to zero. Considering the relationship between network rate and the probability of successfully access, the instantaneous service model in bits per channel use can be described as follows:

(25)

Based on reference [23], this paper quantifies the per channel SC provided by the NOMA-VLC/RF networks to UE_i (i∈{1,2}) as follows:

(26)

3.2. Construction and solution of the SC maximization problem

3.2.1. Optimization problem construction.

Reliability is an important indicator to measure the performance of the communication network. This paper defines SR at UE_i (i∈{1,2}) the probability that a data packet is successfully decoded. The SR at UE_i can be described as follows:

(27)

Since VLC network offers higher data rates than RF network over short distances, it present broad application prospects in wireless communication scenarios [5]. Based on this, we focus on the joint optimization problem of blocklength control and power allocation, under the premise of fixed RF network parameters. The objective is to maximize the SC of NOMA VLC/RF networks, while ensuring statistical delay QoS requirements and SR in multi-UE network. The optimization problem can be expressed as follows:

(28)

where effective bandwidth, denoted as , is defined as the minimum bandwidth resource required to support arrival services while satisfying the fixed QoS requirements. Moreover, represents the target SR in VLC/RF network, denotes a set of positive integers, and highlight the minimum and maximum blocklength of the VLC network, respectively. Moreover, C1 and C2 represent power constraints, C3 expresses the VLC blocklength constraint, C4 denotes the network statistical delay QoS constraint, and C5 is the SR constraint in VLC/RF networks.

3.2.2. Motivation for designing the solution.

The convex optimization is a typical approach to settle the optimization problem. The objective function constructed in this paper incorporates multiple nonlinear components, such as the Q-function, logarithmic function, and exponential function. As a result, obtaining its derivative is not easy, which would increase the difficulty of analyzing its concavity or convexity.

The heuristic-based approaches offer an alternative for settling optimization problems. The solution process relies on extensive stochastic search and parameter adjustment, resulting in relatively high execution cost. Consequently, we propose a resource optimization algorithm for joint blocklength control and power allocation based on grid search, thereby solving this optimization problem while reducing the complexity of model analysis.

3.2.3. Analyzing boundaries of VLC power coefficients and blocklength allocation.

In the process for settling optimization problems utilizing search mechanisms, the undefined search boundaries often lead to a large amount of computation. Therefore, we need to calculate the bounds of the power coefficients and the VLC blocklength to compress the search space, thereby improving the computational efficiency.

Due to the SIC technology characteristics, as long as the SR of UE₁ can be guaranteed, the SR of UE₂ in VLC network can also have the same guarantee. Therefore, the SR in the VLC network should be met , where and denote the SR and the target SR in the VLC network, respectively. Since the decoding error probability of VLC signal is a monotonically decreasing function of SINR [26], when defining the blocklength, the SR at UE₁ is met with the lowest standard. That is, when , the minimum resource configuration can be generated. According to Eq. (21), using the appropriate scaling, we can obtain:

(29)

Thus, the lower bound of the SINR of S₁ decoded at UE₁ can be obtained as follows:

(30)

According to Eq. (6), we can further obtain:

(31)

The rate of short packet transmissions is smaller than the Shannon rate; thus, the SINR of UE_i needs to satisfy .

According to the theory of effective bandwidth and effective capacity, the statistical delay QoS of UE_i is guaranteed if and only if the effective capacity is greater or equal to the effective bandwidth [27]. Changing C3 to can yield that when the UE is in a VLC network, the minimum rate requirements that can guarantee the statistical delay QoS is denoted as follows:

(32)

Thus, the power coefficient by the VLC network to UE₁ is transformed into:

(33)

According to Eq. (33) and constraint C2, the upper bound of the power coefficient of the VLC network to UE₁ is expressed as follows:

(34)

Similarly, when defining the power allocation scheme, the lower bound on the blocklength for VLC network can be derived by meeting the SR requirements at UE with the minimum permissible standard, expressed as follows:

(35)

3.2.4. Solving optimization problem.

Within the derived bounds of blocklength and power coefficient, a two-layer one-dimensional linear search is employed to determine the optimal set of blocklengths and power coefficients α₁* and α₂* that maximize the SC. The specific algorithm process is as follows:

Algorithm 1 Resource optimization algorithm for joint blocklength control and power allocation

Input: θ_i, EB_i (θ_i), , P_R, P_t, , k = 0.

Output: VLC network optimal blocklength , optimal power allocation coefficients α₁*, α₂* for VLC network.

1: Calculate the SINR r_R1, r_R2 of UE₁ and UE₂ under the RF network

2: Calculate the corresponding rate , and the probability of successful transmission of the RF signal

3: For n ← 1 to length(α₁) do// α₁=, from (34) and (31), get the upper and lower bounds of α₁

4:   According to α₁(n), the SINR r₁₁(n), r₁₂(n), r₂₂(n) of UE₁ and UE₂ in the VLC network are calculated

5:   For m ← 1 to length() do// =, from (35), Get lower bound

6:     Calculate the rate (n,m) and signal (n,m) in the VLC network based on (m)

7:     Calculate the SC_i(n,m) of each UE according to the θ_i, and calculate the SC_sum(n,m)

8:     If (SC_i(n,m) EB_i(θ_i)) && then

9:      O(k,:) = [SC_sum(n,m), α₁(n), (m)]; k = k + 1;

10:     End If

11:   End For

12: End For

13: OP* = arg max{O(:, 1)}

14: Return OP* = [SC_sum*, α₁*, , α₂*=1-α₁*]

The search interval covers the entire feasible solution space, and all feasible solutions lie within this interval. For each pair (α₁, ), the corresponding objective function value is determined, and the constraint satisfaction is verified. The proposed algorithm could maximize the sum SC while ensuring the requirements of statistical QoS. It merely requires a sequential traversal of all discrete values to obtain the optimal solution without requiring real-time response. Compared with heuristic-based approaches, the algorithm avoids the potential issue of getting trapped in local optima, thereby ensuring global optimality.

Next, we analyze the complexity of the proposed resource optimization algorithm. The complexity mainly stems from the maximum number of points searched in two one-dimensional search layers, which is determined by the search step sizes (for the power coefficient) and (for the blocklength). Let denote the number of search points for the power coefficient, and is the number of search points for the blocklength. Consequently, the overall algorithmic complexity is O( ×). The value of the step sizes and could affect the accuracy and computational cost of the proposed algorithm. The smaller step sizes yield a denser grid and thus more accurate solutions near the optimum. This comes at the cost of higher computational load. The parameter M defines the search range for the blocklength, and its value could be determined based on system constraints to ensure that the optimal solution lies within the interval, and a larger M increases the computational burden.

3.3. Generalization to Multi-UE Scenario

For the SPC-NOMA-VLC/RF model containing K users (K = X + Y), it is necessary to reanalyze the signal decoding error probability and reassess the SC of VLC/RF networks. The specific steps are as follows:

1. (1) QoS grouping. UE should be classified according to QoS requirements to determine the value of the QoS index within each group.
2. (2) Signal Decoding Process in the VLC/RF systems. The k-th user (UE_k) utilizes SIC to recover its own signal S_k. This process requires UE_k to sequentially decode stronger signals until all signals are decoded or the remaining signal is too weak to decode. Therefore, the condition for successfully decoding the j-th user’s message (j ≤ k) is that the corresponding SINR satisfies:

(36)

where the in the denominator represents the residual inter-user interference.

If the VLC transmission for UE_k fails, the system switches to the RF link. A notable point is that the presence of multi-UE does not affect the decoding process in the RF link.

1. (3) Calculate the probability of successfully access the VLC channel. For the k-th user, the condition for successful communication is that each of the signals S₁, S₂,..., S_k is entirely decoded. Therefore, the probability of successfully access for VLC is the product of the success probabilities of all steps.
2. (4) SC Evaluation for the Multi-UE System. The SC of the multi-UE VLC/RF networks could be evaluated directly using the framework established in Eq. (26), on the premise of reconstruction the instantaneous service rate model for multi-UE based on Eq. (25).

4. Simulation results and analysis

The performance of the proposed resource optimization algorithm is assessed through MATLAB platform, and the computer is equipped with AMD Ryzen 7 8845H processor, 32 GB RAM, and NVIDIA GeForce RTX 4060 GPU. The simulation parameters are shown in Table 1, where the specific values are set referring to [14,28–30].

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Table 1. Simulation parameter table in VLC/RF networks.

https://doi.org/10.1371/journal.pone.0353349.t001

First, to prove the superiority of the proposed scheme, SPC is applied in the NOMA VLC/RF, NOMA VLC, and OMA VLC networks, and the resource optimization schemes are designed. Fig 2 illustrates the impact of the use of various schemes on SR. The VLC transmission power P_t is set to 15 dBm, and the blocklength ranges between 40 and 200 symbols. As illustrated in Fig 2, the SR increases with the blocklength. This is because when the blocklength increases, the decoding error probability becomes smaller, thereby improving the SR. Referring to Fig 2A, it is clear that, under the same blocklength, the performance of the SPC-NOMA VLC/RF scheme is better than that of the SPC-NOMA VLC scheme. This is due to the fact that, when an error occurs in decoding the VLC signal, the designed scheme can promptly use the RF network to provide the UE with auxiliary service. This mechanism improves the SR. Moreover, Fig 2B shows that the performance of the SPC-NOMA VLC scheme is better than that of the SPC-OMA VLC. This finding is due to the NOMA scheme that shares the entire blocklength with different power coefficients, while the OMA scheme should be divided into half of the blocklength, knowing that the NOMA scheme is more reliable.

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Fig 2. The impact of blocklength on the SR under different solutions.

(A) Different scenarios. (B) Different technologies.

https://doi.org/10.1371/journal.pone.0353349.g002

In Fig 2A, “SPC-NOMA-VLC/RF” represents the resource optimization scheme based on SPC and NOMA technologies for VLC/RF networks, while “SPC-NOMA-VLC” denotes the resource optimization scheme based on the SPC and NOMA technologies for VLC network. In Fig 2B, “SPC-OMA-VLC” is the resource optimization scheme based on the SPC and OMA technologies for VLC network.

Fig 3 shows the impact of different schemes on UE service capabilities, where the parameter values are the same as those selected in Fig 2. Referring to Fig 3, when the blocklength L increases, the SC increases first and then decreases. This is because when the blocklength L gets longer, both the probability of successful communication and the SC will be bigger. However, when the blocklength L increases to a certain extent, the probability of successful communication hardly increases anymore, whereas the UE transmission rate starts to gradually decrease, and the SC begins to degrade after reaching the peak. As shown in Fig 3A, the SPC-NOMA VLC/RF scheme can achieve greater SC with a shorter blocklength compared to the scheme without RF assistance. Moreover, Fig 3B shows that the VLC scheme adopting NOMA technology can reach the SC peak more quickly with a shorter blocklength compared to the scheme using OMA technology. A short blocklength could better satisfy the communication equipment with low-latency requirements.

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Fig 3. The impact of blocklength on the SC under different solutions.

(A) Different scenarios. (B) Different technologies.

https://doi.org/10.1371/journal.pone.0353349.g003

To sum up, when the blocklength is short, the VLC/RF scheme with NOMA technology is more suitable to ensure data transmission with high-reliability requirements compared with other schemes. For a clearer demonstration of the proposed scheme superiority in maximizing the SC, the optimal values extracted from Fig 3 are summarized in the Table 2. The largest values of the table are highlighted in bold. As shown in the Table 2, the SPC-NOMA-VLC/RF scheme outperforms both SPC-NOMA-VLC and SPC-OMA-VLC in supporting high capacity transmission under short blocklength conditions.

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Table 2. Performance Comparison of Different Resource Optimization Schemes.

https://doi.org/10.1371/journal.pone.0353349.t002

To validate the superiority of the proposed resource optimization algorithm compared with other methods, we solve the optimization problem utilizing typical heuristic-based approaches including Particle Swarm Optimization (PSO) [31], Grey Wolf Optimization (GWO) [32], and Whale Optimization Algorithm (WOA) [33]. The values of the population sizes and the maximum iteration number are set under the condition of ensuring the convergence of the three algorithms. The purpose is to avoid introducing unfair computational burdens due to excessively large parameter settings. The simulation is shown in Fig 4, and the results show that both the proposed algorithm and the comparison algorithms could obtain the same optimal solution. Subsequently, we analyze the operational efficiency of the algorithm. From Fig 4B, the proposed algorithm could settle the optimization problem requiring significantly less execution time, which proves the advantages of the proposed algorithm.

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Fig 4. The impact of VLC transmission power on the sum SC and the execution time under different algorithms.

(A) Sum SC (B) Execution time.

https://doi.org/10.1371/journal.pone.0353349.g004

The data in Fig 4B are summarized in Table 3, which provides a more specific comparison of the efficiency between the proposed algorithm and heuristic-based algorithms. Table 3 presents the average execution time of each algorithm.

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Table 3. Execution time of different algorithms.

https://doi.org/10.1371/journal.pone.0353349.t003

The proposed algorithm demonstrates an average execution time of approximately 2.3 s from Table 3, which is shorter than that of the other algorithms. Therefore, the proposed algorithm significantly enhances computational efficiency.

Delay is a key metric in network performance evaluation and a fundamental QoS parameter, but it could merely reflect a certain aspect of service quality. From long-term stability perspective, the QoS index provides a more comprehensive reflection of delay performance. Therefore, we analyze the impact of the QoS index on the sum SC for the NOMA VLC/RF networks adopting SPC technology. As shown in Fig 5, when the QoS index is smaller (θ → 0), the SC is greater. This occurs because when the QoS index decreases, the QoS equipment requirements become stricter, the service required by the UE and network capacity increase. It can also be observed that, when the QoS index is unchanged, the greater the VLC transmission power, the bigger the network capacity. The reason is that when the VLC transmission power is set to a higher value, the UE will receive greater power, thereby increasing SC.

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Fig 5. The impact of VLC transmission power on the sum SC under different QoS index.

https://doi.org/10.1371/journal.pone.0353349.g005

Fig 6 presents the impact of target SR of VLC/RF networks, the RF transmission power, and the number of transmitted bits on the sum SC. The VLC transmission power is set to change from 10 dBm to 20 dBm, whereas ranges are in the interval [99.9%, 99.99%, 99.999%]. As shown in Fig 6, a higher VLC transmission power results in a greater the sum SC. Simultaneously, under the same transmission power, a higher target SR leads to a lower sum SC. This is because when the target SR increases, the decoding error probability of UE₁ has increased. The one way to enhance the SR is to increase the allocated power and blocklength. However, this reduces the service obtained by UE₂, leading to an overall degradation in network performance.

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Fig 6. The impact of VLC transmission power on the sum SC under different SR.

https://doi.org/10.1371/journal.pone.0353349.g006

Fig 7 investigates the impact of VLC transmission power on the sum SC under different RF transmission power. The RF transmission power P_R = [5 dBm, 10 dBm, 15 dBm] is set. It is clear that, when the VLC transmission power is fixed, the RF transmission power increases as well as the sum SC. This is because as the RF transmission power rises, the service provided by RF to the equipment becomes bigger. At the same time, it is clear that adjusting the RF transmission power affects slightly the sum SC. As the VLC transmission power increases, the network overall service capability varies substantially. The simulation results prove that the network performance mainly depends on the optimization of VLC parameters in the short packet transmission scenario.

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Fig 7. The impact of VLC transmission power on the sum SC under different RF transmission power.

https://doi.org/10.1371/journal.pone.0353349.g007

Fig 8 displays the impact of VLC transmission power on the blocklength, the sum SC, and decoding error probability. The number of transmission bits for the UE is set to k = 30 and k = 50, respectively. Referring to Fig 8A, it is clear that, under the same transmission bits, the transmission power and the blocklength present a negative correlation trend. As the VLC transmission power grows, the service rate rises, while the blocklength declines. Furthermore, a higher value of k results in a bigger blocklength and improves the sum SC. This is because when the number of transmission bits k increases, a larger blocklength is required to carry data packets, and more service should be supplied to process these data packets, thereby enhancing the network performance. Moreover, Fig 8B demonstrates that the decoding error probability is extremely low with the optimal solution, and the effectiveness of the scheme is validated.

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Fig 8. The impact of VLC transmission power and bits on system performance.

(A) Blocklength and Sum SC. (B) Decoding error probability.

https://doi.org/10.1371/journal.pone.0353349.g008

Furthermore, to validate the extensibility of the algorithm in the multi-UE scenario, we evaluate the sum SC in three configuration scenarios: 2 UEs (1 high-reliability, 1 common), 3 UEs (1 high-reliability, 2 common), and 4 UEs (2 high-reliability, 2 common). Fig 9 illustrates the impact of the variation of VLC transmission power on the sum SC. As the VLC transmission power increases from 10 dBm to 20 dBm, the sum SC under all scenarios rises significantly. This improvement stems from the enhanced SINR at higher power levels, which boosts the service rate and reduces decoding error probability, thereby enhancing the sum SC. Moreover, it can be observed that for a given VLC transmission power, a larger number of UE leads to a higher the sum SC. Although incremental UE reduces the power allocated per UE, the system could serve more data packets simultaneously while still satisfying QoS requirements.

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Fig 9. The impact of VLC transmission power on the sum SC under different user number.

https://doi.org/10.1371/journal.pone.0353349.g009

5. Model limitations and future extensions

We exhibit a resource optimization scheme joint blocklength control and power allocation, assuming the perfect SIC and CSI. Simulation results demonstrate that the proposed optimization scheme achieves greater SC with shorter blocklength while ensuring SR. It is worth noting that the proposed scheme holds applicability and could be further extended to analyze systems performance under the uncertain channel conditions. Specifically, the system model could be reconstructed by introducing residual interference factors and channel estimation errors for the scenarios of imperfect SIC and imperfect CSI. On this basis, the decoding error probability could be analyzed, and then the corresponding resource optimization problem also could be established. By applying the algorithm proposed, it is possible to obtain the optimal solution under the corresponding conditions. In the future work, we would further explore resource optimization problems under imperfect channel conditions to expand and deepen our research value in this field.

6. Conclusion

In this paper, we mainly introduce the resource optimization algorithm for MTC equipment with differentiated QoS requirements in NOMA VLC/RF networks. First, we construct an RF-aided NOMA VLC network model adopting the SPC technology. The successful transmission probability of the UE is derived and the SC of networks is evaluated under NOMA VLC/RF, NOMA VLC, and OMA VLC schemes. Then, under the constraints of UEs statistical delay QoS, SR, and transmission power, an optimization problem with SC maximization is built. We propose a resource optimization algorithm combining blocklength control and power allocation to settle this problem. Finally, the results show that, compared to the NOMA VLC and OMA VLC resource optimization schemes, the proposed NOMA VLC/RF resource optimization scheme could generate greater SC with a shorter blocklength and more suitable performance for high-reliability communication transmission. The presented scheme could provide a theoretical direction for resource optimization in real industrial environments, enabling the identification of optimization space under practical conditions and further guiding system design.

Supporting information