Buffer-Aided Max-Link Relaying for Free Space Optical Communication With Delay Constraints

In this paper, we introduce the concept of buffer-aided cooperative relaying for a free-space optical (FSO) communication system. The proposed FSO system consists of <inline-formula><tex-math notation="LaTeX">$K$</tex-math></inline-formula> decode-and-forward(DF) relays, each equipped with a buffer of size <inline-formula><tex-math notation="LaTeX">$L$</tex-math></inline-formula>. The proposed system undergoes both path loss and atmospheric turbulence (AT) induced fading, and to evaluate the system performance a weak, as well as strong AT region is considered. The Markov chain (MC) approach is used to derive the state transition matrix of the system, which is then further used to model the evolution of buffer states. Analytical expressions of the outage probability and average bit error rate (ABER) are obtained with the help of state transition matrix. The average packet delay of the system is also evaluated with the help of obtained outage probability. The performance of the considered system is analyzed for different values of <inline-formula><tex-math notation="LaTeX">$K$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$L$</tex-math></inline-formula>. The performance is then further compared with the non-buffer-aided (NBA) cooperative relaying for FSO communication systems and it has been observed that the proposed buffer-aided FSO system significantly outperforms the NBA FSO systems for entire range of turbulence. Finally, the impact of <inline-formula><tex-math notation="LaTeX">$L$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$K$</tex-math></inline-formula> on the average packet delay of the system is examined.

areas and offers many advantages over traditional radio frequency (RF) networks, still its usage is limited because of its inferior performance over the links which are far apart from each other. In long-range communications, the performance of FSO degrades mainly due to an atmospheric turbulence (AT) and extreme weather conditions. To, overcome the effects of fading occurring due to AT and extreme weather conditions many fading mitigation techniques like maximum likelihood sequence estimation [2], error-correcting codes [3], and cooperative relaying [4], [5], [6] have been proposed in the literature. Among all the proposed fading mitigation techniques, cooperative communication has gained significant attraction due to its simplicity in implementation [7], [8], [9].

A. Literature Survey
Cooperative communication is a significant tool for obtaining spatial diversity among wireless networks [10]. It is the technique that provides spatial diversity by forming virtual arrays of antenna. In traditional cooperative systems, a source (S) node transmits information to the destination (D) node via a relay (R) node [11], [12] using decode-and-forward (DF) or [13] amplify-and-forward (AF) relaying [14]. In DF relaying R will decode the signal received from S and then forward it to the D while in AF relaying R will amplify the signal received from S then and forward it to the D. To further enhance the performance different configurations in cooperative systems have been proposed [15], [16], [17]. Incremental selective DF protocol for multiple relay systems is given in [15], whereas incremental AF relaying protocol for such system is given in [16], [17]. For FSO relaying networks, DF and AF relaying protocols are proposed in [18] and [19], respectively. In [18], authors discussed the adaptive DF protocol for a log-normal fading scenario, while in [19], authors discussed the AF relaying set-up for the system affected by pointing errors. Many authors have also proposed the cooperative relaying for an RF/FSO based hybrid set-up, in this set-up, one of the links (either S-R or R-D link) is the RF link while the other one is the FSO link. Various scenarios like outdated channel state information (CSI) [20], co-channel interference [21], and pointing errors [22] have been discussed in the literature for the RF/FSO based hybrid set-up.
In a traditional non-buffer-aided (NBA) relaying setup, there is fixed scheduling of data packets to and from the R irrespective of the channel quality [23], [24]; to overcome this drawback, various buffer-aided cooperative systems have been studied in the literature [25], [26], [27], [28], [29], [30], [31], [32]. In [27], This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ a max-max buffer-aided relaying protocol is given, which helps with achieving the full spatial diversity. A max-link selection protocol, which selects a link with the highest channel gain for transmission or reception in a time interval, is discussed in [28]. This protocol can achieve the diversity order of twice the number of relaying nodes, but it results in additional packet delay. A buffer-aided relaying, using relay selection based upon buffer states is provided in [29]. In [30], authors have discussed a scheme that uses the virtues of both max-max and max-link selection protocols. In [31], a cooperative buffer-aided system is proposed in which S-R link is only considered for transmission when R-D link goes into the outage; this helps with reducing the packet delay, at the cost of diversity gain. In [32], authors have proposed a priority based max-link selection protocol, in which transmission or reception is prioritized based upon the buffer status and channel strength.
Few more buffer-aided set-ups for RF relaying considering different configurations like full-duplex (FD) [33], [34] relaying, non-orthogonal-multiplexing-access (NOMA) [35], [36], [37], [38], and system impairments [34], [39], [40], [41] have been discussed in the literature. In [33], the authors considered an FD buffer-aided relaying where only statistical CSI is available at each node. A new relay selection mechanism is developed which only relies on statistical CSI and it then performs the power adaption. An FD buffer-aided system with a relay node suffering from self-interference is given in [34]. In [35], the authors considered a NOMA set-up, in which data is transmitted to the two destinations using NOMA architecture. A relay selection based buffer-aided NOMA is proposed in [36]. In [37] and [38] FD buffer-aided relaying NOMA set-up has been studied. Buffer-aided relaying set-up with outdated CSI is discussed in [39], while in [34], the authors discussed a single relay buffer-aided system in which the relay suffers from self-interference. The authors in [40], dealt with the security aspect of the buffer-aided cooperation and provided a max-ratio link selection scheme, by considering the ratios of all available links in the system. Similarly, in [41], the security aspect of the buffer-aided system is analysed for energy harvesting relays.
When it comes to FSO set-up, [42] and [43] have discussed the buffer-aided relaying for the FSO cooperative set-up. In [42], authors discussed the RF-FSO hybrid set-up, where one buffer is with the FSO unit while the other is shared between FSO and RF units. Authors in [43] proposed the buffer-aided relaying for FSO set-up, where buffers are used only for inter relay cooperation. However in both the set-ups [42], [43], there is no discussion of the link selection based upon the buffer status and link quality. Hence, the advantage of the buffers is not properly utilized in the considered set-ups, which leads to inferior performance.

B. Research Gaps
It is well-known that the FSO links mainly suffer due to AT. The AT is a random phenomenon and highly unpredictable; a high AT can cause the FSO link to go into an outage and interrupt communication. The outage events are more prominent in long-distance FSO links of the order of 1-5 Km. The cooperative FSO systems aim at increasing the range of FSO communication by employing relaying nodes in between the S and D. The simplest cooperative link relies on two cascaded FSO links, also called FSO hops. Both the FSO links/hops get affected by the AT independently, and if one of the two FSO hops observes high AT, then the communication between the S and D gets disrupted.
In the long-distance FSO communication systems, the received signal-to-noise ratio (SNR) is not good to attain a satisfactory error performance and hence quality-of-service (QoS). Therefore, multiple cooperative paths are used in parallel for exploiting the multiple-input multiple-output (MIMO) gains like diversity order and coding gain. However, due to AT we will not be able to exploit the MIMO gain out of multiple relay-based FSO cooperative systems as at any given moment few hops may be in an outage, leading to unsatisfactory performances of their corresponding cooperative links. Hence, the desired MIMO gains may not be attained out of a well-designed practical FSO relaying system. For such a scenario, where the relaying system goes into outage because of the failure of a single link facing extreme AT, the introduction of a buffer into the system might come in handy, as it will help us in selecting the best available link for transmission or reception of data; thereby the effect of AT can be countered. However, in literature, there is no discussion on the selection of max link for the buffer-aided relaying based set-up for FSO networks, even for the simplest case of a single relay having buffer size L.

C. Novelty and Contributions
The introduction of buffers in an RF cooperative system leads to a performance enhancement of the system in terms of throughput, outage, and average bit error rate (ABER). However, when it comes to FSO systems, except for [42] and [43], there is no discussion of buffers in FSO systems. In [42], authors used buffers in a mixed RF-FSO scenario, while in [43] buffers are used only for inter-relay communication. Hence, both [42] and [43] are using buffer in a very limited manner, therefore, the best available FSO link is not being utilized at any given moment, thereby it leads to an unsatisfactory error performance and poor reliability.
Motivated by this, in the proposed system, we are employing buffers in the relays and then further selecting the best available link through the max-link selection protocol. With the help of buffers and max-link selection protocol, we aimed to overcome the detrimental behavior of AT over the FSO relaying systems. Moreover, by using the buffer, we will improve the error performance and as a result the reliability of the FSO relaying system also increases.
The main contributions of the paper can be highlighted as: r We study the generalized buffer-aided FSO relaying set-up employing multiple relays K equipped with an arbitrary L sized buffer.
r We propose a max-link selection protocol for buffer-aided FSO relaying, catering exclusively to the attributes of FSO communications such as peak power and eye-safety constraints. r The effect of the number of relaying nodes and buffer size over the average packet delay is carried out in detail. Furthermore, the trade-off between the communication outage and average packet delay is studied for providing a better understanding of the proposed FSO set-up. The rest of the paper is organized as follows: Section II presents the proposed system model for the K buffer-aided DF cooperative FSO network, and the channel model and link selection criteria for the given set-up are also discussed. Section III analyzes the system performance in terms of outage probability and ABER for strong and weak AT regions. A brief discussion about the average packet delay for the system has been provided in Section III. Furthermore, Section IV demonstrates the numerical results for the proposed system model. Finally, Section V concludes the paper.

II. SYSTEM MODEL
The proposed FSO based buffer-aided cooperative system is shown in Fig. 1. The given system consists of a S node, K DF relay nodes, R k , k ∈ {1, 2, . . . , K} and a D node. For S, we have assumed that there exists K laser transmitters pointing toward each of the relay nodes, also S doesn't have any laser transmitter pointed towards D, hence the direct link is assumed to be absent between S and D. All the relay nodes are equipped with a receive aperture pointing towards S, and a transmit aperture pointing towards D. Moreover, for the given system model, we have assumed that all the relays are placed between S and D. Let us assume that for any relay R k , S to R k distance is denoted by d SR k , similarly, R k to D distance is denoted by

A. Buffer-Aided Relaying
For the given FSO set-up we have assumed that each relay node is equipped with a buffer of size L and the given buffer for any relay node R k is denoted by Q k , at any given instant, the number of data packets in any buffer Q k is denoted by Ψ(Q k ), 0 ≤ Ψ(Q k ) ≤ L. In the beginning, it is assumed that the buffer of all the relay nodes is empty and it gets incremented by one with the arrival of the data packet while it is getting decremented by unity with the departure of a data packet. In general, the protocol for transmission or reception of data through any relay R k is given by: 1) If the buffer status is empty, i.e., Ψ(Q k ) = 0 then R k will receive a data from S, when the electrical signal-to-noise ratio (SNR) for the given S-R k link is greater than a target SNR threshold of γ th . 2) If the buffer status is full, i.e., Ψ(Q k ) = L then R k will forward the modulated data to the D if the electrical SNR for the given R k -D link is greater than a target SNR threshold of γ th . 3) If the buffer status is partially filled, i.e., 0 < Ψ(Q k ) < L then the link (either S-R k or R k -D) having the highest SNR will be selected. Overall, a relay is allowed to transmit or receive data bits when it has its buffer empty.

B. Channel Model
To model the channel, we have considered both AT induced fading and path loss. The path loss of the system is based upon the link distance d i and using [9, eq. (2)] it can be written as where A T and A R are transmitter aperture area and receiver aperture area respectively, σ is the attenuation coefficient which depends upon the visibility, and λ is the optical wavelength of the system. The AT is modeled as follows: 1) Log-normal fading: Let us assume the fading amplitude as h j = exp(Γ j ), using [44, eq (3.112)] it can be described with the help of log-normal probability density function as where Γ j is a Gaussian distributed random variable with a mean μ Γ j (d j ) and variance σ 2 . The given variance can be evaluated with the help of wave number (k), link distance d j and refractive index structure coefficient (C 2 n ) as per [45, eq. (3.109)] as Also, it is assumed that the fading is not attenuating or amplifying the average power of the proposed system, hence we can is the expectation. 2) Negative exponential: Let us assume that the fading amplitude is h j then it can be described for negative exponential fading as where γ j is the average irradiance of the channel.

Remark 1:
In the proposed set-up, we try to demonstrate the benefits of buffer-aided relaying in FSO system. Hence, we will study the impact of buffers in the extremely harsh AT scenario of saturation regime leading to negative exponential fading. Also, in order to check the significance of the buffers over other AT regimes, we have only selected weak AT case leading to lognormal fading scenario.
The proposed system adopts intensity modulation direct detection (IM/DD) based ON-OFF keying (OOK). For the given OOK scheme, if there exists a signal it is denoted by bit 1, while absence of the signal is denoted by bit 0. For any relay R k the received signal is given by where n R k is the additive noise at R k having Gaussian distribution with zero mean and N 0 /2 variance, P SR k is the power allocated to source node, R is the responsivity of the photodetector and it is given as R = ηq/Hf with η as the quantum efficiency of the photodetector, H is the Plank's constant of the photodetector, q is the electron charge and f is the optical frequency. Also, the term g SR k is the channel gain for the SR k link and can be mathematically given by where h SR k is the AT induced fading and L SR k is normalized path loss and can be written as After receiving the signal, R k will either store the data or it will decode it and then modulate it as OOK and forwards it to D as where n D is the additive noise at D and is having a Gaussian distribution with zero mean and N 0 /2 variance, P R k D is the power allocated to relay node, g R k D is the channel gain for R k D link and it can be mathematically given as where h R k D is the AT induced fading and L R k D is the normalized path loss and is given by The expression for electrical SNR of any given link will be The obtained expression of SNR will be further used for the max-link selection.

C. Max Link Selection Scheme
In conventional NBA relaying, for the selection of a link, max-min criteria is generally used which is given by For the given link selection mechanism, the system is getting a diversity of K, i.e., number of relay nodes present in the system. For the buffer-aided relaying set-up, a user gets a liberty of selecting either S-R or R-D link based upon the quality of the links and buffer status hence, for such systems a max link selection protocol is proposed which is defined as By employing the max link selection protocol, a user will get a diversity of twice the number of relay nodes present in the system, hence, with the help of buffer and max link selection protocol cooperative systems had been able to double the diversity. It is noteworthy over here that the proposed max-link scheme is selecting the best available link for transmission/reception at each time slot, thereby helps us in achieving an optimal bit rate.

III. PERFORMANCE ANALYSIS
This section covers the performance analysis of the proposed system. Firstly, with the help of the Markov-chain (MC) approach the state transition matrix (A) of the system is obtained, the obtained state transition matrix is then later used for obtaining the outage probability and ABER of the proposed set-up. At any given time instant, the number of data packets in relay buffers forms a state of MC; for the considered K relay system with the buffer of size L, the total number of states in A will be (L + 1) K , with any state s b given by Next, we will obtain the steady state probability by evaluating the transition probability of one state to another.

A. Outage Analysis
To evaluate the outage performance of the proposed system, we have to find the state transition matrix A of the dimension (L + 1) K × (L + 1) K . In the matrix A, A ab denotes the a, b-th element of the matrix, such that a, b ∈ {1, 2, . . . , (L + 1) K }, A ab also denotes the probability of transition from state s b to s a , such that it can be mathematically given by During the period when no transmission is taking place due to the outage, no change will take place in the state transition, then such element is denoted by A bb . 1) Asymmetric Channels: Let us assume that S-R and R-D are asymmetric channels, i.e. unequal power allocation and independent but non-identical distributions. The term M SR and M RD are number of S-R and R-D links available for the communication. The probability of successful transmission across the link having maximum channel gain among S-R links is given by where γ SR is the SNR of the selected max-link among the available S-R links. Substituting the expression of γ SR into (15) and then further simplifying we get (16) where and is known as power margin for S-R link, here, P S denotes the power allocated to S. Similarly, the probability for unsuccessful transmission will be Furthermore, the probabilities for successful and unsuccessful transmissions over the best R-D link are given as and where and is known as power margin for R-D link, here, P R denotes the power allocated to the selected relay node. The terms h SR and h RD are the fading coefficients for the system which depends upon the AT.
Log-normal fading: Assuming h SR to be a log-normal distributed random variable with 2μ Γ (d SR ) mean and 4σ 2 Γ (d SR ) variance, the probability for successful transmission over S-R link will be where Q(·) denotes the Gaussian q-function. From (16), the probability for unsuccessful transmission is given by Similarly, the probabilities for successful and unsuccessful transmissions over R-D links can be given as and Next let us take X as the link having the highest channel gain among the available S-R links, then the cumulative density function (cdf) of X is given by On further solving (24) we get the final expression of cdf as Differentiating (25) with respect to x we get .
Let Y be the link having highest channel gain among R-D links, then the cdf and probability density function (pdf) of Y can be derived by using (25) and (26) as and Now, since, X and Y are independent we can write Now, the probability of selection of link X for transmission is given by where Similarly, the probability of selection of Y for communication is denoted as where For the log-normal case (29) and (31)  Negative exponential fading: For negative exponential fading, the probabilities of successful and unsuccessful transmissions over the S-R channel is given by (33) and Similarly, for R-D channel we have and For the given case, the cdf of X is given b From, (33), (34) and (37) we get The pdf of X can be obtained by binomially expanding the cdf in (38) first and then differentiating it with respect to x as Similarly, the cdf and pdf of Y can be obtained by using (38) and (39) as and Now, probability of selection of link X is given by where Similarly, probability of selection of Y is denoted as where For the negative exponential case (42) and (44) (46) 2) Symmetric Channels: Let us assume that S-R and R-D are symmetric, i.e., equal power allocations and independent and identical distribution. Assuming that system has M available links for transmission or reception, the probability of successful transmission across the selected max link w is given by Substituting the expression of SNR from (11) into (47), and then further simplifying we get where is the power margin, since the power allocations are same hence the power margins will be same for both S-R and R-D channels. Likewise, the probability for unsuccessful transmission will be The term h w is the AT induced fading, for different fading scenarios we get the expressions for the probability of successful and unsuccessful transmission as given below: Log-normal fading: Assuming h w to be log-normal distributed random variable having mean 2 μ Γ (d w ) and variance 4σ 2 Γ (d w ), we get the expression of successful transmission through the selected best link as Similarly, the probability for the unsuccessful transmission is given by Negative exponential fading: Assuming h w to be negative exponential random variable having mean and variance unity the expression for probability of successful transmission through the selected max link can be given by Also, the probability of unsuccessful transmission through the selected best link for negative exponential fading is given by Considering a simple case of K = 2 and L = 2, the state transition matrix for symmetric channels can be denoted as (54) Moreover, for the asymmetric case, with the help of (21), (23), (29), and (31), the state transition matrix A is obtained for lognormal fading scenario, while for negative exponential fading A can be obtained with the help of (34), (36), (42), and (44). Similarly, for symmetric case, (50) and (51) can be used to obtain A for log-normal scenario, while for negative exponential fading A can be developed with the help of (52) and (53). With the help of A, the steady state probability of the proposed system model can be evaluated. The above developed matrix A exhibits the following properties: r Irreducible: We can transit to every state of the MC irrespective of the state from where we have started.
r Aperiodic: The matrix A possesses a non-zero probability to remain in any state between any time interval.
r Column stochastic: The sum of all the elements in any of the columns will add to unity. Due to aforementioned properties, the steady state probability of the system can be expressed as where π = [π 1 , π 2 , . . . , π (L+1) K ] T with π b = Pr(s b ), b = [1, 1, . . . , 1] T , I is the identity matrix of dimension (L + 1) K × (L + 1) K , and B is the all ones matrix of dimension of (L + 1) K × (L + 1) K . For the proposed relaying model, the system is said to be in outage when it remains in the same buffer state because all the links remain in outage. Considering all the buffer states and then using (55), the outage probability expression of the considered system is given by where diag(A) denotes the row vector of dimension (L + 1) K having all the diagonal elements of A.

B. ABER Analysis
While analyzing the ABER performance of the system we have assumed that the data is transmitted or received only when the instantaneous SNR of any link is greater than the pre-determined threshold value γ th . The instantaneous BER of the IM/DD based ON-OFF keying for max link w is given by where γ w is the SNR of the selected max-link using (11). Applying the Chernoff bound and then using (11) the above expression can be simplified to 1) Log-Normal Fading: Assuming that the system is undergoing log-normal fading then for the M available links the pdf for the max link w is given by On further evaluating the above expression we get Further solving the above expression we get Using (58) and (70) the expression for ABER for the negative exponential fading is given by Applying binomial expansion in the numerator, the above expression can be further simplified to Integrating the above expression we can obtain the final expression of ABER as Now, taking a simple case of K = 2 and L = 2, the error matrix of the proposed FSO system can be expressed as In the matrix E, we can observe that in case of outage, as no transmission is taking place, hence we have put that element to zero. Now, using the matrices E, π and b the overall ABER of the system can be obtained with the help of following expression In the above equation the terms E and π can be obtained from (74) and (55) respectively.

C. Average Packet Delay Analysis
Packet delay refers to the time gap when packet leaves the source and and reaches the destination. As we know for S-R transmission we require only one time slot hence, average packet delay of the whole FSO based set-up can be written as where D k is the average packet delay occurring due to the k-th relay node. Since all the relay nodes have been assumed to be identical hence, the average packet delay for all of them will be same. So, we will analyze the average packet delay for the k-th relay only. Now, using the Little's law [46] the average packet delay for the k-th relay node is expressed as where L k is the average queuing length and τ k is the average throughput. Queuing length for buffer-aided relaying system is expressed by As the probability to select any relay is same hence the average throughput for relay R k is given as where τ is the throughput and with the help of [47] and [48] it can be expressed as where R is the data rate for the given system, since the transmission of data packet from source to destination takes place in two phases hence, we will have R = 1/2. Now, using (78), (79) and (80), (77) can be expressed as Substituting D k from (81) into (76), the final expression of D can be expressed as Substituting the values of P out , ψ l , K and L the average packet delay for the system can be evaluated.

IV. NUMERICAL RESULTS
In this section, we obtain the numerical results for the system in case of log-normal and negative exponential fading scenarios. The outage probability and ABER of the system are obtained and it is verified with the help of the Monte Carlo simulation of 10 6 samples. For the numerical calculations, it is assumed that the system is operating in clear weather with visibility up to 8 Km, the optical wavelength used for the system is λ = 1550 nm. The value of C 2 n is taken as 1 × 10 −14 and atmospheric attenuation is 0.43 dB/Km, i.e., σ ≈ 0.1. All the relay nodes are placed in between the source and destination such that every relay can be assumed to be equidistant from source and destination. Also, S-D distance is taken as 6 Km, i.e. d SD = 6 Km, and the predetermined SNR threshold is taken as 3, i.e. γ th = 3. Fig. 2 depicts the plots for outage probability against the total power of the system for a negative exponential fading scenario wherein the number of relay nodes are varied. We have fixed the buffer size of all the relay nodes to 3, i.e., L = 3  for all relay nodes, the value of K is varied from 2 to 4. The outage probability of the proposed system is obtained and then it is compared with the direct link scenario as well as NBA cooperative relaying scenario. It can be seen from the plot that the proposed system is outperforming the NBA system and a performance gain of approximately 5-7 dB is observed. This improvement in the outage performance of the system is because of the introduction of the buffer in the FSO system. Fig. 3 illustrates the outage probability versus total power for different values of K, when the system is undergoing lognormal fading. The number of the relay nodes in the system is varied from 2 to 4, while the buffer size of each relay node is kept at 3, i.e., L = 3. It can be observed from the Fig. 3 that outage performance keeps on improving as we increase the value of K and P T . The outage performance of the proposed system is also compared with that of the direct (i.e. system having no relay nodes) and NBA cooperative relaying system, and it can be observed from the plots that the proposed buffer-aided system outperforms the NBA system and the performance gain of approximately 2-3 dB is observed for different values of K. It must be noted down that due to the lack of sample size, we have  run Monte Carlo simulations only for 10 6 samples and hence beyond 10 −6 the simulation results are not matching with the analytical one.
In Fig. 4, we have shown the simulated results for the outage performance against the total power of the system for Gamma-Gamma fading scenario. The value of L is fixed at 3, while K is varied from 2 to 4. The values of parameters m 1 and m 2 is taken as per [45, Fig. 3.30] and they have been fixed at m 1 = 4 and m 2 = 1.9 (moderate turbulence region). In order to have a fair comparison we have also simulated the outage performance for NBA scenario and it can be observed from the plots that the proposed system is outperforming NBA relaying scenario for entire range of power, and a performance gain of approximately 4 dB has been observed.
Remark 2: Due to the mathematical complexity we have not analyzed the system for the Gamma-Gamma fading, however in order to provide the system performance for all turbulence regions, we have provided the simulated results for Gamma-Gamma fading for the moderate turbulence region. Fig. 5 shows the plots for outage probability against total power for the negative exponential fading scenario under the  assumption of asymmetric channel. The total power (P T ) is distributed among S and selected relay R as P S = δP T and P R = (1 − δ)P T , respectively. The value of the buffer size is fixed as 2, while the variances of the S-R and R-D channels have been fixed as γ SR = 1 and γ RD = 2, respectively. The plots are obtained for different values of K, and it is observed that the system performance keeps on improving as we increase the value of K. It can also be observed that the performance for equal power allocation is slightly better than that of unequal power allocation, hence, it can be deduced that as we move towards the symmetricity of the channels the performance will keep on improving for same value of K and variances.
Plots for outage probability against buffer size are shown in Fig. 6, for the given figure we have assumed negative exponential fading. We have obtained results for different values of P T and K. From the figure, it can be seen that the performance of the system improves for lower values of L, while for larger values of L the performance saturates, for all the values of K and P T . This observation is because of the fact that when we increase the buffer size the number of states in the transition matrix increases which results in the increase of partially filled states  (i.e. states having the lowest outage probability) and thereby the overall outage probability of the system improves. However, this improvement is minimal for higher values of L and hence for higher L the outage performance almost saturates. Also, as we increase the value of K from 2 to 3 the outage performance improves for the same power levels however the improvement in outage performance is much higher at higher power levels i.e., P T = 20 dB compared to lower power levels, i.e., P T = 10 dB.
Plots for ABER versus SNR for negative exponential fading are shown in Fig. 7. The buffer size for all the relay nodes is fixed as 3, i.e, L = 3. It can be observed from the plots that the ABER performance of the system keeps on improving with the increase in the value of P T , also with the increase in the value of K we are observing a better ABER performance at medium to high SNR range compared to that of lower SNR range. Furthermore, the performance of the proposed system is compared with the NBA system and an improvement of approximately 3 dB is observed. Similar plots of ABER versus SNR for log-normal fading is shown in Fig. 8, in log-normal fading too we observe the similar behavior as negative exponential fading, the system performance is getting improved as the power levels are increased and also with the increase in the value of K.
Plot for ABER versus L for negative exponential fading is shown in Fig. 9. The total power P T is varied from 10-20 dB,  also the number of relay nodes, i.e., K is varied from 2 to 3. It can be observed from the plot that the ABER performance of the system improves for lower values of L, while it is getting saturated as we increase the value of L. The above behavior is due to the fact that with increase in the buffer size, the number of states in the transition matrix increases, it results in the increase of partially filled states (i.e., states having the lowest outage probability) and thereby the ABER of the system improves. However, when the buffer size is increased after certain level the improvement observed is minimal and hence for higher L the outage performance almost saturates. Also, it can be seen that with the increase in the value of K, the improvement in ABER is small at low power levels as compared to high power levels. Fig. 10 shows the plots of average packet delay against K for a fixed power of P T = 15 dB. Plots are obtained for different values of buffer sizes (L = 5, 10, 15). It can be observed from the plot that the average packet delay varies linearly with K and also slope of the plot increases as we increase the value of L. This behavior is observed because as the value of K and L increases the total number of buffers (L × K) in the system increases and the data packet tends to stay more in the buffer as S-R link remains stronger for half of the times. A similar behavior is observed for the plot of average packet delay against L which is shown in Fig. 11. These behavior shows that the value of K and L must be chosen judiciously in order to have a system with lower average packet delay.

V. CONCLUSION
In this paper, we have introduced the concept of buffer-aided relaying for the FSO communication system. A novel FSO system having K DF relay nodes with each relay node equipped with a buffer of size L is proposed. The optical channel is modeled with both path loss and AT based fading and for the analysis, both log-normal as well as negative exponential fading is considered. The system performance is analyzed in terms of outage probability and ABER and the obtained outage and ABER results are plotted against the total power of the system. Moreover, the obtained results are examined for various values of K and L, and then it is compared with the NBA relaying system and the proposed system thoroughly outperforms it for all the values of L and K. The analysis regarding the impact of buffer size on the system performance for different relay combinations and power scenarios is also carried out. A brief discussion about the average packet delay for the proposed system has also been provided, and the impact of K and L on the average packet delay of the system is analysed.