Link Selection in Hybrid RF/VLC Systems
under Statistical Queueing Constraints
Abstract
The codeployment of radio frequency (RF) and visible light communications (VLC) technologies has been investigated in indoor environments to enhance network performances and to address specific qualityofservice (QoS) constraints. In this paper, we explore the benefits of employing both technologies when the QoS requirements are imposed as limits on the buffer overflow and delay violation probabilities, which are important metrics in designing low latency wireless networks. Particularly, we consider a multimechanism scenario that utilizes RF and VLC links for data transmission in an indoor environment, and then propose a link selection process through which the transmitter sends data over the link that sustains the desired QoS guarantees the most. Considering an ONOFF data source, we employ the maximum average data arrival rate at the transmitter buffer and the nonasymptotic bounds on data buffering delay as the main performance measures. We formulate the performance measures under the assumption that both links are subject to average and peak power constraints. Furthermore, we investigate the performance levels when either one of the two links is used for data transmission, or when both are used simultaneously. Finally, we show the impacts of different physical layer parameters on the system performance through numerical analysis.
I Introduction
The evergrowing demand for mobile communications triggered a quest for technical solutions that will support stringent qualityofservice (QoS) constraints. Thanks to the significant advances in white light emitting diodes (LEDs) research, and the availability of an extensive unregulated spectrum, visible light communication (VLC) has emerged as a promising technology. We can utilize LEDs simultaneously for data transmission and illumination, since they have many unique aspects compared to the other communication technologies [1]. Moreover, we can improve data security because light does not penetrate the surrounding walls. We can also sustain an allimportant green agenda and minimize the installation costs because we do not require an extensive infrastructure. Nevertheless, attention must be paid to certain limitations and challenges in VLC systems, e.g., smaller coverage, strong dependence on lineofsight components and achievable rates that vary with spatial fluctuations [2]. In order to overcome these constraints, researchers proposed hybrid RF/VLC systems [1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], where end users can benefit from the wide coverage area that RF systems support and the stable rates that VLC systems provide. Such networks are practically feasible as RF and VLC systems can coexist without causing interference on each other and operate in the same environment, such as offices and rooms.
Comparing hybrid RF/VLC systems with systems that employ either RF or VLC only, the authors in [3, 4, 14, 7, 1] demonstrated a remarkable increase in data transmission throughput, energy efficiency and delay performance in hybrid RF/VLC systems. Moreover, the authors in [5, 6] projected a hybrid system in which they use VLC links for downlink communication and RF links for uplink communication. In such a system, the authors in [8, 9, 10] and the ones in [12, 13, 15] investigated handover and load balancing mechanisms, respectively. Alternatively, considering an outdoor environment, the authors in [16] studied a pointtopoint transmission scenario in which the system can switch between RF links and VLC links after comparing the signaltonoise ratio levels in each link. Regarding the same system setting, the authors in [17] assumed that both RF and VLC links have the same transmission rates, and then proposed a diversitybased transmission scheme such that the transmitter sends data by employing both links simultaneously.
The aforementioned studies analyzed the hybrid RF/VLC systems mostly from the physical layer perspective, i.e., they did not concentrate on the data link layer metrics, such as limits on the buffer overflow and buffering delay probabilities, as much as needed. Noting the dramatic increase in the demand for reliable delaysensitive services in recent years (mobile video traffic making up of the total global data traffic at the end of [18]), in addition to the physical layer performance metrics, we need QoS metrics that can be a crosslayer analysis tool between the physical layer features and the performance levels in datalink layer. In this context, the authors in [19, 20, 21, 22, 23, 24, 25] performed crosslayer analysis between physical and datalink layers in many different RF scenarios. Regarding a Markovian data arrival process at a transmitter buffer and a statistically varying data service process in a wireless channel, the authors in [25] characterized the maximum average data arrival rate at the transmitter buffer with the presence of statistical constraints on the buffer overflow probability. On the other hand, to the best of our knowledge, there are only a few studies that investigated crosslayer performance levels in VLC systems. For example, the authors employed effective capacity as a performance measure in resource allocation schemes in VLC systems [26] and heterogeneous networks composed of VLC and RF links [27]. Here, we note that the authors in [27] concentrated on the case of constant data arrival rates at the transmitter buffer, which is not realistic in certain practical settings. For more details in effective capacity, we refer to [20].
In this paper, assuming an ONOFF modeled data arrival process at the transmitter buffer and that the transmitter can use both RF and VLC channels for data transmission, we investigate the performance gains achieved by a hybrid RF/VLC system. This operates under statistical QoS constraints, which are inflicted as limits on the buffer overflow and delay violation probabilities. We perform a crosslayer analysis of these hybrid systems in physical and datalink layers. We employ first the maximum average data arrival rate at the transmitter buffer considering the asymptotic buffer overflow probability approximation, and then nonasymptotic buffering delay violation probability as the main performance measures. We propose a mathematical toolbox to system designers for performance analysis in hybrid RF/VLC systems that work under low latency conditions. To summarize, the main contributions of this paper are as follows:

Assuming that both RF and VLC links are subject to average and peak power constraints, we express the maximum average data arrival rate at the transmitter that the data service process from the transmitter to the receiver can support under QoS constraints when either the RF or VLC link is used, or both links are simultaneously used for data transmission.

We propose three different link usage strategies. We base two of the proposed strategies on the assumption that the receiver does not have a multihoming capability, thus data transmission is possible over only one link, either the RF or VLC link. In the third strategy, we assume that link aggregation is possible and data can be transmitted over both links simultaneously following a power sharing policy.

We obtain the nonasymptotic data backlog and buffering delay violation probability bounds considering the proposed link usage strategies.
Particularly, we provide a rudimentary model for multimechanisms in communication systems that operate under QoS constraints. We employ RF and VLC links as two different mechanisms. Our model can be easily invoked in settings with more than two different mechanisms as well. The reason behind multimechanisms in communications is to boost performance levels through the increased degree of freedom. Therefore, in order to introduce our model smoothly and make it easier for readers to understand our objective, we also benefit from the existing literature in RF and VLC studies. However, to the best of our knowledge, the analytical framework provided in this paper, in which we investigate the QoS performance, is not addressed in other studies. One aspect of this hybrid system is that VLC links provide timeinvariant transmission rates, while RF links provide rates that vary over time. A communication setting that depends solely on an RF link may suffer low transmission rates, and have longer data backlogs in the transmitter buffer. However, a communication setting that can utilize both RF and VLC links, for instance, can take advantage of the constant transmission rate in the VLC link when the transmission rate in the RF link falls below a certain level. The rest of the paper is organized as follows. We introduce the hybrid RF/VLC system in Section II. We provide detailed descriptions of both RF and VLC channels. In Section III, we provide the performance analysis and the link selection process. In Section IV, we substantiate our results with numerical demonstrations. We conclude the paper in Section V and relegate the proofs to the Appendix.
Ii System Model
We consider a network access controller that provides a connection to a user^{1}^{1}1The analytical framework provided in this paper can easily be extended to a multiuser scenario. For more details, we refer to Remark 4 in Section III., through either an RF access point or a VLC access point, which are positioned at different locations in an indoor environment as seen in Figure 1. Herein, we assume a downlink scenario, i.e., the network access controller acts as a transmitter and the user acts as a receiver. In the sequel, we use transmitter and receiver instead of network access controller and user, respectively. Initially, the transmitter receives data from a source (or sources) and stores it as packets in its buffer. Subsequently, it sends the data packets in frames of seconds to the receiver following a given transmission strategy. The receiver is considered to be equipped with an RF frontend and a photodiode. We also note that the VLC coverage area is generally smaller than the RF coverage area, and that they overlap^{2}^{2}2This model is also considered in [9].. Finally, we consider a powerlimited system and assume that the network controller is constrained by a fixed average power budget, denoted as , for data transmission. In the sequel, we initially introduce the RF and VLC channels, and then describe the data source model.
Iia RF Channel Model
During the data transmission in the flatfading RF channel, the inputoutput relation at time instant is expressed as
(1) 
where and are the complex channel input and output at the RF access point of the transmitter and the RF frontend of the receiver, respectively. The complex channel input is subject to an average power constraint, , i.e., , and a peak power constraint, , i.e., . Above, is the complex channel fading gain with an arbitrary distribution having a finite average power, i.e., . Furthermore, we consider a blockfading channel and assume that the fading gain stays constant during one transmission frame ( seconds), i.e., , where the available bandwidth is Hz in the channel and is the channel fading gain in the time frame. Note that there are symbols transmitted in one time frame. Moreover, the fading gain changes independently from one frame to another. Meanwhile, is the additive noise at the RF frontend of the receiver, which is a zeromean, circularly symmetric complex Gaussian random variable with variance , i.e., . The noise samples are assumed to be independent and identically distributed.
We assume that a reliable data transmission exists as long as the transmission rate in the channel is lower than or equal to the instantaneous mutual information between the channel input and output^{3}^{3}3This assumption is based on the known result in the literature that transmitting data at rates less than or equal to the instantaneous mutual information has a high reliability, thus the decoding error is negligible [28, Eq. (10)] [29].. In particular, when the transmission rate in the time frame (i.e., bits per frame) is lower than or equal to the instantaneous mutual information (i.e., bits per frame) between the channel input and the channel output , a reliable data transmission occurs and bits are decoded correctly by the receiver. Here, we assume that is large enough so that the decoding error probability is negligible when . In [30], a lower bound on the maximum mutual information (or channel capacity) is provided, where the input has a twodimensional circularly truncated Gaussian distribution. Therefore, we set the instantaneous data transmission rate to the lower bound and assume that the input has a twodimensional circularly truncated Gaussian distribution. Specifically, we have a reliable transmission in the time frame when
(2)  
where and are the solutions of the following equations:  
(2a)  
and  
(2b) 
Above, is timedependent and changes from one time frame to another because the maximum instantaneous mutual information in each frame is a function of the channel fading gain.
IiB VLC Channel Model
We assume that the transmitter employs intensity modulation/direct detection. Principally, the VLC access point of the transmitter is equipped with an LED and the data is modulated on the intensity of the emitted light. The receiver that collects light using a photodiode generates an electrical current or voltage proportional to the intensity of the received light. Besides, we know that VLC channels are typically composed of lineofsight as well as multipath components. However, the majority of the collected energy at photodiodes (more than %) comes from the lineofsight components in typical indoor scenarios [31]. Therefore, we can assume that the VLC channel is flat with a dominant lineofsight component [32, 33, 34], and the channel gain does not vary during the data transmission as long as the receiver is stationary^{4}^{4}4Smallscale variations in VLC channels (i.e., fading) is mitigated since the area of a photodiode is much larger than the light wavelength [35, Sec. 2.5]. Thus, VLC channels are known as timeinvariant. This fact is almost true regardless of the frame duration or the user being stationary or mobile in indoor environments, because the users are either stationary or move very slowly.. Accordingly, the inputoutput relation in the VLC channel between the VLC access point of the transmitter and the photodiode of the receiver at time instant is given as follows:
(3) 
where and are the realvalued channel input and output, respectively. Above, is the opticaltoelectrical conversion efficiency (or detector responsivity) of the photodiode in amperes per watt and is the additive noise at the photodiode of the receiver, which is a zeromean, real Gaussian random variable with variance , i.e., . The noise samples are independent and identically distributed. Moreover, is the timeinvariant optical channel gain.
Recall that the data transmitted over the VLC link is modulated on the light that illuminates the environment. Hence, assuming that the operation range of the radiated optical power is limited between and when the light is on, we modulate the data between the power levels and , i.e., . As a result, the data bearing symbol, is limited as follows: . Hence, we can reexpress the inputoutput relation in (3) as
(4) 
Now, assuming that the expected value of is bounded by , i.e., , and that the available bandwidth in the optical channel is Hz, we set the transmission rate in the channel in bits per frame to the lower bound on the channel capacity, which is defined as follows [36]:
(5)  
when , and  
(6) 
when , where is the unique solution to
(7) 
Above, and are constant values, because the mutual information in the VLC link does not change by time, i.e., due to the strong lineofsight channel component, the channel gain, , does not change. As for the input distribution, we refer to [36, Eq. 42].
IiC Source Model
Regarding the data arrival process at the transmitter buffer, we consider a twostate discretetime Markov process^{5}^{5}5We can project certain data arrival models on ONOFF Markov processes. For instance, voice sources are generally modeled with ON and OFF states [37]. with ON and OFF states in each time frame. When the source is in the ON state in one time frame, the data from a source (or sources) arrives at the transmitter buffer. In the ON state, we consider a constant data arrival rate, i.e., bits per frame. The number of bits arriving at the transmitter buffer is zero in the OFF state. As shown in Fig. 2, the transition probability from the ON state to the OFF state is denoted by and the transition from the OFF state to the ON state is denoted by . The probability of staying in the ON state is and the probability of staying in the OFF state is . Hence, the state transition matrix becomes
(8) 
Now, let and be the steadystate probabilities of the data arrival process being in the ON and OFF states, respectively, where . Then, we have the following equality: . Subsequently, we have and , and hence, the average data arrival rate at the transmitter buffer is bits per frame. We finally note that the following analysis can easily be extended to other source models.
Iii Performance Analysis
In this section, we investigate the performance levels that the aforementioned system achieves by opportunistically exploiting the RF and VLC channels for data transmission. Herein, because the data is initially stored in the transmitter buffer before transmission, we assume that certain constraints are applied on the amount of the data in the buffer and the buffering delays. Therefore, we examine the system under QoS constraints that are associated with buffer overflow (data backlog) and buffering delay, and express the decay rate of the tail distribution of the queue length as [19, Eq. (63)]
(9) 
where is the stationary queue length, (see Fig. 1), and is the buffer overflow threshold. Above, denotes the decay rate of the tail distribution of the data backlog, . Accordingly, we can approximate the buffer overflow probability for a large threshold, , as . Notice that the buffer overflow probability decays exponentially with a rate controlled by , which is also defined as the QoS exponent. Basically, larger implies stricter QoS constraints, whereas smaller corresponds to looser constraints.
Recall that the outgoing service from the transmitter queue is , given in (2), when the data is sent through the RF channel and it is , given in (5) and (6), when the data is sent through the VLC channel, while the data arrival is ONOFF Markov process with bits per frame in the ON state and zero bits in the OFF state. Hence, assuming that the buffer size is infinite and considering independent data arrival and workconserving data service processes, there exist unique and such that and , respectively, where , and are the asymptotic logmoment generating functions of the total amount of bits arriving at the transmitter buffer, the total service from the transmitter in the RF channel and the total service from the transmitter in the VLC channel, respectively [38, Theorem 2.1]. In particular, the asymptotic logmoment generating functions for any are
(10)  
(11) 
We refer to [39, Example 7.2.7] for obtaining the logmoment generating functions. Using , we can express the maximum average data arrival rate at the transmitter buffer that the service process in the RF channel can sustain for any as
(12) 
where is given in (2). For the derivation of , we refer to Appendix A. Likewise, using and following the steps in Appendix A, we can also express the maximum average data arrival rate at the transmitter buffer that the service process in the VLC channel can sustain for any as
(13) 
where is given in (5) and (6) accordingly with the relation between and . Moreover, in the special case where and , the expressions in (III) and (III) provide the effective capacity, which is the maximum sustainable constant data arrival rate by the channel process given the QoS constraints [20]. In another special case where , i.e., the statetransitions are independent of the past and current states, we have
(14) 
Iiia Link Selection Policy
In this section, we focus on the channel selection process that the transmitter employs. We set the maximum average data arrival rate under QoS constraints as the objective in the channel selection process. In particular, the transmitter chooses the channel in which the service process maximizes the average data arrival rate at the transmitter buffer. Notice that the transmission rate in the VLC channel is constant, whereas the transmission rate in the RF channel varies due to the changes in the channel fading gain. Now, due to the fact that the channel fading gains are known by the receiver as well as the transmitter, we provide the following proposition.
Proposition 1
In the aforementioned RF/VLC system, the transmitter sends data to the receiver over the VLC link when the following condition for a given QoS exponent, , holds:
(15) 
where
(16) 
Proof: See Appendix B.
Proposition 1 states that if the maximum attainable transmission rate in the VLC channel is greater than the righthand side of (15), the transmitter should perform transmission over the VLC link because the statistical variations in the RF channel deteriorates the buffer stability. Meanwhile, in the special case when , we reexpress (15) as
(17) 
Specifically, the constant rate in the VLC channel should be greater than the effective capacity of the RF channel such that the VLC channel is chosen for data transmission. In the following, we present two transmission strategies, such that (i) the data is transmitted over the link with the highest instantaneous transmission rate in one time frame, and (ii) the data is transmitted over both links simultaneously.
HybridType I Transmission Strategy
In the above analysis, we obtain the performance levels when the transmitter chooses either of these two channels for data transmission following a link selection process based on the maximum average data arrival rates that the service processes in the channel can support. On the other hand, if there exists a fast and stable handover mechanism between the transmitter and the receiver, the transmitter will forward the data to the receiver over the link that provides the maximum lower bound on the instantaneous mutual information in the corresponding channel. For instance, when the lower bound on the instantaneous mutual information in the RF channel in the time frame, , is greater than the lower bound on the instantaneous mutual information in the VLC channel, , i.e., , the transmitter sends the data over the RF link in the corresponding time frame only. Otherwise, it prefers sending the data over the VLC link. Respectively, we can establish the channel selection criterion as follows: The transmitter sends the data over the RF link when
(18) 
Otherwise, it sends the data over the VLC link. The aforementioned selection test can also be considered as the outage condition in the RF channel, i.e., the RF link is in outage when . Noting that the channel fading gain changes independently from one time frame to another in the RF channel, the logmoment generating function of the service process becomes
(19) 
where is the conditional expectation given that , i.e., , where is the probability density function of . Hence, the maximum average data arrival rate at the transmitter buffer that the hybrid service process can sustain under the QoS constraints specified by becomes
(20) 
where .
HybridType II Transmission Strategy
Different than the aforementioned protocols, we consider a transmitter that sends data over both links simultaneously in each frame. We assume that the receiver has a multihoming capability. We further assume a multiplexingbased transmission scheme such that the data streams transmitted over the RF and VLC links are different and independent from each other. Indeed, this scenario is feasible since the light and RF waves do not cause interference on each other. We further assume a power allocation policy between the two links, i.e., the average power constraint in the RF link is set to , and the one in the VLC link is set to , where is the total average power constraint, and . Then, the instantaneous transmission rate in the RF channel in the time frame becomes
(21)  
where and are the solutions of (2a) and (2b). 
Similarly following (5)–(6), the transmission rate in the VLC channel becomes
(22)  
when , and  
(23) 
when , where is the unique solution of (7). It follows that the total transmission rate in each frame is the sum of the transmission rates in both links, i.e., . Then, the logmoment generating function can be readily expressed as
(24) 
and the maximum average data arrival rate is equal to
(25) 
where . We finally remark that the optimal value of that maximizes the sum transmission rate, i.e., , can be obtained numerically.
Remark 1
Employing the aforementioned strategies requires the perfect knowledge of both RF and VLC channels at the transmitter side in each frame. We assume that the channel estimation is performed at the receiver, and forwarded to the transmitter in a delayfree and errorfree feedback channel. This increases the signaling overhead. Moreover, applying HybridType II Transmission Strategy increases the implementation complexity because power sharing should be performed in each transmission frame. From the implementation perspective, exploiting HybridType II Transmission Strategy is limited because the receiver should have a multihoming capability that enables it to perform link aggregation and receive data from different transmission technologies simultaneously. On the other hand, the maximum average data arrival rate is a steadystate performance measure. Thus, the selection process explained in Proposition 1 is considered as a largetimescale operation, which can be performed over periods of multiple transmission frames, and therefore, the implementation complexity decreases. Such a largetimescale operation is also proposed in [40].
Remark 2
The selection process can also be employed in cases the access point controller has to choose between more than two transmission links. For example, let there be transmission links, and let {} be the maximum average data arrival rates at the transmitter buffer sustained by these links. Then, Proposition 1 in the paper will be updated with the solution of the following maximization problem:
(26) 
Similarly, let {} be the instantaneous transmission rates provided by each link in the time frame. Then, the link selection criterion in HybridType I Transmission Strategy will be updated with the solution of the following maximization problem:
(27) 
Subsequently, the logmoment generating function of the service process will be
(28) 
where is the probability that the transmitter chooses link for data transmission. Finally, if the receiver has a multihoming capability, HybridType II Transmission Strategy can be applied with power sharing to maximize the total transmission rate in each frame.
IiiB Impacts of Handover Delay
Handover delay occurs when the transmitter moves from one link to the other, which is the case in HybridType I Transmission Strategy. In this strategy, data transmission in one time frame is performed over the link that provides the maximum instantaneous transmission rate in that frame. Particularly, the transmitter switches from one link to the other or stay in the same link at the end of any time frame after comparing the instantaneous transmission rates in both links.
Now, given that denotes the duration of one single handover phase, let us initially assume that the frame duration is larger than the handover phase, i.e., . For the sake of simplicity, we divide the frame duration into subframes that are equal to , i.e., , where and . Particularly, a series of subframes of data transmission phase is followed by one subframe of handover process if the transmitter changes the transmission link, or by another series of subframes of data transmission phase if the transmitter stays in the same transmission link. For an analytical representation, we model the buffer activity at the end of each subframe as a discretetime Markov process. As shown in Fig. 3, we have states. The first states, i.e., {State 1,,State }, represent the data transmission subframes in the VLC link, and State represents the handover process from the VLC link to the RF link. Similarly, the subsequent states, i.e., {State ,,State }, represent the data transmission subframes in the RF link, and State represents the handover process from the RF link to the VLC link. Notice that the state transition probability from State to State is 1 for and for because the data transmission in each link is completed in subframes and the link change may occur at the end of the and subframes. On the other hand, in State , either the transmitter changes the link and the system enters State with probability , or the transmitter stays in the same link and the system enters State 1 with probability . Similarly, in State , either the transmitter changes the link and the system enters State with probability , or the transmitter stays in the same link and the system enters State with probability . Finally, the system moves from State to State and from State to State 1 with probability 1 because at the end of one handover phase, the transmitter starts data transmission.
As seen in (18), if , the transmitter sends the data over the RF link. Otherwise, it sends the data over the VLC link. Therefore, for this specific case, we have and . Then, we can express the transition matrix as
(29) 
where is the state transition probability from State to State . Then, we can recharacterize the logmoment generating function of the hybrid system for any , which is provided in (19), as follows [41, Chap. 7, Example 7.2.7]:
(30) 
where is the spectral radius of the matrix , and is a diagonal matrix whose components are the moment generating functions of the processes in states. Notice that the transmitted bits are removed from the transmitter buffer only at the ends of the and frames. Therefore, the moment generating functions are and in the and frames, respectively. However, there are no bits removed in the other states, i.e., the service rates in the other states are effectively zero. Hence, the moment generating functions are 1 in other states. Moreover, the unique QoS exponent, , is obtained when .
IiiC Nonasymptotic Bounds
The aforementioned results provide the performance analysis in the steadystate. Particularly, the analysis is obtained when the number of time frames is very large. On the other hand, nonasymptotic bounds regarding the statistical queueing and delay characterizations at the transmitter buffer are of interest for system designers as well. Therefore, we address the framework of Network Calculus [39, 42, 43], and consider [44, Theorem 2], which states that a minimal bound on the queue length can be found for a given buffer overflow probability. Particularly, given the RFbased data service process in Section IIA and the twostate Markov modeled data arrival process in Section IIC, the buffer threshold, , is expressed as
(31) 
for a given buffer overflow probability, , where
(32) 
and
(33) 
with
(34) 
Above, the buffer violation probability is , and and are free parameters. Notice also that the logmoment generating function in (32), i.e., , is timeinvariant because the service process is memoryless, whereas the logmoment generating function in (33), i.e., , is timevariant because the arrival process depends on its current state. Moreover, we remark that when goes to infinity, we have , where is expressed in (10). Herein, we refer to [44] for further calculation details in (32) and (33).
Likewise, when the VLCbased data service process in Section IIB is employed, we have
(35) 
and
(36) 
Notice that in (35) is minimized when . Moreover, because cannot be smaller than zero, we have and . Therefore, when the VLCbased service channel is chosen as the service process, we have
(37) 
Now, assuming a fast and stable handover mechanism between the transmitter and the receiver, and a service channel selection process as described in HybridType I Transmission Strategy, we can characterize the delay bound as follows: where
(38) 
and is given in (19).
Remark 3
Let us assume a firstcome firstserved protocol exists at the transmitter buffer. Then, the minimal bound on the buffering delay is expressed as follows [44, Theorem 1]:
(39) 
when the RFbased service process, or the VLCbased service process, or HybridType I Transmission Strategy is employed.
Remark 4
We consider a userrelated performance measure, i.e., the maximum average data arrival rate at the data buffer, and formulate the link selection by employing the transmission rates provided by the RF and VLC links. Our analytical framework can easily be extended to a more general multiuser scenario by regarding the rate allocations for each user in the transmission links and the receiveroriented data arrival processes at the transmitter buffer. In this regard, we refer to Fig. 7 in Section IV, where we employ frequencydivision multiple access (FDMA) and timedivision multiple access (TDMA) protocols for numerical illustrations. Moreover, our paper is different than [27, 13, 40, 45]. The system sum throughput is maximized in [27, 13], and the system average power consumption is minimized in [40, 45]. In these studies, a framework in which a joint resource allocation and link assignment process is employed is not provided, and the optimization problems are in principle mixed integer and nonlinear programming problems, which are mathematically intractable. Therefore, the main optimization problems are decomposed into solvable subproblems, and iterative algorithms are provided.
Remark 5
The link selection process can easily be adopted into scenarios where the receiver is mobile. All that one needs is to consider the logmoment generating function in the VLC link for changing transmission rates, i.e., , and base the channel selection process on the link that increases the maximum average data arrival rate at the transmitter buffer. In HybridType I Transmission Strategy and HybridType II Transmission Strategy, because the channel fading gains in the RF and VLC links are instantaneously known at the transmitter in each time frame, the aforementioned analysis will not be different than what we currently have in the paper even if the transmission rates vary due to mobility. We also note that the user mobility is normally low in indoor scenarios. In this regard, we refer to [46].
Iv Numerical Results
VLC System  

LED half intensity viewing angle,  PD field of view (FOV),  
PD physical area,  Channel bandwidth,  MHz  
PD opt.toelect. conversion efficiency,  A/W  PD optical concentrator gain,  
Vertical distance,  m  Noise power spectral density,  Hz 
RF System  
Channel bandwidth,  MHz  Path loss exponent,  
Rician factor,  dB  Lognormal standard deviation,  dB 
Ambient Temperature,  K 
In this section, we present numerical results that substantiate our theoretical findings. Unless otherwise specified, we set the transmission time frame to milliseconds, i.e., . We assume that the LED at the VLC access point of the transmitter has a Lambertian radiation pattern, and that the transmitter and receiver planes are parallel to each other. We further assume that the transmitter is directed downwards, while the receiver is directed upwards, as depicted in Fig. 4. However, our theoretical results can easily be adopted into different positional settings. Herein, the lineofsight channel gain, i.e., , is given as follows [47]:
(40) 
where is the surface area of the photodiode, is the distance between the LED at the transmitter and the photodiode at the receiver, and is the normal distance between the transmitting and receiving planes. Moreover, is the optical concentrator gain at the photodiode, is the angle of incidence, and is the field of view, i.e., the maximum angle at which the light emitted by the LED is detected by the photodiode. In addition, is the Lambertian index, where is the LED half intensity viewing angle, and if and otherwise. Finally, the thermal noise power at the photodiode is , where is the noise power spectral density.
Regarding the RF channel, we consider a Rician fading distribution with the Rician factor, , where the channel realizations, , are independent and identically distributed, circularly symmetric complex Gaussian random variables with mean and variance . Setting to a reasonable value, we can reflect the channel characteristics in millimeter wave range communications as well [2]. Here, is the largescale path loss in decibels as a function of the distance between the RF access point at the transmitter and the RF frontend at the receiver, , and it is given by [48]
(41) 
where dB is the path loss at a reference distance, m, and an operating frequency of GHz. In addition, is the path loss exponent and represents the shadowing effect, which is assumed to be a zeromean Gaussian random variable with a standard deviation expressed in decibels^{6}^{6}6Empirical values of , , and in different indoor scenarios were provided in [48, 49, 50]. For instance, the value of ranges from to , while varies between dB and dB inside buildings [48].. Finally, the thermal noise power at the RF frontend of the receiver is , where is the Boltzmann constant and is the ambient temperature [14]. Table I summarizes the simulation parameters, unless otherwise stated. Finally, setting the average transmission power constraint to in all the transmission strategies, we define as the averagetopeak power ratio in the RF and VLC links, i.e., .
Iva Transmission Strategies
We consider the scenario in which a transmitter has a VLC access point and an RF access point, as shown in Fig. 1. The receiver is located at a distance m from the VLC access point, i.e., the user is located at a horizontal distance m from the cell center, where and m, and at a distance m from the RF access point. In Fig. 5, we plot the maximum average data arrival rates at the transmitter buffer, and , as functions of the average power constraint, , with different averagetopeak power ratios, i.e, , when the VLC and RF links are employed, respectively. We have the results for in Fig. 5(a) and for in Fig. 5(b). We observe that the maximum average data arrival rates increase faster with the increasing average power constraint in the VLC link than they do in the RF link. For instance, when and , the maximum average data arrival rate increases more than bits per frame with increasing from dBm to dBm in the VLC link, whereas it increases bits per frame in the RF link. We observe the same behavior when the peak power constraint increases. We can explain this result with the stochastic nature of the transmission rates in the RF channel. Particularly, when the instantaneous transmission rate in the RF channel becomes very low, more data packets are accumulated in the transmitter buffer. Therefore, in order to sustain the QoS constraints, the transmitter buffer should accept data at lower arrival rates. On the other hand, because the transmission rate in the VLC channel is constant, the maximum average data arrival rate increases almost linearly with the increasing transmission rate in the VLC channel. Moreover, the performance in the RF link is better than the VLC link when the average power constraint is lower, and the performance in the VLC link is better than the RF link when the average power constraint is higher.
In Fig. 6, we plot and as functions of the QoS exponent, , for different values of and when . We have dBm in Fig. 6(a) and dBm in Fig. 6(b). The performance in the RF link is better than the VLC link when is low, whereas the performance in the RF link decreases faster with increasing and becomes less than the performance in the VLC link. In other words, the stochastic nature of the RF channel prevents the RF link from supporting data arrival rates at the transmitter buffer when the QoS constraints are stringent. Indeed, the maximum average data arrival rate that the RF link supports approaches zero exponentially with increasing regardless of the average power constraint and the source statistics, i.e., and . However, the RF link can support higher data arrival rates if the QoS constraints are looser. We further observe that increasing (or decreasing ), results in better performance values in the VLC and RF links because the steadystate probability of the ON state, i.e., , and the average arrival rate at the transmitter buffer increase. However, the effect of the source statistics on the performance values is much less than that of the QoS constraints, especially in the RF link. In other words, the randomness in the service process has a higher impact on the system performance than the randomness in the arrival process has.
In typical indoor scenarios, VLC and RF access points serve multiple receivers. This is applied by sharing available resources (i.e., power, time, and bandwidth) among the served receivers. Herein, we assume that the transmitter employs the commonly known FDMA and TDMA schemes in both links. In Fig. 7, we plot the maximum average data arrival rate per user given that all the users are uniformly positioned within the coverage area of the VLC access point. We observe that the performance per user in the RF link is generally much higher than the performance per user in the VLC link when the number of receivers is above 4. Basically, the system can serve more users in the RF link than the VLC link when the QoS constraints are of interest. In addition, the results in Fig. 7 agree with the results in Fig. 5, where the performance of the VLC link is highly affected by the decreasing average power constraint. We finally see that with the decreasing LED viewing angle, the performance in the VLC link becomes better because the transmission power is concentrated in smaller areas. Notice also that the VLC channel gain is affected by through the Lambertian index. Herein, we show the performance sensitivity of the RF and VLC links to the allocated transmission resources such as power, time, and bandwidth, given that the available resources are equally shared among the users. We also show that our framework can easily be invoked in a multiuser scenario.
We further explore the system performance with respect to the receiver location. We set as the Cartesian coordinates of the VLC access point, as the coordinates of the RF access point, and as the coordinates of the receiver, where . Particularly, we consider the following strategies:

RFonly Strategy: The transmitter sends data over the RF link only, and the maximum average arrival rate, , is expressed in (III).

VLConly Strategy: The transmitter sends data over the VLC link only, and the maximum average arrival rate, , is expressed in (III).

Hybridtype I Transmission Strategy: The transmitter sends data over the link that provides the highest transmission rate, and the maximum average arrival rate, , is expressed in (20).

Hybridtype II Transmission Strategy: The transmitter sends data over both links simultaneously following a power allocation policy to maximize the transmission rate, and the maximum rate, , is expressed in (25).
In Fig. 8, we plot the maximum average arrival rate as a function of the and coordinates of the receiver location for different QoS constraints, where and . As seen in Fig. 8(a) and Fig. 8(e), the position of the receiver does not impact the performance levels in the RF link necessarily, i.e., the performance level stays almost constant when the receiver stays in the defined range. However, the performance levels in the other strategies are affected by the position of the receiver, and the maximum average data arrival rate increases as the receiver gets closer to the point because the constant transmission rate from the VLC access point to the receiver increases and the stochastic nature of the RF link is mitigated more with the increasing rate in the VLC link. As seen in Fig. 8(b) and Fig. 8(f), the maximum average data arrival rate goes to zero in VLConly Strategy as the receiver goes out of the coverage area of the VLC access point. Similarly, as seen in Fig. 8(c), Fig. 8(d), Fig. 8(g) and Fig. 8(h), the maximum average data arrival rate becomes equal to the one in RFonly Strategy as the receiver goes out of the the coverage area of the VLC access point. Furthermore, Hybridtype I Transmission Strategy provides higher performance levels than RFonly Strategy and VLConly Strategy do because the transmitter, employing Hybridtype I Transmission Strategy, sends the data over the RF link when the instantaneous transmission rate in the RF link is higher than the rate in the VLC link, and mitigates the lower transmission rates in the RF link by sending the data in the VLC link. Finally, Hybridtype II Transmission Strategy outperforms all the other strategies. However, the performance gap between Hybridtype I Transmission Strategy and Hybridtype II Transmission Strategy is not necessarily large. Hence, it is more advantageous to employ Hybridtype I Transmission Strategy in order to avoid the hardware complexity that follows the addition of multihoming capability in Hybridtype II Transmission Strategy.
In Fig. 9, assuming that the handover process causes a transmission delay, where the handover process takes seconds for and , we plot the maximum average data arrival rate in Hybridtype I Transmission Strategy, , as a function of considering different user locations. Noting that smaller means a longer handover period, we observe that the transmission performance is highly affected by the handover process. With increasing , the performance levels approach the values that are obtained when there is no handover delay. Moreover, the maximum average data arrival rates are higher in Fig. 9(a) than in Fig. 9(b), because the constant transmission rate in the VLC link is higher when the user is at the center. Subsequently in Fig. 10, we plot the maximum average data arrival rates as functions of the vertical distance between the VLC access point and the receiver. We set the position of the receiver to , i.e., the horizontal distance to the VLC cell center is m. When m, the performance level in the VLC link is zero because the cell area is very small and does not cover the point where the user stands, i.e., m, and is the cell radius. The performance levels in all strategies except RFonly Strategy increase up to a value, and then decrease with increasing . This is because the increase in the LED viewing angle is relatively less than the increase in at the beginning. Therefore, the user is effectively getting closer to the cell center and having more rate in the VLC link. In other words, the gain achieved by getting closer to the cell center is higher than the expected degradation due to the increasing cell radius. However, with increasing beyond a certain value, the user gets far away from the VLC access point, and hence, the radiated power spreads over more area, which eventually leads to decreased transmission rates in the VLC link. Therefore, the gain in the VLC link vanishes as the distance to the VLC access point becomes larger.
IvB Nonasymptotic Delay Bounds
In the aforementioned results, we analyze the system performance in the steadystate. In the following, we provide results regarding the nonasymptotic bounds, i.e., the bounds on the buffering delay experienced by the data in the transmitter buffer. Particularly, we plot the delay bound as a function of the state transition probability from the OFF state to the ON state in the data arrival process, , for different values, where is the state transition probability from the ON state to the OFF state in the data arrival process. We set the average data arrival rate, , to a value very close to the average data service (transmission) rate in the transmission channel. We note that the average data service rate in the transmission channel depends on the chosen transmission strategy. The delay bound is the highest in RFonly Strategy as seen in Fig. 11(a), whereas it is the lowest in VLConly Strategy as seen in Fig. 11(b). However, the arrival rate that RFonly Strategy supports is higher than the rate that VLConly Strategy supports. More interestingly, the hybrid strategies can support higher arrival rates with less delay bounds, and Hybridtype II Transmission Strategy outperforms all the others. Herein, the system takes advantage of the occasional higher rates in the RF links, and mitigates the lower rates in the RF link by the constant transmission rate in the VLC link. Moreover, increasing and decreasing cause the delay bound to increase.
Finally, we explore the effects of the data arrival rate, , on the delay bound performance in Fig. 12. We set and , and consider different average power constraints, i.e., , , and dBm. The delay bounds increase asymptotically as the average arrival rate approaches the average data service rates in the channels, because when the average data arrival rate is greater than the average data service rate in one channel, the system becomes unstable, and long buffering periods are expected. Moreover, as seen in Fig. 12(b), the delay bounds are the minimum in VLConly Strategy but the ranges of the average data arrival rates are smaller than the other strategies, and as seen in Fig. 12(a), the delay bounds are the maximum in RFonly Strategy. However, the hybrid strategies again outperform the others i.e., the hybrid strategies take advantage of the VLC link when the rate in the RF link goes down drastically, and utilize the RF link when the channel conditions are better again. While the VLC link provides stability and decreases the delay bounds, the RF link increases the range of average data arrival rate that can be supported. Finally, in Fig. 12 we see that increasing the average power can potentially improve the system performance in terms of the buffering delay.
V Conclusions
In this paper, we have analyzed the performance of a hybrid RF/VLC system when statistical QoS constraints are inflicted as limits on the buffer overflow and delay violation probabilities. We have provided a crosslayer analysis regarding physical and datalink layers by employing the maximum average arrival rate at the transmitter buffer and the nonasymptotic delay bounds as the main performance measures. We have proposed and analyzed three strategies in which RF and VLC links are utilized for data transmission. We have further formulated the performance levels achieved by each of the proposed strategies. Through numerical results, we have shown that RF technology can be beneficial when there are lower average power constraints and/or looser QoS requirements. Moreover, we have shown that utilizing the VLC technology for data transmission, either alone or in a hybrid transmission strategy, can potentially enhance the system performance in terms of delay performance. It lowers the buffering delay bounds, when compared to the RF technology. Particularly, when data arrival rates at the transmitter buffer is low, VLC links provide lower queueing delays than RF links do, but RF links support higher data arrival rates at the transmitter buffer.
a Derivation of in (Iii)
Recall that is the data arrival rate at the buffer when the data source is in the ON state, and that is the steadystate probability of the ON state. Thus, the average data arrival rate is . Moreover, we have in steadystate, i.e.,