Delay Jitter Analysis for VLC Under Indoor Industrial Internet of Things Scenarios

The delay jitter in the visible light communication (VLC) network is analyzed in this work. First, the delay jitter under single-user and multi-user scenarios is investigated and a closed-form expression is obtained, whose accuracy is verified by simulation. Second, through analyzing the delay jitter under two typical indoor light emitting diode (LED) layouts, we find that the mean user delay jitter under cellular layout is a little lower than that of square layout. Then, we study the delay jitter for mobile users, and the relationship among user movement speed, arrival rate and delay jitter in different scenarios. The delay jitter performance is evaluated assuming perfect coordinated multiple points (CoMP) transmission strategy in VLC network. Simulation results show that in the context of VLC, static users can experience an average delay jitter of up to <inline-formula><tex-math notation="LaTeX">$10^{-6}$</tex-math></inline-formula> s, while mobile users can experience an average delay jitter of up to <inline-formula><tex-math notation="LaTeX">$10^{-5}$</tex-math></inline-formula> s. Therefore, VLC network can provide users with good delay jitter performance and customized services to meet the higher service quality requirements of future users.

and fixed monitors require real-time communication, whereas mobile devices such as robotic arms and mobile robots rely on motion-based communication.Thus, our study focuses on analyzing delay jitter in both static and mobile devices within IIoT environments.
Private networks are commonly deployed in IIoT applications to facilitate user communication.These networks are customized to serve specific devices within the end-user organization, thereby addressing concerns regarding the impact of public users on device connectivity, throughput, and other network performance metrics.They offer advantages such as customized security measures and seamless integration with other business systems within industrial enterprises [4].Generally, private networks provide ample resources to users, including high-bandwidth services for industrial enterprises, and ensure reliable, low-latency communications.This meets strict requirements for coverage and capacity, ensuring consistent high quality and reliability [5].Due to the plentiful communication resources of these networks, user communication demands are well supported, usually keeping the system in a state of low load.
Visible light communication (VLC), utilizing light-emitting diodes (LEDs), presents a promising alternative to radio frequency (RF) communication in scenarios sensitive to electromagnetic interference and radiation [17], [18], [19].VLC offers energy-efficient, low-cost solutions with satisfactory area spectral efficiency (ASE) [20].Consequently, researching delay jitter performance has become a key driver for large-scale applications in wireless communication, necessitating further exploration of model-related studies in VLC.
Several studies have investigated delay jitter performance in various communication systems.Hammad et al. [21] introduced a fundamental analytical model for Internet Protocol (IP) networks and provided an approximate analytical expression for end-to-end delay jitter.Dahmouni [22] focused on the mathematical modeling of packet delay jitter in queuing systems, emphasizing correlation relationships at the data arrival point.Liang et al. [23] analyzed jitter-free transmission under variable bit-rate (VBR) channels, considering trade-offs among initial playback delay, receiver buffer size, and jitter-free probability.Fulton et al. [24] conducted an analysis of delay jitter in tagged flows within a link shared by multiple streams.Additionally, Dahmouni [25] employed a fast delay jitter calculation model to address routing issues for flows affected by jitter or delay.
Furthermore, Huremovic et al. [26] proposed an analytical model for delay jitter using the interrupted Poisson process (IPP) for incoming traffic.Megha et al. [27] investigated the problem of traffic splitting for real-time applications in the uplink of a multi-access system based on cellular networks, providing a detailed analysis of its delay jitter performance.Yu et al. [28] explored the inclusion of jitter reduction in Industrial Wireless Sensor and Actuator Networks scheduling schemes, proposing a specific scheme to mitigate jitter.Althoubi et al. [29] introduced a queuing model applicable to edge sensor networks and data center networks, estimating average end-to-end delay and predicting delay jitter based on this model.
Currently, research primarily focuses on the fundamental investigation of delay jitter performance in IP networks, with a predominant emphasis on studying delay jitter performance under the M/M/1 queuing model.Limited research has been conducted on delay jitter under VLC and RF channels.Furthermore, there is a lack of studies that provide theoretical derivation to analyze system performance under actual channel conditions.Hence, in this study, we establish a cross-layer model and derive a mathematical formula for the delay jitter performance of VLC users.We compare and analyze the benefits of VLC and RF communication in terms of delay jitter.Our concise formulaic results enable rapid evaluation of whether network delay jitter performance meets user requirements before deploying the network.This study lays the foundation for advancing improvements in VLC custom services in 6th generation (6G) networks.
In this study, we focus on examining the delay jitter of static and mobile devices in the context of IIoT.We specifically consider scenarios with lighter loads in the industrial IoT context, which aligns with real-world situations.
The remainder of this paper is organized as follows.Section II introduces the system model.Section III derives the transmission delay jitter under VLC for static users and provides a closedform expression.Section IV presents the rate correction algorithm for the mobile user scenario in a VLC network.Section V evaluates the transmission delay jitter in single and multi-user scenarios, and presents the numerical results of delay jitter for both static and mobile user scenarios.Section VI describes the experimental setup and presents the results of delay jitter in a VLC network.Finally, Section VII concludes the work.

A. Physical-Layer Characteristics
We consider an indoor VLC scenario where multiple LEDs are mounted on the ceiling, and each user terminal is equipped with a photodiode (PD) for detecting the optical signal.The LEDs and PDs are parallel to the horizontal plane.The bandwidth of the VLC system is B v [Hz].
Since the reflection component is significantly weaker than the line-of-sight (LOS) component, we only consider LOS link.Assuming Lambertian radiation pattern, the optical channel gain from the LED to the PD is given by [30] where m = −1/ log(cos θ 1/2 ) denotes the Lambertian order; θ 1/2 denotes the half-power angle of the LED; θ denotes the irradiance angle of LED; ϕ denotes the incidence angle of PD; d denotes the distance between the transmitter and receiver, T (ϕ) denotes the optical filter gain; f (ϕ) denotes the optical concentrator gain; ϕ FOV denote the FOV of PD; rect(•) is the rectangular function.
Since the LED and PD are both parallel to the horizontal plane, we have cos(θ) = cos(ϕ) = L/d, where L is the transceiver height and d is the transceiver distance.Assuming sufficiently large φ FOV , we have where C is a constant.Let σ 2 v denote the noise power, P denote the optical power of the transmitted signal, γ denote the PD photoelectric conversion coefficient.The achievable rate in VLC (also known as the lower bound of VLC channel capacity) can be written as [31]

B. MAC Layer Characteristics
Assume that the incoming packet requests are stored in the transmitter-side buffer.Furthermore, arrival interval δ between any consecutive requests is independent of each other and obeys the following exponential distribution: where λ > 0 denotes the arrival rate.
The packet arrival and serving process is characterized as follows.The arrival time of the k-th packet request is t k .The interval between the arrival of the k-th packet and that of the During the process of packet transmission, the random arrival of packets can lead to packet queuing.The leaving time of the k-th packet request is r k .The duration of the k-th packet request waiting in the queue is W k = r k − t k .The service duration for the k-th packet is S k .The transmission duration of the k-th packet is T k = S k + W k .The queuing process is illustrated in Fig. 1.The average service time per packet is E[S k ] = 1/μ, and then the ratio of the average arrival rate to the average service rate is ρ = λ/μ.We assume ρ 1 in this work.

A. Delay Jitter Model
The delay jitter is defined as the mean absolute difference in transmission duration of consecutive packets, i.e., According to the delay jitter analysis model and the queuing process illustrated in Fig. 1, waiting duration of the (k + 1)-th packet is given by [21] Then, we have Equation ( 7) can be further rewritten as Based on the above definition, the mean delay jitter is given by [22] where f S (s) denotes the probability density function (PDF) of service time, and f T (t) denotes the PDF of transmission time.
In VLC network, due to directional and stable transmission link gain, the queuing system of a static user can be modeled as an M/D/1 system.User transmission rate R v can be determined by its position and (3).The service time of packet request is given by where L f is the packet length and B is the bandwidth allocated to the user.
In both single-user and multi-user scenarios, each user requires transmitting a significant amount of random packets.In the single-user scenario, the entire available bandwidth of the system is dedicated to a single user, allowing for a direct analysis of the delay jitter associated with multiple packets for this specific user.In the multi-user scenario, the communication bandwidth assigned to each user is dependent on the total number of users to mitigate interference during simultaneous transmissions.This work investigates the impact of bandwidth allocation on the delay jitter of users in a multi-user environment.
Assume that the users are uniformly distributed in the communication area with density n [m −2 ].The average number of users under a single base station (BS) within an effective area A [m 2 ] is given by N = ceil(n • A), where ceil(•) denotes the rounding-up operation.

B. Delay Jitter Analysis for Static Users in VLC Network
Assume there are N users, and each user has multiple randomly arriving packets for transmission.In a multi-user scenario, frequency division is employed to ensure no interference for signal transmission on each sub-channel with bandwidth B = B v N .The service time can be calculated as s = In particular, when there is only one user, we have N = 1.
Since the service time of a static user is fixed, the delay jitter can be expressed as (11) where ŝ represents the packet service time under distance r from the center of the LED coverage circle.For a single static user, since its service time is fixed, we have Proposition 1: For the case with ρ 1, the user waiting time PDF can be approximated as ) Proof: Note that the user waiting time is 0 with probability 1 − ρ.Given sufficiently small probability ρ, we assume that the likelihood of more than two users in the buffer is negligible.
From (4), the cumulative distribution function (CDF) of the arrival interval between packets is Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
When the packet arrival interval is less than the service time 1/μ, substituting δ = 1 μ into (14) gives Under assumption ρ 1, (15) approximates to ρ.From this, we infer that the user waiting time is non-zero primarily when there is only one packet in the buffer, since the waiting occurs when the arrival interval between packets is less than the service time.
Let f b (w) denote the waiting time PDF when the system is busy.In this scenario, the user waiting time can be expressed as w = 1 μ − δ, leading to the following approximate expression: Let U (•) denote the step function and δ(•) denote the impulse function.The waiting time PDF can be approximated as Note that λ = ρ • μ, where ρ represents the probability that the system is busy.
When the system is idle with probability 1 − ρ, the corresponding PDF of the waiting time can be approximated as Based on the above analysis, for sufficiently small ρ 1, the waiting time PDF can be approximated as Based on the above jitter analysis of static users, incorporating ( 12) and ( 13) into (11), we have the following results.
Proposition 2: For ρ 1, the expectation of delay jitter under service time s and arrival rate λ can be approximated as Proof: According to (12), PDF f W takes values on (0, s) and PDF f T takes values on (s, 2s).Thus, we divide the range into three parts (0, s), (s, 2s), (2s, 3s) for integration.We have We evaluate the closed-form expression for the two integral parts.The first part is given by The second part is given by Based on the ( 21) and ( 22), we have From ( 23), we can observe the impact of cross-layer parameters on delay jitter performance.The delay jitter is related to service time s in the physical layer and packet arrival rate λ in the MAC layer.These two variables collectively contribute to the overall delay jitter performance.
When the users move away from the LOS area of one LED, they switch to the LOS area of another LED [32].Such transition can introduce variations in transmission time, thereby increasing the delay jitter.In this work, we assume sufficiently smooth handover to another LED, and investigate the limit of delay jitter due to the link gain variation.

C. Average Delay Jitter of VLC Network
According to (11), the average jitter can be expressed as Since service time s depends on distance r between each user and the center of LED coverage circle, we can obtain PDF f r (r) of distance r assuming uniform user PDF.Specifically, we obtain PDF , the PDF of channel gain can be derived as Then, the PDF of ε can be derived as the PDF of R v is given by and the PDF of S can be derived as In this study, we investigate an indoor VLC network with multiple LEDs to provide full-coverage for indoor users.We consider two specific types of LED layouts: cellular layout and square layout, as shown in Fig. 2. PDF f r in ( 25) is derived as follows.The effective communication area of each LED under cellular and square layout are shown in Fig. 3(a) and (b), respectively.Let d denote the distance between LEDs, and R c denote the transmission distance of the LEDs.Assuming uniform distribution of users.The effective square area under square layout is d 2 , and the PDF of each user distance from the LED center under square layout is given by where F S (r) represents the CDF of user's distance r from the LED center in Fig. 3(b), given by  Under cellular LED layout, the effective area of the cellular is The PDF of each user's distance from the LED center is expressed as where F C (r) represents the CDF of user's distance r from the LED center in Fig. 3(a), given by From ( 29) and ( 31), we can derive the PDF of user distance r, within the effective area under the square and cellular network layouts.Next, by substituting ( 29) and ( 31) into (25), we obtain the PDF of the link gain.Subsequently, the obtained link gain PDF is substituted into ( 26)- (28) to derive the service time PDF for both the square and cellular network layouts.Finally, by substituting the service time PDF into (24), we can calculate the mean delay jitter in the effective area for both the square and cellular network layouts.

IV. DELAY JITTER ANALYSIS FOR MOBILE USERS
For mobile terminals, the location and achievable transmission rate are not fixed, an analytical solution is not tractable.Thus, we resort to a numerical solution.Assume fixed trajectory and known speed, such that the location and achievable rate Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

TABLE I MAIN PARAMETERS IN SIMULATION
are prior knowledge.Also assume that the adjacent LEDs can perform simultaneous coordinate transmission, such that the delay jitter due to serving LED switch is negligible.

A. Rate Correction for Mobile Terminals
Assume that for packet j, the achievable rate at the transmission instant is R j , and the achievable rate when the service is completed is R j .Then, the transmission rate is Again, if N users are served by one LED, the bandwidth is equally split among the entire bandwidth, and the delay jitter can be obtained through Algorithm 1.

B. Calculation of Delay Jitter
Since the delay jitter of a mobile user is influenced by factors such as movement speed v and direction, it cannot be directly derived from (11).Therefore, we adopt numerical simulation to obtain the average delay jitter of the whole motion process of mobile users.First, we calculate the absolute value of the difference between the end-to-end delay of two adjacent packets to obtain the delay jitter value.Let N p denote the cumulative number of packets.Subsequently, the average delay jitter is calculated based on the N p − 1 delay jitter values obtained after transmitting N p packets throughout the entire motion process.The mean delay jitter is given by J = 1 Hence, in the context of numerical calculations for user delay jitter, our primary objective is to acquire the end-to-end delay of each packet.The end-to-end delay encompasses both queuing delay and transmission delay.The queuing delay can be obtained by recording packet arrival time t j and packet departure queue time r j during the simulation.To calculate the transmission delay, it is necessary to record the location of the packet when it initiates service.Subsequently, the transmission rate of the packet is computed using Algorithm 1, enabling the determination of the transmission delay based on the packet length L f .

V. NUMERICAL RESULTS
We numerically evaluate the delay jitter for both VLC and compare it with that of RF communication system.The delay jitter analysis for users in RF network is presented in Appendix A. To ensure a fair comparison of delay jitter, we set the same Algorithm 1: Calculation of Delay Jitter for Users With Uniform Rectilinear Motion.
1: Input the j-th and (j + 1)-th packets at the time of arrival t j and t j+1 , moment r j at which the j-th packet to leave, movement speed v and packet length L f .2: Obtain position x j corresponding to the mobile user at the r j moment.3: Get the corresponding transmission rate R S j = min(R j , R j ).4: Obtain the j-th packet service completion time r j + S j , where r j+1 = t j + S j , 7: r j+1 = t j+1 , 10: ). 13: Obtain the (j + 1)-th packet service time ), and the following delay jitter total electrical transmission power for both VLC LEDs and RF BS.We have configured the optical power (peak power) of the LED under VLC to be 89mW, with an average power P/2 of approximately 44.5mW, equivalent to about 16.7dBm.Assuming a wall-plug efficiency of 10% and a power loss of 10dB, we also set the average power of the RF BS to 28dBm.Consequently, while maintaining consistent electrical power levels, we conducted an analysis of the system's delay jitter performance.Furthermore, we ensure that the mounting height of the LEDs aligns with that of the RF BS.In RF communication networks, the BSs are conventionally located at the center of the coverage range.Throughout the simulation, typical bandwidth configurations are adopted for both VLC and RF bands [33], [34].Unless stated otherwise, the main simulation parameters are listed in Table I.
The PDF of packet request arrival intervals is derived from (4), and 10 5 arrival events are randomly generated for numerical evaluation.

A. Delay Jitter Results for Single Static User
Consider a scenario with a single VLC transmitter and a single static receiver.The single user is uniformly distributed in an circle with radius of 3 meters.In Fig. 4, we plot the relationship between delay jitter and transmission rate of a single static user for packet request arrival rate λ ∈ {100, 500, 900}.In the simulation, we generated 10 5 arrival events randomly based on the (4).The delay jitter results in Fig. 4(a) show that lower packet request arrival rate leads to lower delay jitter in VLC, and the delay jitter decreases with the transmission rate.Moreover, it can Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.be seen from Fig. 4(a) that the difference between the simulated delay jitter and the theoretical delay jitter is very small.
For RF communication, we assume that the small-scale fading follows the Rice distribution.We adopt acceptance-rejection method to generate 10 5 service slots [35].It can be seen from Fig. 4(b) that a lower packet request arrival rate corresponds to a lower delay jitter.Furthermore, the delay jitter of a static user decreases as the transmission rate increases.Notably, the theoretically calculated delay jitter value is a bit than the simulated one.
At the same packet arrival rate, it is observed that the static user experiences lower delay jitter under VLC, attributable to the constant transmission rate from (11).Conversely, the presence of small-scale fading in RF communication results in larger delay jitter.

B. Delay Jitter Results for Multiple Static Users
Assuming a uniform distribution of users within the network depicted in Fig. 3, each circle has a radius of 3 meters.Let the density of uniformly distributed users be n = 3 [/m 2 ].By adjusting the spacing of the LEDs, a comparison is provided to assess the impact of cellular network layout and square network layout on user delay jitter performance within the same effective area.Fig. 5 demonstrates that the cellular network layout yielded lower delay jitter for users compared to the square network layout.This can be attributed to the more concentrated distribution of user-LED distances in the cellular network layout, resulting in a shorter average distance between users and LEDs and enabling users to achieve higher transmission rates.
Furthermore, we employ a closed-form expression to estimate the average delay jitter under both cellular and square layouts, which validated the accuracy of our approach.By substituting the service time S in the multi-user VLC network into (19), we obtain the relationship between delay jitter and the distance r between the user and the center of the LED coverage circle, as depicted in Fig. 6.It can be observed that, under a constant λ, the delay jitter increases with the user's distance r from the center   due to lower transmission rates at the edge of the LED coverage circle, resulting in longer service times S. Additionally, at the same distance r from the center, the delay jitter increases with λ.For static users, the arrival rate of packets at the MAC layer becomes the primary factor influencing user delay jitter, as the user's transmission rate remains unchanged.Fig. 7(a) and (b) depict the PDF of delay jitter at different λ values.It can be observed that when λ is small, the delay PDF is approximately Gaussian.As λ increases, the variance of delay jitter increases, and a long-tail characteristic appears, resulting in an increase in average delay jitter.
To better deploy VLC network and select LED devices, we also investigate the effects of LED spacing, LED radiation angle, and LED power on delay jitter.Fig. 8(a)-(b) show the delay jitter under different LED parameters.It can be seen that the delay jitter increases with increasing LED pitch and the half-power angle of the LED, and decreases as LED power increases.
Next, we conduct simulations for different PD sizes.Under constant LED parameters, numerical results in Fig. 9 show that the lower delay jitter gets lower as the PD size increases.
In RF communication, the selection of the number of antennas in the network is crucial to meet user delay jitter performance.Therefore, we studied the effect of the number of antennas on  where each antenna has the same power, and the delay jitter experienced by static users.The findings indicate that a higher number of spatially divided zones correspond to lower delay jitter.This can be attributed to the increased allocation of communication bandwidth for each user within the expanded zones, resulting in decreased delay jitter.

C. Delay Jitter Results for Mobile Users in a Single Cell
In the context of IIoT, there are certain communication devices that follow predefined motion trajectories and require high performance in terms of delay jitter.Consequently, we have conducted a comprehensive investigation on the delay jitter performance for such motion users.In this study, we assume that the coverage area of each LED is a circular area with a radius of 3m.Furthermore, we consider the scenario where users move from one end to the other along the diameter of the VLC coverage circle at a velocity denoted as v. Notably, the achievable communication speed for users exhibits an initial increase followed by a subsequent decrease within their motion trajectory.In the experiments, we set the velocity values to v = 2, 4, 6, 8, 10 m/s.To obtain the average delay jitter experienced by users with uniform linear motion in the VLC network, we repeat the simulation of user motion trajectories for a total of 10 5 times.
From Fig. 11(a), it can be observed that for a fixed movement rate v, the delay jitter decreases with arrival rate λ.This trend is contrary to the observation for static users.The observed results are attributed to user movement, wherein a lower packet arrival rate λ leads to a longer interval I j between the arrival moments of two adjacent packets.Such difference in arrival times results in variations in transmission rates and consequently larger delay jitter.
Consider the scenario with N users, where the bandwidth allocation for each user is B/N .Firstly, we determine the number of users within the coverage circle of the LED.Subsequently, we calculate the communication bandwidth for each user.The delay jitter for multiple mobile users is depicted in Fig. 11(b).
When evaluating the delay jitter under multiple mobile users in RF network, the bandwidth allocated to each user is B r /N r , where N r is the number of users in the RF communication network.Assume that the user density is n = 3 [/m 2 ].Fig. 12(b) illustrates the delay jitter at various user movement speeds v and packet arrival rates λ.It is observed that given fixed movement rate v, the delay jitter increases with arrival rate of λ, which does not change significantly with movement speed due to small-scale fading in RF communication network.For multi-user case, the degradation in jitter performance of mobile users in RF communication networks can be attributed to two main factors.Firstly, there are more users in RF communication network compared with VLC network, and the frequency bands can be shared among multiple users, which significantly reduces the transmission rate compared to single-user case.Secondly, the small-scale fading in RF communication network introduces instability in packet transmission rates.
In a single-user scenario, a higher packet transmission rate reduces the impact on packet arrival rates and the transmission time jitter, as shown in Fig. 12(a).Conversely, increasing the number of users leads to reduced bandwidth for each user, thereby decreasing the packet transmission rate and increasing the service time jitter.Consequently, a higher packet arrival rates significantly increases the delay jitter.
In Figs.11(a) and 12(a), the bandwidth of 200MHz in RF communication network results in high transmission rates for single user in a cell.However, the VLC link experiences significant attenuation, resulting in lower link gain and thus lower transmission rates.Additionally, the transmission rate in a VLC network is significantly affected by the distance, and user movement within the VLC coverage area leads to substantial variations in the transmission rates of consecutive data packets.Thus, under moving user conditions, the delay jitter for VLC is higher compared with that for RF communication.

D. Delay Jitter Results in IIoT Factory Scenario
In the practical IIoT scenario, the construction of VLC network is impacted by the characteristics of overlapping areas on user delay jitter performance.Therefore, we conduct a comprehensive analysis of user delay jitter in a typical IIoT factory setting.Following the standard definition of an indoor IIoT factory by 5G-ACIA (Alliance for Connected Industries and Automation), the size of the indoor room is 78m ×180 m in Fig. 13.The LEDs are arranged on the ceiling, with a height of 5m from the ground, and a spacing of 6m.Our focus is on investigating the delay jitter among multiple users within a square VLC layout, chosen due to the convenience of deploying a square VLC network within a rectangular IIoT Factory.
To ensure there are no communication blind spots in the factory, the center of the outermost LED circle is positioned 3m from the edge of the factory, as illustrated in Fig. 13(a).In total, there are 390 LEDs distributed across 13 rows and 30 columns of the room.In the framework of VLC, a single user experiences the lowest transmission rates 17.09Mbps at the vertices of each LED's effective coverage area.The highest transmission rate is 29.82Mbps observed at the center of these coverage areas.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.In contrast, in RF environments, assume that each zone is non-overlapping for RF communication network, enabling the utilization of a 200MHz frequency band for each zone.The minimum transmission rate for a single user is 30.89Mbps at the corner of the factory, while the maximum transmission rate is 2.858Gbps at the center of the factory.
We introduce coordinated multiple points (CoMP) in the context of VLC network, which adopts a strategy of coordinated transmission from multiple LEDs to users in overlapping areas.The frequency division multiple access (FDMA) scheme is applied in the VLC network.The available frequency band B l for each LED is determined as B l = B r /4 due to the overlap cells, where B r represents the total available frequency band.Additionally, the available frequency band for each user is calculated as B l /N , where N denotes the total number of users.
Numerical simulation is applied for analyzing static users delay jitter in VLC network.Assuming that the users in each area are uniformly distributed with density n = 3 [/m 2 ], we count the number of users under each area.The location of the user determines the collaborative communication mode and transmission rate.Further assume that the interval between the arrival of the packet follows the negative exponential distribution.Given fixed packet length L f , the delay jitter of the static user at each position in each area can be obtained through simulation.Fig. 15 shows the delay jitter under packet arrival rate λ ∈ {10, 50, 90} in VLC square network.The delay jitter results show that smaller packet request arrival rate leads to smaller delay jitter in VLC, and the delay jitter decreases with the transmission rate.
As depicted in Fig. 15, the VLC network demonstrates lower delay jitter than the RF network at a smaller packet arrival rate.However, as packet arrival rate λ increases, the delay jitter performance of the RF network and the VLC network becomes comparable.This is attributed to the adoption of CoMP and FDMA technologies in VLC network, which result in a reduction of available communication bandwidth.Therefore, when the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.packet arrival rate is large, the delay jitter performance of the VLC network is comparable to that of the RF network.On the other hand, in the RF network, the correlation between delay jitter and packet arrival rate λ is small, because the variation of transmission rate of the RF network primarily stems from small-scale fading.In order to compare the effect of different antenna numbers on the delay jitter performance of static users in RF under equal total power conditions, Fig. 14 illustrates the delay jitter performance in RF IIoT factory scenario.Various delay jitters are observed among users in the scene under different sector angles.It can be observed that, at the same packet arrival rate, the delay jitter of users increases as the sector angle increases.Such increase in delay jitter can be attributed to the larger sector angle, which results in a higher number of users within the sector.Consequently, the communication bandwidth allocated to each user decreases, thereby influencing the increase in delay jitter.
Numerical simulation is conducted to investigate the delay jitter experienced by mobile users.We assume a fixed user movement speed, and the user movement direction is uniformly distributed on the interval [0, 2π], updating at a period of 1 s.The interval of user arrival packets follows a negative exponential distribution.The transmission rate of each packet is determined by the user position at each start of transmission, which can lead to unstable transmission due to user's movement.To solve this issue, we adopt the minimum transmission rate searched throughout the entire packet transmission duration, denoted as R min , within a certain radius circle centered on the user current location at the time of transmission.The search radius is L f /R min .
Fig. 16(a) shows the delay jitter under different user movement speeds with packet arrival rate λ ∈ {10, 50, 90} in VLC square network.It is shown that lower packet request arrival rate λ leads to lower delay jitter in VLC, and the delay jitter decreases with the transmission rate.
In order to ensure fairness between VLC network and RF network, we set the BS height H net = 5m in RF network, and the average power of the BS is the same as the sum of the average power of VLC square network.In addition, space division multiple access (SDMA) also adopted in RF network.In the RF scenario, we assume an ideal transmission scenario for antenna interference-free regions, whereby the antenna's side lobes are small enough to ensure no interference between various Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.sectors.Assuming that the antennas exhibit perfect beamforming characteristics, we set 36 sectors to achieve a sector angle of 10 • per sector in Fig. 13(b).Furthermore, we ensure that the total average power of all antennas is equivalent to that of the LEDs in the VLC network.
Due to the differences in the layout of RF network in IIoT scenarios, the distance PDF involving 9 sectors in the theoretical analysis becomes quite complex.Therefore, we adopt numerical simulation.For simulating static users in RF network, we employ the same simulation approach as that adopt for static users in VLC network.Mobile users in the RF network move also in the same manner as mobile users in the VLC network.Fig. 16(b) shows the delay jitter of mobile users in the RF network which fluctuates greatly.In addition, there is no explicit relationship between the user's movement speed v and packet arrival rate λ and delay jitter, which is mainly caused by small-scale fading in the RF channel.Note that such results are obtained assuming perfect environment for RF communication.Any metal obstacles may cause significant shadowing and thus delay jitter degradation.
In addition, by comparing Fig. 16(b) with Fig. 15, we can see that in the RF network, the delay jitter of a static user is greater than that of a mobile user.Moreover, when the mobile users was closer to the BS, which allowed the packets to be transmitted at a higher transmission rate, the delay jitter value can be reduced.

VI. REAL LINK GAIN EVALUATION
We also conduct actual measurements to obtain the gain of the visible light channel in real scene.Additionally, we perform offline tests to obtain the delay jitter values at a specific user packet arrival rate.These experimental results provide verification of the VLC channel's supportability in terms of its ability to deliver extreme delay jitter performance.
Fig. 17 illustrates the experimental VLC configuration.The LED is positioned on the ceiling, approximately 2.2 meters away from the slide, while the avalanche photodiode (APD) is installed on the terminal that moves in a straight line along the slide.While both APD and PD can convert optical signals into electrical  We generate squared waveform with a peak-to-peak voltage of 8V and a frequency of 500kHz through an arbitrary waveform generator (AWG).Then, the electrical signal is fed into the bias-T which provides a certain DC bias voltage for LED to ensure the normal illumination.The APD receives the signal and is connected to the oscilloscope.We record the peak-to-peak value of the received signal at each position of the slide rail, and obtain the link gain value of each location on the slide rail under the LED coverage circle.We also test the noise variance in the experimental scenario.Specifically, we connect the APD with the oscilloscope, record the oscilloscope waveform to process the noise variance when the APD is powered on.It is observed that the noise power in the experimental environment is 3.829 × 10 −8 .
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Based on the achievable rate estimated in the above experimental scenario, we simulate the delay jitter of a single static user, as shown in Fig. 18(a), where the x-axis represents the distance coordinate between the user and the LED coverage center.It can be observed that lower distance to the LED coverage center leads to lower user delay jitter.
Assume that the user are uniformly distributed with density n = 3 [/m 2 ].We carry out the delay jitter simulation under multiple static users at different positions on the slide, as shown in Fig. 18(b).Similarly, smaller delay jitter is observed near the LED coverage center from Fig. 18(b).Compared with the delay jitter results under a single static user, reduced communication bandwidth leads to a lower communication rate and increased delay jitter.
We evaluate the delay jitter of a single mobile user based on the link gain obtained in the experimental scenario, as shown in Fig. 19(a).The user delay jitter with movement speed v under different arrival rates λ shows similar trend as that in Fig. 11(a The largest influence is the difference in parameter noise power settings, where lower noise power in simulation leads to higher communication rate and lower delay jitter value. Finally, we evaluate the delay jitter for multiple mobile users based on the link gain obtained in the experimental scenario, as shown in Fig. 19(b).We can clearly see that the delay jitter is larger than that in single-mobile user scenario from Fig. 19(a).It is observed that given fixed movement speed v, the delay jitter decreases with arrival rate λ.We can also observe that the delay jitter increases with movement speed v under the condition of fixed λ.

VII. CONCLUSION
In this work, we have developed a comprehensive channel model for indoor VLC and RF networks, while also establishing a queuing model for analyzing the delay jitter.Firstly, we have analyzed static user delay jitter under VLC and RF networks and obtained the closed-form expression of delay jitter under VLC network.Secondly, we have analyzed mobile user delay jitter under VLC and RF networks.Numerical results have shown that delay jitter under VLC has advantages in dense users, and can provide service quality assurance for users in IIoT scenarios.In view of the advantages of the VLC network, we have derived the expression of delay jitter under the VLC square network and cellular network, and analyzed the influence of various LED parameters on delay jitter.Finally, we have verified the user delay jitter under VLC network through experiments.
This work comprehensively considers the delay jitter under the user's static or motion conditions, and also considers the influence of LED layout and LED parameters on delay jitter performance in VLC.The insights gained from this study provide valuable guidance for the future implementation of VLC in industrial scenarios.

A. RF Physical-Layer Transmission
We consider an RF network layout comprising a single circular macro-cell with one BS.The area of interest is a circular region with radius ρ RF .There are no obstructions between the BS and the user, and the BS is positioned at the center of the circular area.The RF network operates within the 5GHz band, utilizing around 200MHz frequency resources for all RF users.FDMA approach is adopted to serve our users.
The path loss is calculated as [36], where α is the path loss coefficient, d is the distance between the transmitter and the receiver in the unit of meters, d 0 is the reference distance taken as 1m.
Assume that the signal between RF BS in 5GHz band and the user follows Rice distribution, which is primarily determined by Rice factor K and total power Ω.Let ρ denote the distance between the user and the RF BS.The total power is the sum of the direct radial power v 2 and other multipath link power 2σ 2 , and the BS in the 5GHz RF band and the user's omnidirectional antenna gain are normalized to 1.The received signal amplitude value |h l | follows Rice distribution as follows, Then, the PDF of the received signal amplitude is x Ω(ρ) exp −K − (K + 1)x 2 Ω(ρ) where I 0 (•) represents the modified Bessel function of the first kind with order 0. Assuming that the data transmission occurs in the static fading state, the baseband signal at moment m is expressed as y r (m) = x r (m)h r (m) + w r (m), (36) where x r (m) and y r (m) are the complex channel inputs and outputs of the RF access point, respectively.The channel gain {h r (m)} represents fading processes, and {w r (m)} is the additive noise, which is independent of each other satisfying complex Gaussian distribution CN (0, N 0 ).Assume block fading channel, where the channel fading gain remains constant for the duration of packet transmission.We further assume that the transmission power of the RF signal is limited by the average power P avg and peak power P peak [37], i.e., E{|x r (m)| 2 } ≤ P avg and |x r (m)| 2 ≤ P peak .In work [38], a lower bound on the maximum mutual-information entropy is given, which is set to the data transfer rate for reliable transmission.In the l-th packet transmission, the transmission rate is expressed as [

B. Mean Delay Jitter for Static Users in RF Network
For a specific location within the RF network, the delay jitter experienced by a static user at distance ρ from the center of RF coverage circle can be mathematically expressed as (39) Due to small scale fading, the transmission rate is not constant.We first derive the PDF of user communication rate based on the channel gain PDF under static users, and then derive the PDF of service time.
The channel gain follows Rice distribution with PDF given in (35).The PDF of transmission rate is expressed as Substituting (41) into (40), we can derive the theoretical expression for delay jitter in the RF network.However, due to the impact of channel gain following Rice distribution, the subsequent calculation involves a series of integral operations, resulting in high computational complexity.As a result, we Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
approximate the service time PDF with a Gaussian distribution.The fitted form is expressed as where the values of constants a, b and c in Gaussian approximation can be obtained through simulations.In the simulation process, the accuracy of Gaussian approximation can be validated.
For the analysis of multi-user delay jitter in RF network, we assume that users are uniformly distributed.The PDF of the distance between each user and the RF BS is expressed as According to the above PDF of the distance between the user and the BS of RF network, we can obtain the mean delay jitter for RF communication according to (39) and (43) as

Fig. 4 .
Fig. 4. Delay jitter for a single static user versus communication rate in (a) VLC network and (b) RF network.

Fig. 5 .Fig. 6 .
Fig. 5. Delay jitter versus packet arrival rate for cellular layout and square layout in VLC network.

Fig. 7 .
Fig. 7. PDF of jitter for multiple satic users under arrival rate in (a) cellular and (b) square network layouts.

Fig. 8 .
Fig. 8. Delay jitter for multiple satic users versus arrival rate under different (a) LED spacings, (b) LED half-power angles, and (c) LED powers.

Fig. 9 .
Fig. 9.The delay jitter for multiple static users versus the arrival rate under different PD sizes.

Fig. 10 .
Fig. 10.Delay jitter in RF network versus the number of antennas.

Fig. 11 .
Fig. 11.Delay jitter in VLC network versus move speed for (a) a single user and (b) multiple users.

Fig. 12 .
Fig. 12. Delay jitter in RF network versus move speed for (a) a single user and (b) multiple users.

Fig. 15 .
Fig. 15.Delay jitter for static users under IIoT factory environments versus packet arrival rate in VLC and RF networks.

Fig. 16 .
Fig. 16.Delay jitter for mobile multiple users scenario versus movement speed in (a) VLC network and (b) RF network.

Fig. 17
Fig. 17.The experimental configuration based on the slide rail in the laboratory.
Fig. 17.The experimental configuration based on the slide rail in the laboratory.

Fig. 18 .
Fig. 18.Delay jitter versus user position for (a) a single static user and (b) multiple static users in real VLC network.

Fig. 19 .
Fig. 19.Delay jitter versus movement speed for (a) a single static user and (b) multiple static users in real VLC network.
). Different delay jitter values in Figs.11(a) and 19(a) are caused by different parameter settings in the experiment and simulation.

2 −
1}.Therefore, the PDF of service time is expressed as 37]R r = B r log 2 1 + 2 |h l | 2 B ris the available bandwidth in the RF system.Parameters a and b are determined by P avg and P peak as where