Open Access

A new scenario of triple-hop mixed RF/FSO/RF relay network with generalized order user scheduling and power allocation

EURASIP Journal on Wireless Communications and Networking20162016:260

https://doi.org/10.1186/s13638-016-0751-9

Received: 11 June 2016

Accepted: 6 October 2016

Published: 28 October 2016

Abstract

This paper proposes and evaluates the performance of multiuser (MU) triple-hop mixed radio frequency (RF)/free-space optical (FSO) relay network with generalized order user scheduling. An important example on the applicability of this scenario is in cellular networks where two sets of various users are communicating with their own base stations (BSs) over RF links and these BSs are connected together via an FSO link. The considered system includes K 1 sources or users, two decode-and-forward (DF) relays, and K 2 destinations or users. The sources and destinations are connected with their relay nodes through RF links, and the relays are connected with each other through an FSO link. To achieve MU diversity, the generalized order user scheduling is used on the RF hops to select among sources and destinations. In the analysis, the RF channels are assumed to follow the Rayleigh fading model and the FSO channel is assumed to follow the Gamma-Gamma fading model including the effect of pointing errors. Closed-form expressions are derived for the outage probability, average symbol error probability (ASEP), and ergodic channel capacity. Moreover, in order to gain more insight onto the system behavior, the system is studied at the signal-to-noise ratio (SNR) regime whereby the diversity order and coding gain are provided and studied. The asymptotic results are used to obtain the optimum transmission power of the system. Monte Carlo simulations are given to validate the achieved exact and asymptotic results. The results show that the diversity order and coding of the proposed scenario are determined by the worst link among the three links. Also, results illustrate the effectiveness of the proposed power allocation algorithm in enhancing the system performance compared to the case with no power allocation.

Keywords

Mixed RF/FSO/RF relay networkGeneralized order user schedulingMultiuser diversityRayleigh fadingGamma-Gamma fadingPower allocation

1 Introduction

The free-space optical (FSO) communication has been recently proposed as an efficient means to deal with the “last-mile” problem in wireless networks [1]. In such systems, the data transmission takes place between an optical transmitter and a receiver located, for example, on high buildings, separated by several hundred meters. Having the ability to operate on unlicensed optical beams and the potential for providing broadband communication capacity, the FSO communications represent a cost-effective alternative and/or a complement to radio frequency (RF) counterparts. In addition, features such as high security, flexibility, rapid deployment time, and rigidity to RF interference have made FSO communications appealing for emergency situation recovery and military applications [2].

Cooperative relay networks have recently attracted the attention of many researchers as an efficient solution for the multipath fading problem in wireless communications [3]. Using relays in wireless networks aims to provide diversity, widen the coverage area, and reduce the need for high-power transmitters. In such networks, a relay node or a set of relays help a source node in sending its message to the destination via either an amplify-and-forward (AF) scheme or a decode-and-forward (DF) scheme. Despite its requirement for more signal processing, the DF relaying protocol gives better results compared to the AF protocol, especially at low signal-to-noise ratio (SNR) values. Recently, a mixture of relay and FSO networks has been introduced in the literature aiming to increase the coverage distance of FSO networks which is usually limited to a few hundred meters in realistic conditions due to atmospheric turbulence condition effects [4]. In such networks, the source message is transmitted to a relay node over an RF link and then forwarded to the destination over an FSO link. Having relays in wireless networks helps in increasing the communication distance as well as in providing diversity.

The new scenario of mixed RF/FSO relaying network can be also used for user multiplexing where multiple users with only RF capability can be multiplexed into a single FSO link [5]. The RF/FSO relay communication has the ability to fill the connectivity gap between the last-mile network and the backbone network as in developing countries where the last-mile connectivity can be delivered via high-speed FSO links [6]. Such mixed relaying scheme conserves economic resources by avoiding unnecessary modifications to the current mobile devices and, at the same time, saves bandwidth by utilizing FSO capabilities. These attractive features of mixed RF/FSO relay networks make them a strong candidate for current and soon-to-come wireless networks.

The FSO relaying networks with single relay have been studied in the literature under various conditions [7, 8]. The outage performance of AF and DF FSO relaying networks over log-normal fading channels was studied in [7] assuming the presence of a direct link between the source and the destination. The log-normal fading model is usually used to model the FSO links assuming weak atmospheric turbulence conditions, whereas the Gamma-Gamma fading model is more accurate and can be used to model the FSO links under both weak and strong turbulence conditions. The performance of FSO relay networks over Gamma-Gamma fading channels was studied in [8]. The exact outage and error probabilities of two-way FSO relay networks were derived in addition to the derivation of an approximate expression for the symbol error probability. The effect of pointing errors was combined with the turbulence-induced fading as one channel statistic in studying the performance of dual-hop mixed RF/FSO relay networks in [5].

In the area of parallel FSO relaying, the authors in [9, 10] studied the performance of dual-hop FSO networks over log-normal channels for DF and AF schemes, respectively. The performance of dual-hop FSO selective relaying network where the source message is forwarded to the destination along the direct link or along the best relay was studied in [11]. Closed-form and asymptotic expressions were derived for the bit error probability assuming Rayleigh and log-normal fading channels. A key paper which provides some new exact and approximate statistics of the sum of Gamma-Gamma variates and their application in RF and FSO DF relay networks was presented in [12]. The outage performance of channel state information (CSI)-assisted and semi-blind AF opportunistic FSO relay networks was studied in [13] assuming composite channels.

Recently, the scenario of mixed RF/FSO relay networks with multiple users has induced several researchers to turn their attention to work on this hot topic. In [5, 14], the outage and error probabilities in addition to channel capacity of dual-hop multiuser DF and fixed-gain AF mixed RF/FSO relay networks were derived and analyzed, respectively. Despite the presence of multiple users, only one user was assumed to communicate with a relay node through an RF link and the relay was assumed to be connected with a destination through a Gamma-Gamma-modeled FSO channel with pointing errors. No multiuser diversity was achieved in that study. In [15], Miridakis et al. studied the outage and error probability performance of multiuser dual-hop DF mixed RF/FSO relay network with the V-BLAST technique. A resource allocation scheme for multiuser mixed RF/FSO relay network was proposed in [16], where the data of users on the RF hop are generated according to a zero-mean rotationally invariant complex Gaussian distribution. The authors claimed the effectiveness of the proposed link allocation protocol even in the conditions where the FSO link is affected by severe atmospheric conditions. The area of hybrid RF-FSO networks has been recently of interest for many researchers. In [17], considering the cases with and without hybrid automatic repeat request (HARQ) and joint transmission and reception of the RF and FSO messages, the authors derived closed-form expressions for the message decoding probabilities, the throughput, and the outage probability of the RF-FSO setups. The same scenario was also studied by the same authors in [18] but with consideration to the effect of adaptive power allocation on the system throughput and outage probability.

Most recently, the performance of the multiuser mixed RF/FSO relay network with outdated channel information and power allocation has been presented in [19]. Opportunistic scheduling where the user of the best RF channel is selected to send its message to the relay node was used. A generalization to the work in [19] was considered in [20], where the user of the N th best RF channel is selected to send its message to the relay node in the first communication phase. Closed-form expressions for the outage and symbol error probabilities were derived, in addition to channel capacity with the effect of outdate channel information. In [21], the security analysis of multiuser mixed RF/FSO relay networks was analyzed. The paper studied the effect of a single passive eavesdropper attack on the performance of mixed RF/FSO relay network with multiple users and multiple antennas relay. The RF links and FSO link were assumed to follow the Nakagami-m and Gamma-Gamma fading models, respectively, with consideration to the effect of pointing errors on the FSO link. The authors derived closed-form expressions for the outage probability, average symbol error probability (ASEP), and channel capacity as reliability performance measures for the authorized mixed RF/FSO relay network and closed-from expression for the intercept probability as a security measure. Asymptotic expressions were also derived for the outage probability at high SNR values and used for achieving the optimum transmission powers of the selected user and relay node, where opportunistic scheduling was used to select among the users of the first hop.

Most of the previous studies considered the scenario of the dual-hop mixed RF/FSO relay network. This scenario could be seen in applications where multiple users communicate with a relay node via RF links and the relay forwards their massages to a base station (BS) over an FSO link. Also, such a scenario can be seen in indoor applications where multiple users communicate with an access point that is in turn connected to a macro BS via an FSO link [16]. Another important scenario which can be seen in practice is the triple-hop mixed RF/FSO/RF relay network. Example of such applications are as follows: (1) in cellular networks where two sets of various users communicate with their own BSs over RF links and these BSs are connected together via an FSO link and (2) in indoor applications where two sets of users communicate with their access points inside two buildings and these access points are connected via an FSO link. The same setup of triple-hop relay network can be also used in other types of networks such as in mixed mmWave RF/FSO/mmWave RF relaying network, mixed RF/visible light communication (VLC)/RF relaying network, and mixed VLC/RF/VLC relaying network. To exploit the presence of multiple users and achieve the multiuser diversity in such networks, a single user can be selected/scheduled among the available users and allowed to conduct its transmission. The opportunistic scheduling is among the well-known and efficient user selection schemes that are usually used to select among the users. In this scheme, the user with the best channel is always allowed to conduct its transmission in a downlink or an uplink scenario. Also, this scheme is usually used to achieve the maximum sum-rate capacity in wireless networks. A generalization to the opportunistic user scheduling is the generalized order user scheduling, where the user of the N th best channel is selected for conducting its communication. This scheme is applicable in situations where the scheduling unit fails in error in selecting the best user among the available users due to error in estimating the users’ channels. More papers can be found in the literature on mixed or hybrid networks with and without relay nodes [2224]. Also, it is worthwhile to mention here that the scenario of triple-hop relaying was already considered in literature by many researchers but for one type of links [25, 26].

In this paper, we introduce the new scenario of triple-hop mixed RF/FSO/RF relay network with the generalized order user scheduling scheme to select among the users of the first and third RF links. The considered system includes K 1 sources or users, two DF relays, and K 2 destinations. The sources and destinations are connected with their relay nodes through the RF links, and the relays are connected with each other through an FSO link. Using the generalized order user scheduler, the source with the \(N_{1}^{\text {th}}\) best SNR among the available sources is allowed to communicate with the first relay node. Also, using the same scheduling criterion, the destination which has the \(N_{2}^{\text {th}}\) best SNR is selected to receive its message from the second relay. Furthermore, the RF links are assumed to follow the Rayleigh fading model and the FSO link is assumed to follow the Gamma-Gamma fading model with the effect of pointing errors. Closed-form expressions are derived for the outage probability, ASEP, and ergodic channel capacity. Moreover, the system performance is studied at the high SNR regime, where approximate expressions for the outage probability, the diversity order, and coding gain are derived and analyzed. Furthermore, the asymptotic results are used to obtain the optimum transmission powers of the selected user on the first hop, the first relay, and the second relay. Some simulation and numerical examples are provided to study the effect of the number of users and order of selected users on both the first and third hops, atmospheric turbulence parameters, pointing errors, and power allocation on the system performance.

The rest of this paper is organized as follows. Section 2 presents the system and channel models. The exact performance analysis is evaluated in Section 3. Section 4 provides the asymptotic outage performance analysis and power allocation. Some simulation and numerical results are presented and discussed in Section 5. Finally, conclusions are given in Section 6.

2 System and channel models

Consider a triple-hop mixed RF/FSO/RF relay network consisted of K 1 sources on the first hop U k (k=1,…,K 1), two un-coded type DF relays R i (i=1,2), and K 2 destinations on the third hop D j (j=1,…,K 2), as shown in Fig. 1. The sources are assumed to be connected with the first relay node through RF links; this relay is connected with another relay through an FSO link, and finally, the second relay is connected with the destinations through RF links. It is assumed that each user is equipped with a single antenna: the first relay is equipped with a single antenna and a single photo-aperture transmitter; the second relay is equipped with a single photo detector and a single antenna, and each destination is equipped with a single antenna. The direct links between the sources and destinations are assumed to be in deep fade, and hence, they are not considered in the analysis of this paper. Also, we assume block fading model where the channel coefficient stays constant over an entire block of communication. The communication is assumed to operate in a half-duplex mode and to be conducted over three phases: selected user USel→R1, R1→R2, and R2→DSel. The received signal at R1 from the kth user can be expressed as
$$ y_{k,\textsf{r}_{1}}=\sqrt{P_{k}}h_{k,\textsf{r}_{1}}x_{k,\textsf{r}_{1}}+n_{\textsf{r}_{1}}, $$
(1)
Fig. 1

Multiuser triple-hop mixed RF/FSO/RF relay network with generalized order user scheduling

where P k is the transmit power of the kth user, \(\phantom {\dot {i}\!}h_{k,\textsf {r}_{1}}\) is the channel coefficient of the \(\textsf {U}_{k}\rightarrow \textsf {R}_{1}\) link, \(\phantom {\dot {i}\!}x_{k,\textsf {r}_{1}}\) is the transmitted symbol of U k with \(\mathbb {E}\{|{x_{k,\textsf {r}_{1}}}|^{2}\}=1\), and \(n_{\textsf {r}_{1}}\thicksim \mathcal {N}(0, N_{01})\) is an additive white Gaussian noise (AWGN) term, where \(\mathbb {E}\{\cdot \}\) is the mathematical expectation.

Using (1), the SNR at R1 due to U k can be written as
$$\begin{array}{*{20}l} \gamma_{\textsf{U}_{k},\textsf{R}_{1}}&=\frac{P_{k}}{N_{01}}|h_{k,\textsf{r}_{1}}|^{2}. \end{array} $$
(2)
According to the generalized order user scheduling, the source with the \(N_{1}^{\text {th}}\) best \(\phantom {\dot {i}\!}\gamma _{\textsf {U}_{k},\textsf {R}_{1}}\) or equivalently, the \(N_{1}^{\text {th}}\) largest \(|h_{k,\textsf {r}_{1}}|^{2}\phantom {\dot {i}\!}\) among the other sources is selected to transmit its message to R1 in the first communication phase. In other words, the source is selected such that \(\gamma _{\textsf {U}_{\text {Sel}},\textsf {R}_{1}}=\underset {\substack {k}}{N_{1}^{\text {th}}\ \text {max}}\left \{\gamma _{\textsf {U}_{k},\textsf {R}_{1}}\right \}\). The subcarrier intensity modulation (SIM) scheme is employed at the relay R1, where a standard RF coherent/noncoherent modulator and demodulator can be used for transmitting and recovering the source data [2729]. At R1, after filtering by a bandpass filter (BPF), a direct current (DC) bias is added to the filtered RF signal to ensure that the optical signal is non-negative. Then, the biased signal is sent to a continuous wave laser driver. The retransmitted optical signal at R1 is written as [4]
$$ y_{\textsf{r}_{1}}^{\text{Opt}}=\sqrt{P_{\text{Opt}}}(1+\mathcal{M}y_{\text{Sel},\textsf{r}_{1}}), $$
(3)
where P Opt denotes the average transmitted optical power and it is related to the relay electrical power P r by the electrical-to-optical conversion efficiency η 1 as \(P_{\text {Opt}}=\eta _{1}P_{\textsf {r}_{1}}\phantom {\dot {i}\!}\), \(\mathcal {M}\) denotes the modulation index, and \(\phantom {\dot {i}\!}y_{\text {Sel},\textsf {r}_{1}}\) is the RF received signal at R1 from the selected source. The optical signal at R2 received from R1 at the second phase of communication can be expressed as
$$ \begin{aligned} y_{\textsf{r}_{1},\textsf{r}_{2}}=g_{\textsf{r}_{1},\textsf{r}_{2}}\left\{\sqrt{P_{\text{Opt}}}\left[1+\mathcal{M} \left(\sqrt{P_{\text{Sel}}}h_{\text{Sel}, \textsf{r}_{1}}x_{\text{Sel},\textsf{r}_{1}}+n_{\textsf{r}_{1}}\right)\right]\right\}+n_{\textsf{r}_{2}}, \end{aligned} $$
(4)
where \(n_{\textsf {r}_{2}}\thicksim \mathcal {N}(0, N_{02})\) is an AWGN term at R2. Moreover, the channel coefficient of the R1→R2 link which is given by \(\phantom {\dot {i}\!}g_{\textsf {r}_{1},\textsf {r}_{2}}\) is modeled as \(\phantom {\dot {i}\!}g_{\textsf {r}_{1},\textsf {r}_{2}}=g_{a}g_{f}\), where g a and g f are the average gain and the fading gain of the FSO link, respectively, and are given by [30]
$$\begin{array}{*{20}l} \left\{ \begin{array}{ll} g_{a}=\left[\text{erf}\left(\frac{\sqrt{\pi}q}{\sqrt{2}\phi d^{\text{FSO}}}\right)\right]^{2}\times 10^{-\kappa d^{\text{FSO}}/10}, \\ g_{f}\sim \text{GGamma}(\alpha,\beta), \\ \end{array} \right. \end{array} $$
(5)
where q is the aperture radius, ϕ is the divergence angle of the beam, d FSO is the distance between the FSO transmitter and the receiver, κ is the weather-dependent attenuation coefficient, and GGamma(α,β) represents a Gamma-Gamma random variable with parameters α and β. Assuming spherical wave propagation, the parameters α and β in the Gamma-Gamma distribution which represent the fading turbulence conditions are related to the physical parameters as follows [31]:
$$\begin{array}{*{20}l} &\alpha=\left[\exp\left\{\frac{0.49\vartheta^{2}}{\left[1+0.18\xi^{2}+0.56\vartheta^{12/5}\right]^{7/6}}\right\}-1\right]^{-1}, \end{array} $$
(6)
$$\begin{array}{*{20}l} &\beta=\left[\exp\left\{\frac{0.51\vartheta^{2}[1+0.69\vartheta^{12/5}]^{-5/6}}{\left[1+0.9\xi^{2}+0.62\xi^{2}\vartheta^{12/5}\right]^{5/6}}\right\} -1\right]^{-1}, \end{array} $$
(7)

where \(\vartheta ^{2}=0.5{C_{n}^{2}}\varsigma ^{7/6}(d^{\text {FSO}})^{11/6}\), ξ 2=ς q 2/d FSO, and ς=2π/λ FSO. Here, λ FSO is the wavelength and \({C_{n}^{2}}\) is the weather-dependent index of refraction structure parameter.

When the DC component is filtered out at R2 and an optical-to-electrical conversion is performed and assuming \(\mathcal {M}=1\), the received signal can be expressed as follows:
$$ y_{\textsf{r}_{1},\textsf{r}_{2}}=g_{\textsf{r}_{1},\textsf{r}_{2}}\sqrt{P_{\text{Ele}}}\left(\sqrt{P_{\text{Sel}}}h_{\text{Sel}, \textsf{r}_{1}}x_{\text{Sel},\textsf{r}_{2}}+n_{\textsf{r}_{1}}\right)+n_{\textsf{r}_{2}}, $$
(8)

where \(\phantom {\dot {i}\!}P_{\text {Ele}}=\eta _{2} P_{\text {Opt}}=\eta _{1}\eta _{2} P_{\textsf {r}_{1}}\) is the electrical power received at R2 with η 2 is the optical-to-electrical conversion efficiency.

From (8), the SNR at R2 can be written as
$$\begin{array}{*{20}l} \gamma_{\textsf{R}_{2}}=\frac{\gamma_{\textsf{U}_{\text{Sel}},\textsf{R}_{1}}\gamma_{\textsf{R}_{1},\textsf{R}_{2}}}{\gamma_{\text{Sel}, \textsf{R}_{1}}+\gamma_{\textsf{R}_{1},\textsf{R}_{2}}+1}, \end{array} $$
(9)

where \(\gamma _{\textsf {U}_{\text {Sel}},\textsf {R}_{1}}=\frac {P_{ \text {Sel}}}{N_{01}}|h_{\text {Sel},\textsf {r}_{1}}|^{2}\) and \(\gamma _{\textsf {R}_{1},\textsf {R}_{2}}=\frac {\eta _{1}\eta _{2}P_{\textsf {r}_{1}}}{N_{02}}|g_{\textsf {r}_{1},\textsf {r}_{2}}|^{2}\), where \(P_{\textsf {r}_{1}}\phantom {\dot {i}\!}\) is the transmit power at R1.

The SNR in (9) can be rewritten using the standard approximation \(\phantom {\dot {i}\!}\gamma _{\textsf {R}_{2}}\cong \text {min}(\gamma _{\textsf {U}_{\text {Sel}},\textsf {R}_{1}},\gamma _{\textsf {R}_{1},\textsf {R}_{2}})\) as [5, 14]
$$\begin{array}{*{20}l} \gamma_{\textsf{R}_{2}}=\frac{\gamma_{\textsf{U}_{\text{Sel}},\textsf{R}_{1}}\gamma_{\textsf{R}_{1},\textsf{R}_{2}}} {\gamma_{\textsf{U}_{\text{Sel}},\textsf{R}_{1}}+\gamma_{\textsf{R}_{1},\textsf{R}_{2}}}. \end{array} $$
(10)
The signal received at D j from R2 in the third phase of communication can be written as
$$ y_{\textsf{r}_{2},\textsf{d}_{j}}=\sqrt{P_{\textsf{r}_{2}}}h_{\textsf{r}_{2},{j}}x_{\textsf{d}_{j}} +n_{\textsf{d}_{j}}, $$
(11)

where \(P_{\textsf {r}_{2}}\phantom {\dot {i}\!}\) is the transmit power at R2, \(\phantom {\dot {i}\!}h_{\textsf {r}_{2},{j}}\) is the channel coefficient of the R2→D j link, \(x_{\textsf {d}_{j}}\phantom {\dot {i}\!}\) is the transmitted symbol of d j with \(\mathbb {E}\{|x_{\textsf {d}_{j}}|^{2}\}=1\), and \(n_{\textsf {d}_{j}}\thicksim \mathcal {N}(0, N_{03})\) is an AWGN term.

Using (11), the SNR at D j can be written as
$$\begin{array}{*{20}l} \gamma_{\textsf{R}_{2},\textsf{D}_{j}}&=\frac{P_{\textsf{r}_{2}}}{N_{03}}|h_{\textsf{r}_{2},{j}}|^{2}. \end{array} $$
(12)

According to the generalized order user scheduling, the destination with the \(N_{2}^{\text {th}}\) best \(\gamma _{\textsf {R}_{2},\textsf {D}_{j}}\phantom {\dot {i}\!}\) or equivalently, the \(N_{2}^{\text {th}}\) largest \(\phantom {\dot {i}\!}|h_{\textsf {r}_{2},{j}}|^{2}\) among the other destinations is selected to receive its message from R2 in the third communication phase. In other words, the destination is selected such that \(\gamma _{\textsf {R}_{2},\textsf {D}_{\text {Sel}}}=\underset {\substack {j}}{N_{2}^{\text {th}}\ \text {max}}\left \{\gamma _{\textsf {R}_{2},\textsf {D}_{j}}\right \}\).

We assume that the channel coefficients of the RF links \(\phantom {\dot {i}\!}h_{k,\textsf {r}_{1}} (k = 1 = \cdots = K_{1})\) and \(h_{\textsf {r}_{2},{j}} (j= 1 = \cdots = K_{2})\phantom {\dot {i}\!}\) follow the Rayleigh fading model and hence, the channel gains \(\phantom {\dot {i}\!}|{h_{k,\textsf {r}_{1}}}|^{2}\) and \(|{h_{\textsf {r}_{2},j}}|^{2}\phantom {\dot {i}\!}\) are exponential distributed random variables with mean powers \(\phantom {\dot {i}\!}\Omega _{k,\textsf {r}_{1}}\) and \(\phantom {\dot {i}\!}\Omega _{\textsf {r}_{2},j}\), respectively. Therefore, the probability density functions (PDFs) of \(\phantom {\dot {i}\!}\gamma _{{\textsf {U}_{k}},\textsf {R}_{1}}\) and \(\phantom {\dot {i}\!}\gamma _{\textsf {R}_{2},{\textsf {D}_{j}}}\) are, respectively, given by \(f_{\gamma _{{\textsf {U}_{k}},\textsf {R}_{1}}}(\gamma)=\lambda _{k,\textsf {r}_{1}}\exp \left (-\lambda _{k,\textsf {r}_{1}}\gamma \right)\), where \(\lambda _{k,\textsf {r}_{1}}=1/\bar {\gamma }_{k,\textsf {r}_{1}}\) with \(\bar {\gamma }_{k,\textsf {r}_{1}}=\frac {P_{k}}{N_{0}}\mathbb {E}\{|{h_{k,\textsf {r}_{1}}}|^{2}\}=\frac {P_{k}}{N_{01}}\Omega _{k,\textsf {r}_{1}}\) and \(f_{\gamma _{\textsf {R}_{2},{\textsf {D}_{j}}}}(\gamma)=\lambda _{\textsf {r}_{2},j}\exp \left (-\lambda _{\textsf {r}_{2},j}\gamma \right)\), where \(\lambda _{\textsf {r}_{2},j}=1/\bar {\gamma }_{\textsf {r}_{2},j}\) with \(\bar {\gamma }_{\textsf {r}_{2},j}=\frac {P_{k}}{N_{03}}\mathbb {E}\{|{h_{\textsf {r}_{2},j}}|^{2}\}=\frac {P_{\textsf {r}_{2}}}{N_{03}} \Omega _{\textsf {r}_{2},j}\). On the other hand, it is assumed that the FSO link experiences a unified Gamma-Gamma fading model including the pointing errors effect whose SNR PDF is given by [5]
$$ { \begin{aligned} f_{\gamma_{\textsf{R}_{1},\textsf{R}_{2}}}(\gamma)=\frac{\zeta^{2}}{r\gamma\Gamma(\alpha)\Gamma(\beta)} \mathrm{G}^{3,0}_{1,3} \left[ \begin{array}{c|c} & \zeta^{2}+1\\[-4pt] \alpha\beta\left(\lambda_{\textsf{r}_{1},\textsf{r}_{2}}\gamma\right)^{\frac{1}{r}} & \\[-4pt] & \zeta^{2},\alpha,\beta\\ \end{array}\right], \end{aligned}} $$
(13)

where ζ is the ratio between the equivalent beam radius at the receiver and the pointing error displacement standard deviation (jitter) at the receiver (i.e. when ζ, we get the non-pointing error case) [5], r is the parameter defining the type of detection technique (i.e. r=1 represents heterodyne detection and r=2 represents intensity modulation (IM)/direct detection (DD)), α and β are the fading parameters related to the atmospheric turbulence conditions with lower values indicating severe atmospheric turbulence conditions, Γ(.) is the Gamma function as defined in [32, Eq. (8.310)], \(\lambda _{\textsf {r}_{1},\textsf {r}_{2}}=1/\bar {\gamma }_{\textsf {r}_{1},\textsf {r}_{2}}\) with \(\bar {\gamma }_{\textsf {r}_{1},\textsf {r}_{2}}=\frac {\eta _{1}\eta _{2}P_{\textsf {r}_{1}}}{N_{02}}\mathbb {E}\{| {g_{\textsf {r}_{1},\textsf {r}_{2}}}|^{2}\}=\frac {\eta _{1}\eta _{2}P_{\textsf {r}_{1}}}{N_{02}}\mu _{\textsf {r}_{1},\textsf {r}_{2}}\), and G(.) is the Meijer G-function as defined in [32, Eq. (9.301)].

The end-to-end (e2e) SNR at the selected destination can be written using the standard approximation \(\phantom {\dot {i}\!}\gamma _{\textsf {D}}\cong \text {min}(\gamma _{\textsf {R}_{2}},\gamma _{\textsf {R}_{2},\textsf {D}_{\text {Sel}}})\) as [5, 14]
$$\begin{array}{*{20}l} \gamma_{\textsf{D}}=\frac{\gamma_{\textsf{R}_{2}}\gamma_{\textsf{R}_{2},\textsf{D}_{\text{Sel}}}} {\gamma_{\textsf{R}_{2}}+\gamma_{\textsf{R}_{2},\textsf{D}_{\text{Sel}}}}. \end{array} $$
(14)

Achieving the system performance measures requires obtaining the statistics of the e2e SNR provided in (14).

3 Exact performance analysis

In this section, we derive the exact outage probability, ASEP, and channel capacity of the considered system.

3.1 Outage probability

The outage probability is an important performance metric in wireless communications and defined as the probability that the SNR at the selected destination goes below a predetermined outage threshold γ out, i.e., P out=Pr[γ Dγ out], where Pr[.] is the probability operation and γ out is a predetermined outage threshold or, equivalently, the system is unable to achieve adequate reception which is common to occur in any communication system. The outage probability is also equivalent to other metrics which is the outage capacity (different way of looking into the system quality), where for any given rate and outage probability level, there is an outage capacity associated with it, with the interpretation that when the system is not in outage (which occurs with probability 1−P out), this particular transmission rate can be supported [33]. Clearly, the outage probability can be obtained from the cumulative distribution function (CDF) of the e2e SNR as \(P_{\textsf {out}}=F_{\gamma _{\textsf {D}}}(\gamma _{\textsf {out}})\phantom {\dot {i}\!}\). This CDF can be written in terms of CDFs of the three hops’ SNRs as [34]
$$\begin{array}{*{20}l} F_{\gamma_{\textsf{D}}}(\gamma)=1-&\left\{\left(1-F_{\gamma_{\textsf{U}_{\text{Sel}},\textsf{R}_{1}}}(\gamma)\right) \left(1-F_{\gamma_{\textsf{R}_{1},\textsf{R}_{2}}}(\gamma)\right)\right.\\&\left.\;\times\left(1-F_{\gamma_{\textsf{R}_{2},\textsf{D}_{\text{Sel}}}}(\gamma)\right)\right\}, \end{array} $$
(15)
where \(F_{\gamma _{\textsf {U}_{\text {Sel}},\textsf {R}_{1}}}(\gamma)\), \(F_{\gamma _{\textsf {R}_{1},\textsf {R}_{2}}}(\gamma)\), and \(F_{\gamma _{\textsf {R}_{2},\textsf {D}_{\text {Sel}}}}(\gamma)\) are the CDFs of first hop, second hop, and third hop SNRs, respectively. It is clear from (15) that the system gets in outage once at least one of the three hops gets in outage or, equivalently, the SNR of that hop becomes less than γ out. With a direct expansion, the CDF in (15) can be rewritten in a more detailed form as follows:
$$ {\begin{aligned} F_{\gamma_{\textsf{D}}}(\gamma)=&\,F_{\gamma_{\textsf{U}_{\text{Sel}},\textsf{R}_{1}}}(\gamma) +F_{\gamma_{\textsf{R}_{1},\textsf{R}_{2}}}(\gamma)+F_{\gamma_{\textsf{R}_{2},\textsf{D}_{\text{Sel}}}} (\gamma)\\&-F_{\gamma_{\textsf{U}_{\text{Sel}},\textsf{R}_{1}}}(\gamma)F_{\gamma_{\textsf{R}_{1},\textsf{R}_{2}}} (\gamma)-F_{\gamma_{\textsf{U}_{\text{Sel}},\textsf{R}_{1}}}(\gamma)F_{\gamma_{\textsf{R}_{2},\textsf{D}_{\text{Sel}}}}(\gamma)\\ &-F_{\gamma_{\textsf{R}_{1},\textsf{R}_{2}}}(\gamma)F_{\gamma_{\textsf{R}_{2},\textsf{D}_{\text{Sel}}}}(\gamma) +F_{\gamma_{\textsf{U}_{\text{Sel}},\textsf{R}_{1}}}(\gamma)F_{\gamma_{\textsf{R}_{1},\textsf{R}_{2}}}(\gamma)\\ &\times F_{\gamma_{\textsf{R}_{2},\textsf{D}_{\text{Sel}}}}(\gamma). \end{aligned}} $$
(16)
In order to calculate (16), the CDF of each hop of the three hops needs to be obtained first as follows.
  • First hop link

    Using the generalized order user selection, the PDF \(F_{\gamma _{\textsf {U}_{\text {Sel}},\textsf {R}_{1}}}(\gamma)\) can be written for independent identically distributed sources’ channels (\(\phantom {\dot {i}\!}\lambda _{1,\textsf {r}_{1}}=\lambda _{2,\textsf {r}_{1}}=\ldots =\lambda _{K_{1},\textsf {r}_{1}}=\lambda _{u,\textsf {r}_{1}}\)) as [35]
    $$ \begin{aligned} f_{\gamma_{\textsf{U}_{\text{Sel}},\textsf{R}_{1}}}(\gamma)&={K_{1}-1 \choose N_{1}-1}K_{1}f_{\gamma_{\textsf{U},\textsf{R}_{1}}}(\gamma)\left(F_{\gamma_{\textsf{U}, \textsf{R}_{1}}}(\gamma)\right)^{K_{1}-N_{1}}\\&\quad\times\left(1-F_{\gamma_{\textsf{U},\textsf{R}_{1}}}(\gamma)\right)^{N_{1}-1}, \end{aligned} $$
    (17)

    where \(f_{\gamma _{\textsf {U},\textsf {R}_{1}}}(\gamma)\) and \(f_{\gamma _{\textsf {U},\textsf {R}_{1}}}(\gamma)\) are the PDF and CDF of a source channel’s SNR at the first hop, respectively, which are given for the Rayleigh fading channels by \(f_{\gamma _{\textsf {U},\textsf {R}_{1}}}(\gamma)= \lambda _{u,\textsf {r}_{1}}\exp \left (-\lambda _{u,\textsf {r}_{1}}\gamma \right)\) and \(F_{\gamma _{\textsf {U},\textsf {R}_{1}}}(\gamma)=1-\exp \left (-\lambda _{u,\textsf {r}_{1}}\gamma \right)\), and N 1 is the order of the selected source. In other words, the PDF in (17) represents the PDF of the \(N_{1}^{\text {th}}\) best SNR or, equivalently, the source of the best \(N_{1}^{\text {th}}\) SNR is selected by the first relay.

    Upon substituting these statistics in (17) and using the binomial rule, the PDF in (17) can be rewritten as
    $$\begin{array}{*{20}l} f_{\gamma_{\textsf{U}_{\text{Sel}},\textsf{R}_{1}}}(\gamma)&=K_{1}\lambda_{u,\textsf{r}_{1}}{K_{1}-1 \choose N_{1}-1}\sum_{k=0}^{K_{1}-N_{1}}{K_{1}-N_{1} \choose k}\\&\quad\times(-1)^{k}\exp\left(-\left(k+N_{1}\right) \lambda_{u,\textsf{r}_{1}}\gamma\right). \end{array} $$
    (18)
    Integrating, (18) using \(\int _{0}^{\gamma }f_{\gamma _{\textsf {U}_{\text {Sel}},\textsf {R}_{1}}}(t)dt\), we get
    $$\begin{array}{*{20}l} F_{\gamma_{\textsf{U}_{\text{Sel}},\textsf{R}_{1}}}(\gamma)&=K_{1}{K_{1}-1 \choose N_{1}-1}\sum_{k=0}^{K_{1}-N_{1}}\frac{{K_{1}-N_{1} \choose k}(-1)^{k}}{\left(k+N_{1}\right)}\\ &\quad\times[1-\exp\left(-\left(k+N_{1}\right) \lambda_{u,\textsf{r}_{1}}\gamma\right)]. \end{array} $$
    (19)
  • Second hop link

    The CDF \(F_{\gamma _{\textsf {R}_{1},\textsf {R}_{2}}}(\gamma)\) can be obtained by integrating the PDF in (13) using \(\int _{0}^{\gamma }f_{\gamma _{\textsf {R}_{1},\textsf {R}_{2}}}(t)dt\) and with the help of Eq. 07.34.21.0003.01 in [36] to get
    $$ \begin{aligned} F_{\gamma_{\textsf{R}_{1},\textsf{R}_{2}}}(\gamma)=A \mathrm{G}^{3r,1}_{r+1,3r+1} \left[ \begin{array}{c|c} & 1,\chi_{1}\\[-4pt] \frac{B}{\bar{\gamma}_{\textsf{r}_{1}, \textsf{r}_{2}}}\gamma & \\[-4pt] &\chi_{2},0 \end{array}\right], \end{aligned} $$
    (20)

    where \(A=\frac {r^{\alpha +\beta -2}\zeta ^{2}}{(2\pi)^{r-1}\Gamma (\alpha)\Gamma (\beta)}\), \(B=\frac {(\alpha \beta)^{r}}{r^{2r}}\), \(\chi _{1}=\frac {\zeta ^{2}+1}{r},\ldots, \frac {\zeta ^{2}+r}{r}\) comprises of r terms, and \(\chi _{2}=\frac {\zeta ^{2}}{r},\ldots,\frac {\zeta ^{2}+r-1}{r}, \frac {\alpha }{r}, \ldots,\frac {\alpha +r-1}{r},\frac {\beta }{r},\ldots,\frac {\beta +r-1}{r}\) comprises of 3r terms. Note that introducing these parameters primarily aims to simplify the calculations and expressions of the paper.

  • Third hop link

    Similar to the first hop link, the PDF and CDF of the SNR of the selected destination can be, respectively, written as
    $$\begin{array}{*{20}l} f_{\gamma_{\textsf{R}_{2},\textsf{D}_{\text{Sel}}}}(\gamma)&=K_{2}\lambda_{\textsf{r}_{2},u}{K_{2}-1 \choose N_{2}-1}\sum_{j=0}^{K_{2}-N_{2}}{K_{1}-N_{1} \choose j}\\ &\quad\times(-1)^{j}\exp\left(-\left(j+N_{2}\right)\lambda_{\textsf{r}_{2},u}\gamma\right), \end{array} $$
    (21)
    $$\begin{array}{*{20}l} F_{\gamma_{\textsf{R}_{2},\textsf{D}_{\text{Sel}}}}(\gamma)&=K_{2}{K_{2}-1 \choose N_{2}-1}\sum_{j=0}^{K_{2}-N_{2}}\frac{{K_{2}-N_{2} \choose j}(-1)^{j}}{\left(j+N_{2}\right)}\\ &\quad\times[1-\exp\left(-\left(j+N_{2}\right)\lambda_{\textsf{r}_{2},u}\gamma\right)], \end{array} $$
    (22)

    where the users on the third hop have been assumed to have independent identical distributed channels in (21) and (22), that is, (\(\phantom {\dot {i}\!}\lambda _{\textsf {r}_{2},1}=\lambda _{\textsf {r}_{2},2}=\ldots =\lambda _{\textsf {r}_{2},K_{2}}= \lambda _{\textsf {r}_{2},u}\)). Again, the PDF in (21) represents the PDF of the \(N_{2}^{\text {th}}\) best SNR or, equivalently, that the destination of the \(N_{2}^{\text {th}}\) best SNR is selected by the second relay.

    Upon substituting (19), (20), and (22) in (15) and after some simplifications, we get
    $$ \begin{aligned} F_{\gamma_{\textsf{D}}}(\gamma)&={K_{1}-1 \choose N_{1}-1}K_{1}\sum_{k=0}^{K_{1}-N_{1}}\frac{{K_{1}-N_{1} \choose k}(-1)^{k}} {(k+N_{1})} \left\{1-\exp\left(-\tau_{1}\gamma\right)-A{\vphantom{\left(\mathrm{G}^{3r,1}_{r+1,3r+1} \left[ \begin{array}{c|c} & {1,\chi_{1}} \\[-4pt] \delta_{0}\gamma & \\[-4pt] & {\chi_{2},0} \\ \end{array}\right] \left[1-\exp\left(-\tau_{1}\gamma\right)\right]\right)}}\right.\\&\left.\quad\times\left(\mathrm{G}^{3r,1}_{r+1,3r+1} \left[ \begin{array}{c|c} & {1,\chi_{1}} \\[-4pt] \delta_{0}\gamma & \\[-4pt] & {\chi_{2},0} \\ \end{array}\right] \left[1-\exp\left(-\tau_{1}\gamma\right)\right]\right)\right\}\\ &\quad +{K_{2}-1 \choose N_{2}-1}K_{2}\sum_{j=0}^{K_{2}-N_{2}}\frac{{K_{2}-N_{2} \choose j}(-1)^{j}}{(j+N_{2})} \left\{1-\exp\left(-\tau_{2}\gamma\right){\vphantom{\left(\mathrm{G}^{3r,1}_{r+1,3r+1} \left[ \begin{array}{c|c} & 1,\chi_{1}\\[-4pt] \delta_{0}\gamma & \\[-4pt] & \chi_{2},0 \\ \end{array}\right] \left[1-\exp\left(-\tau_{2}\gamma\right)\right]\right)}}\right.\\&\left.\quad-A\left(\mathrm{G}^{3r,1}_{r+1,3r+1} \left[ \begin{array}{c|c} & 1,\chi_{1}\\[-4pt] \delta_{0}\gamma & \\[-4pt] & \chi_{2},0 \\ \end{array}\right] \left[1-\exp\left(-\tau_{2}\gamma\right)\right]\right)\right\}\\ & \quad -{K_{1}-1 \choose N_{1}-1}K_{1}\sum_{k=0}^{K_{1}-N_{1}}\frac{{K_{1}-N_{1} \choose k}(-1)^{k}}{(k+N_{1})}{K_{2}-1 \choose N_{2}-1}K_{2}\\&\quad\times\sum_{j=0}^{K_{2}-N_{2}}\frac{{K_{2}-N_{2} \choose j}(-1)^{j}}{(j+N_{2})} \left\{{\vphantom{\sum_{0}^{1}}}1-\exp\left(-\tau_{1}\gamma\right)-\exp\left(-\tau_{2}\gamma\right)\right.\\ & \quad +\exp\left(-[\tau_{1}+\tau_{2}]\gamma\right)- A \left({\vphantom{\sum_{0}^{1}}}\mathrm{G}^{3r,1}_{r+1,3r+1}\right. \left[ \begin{array}{c|c} & 1,\chi_{1}\\[-4pt] \delta_{0}\gamma & \\[-4pt] & \chi_{2},0 \\ \end{array}\right]\\ &\quad\left.\left.\times\left[1-\exp\left(-\tau_{1}\gamma\right)-\exp\left(-\tau_{2}\gamma\right) +\exp\left(-[\tau_{1}+\tau_{2}]\gamma\right)\right]{\vphantom{\sum_{0}^{1}}}\right){\vphantom{\sum_{0}^{1}}}\right\}\\ &\quad +A\mathrm{G}^{3r,1}_{r+1,3r+1} \left[ \begin{array}{c|c} &1,\chi_{1} \\[-4pt] \delta_{0}\gamma & \\[-4pt] & \chi_{2},0 \\ \end{array}\right], \end{aligned} $$
    (23)

    where \(\tau _{1}=(k+N_{1})\lambda _{u,\textsf {r}_{1}}\phantom {\dot {i}\!}\), \(\delta _{0}=\frac {B}{\bar {\gamma }_{\textsf {r}_{1},\textsf {r}_{2}}}\phantom {\dot {i}\!}\), and \(\tau _{2}=(j+N_{2})\lambda _{\textsf {r}_{2},u}\phantom {\dot {i}\!}\).

    The CDF in (23) represents an important statistic to the e2e SNR γ D and allows to derive several performance measures in closed-form expressions as will be seen in the next sections of the paper. Up to now, the outage probability can be obtained by replacing γ by γ out in (23).

3.2 Average symbol error probability

For evaluating the ASEP, we use the CDF-based method where the ASEP can be expressed in terms of the CDF of γ D as [37]
$$\begin{array}{*{20}l} \text{ASEP}=\frac{a\sqrt{b}}{2\sqrt{\pi}}\int_{0}^{\infty}\frac{\exp\left(-b\gamma\right)}{\sqrt{\gamma}}F_{\gamma_{\textsf{D}}}(\gamma) d\gamma, \end{array} $$
(24)

where a and b are modulation-specific parameters. Note that we adopt the SIM scheme and hence the known digital modulation techniques can be used, such as phase shift keying (PSK) [38]. Therefore, the error probability computing method (24) used in RF wireless communication systems can be used to evaluate the error probability performance in FSO systems.

Upon substituting (23) in (24) and using Eq. 7.813.1 in [32] and Eq. 3.381.4 in [32], we get
$$ \begin{aligned} \text{ASEP} & =\frac{a\sqrt{b}}{2\sqrt{\pi}}\left\{{K_{1}-1 \choose N_{1}-1}K_{1}\sum_{k=0}^{K_{1}-N_{1}}\frac{{K_{1}-N_{1} \choose k}(-1)^{k}}{(k+N_{1})} \left(\frac{\Gamma(1/2)}{b^{\frac{1}{2}}}\right.\right.\\ &\quad-\frac{\Gamma(1/2)}{(b+\tau_{1})^{\frac{1}{2}}}-A \left[{\vphantom{\sum_{0}^{1}}}b^{-\frac{1}{2}}\mathrm{G}^{3r,2}_{r+2,3r+1}\right. \left[ \begin{array}{c|c} &\frac{1}{2},1,\chi_{1}\\[-4pt] \frac{\delta_{0}}{b} & \\[-4pt] & \chi_{2},0\\ \end{array}\right]\\ & \quad -(b+\tau_{1})^{-\frac{1}{2}}\mathrm{G}^{3r,2}_{r+2,3r+1} \left.\left.\left[ \begin{array}{c|c} &\frac{1}{2},1,\chi_{1} \\[-4pt] \frac{\delta_{1}}{(b+\tau_{1})} & \\[-4pt] & \chi_{2},0\\ \end{array}\right]\right]\right) +{K_{2}-1 \choose N_{2}-1}\\ &\quad\times K_{2}\sum_{j=0}^{K_{2}-N_{2}}\frac{{K_{2}-N_{2} \choose j}(-1)^{j}}{(j+N_{2})} \left(\frac{\Gamma(1/2)}{b^{\frac{1}{2}}}-\frac{\Gamma(1/2)}{(b+\tau_{2})^{\frac{1}{2}}}\right.\\ & \quad -A\left[b^{-\frac{1}{2}}\mathrm{G}^{3r,2}_{r+2,3r+1} \left[ \begin{array}{c|c} & \frac{1}{2},1,\chi_{1} \\[-4pt] \frac{\delta_{0}}{b} & \\[-4pt] & \chi_{2},0\\ \end{array}\right] -(b+\tau_{2})^{-\frac{1}{2}}\mathrm{G}^{3r,2}_{r+2,3r+1}\right.\\ &\quad\left.\left.\times\left[ \begin{array}{c|c} & \frac{1}{2},1,\chi_{1}\\[-4pt] \frac{\delta_{0}}{(b+\tau_{2})} & \\[-4pt] &\chi_{2},0\\ \end{array}\right]\right]\right)- {K_{1}-1 \choose N_{1}-1}K_{1}\sum_{k=0}^{K_{1}-N_{1}}\frac{{K_{1}-N_{1} \choose k}}{(k+N_{1})}\\ & \quad \times(-1)^{k}{K_{2}-1 \choose N_{2}-1}K_{2}\sum_{j=0}^{K_{2}-N_{2}}\frac{{K_{2}-N_{2} \choose j}(-1)^{j}}{(j+N_{2})} \left(\frac{\Gamma(1/2)}{b^{\frac{1}{2}}}\right.\\&\quad-\frac{\Gamma(1/2)}{(b+\tau_{1})^{\frac{1}{2}}} -\frac{\Gamma(1/2)}{(b+\tau_{2})^{\frac{1}{2}}}+\frac{\Gamma(1/2)}{(b+\tau_{1}+\tau_{2})^{\frac{1}{2}}}\\ & \quad - A\left[b^{-\frac{1}{2}}\times\mathrm{G}^{3r,2}_{r+2,3r+1} \left[\!\! \begin{array}{c|c} & \frac{1}{2},1,\chi_{1}\\[-4pt] \frac{\delta_{0}}{b} & \\[-4pt] &\chi_{2},0\\ \end{array}\!\!\right] -(b+\tau_{1})^{-\frac{1}{2}}\mathrm{G}^{3r,2}_{r+2,3r+1}\right.\\& \quad\times\left[ \begin{array}{c|c} & \frac{1}{2},1,\chi_{1}\\[-4pt] \frac{\delta_{0}}{(b+\tau_{1})} & \\[-4pt] & \chi_{2},0\\ \end{array}\right] -\mathrm{G}^{3r,2}_{r+2,3r+1} \left[ \begin{array}{c|c} & \frac{1}{2},1,\chi_{1}\\[-4pt] \frac{\delta_{0}}{(b+\tau_{2})} & \\[-4pt] & \chi_{2},0\\ \end{array}\right] \\ & \quad\times(b+\tau_{2})^{-\frac{1}{2}} +\mathrm{G}^{3r,2}_{r+2,3r+1} \left[ \begin{array}{c|c} & \frac{1}{2},1,\chi_{1}\\[-4pt] \frac{\delta_{0}}{(b+\tau_{1}+\tau_{2})} & \\[-4pt] &\chi_{2},0\\ \end{array}\right]\\& \quad\times\left.\left.\left.{\vphantom{\left[ \begin{array}{c|c} & \frac{1}{2},1,\chi_{1}\\[-4pt] \frac{\delta_{0}}{b} & \\[-4pt] & \chi_{2},0\\ \end{array} \right]}}(b+\tau_{1}+\tau_{2})^{-\frac{1}{2}}\right]\right) +Ab^{-\frac{1}{2}}\mathrm{G}^{3r,2}_{r+2,3r+1} \left[ \begin{array}{c|c} & \frac{1}{2},1,\chi_{1}\\[-4pt] \frac{\delta_{0}}{b} & \\[-4pt] & \chi_{2},0\\ \end{array} \right]\right\}. \end{aligned} $$
(25)

3.3 Ergodic channel capacity

It is well known that the atmospheric turbulence over the FSO links is slow in fading and since the coherence time of the channel is in the order of milliseconds (ms), the turbulence-induced fading remains constant over a large number of transmitted bits [39]. Furthermore, including the effects of the pointing error in our paper makes the signal fluctuate at a very high rate. Because the coherence time of the FSO fading channel is in the order of milliseconds, a single fade can obliterate millions of bits at gigabits/second data rates and therefore, the average (i.e., ergodic) capacity of the channel represents the best achievable capacity of an optical wireless link which is our focus in this work to ensure the long-term ergodic properties of the turbulence process [40]. Using the PDF-based method, the ergodic capacity can be expressed in terms of the PDF of γ D as
$$\begin{array}{*{20}l} {C}=\frac{1}{\text{ln}(2)}\int_{0}^{\infty}\text{ln}(1+\gamma)f_{\gamma_{\textsf{D}}}(\gamma) d\gamma. \end{array} $$
(26)
It is clear that evaluating (26) requires the evaluation of \(f_{\gamma _{\textsf {D}}}(\gamma)\) first. Upon differentiating (23) with respect to γ and using Eq. 07.34.20.0001.01 in [36], we get the following:
$$ \begin{aligned} f_{\gamma_{\textsf{D}}}(\gamma)& ={K_{1}-1 \choose N_{1}-1}K_{1}\sum_{k=0}^{K_{1}-N_{1}}\frac{{K_{1}-N_{1} \choose k}(-1)^{k}}{\tau_{1}} \left(\tau_{1}\exp\left(-\tau_{1}\gamma\right){\vphantom{\left[ \begin{array}{c|c} & 1,\chi_{1}\\[-4pt] \delta_{0}\gamma & \\[-4pt] & \chi_{2},0\\ \end{array}\right]}}\right.\\ &\quad-A\left\{\frac{1}{\delta_{0}\gamma}\mathrm{G}^{3r,0}_{r,3r} \left[ \begin{array}{c|c} & \chi_{1}\\[-4pt] \delta_{0}\gamma & \\[-4pt] & \chi_{2}\\ \end{array}\right] \left[1-\exp\left(-\tau_{1}\gamma\right)\right]\right.\\ & \quad +\tau_{1}\left.\left.\exp\left(-\tau_{1}\gamma\right)\mathrm{G}^{3r,1}_{r+1,3r+1} \left[ \begin{array}{c|c} & 1,\chi_{1}\\[-4pt] \delta_{0}\gamma & \\[-4pt] & \chi_{2},0\\ \end{array}\right]\right\}\right)\\ &\quad + {K_{2}-1 \choose N_{2}-1}K_{2}\sum_{j=0}^{K_{2}-N_{2}}\frac{{K_{2}-N_{2} \choose j}(-1)^{j}}{\tau_{2}}\left(\tau_{2}\exp\left(-\tau_{1}\gamma\right){\vphantom{\left[ \begin{array}{c|c} & 1,\chi_{1}\\[-4pt] \delta_{0}\gamma & \\[-4pt] & \chi_{2},0\\ \end{array}\right]}}\right.\\ & \quad -A\left\{\frac{1}{\delta_{0}\gamma}\times\left[1-\exp\left(-\tau_{2}\gamma\right)\right]\mathrm{G}^{3r,0}_{r,3r} \left[ \begin{array}{c|c} & \chi_{1}\\[-4pt] \delta_{0}\gamma & \\[-4pt] & \chi_{2}\\ \end{array}\right]\right.\\ &\quad+\tau_{2}\exp\left(-\tau_{2}\gamma\right)\mathrm{G}^{3r,1}_{r+1,3r+1} \left.\left.\left[ \begin{array}{c|c} & 1,\chi_{1}\\[-4pt] \delta_{0}\gamma & \\[-4pt] & \chi_{2},0\\ \end{array}\right] \right\}\right)\\ &\quad-{K_{1}-1 \choose N_{1}-1}K_{1}\sum_{k=0}^{K_{1}-N_{1}}\frac{{K_{1}-N_{1} \choose k}}{\tau_{1}}\\ & \quad \times(-1)^{k}{K_{2}-1 \choose N_{2}-1}K_{2}\sum_{j=0}^{K_{2}-N_{2}}\frac{{K_{2}-N_{2} \choose j}(-1)^{j}}{\tau_{2}}\\ &\quad\times\left({\vphantom{\sum_{0}^{0}}}\tau_{1}\exp\left(-\tau_{1}\gamma\right)+\tau_{2}\exp\left(-\tau_{2}\gamma\right) -\left[\tau_{1}+\tau_{2}\right]\right.\\ &\qquad\exp\left(-\left[\tau_{1}+\tau_{2}\right]\gamma\right) -A\left\{\frac{1}{\delta_{0}\gamma}\left[1-\exp\left(-\tau_{1}\gamma\right)\right.\right.\\ &\qquad\left.-\exp\left(-\tau_{2}\gamma\right) +\exp\left(-\left[\tau_{1}+\tau_{2}\right]\gamma\right){\vphantom{\frac{1}{\delta_{0}\gamma}}}\right] \mathrm{G}^{3r,0}_{r,3r} \left[ \begin{array}{c|c} & \chi_{1}\\[-4pt] \delta_{0}\gamma & \\[-4pt] & \chi_{2}\\ \end{array}\right]\\ & \qquad+\mathrm{G}^{3r,1}_{r+1,3r+1} \left[ \begin{array}{c|c} & 1,\chi_{1}\\[-4pt] \delta_{0}\gamma & \\[-4pt] & \chi_{2},0 \end{array}\right] \times\left[\tau_{1}\exp\left(-\tau_{1}\gamma\right)\right.\\ &\qquad+\tau_{2}\exp\left(-\tau_{2}\gamma\right) -\left[{\vphantom{\sum_{0}^{0}}}\tau_{1}+ \left.\left.\left.\tau_{2}\right]\exp\left(-\left[\tau_{1}+\tau_{2}\right]\gamma\right){\vphantom{\sum_{0}^{0}}}\right]\right\}\right)\\ &\quad+\frac{A}{\gamma}\mathrm{G}^{3r,0}_{r,3r} \left[ \begin{array}{c|c} & \chi_{1}\\ \delta_{0}\gamma & \\ & \chi_{2}\\ \end{array}\right]. \end{aligned} $$
(27)
Upon substituting (27) in (26) and using \(\text {ln}(1+\gamma)=\mathrm {G}^{1,2}_{2,2} \left [ \begin {array}{c|c} & 1,1\\[-4pt] \gamma &\\[-4pt] & 1,0\\ \end {array}\right ]\) in integrals which include Meijer G-function, we get
$$ \begin{aligned} C & =\frac{1}{\text{ln}(2)}\left\{{K_{1}-1 \choose N_{1}-1}K_{1}\sum_{k=0}^{K_{1}-N_{1}}\frac{{K_{1}-N_{1} \choose k}(-1)^{k}}{\tau_{1}} \left(\tau_{1}\mathcal{I}_{1}{\vphantom{\frac{1}{\delta_{0}}}}\right.\right.\\&\quad\left.-A\left\{\frac{1}{\delta_{0}}\left[\mathcal{I}_{2}-\mathcal{I}_{3}\right]+\tau_{1}\mathcal{I}_{4}\right\}\right)+{K_{2}-1 \choose N_{2}-1}K_{2}\\ & \quad \times\sum_{j=0}^{K_{2}-N_{2}}\frac{{K_{2}-N_{2} \choose j}(-1)^{j}}{\tau_{2}}\left(\tau_{2}\mathcal{I}^{'}_{1}-A\left\{\frac{1}{\delta_{0}} \left[\mathcal{I}_{2}-\mathcal{I}^{'}_{3}\right]+\tau_{2}\mathcal{I}_{4}^{'}\right\}\right)\\&\quad-K_{1}{K_{1}-1 \choose N_{1}-1}\sum_{k=0}^{K_{1}-N_{1}}\frac{{K_{1}-N_{1} \choose k}(-1)^{k}}{\tau_{1}} \times{K_{2}-1 \choose N_{2}-1} K_{2}\sum_{j=0}^{K_{2}-N_{2}}\\ & \quad \times\frac{{K_{2}-N_{2} \choose j}(-1)^{j}}{\tau_{2}}\left(\tau_{1}\mathcal{I}_{1}+\tau_{2}\mathcal{I}^{'}_{1}-\left[\tau_{1}+\tau_{2}\right]\mathcal{I}_{5}{\vphantom{\frac{1}{\delta_{0}}}}\right.\\ &\quad-A\left\{\frac{1}{\delta_{0}}\left[\mathcal{I}_{2}-\mathcal{I}_{3}-\mathcal{I}_{3}^{'}\right]+\tau_{1}\mathcal{I}_{4} +\tau_{2}\mathcal{I}_{4}^{'}+\frac{1}{\delta_{0}}\mathcal{I}_{6}\right.\\&\quad\!\!\left.\left.\left.{\vphantom{\frac{1}{\delta_{0}}}}- \left[\tau_{1}+\tau_{2}\right]\mathcal{I}_{7}\right\}\right)+\frac{1}{\delta_{0}}\mathcal{I}_{2}{\vphantom{\sum_{k=0}^{K_{1}-N_{1}}}}\right\}, \end{aligned} $$
(28)
where
$$\begin{array}{*{20}l} \mathcal{I}_{1}=\int_{0}^{\infty}\text{ln}(1+\gamma)\exp\left(-\tau_{1}\gamma\right)d\gamma, \end{array} $$
(29)
$$ \begin{aligned} \mathcal{I}_{2}=\int_{0}^{\infty}\gamma^{-1}\mathrm{G}^{1,2}_{2,2} \left[ \begin{array}{c|c} & 1,1\\[-4pt] \gamma & \\[-4pt] & 1,0\\ \end{array}\right] \mathrm{G}^{3r,0}_{r,3r} \left[ \begin{array}{c|c} & \chi_{1}\\[-4pt] \delta_{0}\gamma & \\[-4pt] & \chi_{2} \end{array}\right] d\gamma, \end{aligned} $$
(30)
$$ \begin{aligned} \mathcal{I}_{3}=\int_{0}^{\infty}\gamma^{-1}\exp\left(-\tau_{1}\gamma\right)\mathrm{G}^{1,2}_{2,2} \left[ \begin{array}{c|c} & 1,1\\[-4pt] \gamma & \\[-4pt] & 1,0 \end{array}\right] \mathrm{G}^{3r,0}_{r,3r} \left[ \begin{array}{c|c} & \chi_{1}\\[-4pt] \delta_{0}\gamma & \\[-4pt] & \chi_{2} \end{array}\right] d\gamma, \end{aligned} $$
(31)
$$ \begin{aligned} \mathcal{I}_{4}=\int_{0}^{\infty}\exp\left(-\tau_{1}\gamma\right)\mathrm{G}^{1,2}_{2,2} \left[ \begin{array}{c|c} & 1,1\\[-4pt] \gamma & \\[-4pt] & 1,0 \end{array}\right] \mathrm{G}^{3r,1}_{r+1,3r+1} \left[\! \begin{array}{c|c} & 1,\chi_{1} \\[-4pt] \delta_{0}\gamma & \\[-4pt] & \chi_{2},0 \end{array}\!\right] d\gamma, \end{aligned} $$
(32)
$$\begin{array}{*{20}l} \mathcal{I}_{5}=\int_{0}^{\infty}\text{ln}(1+\gamma)\exp\left(-\left[\tau_{1}+\tau_{2}\right]\gamma\right)d\gamma, \end{array} $$
(33)
$$ \begin{aligned} \mathcal{I}_{6}&=\int_{0}^{\infty}\gamma^{-1}\exp\left(-\left[\tau_{1}+\tau_{2}\right]\gamma\right)\mathrm{G}^{1,2}_{2,2} \left[ \begin{array}{c|c} & 1,1\\[-4pt] \gamma & \\[-4pt] & 1,0 \end{array}\right]\\ &\quad\times\mathrm{G}^{3r,0}_{r,3r} \left[ \begin{array}{c|c} & \chi_{1}\\[-4pt] \delta_{0}\gamma & \\[-4pt] & \chi_{2} \end{array}\right] d\gamma, \end{aligned} $$
(34)
$$ \begin{aligned} \mathcal{I}_{7}&=\int_{0}^{\infty}\exp\left(-\big[\tau_{1}+\tau_{2}\big]\gamma\right)\mathrm{G}^{1,2}_{2,2} \left[ \begin{array}{c|c} & 1,1\\[-4pt] \gamma &\\[-4pt] 1,0 \end{array}\right]\\& \quad\times\mathrm{G}^{3r,1}_{r+1,3r+1} \left[ \begin{array}{c|c} & 1,\chi_{1}\\[-4pt] \delta_{0}\gamma & \\[-4pt] & \chi_{2},0 \end{array}\right] d\gamma, \end{aligned} $$
(35)

where \(\mathcal {I}_{1}^{'}=\mathcal {I}_{1}\), \(\mathcal {I}_{3}^{'}=\mathcal {I}_{3}\), and \(\mathcal {I}_{4}^{'}=\mathcal {I}_{4}\) with replacing k by j, N 1 by N 2, and \(\phantom {\dot {i}\!}\lambda _{u,\textsf {r}_{1}}\) by \(\lambda _{\textsf {r}_{2},u}\phantom {\dot {i}\!}\), respectively.

The integrals \(\mathcal {I}_{1}\) and \(\mathcal {I}^{'}_{1}\) can be obtained using Eq. 4.337.2 in [32], and the other integrals can be obtained with the help of the integral properties of Meijer G-function ([36, Eq. 07.34.21.0011.01] and [36, Eq. 07.34.21.0081.01]). Upon doing these integrations, we get the following:
$$ \begin{aligned} C &=\frac{1}{\text{ln}(2)}\left\{{K_{1}-1 \choose N_{1}-1}K_{1}\sum_{k=0}^{K_{1}-N_{1}}\frac{{K_{1}-N_{1} \choose k}(-1)^{k}}{\tau_{1}}\left(-\exp\left(\tau_{1}\right)\mathrm{E}_{\mathrm{i}}(-\tau_{1}){\vphantom{\left[ \begin{array}{c|c} & 0, 1, \chi_{1}\\[-4pt] \delta_{0} & \\[-4pt] & \chi_{2},0 \end{array}\right]}}\right.\right.\\ &\quad- A\left\{\frac{1}{\delta_{0}}\left[\mathrm{G}^{3r+2,1}_{r+2,3r+2} \left[ \begin{array}{c|c} & 0, 1, \chi_{1}\\[-4pt] \delta_{0} & \\[-4pt] & \chi_{2},0 \end{array}\right]-\mathrm{G}^{0,1:1,2:3r,0}_{1,0:2,2:r,3r}\right.\right.\\ & \quad \left.\times\left[ \begin{array}{c|c|c|c} & 1 & 1,1 & \chi_{1}\\[-4pt] \frac{1}{\tau_{1}},\delta_{0}\tau_{1} &&&\\[-4pt] & - & 1,0 & \chi_{2}\\ \end{array}\right]\right] \mathrm{G}^{0,1:1,2:3r,1}_{1,0:2,2:r+1,3r+1} \\&\quad\times\left.\left.\!\left[ \begin{array}{c|c|c|c} & 2 & 1,1 & 1,\chi_{1} \\[-4pt] \frac{1}{\tau_{1}},\delta_{0}\tau_{1} &&&\\[-4pt] & - & 1,0 & \chi_{2},0 \\ \end{array}\right] \times(\tau_{1})^{2}\right\}\right)+{K_{2}-1 \choose N_{2}-1}K_{2}\\ & \quad \times\sum_{j=0}^{K_{2}-N_{2}}{K_{2}-N_{2} \choose j}(-1)^{j}(\tau_{2})^{-1}\left(-\exp\left(\tau_{2}\right)\mathrm{E}_{\mathrm{i}}(-\tau_{2}){\vphantom{\left[ \begin{array}{c|c} & 0, 1, \chi_{1}\\[-4pt] \delta_{0} & \\[-4pt] & \chi_{2}, 0 \end{array}\right]}}\right.\\ &\quad- A\left\{\frac{1}{\delta_{0}}\left[\mathrm{G}^{3r+2,1}_{r+2,3r+2} \left[ \begin{array}{c|c} & 0, 1, \chi_{1}\\[-4pt] \delta_{0} & \\[-4pt] & \chi_{2}, 0 \end{array}\right]-\mathrm{G}^{0,1:1,2:3r,0}_{1,0:2,2:r,3r}\right.\right.\\ & \left.\quad\times\left[ \begin{array}{c|c|c|c} & 1 & 1,1 & \chi_{1}\\[-4pt] \frac{1}{\tau_{2}},\delta_{0}\tau_{2} &&&\\[-4pt] & - & 1,0 & \chi_{2}\\ \end{array}\right]\right] +(\tau_{2})^{2}\mathrm{G}^{0,1:1,2:3r,1}_{1,0:2,2:r+1,3r+1}\\ &\quad\times\left.\left.\!\left[ \begin{array}{c|c|c|c} & 2 & 1,1 & 1,\chi_{1}\\[-4pt] \frac{1}{\tau_{2}},\delta_{0}\tau_{2} &&& \\[-4pt] & - & 1,0 & \chi_{2},0 \end{array}\right] \right\}\right) -{K_{1}-1 \choose N_{1}-1}K_{1}\\ & \quad \times\sum_{k=0}^{K_{1}-N_{1}}\frac{{K_{1}-N_{1} \choose k}(-1)^{k}}{\tau_{1}}{K_{2}-1 \choose N_{2}-1}K_{2}\sum_{j=0}^{K_{2}-N_{2}}(-1)^{j} \frac{{K_{2}-N_{2} \choose j}}{\tau_{2}}\\&\quad\times\left({\vphantom{\sum_{0}^{0}}}-\exp\left(\tau_{1}\right) \mathrm{E}_{\mathrm{i}}(-\tau_{1})-\exp\left(\tau_{2}\right)\mathrm{E}_{\mathrm{i}} (-\tau_{2})+\exp\left(\tau_{1}+\tau_{2}\right) \right.\\ & \quad \times\mathrm{E}_{\mathrm{i}}(-(\tau_{1}+\tau_{2}))- A\!\left\{\!\frac{1}{\delta_{0}}\!\!\left[\!\mathrm{G}^{3r+2,1}_{r+2,3r+2} \left[ \begin{array}{c|c} & 0, 1, \chi_{1}\\[-4pt] \delta_{0} & \\[-4pt] &\chi_{2},0 \end{array}\right]-\mathrm{G}^{0,1:1,2:3r,0}_{1,0:2,2:r,3r}\right.\right.\\&\quad\times \left[ \begin{array}{c|c|c|c} & 1 & 1,1 & \chi_{1}\\[-4pt] \frac{1}{\tau_{1}},\delta_{0}\tau_{1} &&&\\[-4pt] & - & 1,0 & \chi_{2} \end{array}\right]-\mathrm{G}^{0,1:1,2:3r,0}_{1,0:2,2:r,3r} \left[ \begin{array}{c|c|c|c} & 1 & 1,1 & \chi_{1} \\[-4pt] \frac{1}{\tau_{2}},\delta_{0}\tau_{2} &&&\\[-4pt] & - & 1,0 & \chi_{2} \end{array}\right]\\ & \quad +\mathrm{G}^{0,1:1,2:3r,0}_{1,0:2,2:r,3r} \left.\left[ \begin{array}{c|c|c|c} & 1 & 1,1 & \chi_{1}\\[-4pt] \frac{1}{\left[\tau_{1}+\tau_{2}\right]},\delta_{0}\left[\tau_{1}+\tau_{2}\right] &&&\\[-4pt] & - & 1,0 & \chi_{2} \end{array}\right]\right] +(\tau_{1})^{2}\\ & \quad \times\mathrm{G}^{0,1:1,2:3r,1}_{1,0:2,2:r+1,3r+1} \left[ \begin{array}{c|c|c|c} & 2 & 1,1 & 1,\chi_{1}\\[-4pt] \frac{1}{\tau_{1}},\delta_{0}\tau_{1}&&&\\[-4pt] & - & 1,0 & \chi_{2},0 \end{array}\right]\\ &\quad+(\tau_{2})^{2}\mathrm{G}^{0,1:1,2:3r,1}_{1,0:2,2:r+1,3r+1}\left[ \begin{array}{c|c|c|c} & 2 & 1,1 & 1,\chi_{1}\\[-4pt] \frac{1}{\tau_{2}},\delta_{0}\tau_{2} &&&\\[-4pt] & - & 1,0 & \chi_{2},0 \end{array}\right] -\left[\tau_{1}+\tau_{2}\right]^{2}\\ & \quad\times\mathrm{G}^{0,1:1,2:3r,0}_{1,0:2,2:r,3r} \left.\left[ \begin{array}{c|c|c|c} & 1 & 1,1 & \chi_{1}\\[-4pt] \frac{1}{[\tau_{1}+\tau_{2}]},\delta_{0}[\tau_{1}+\tau_{2}]&&&\\[-4pt] & - & 1,0 & \chi_{2} \end{array}\right] \right\}\\&\quad+\frac{1}{\delta_{0}}\mathrm{G}^{3r+2,1}_{r+2,3r+2} \left.\left[ \begin{array}{c|c} & 0, 1, \chi_{1}\\[-4pt] \delta_{0} \\[-4pt] &\chi_{2}, 0 \end{array}\right]\right\}, \end{aligned} $$
(36)

where Ei(.) is the exponential integral function defined by Eq. 8.211.1 in [32], and G[Z 1,Z 2|.|.|.] is the extended generalized bivariate Meijer G-function. Note that in order to evaluate the expression in (36), a Mathematica code similar to the one provided in Table 2 in [41] has been used here.

4 Asymptotic outage performance and power allocation

Due to complexity of the achieved expressions in the previous section, it is hard to easily study the effect of various system parameters and get more insights about the system performance. Therefore, we see that it is important to derive simple expressions for the outage probability which will be helpful in achieving more insights about the system behavior. These expressions will be used also in allocating the transmission power for the transmitting nodes of the system (first hop source’s power, second hop relay’s power, and third hop relay’s power).

4.1 Asymptotic outage probability

The outage probability can be written at the high SNR regime as \(\phantom {\dot {i}\!}P_{\textsf {out}}\simeq (G_{\textsf {c}}SNR)^{-G_{\textsf {d}}}\), where G c and G d are the coding gain and diversity order of the system, respectively [42]. Obviously, G c represents the horizontal shift in the outage probability performance relative to the benchmark curve \(\phantom {\dot {i}\!}(\text {SNR})^{-G_{\textsf {d}}}\) and G d refers to the increase in the slope of the outage probability vs SNR curve. Therefore, the parameters on which the diversity order depends will affect the slope of the outage probability curves and the parameters on which the coding gain depends will affect the position of the curves. Obtaining the outage probability in this simple form allows us to easily study and know the effect of each system parameter on the system performance instead of dealing with the long/complex expressions derived in Section 3. Notice that such an accurate approximation has been widely used in the conventional cooperative diversity systems.

Here, we consider the case of identical sources’ channels \(\phantom {\dot {i}\!}(\lambda _{1,\textsf {r}_{1}}=\lambda _{2,\textsf {r}_{1}}=\cdots =\lambda _{K_{1},\textsf {r}_{1}}=\lambda _{u,\textsf {r}_{1}})\) and identical destinations’ channels \(\phantom {\dot {i}\!}(\lambda _{\textsf {r}_{2},1}=\lambda _{\textsf {r}_{2},2}=\cdots =\lambda _{\textsf {r}_{2},K_{2}}=\lambda _{\textsf {r}_{2},u})\). Again, we follow the same procedure that we followed before in Section 3 in obtaining the outage probability of the proposed scenario by dealing with the approximate CDF of each hop separately and then calculating the approximate CDF of the e2e SNR.

4.1.1 First hop link

Upon using the Taylor series representation of the exponential term in the CDF \(F_{\gamma _{\textsf {U},\textsf {R}_{1}}}(\gamma)=1-\exp \left (-\lambda _{u,\textsf {r}_{1}}\gamma \right)\), we get \(F_{\gamma _{\textsf {U},\textsf {R}_{1}}}(\gamma)\approx 1-\big [1-(\lambda _{u,\textsf {r}_{1}}\gamma) +\frac {(\lambda _{u,\textsf {r}_{1}}\gamma)^{2}}{2!}-\frac {(\lambda _{u,\textsf {r}_{1}}\gamma)^{3}}{3!} +\frac {(\lambda _{u,\textsf {r}_{1}}\gamma)^{4}}{4!}-\cdots \big ]\), which for high values of \(\bar {\gamma }_{u,\textsf {r}_{1}}\) (\(\bar {\gamma }_{u,\textsf {r}_{1}} \rightarrow \infty \)) simplifies to \(\lambda _{u,\textsf {r}_{1}}\gamma \phantom {\dot {i}\!}\) and hence, the PDF \(f_{\gamma _{\textsf {U},\textsf {R}_{1}}}(\gamma)\) simplifies to \(\lambda _{u,\textsf {r}_{1}}\phantom {\dot {i}\!}\). Upon substituting these statistics in (18) and integrating the result using \(\int _{0}^{\gamma }f_{\gamma _{\textsf {U}_{\text {Sel}},\textsf {R}_{1}}}(t)dt\), we get
$$\begin{array}{*{20}l} F_{\gamma_{\textsf{U}_{\text{Sel}},\textsf{R}_{1}}}(\gamma)\simeq&{K_{1}-1 \choose N_{1}-1}K_{1}(\lambda_{u,\textsf{r}_{1}})^{K_{1}-N_{1}+1}\\ &\times\sum_{k=0}^{N_{1}-1}\frac{{N_{1}-1 \choose k}(-1)^{k}(\lambda_{u,\textsf{r}_{1}})^{k}}{(k+K_{1}-N_{1}+1)}\gamma^{k+K_{1}-N_{1}+1}. \end{array} $$
(37)
The CDF in (37) is still dominant for the first term of the summation and hence, it can be further simplified to
$$\begin{array}{*{20}l} F_{\gamma_{\textsf{U}_{\text{Sel}},\textsf{R}_{1}}}(\gamma)\simeq&{K_{1}-1 \choose N_{1}-1}K_{1}(\lambda_{u,\textsf{r}_{1}})^{K_{1}-N_{1}+1}\frac{\gamma^{K_{1}-N_{1}+1}}{(K_{1}-N_{1}+1)}. \end{array} $$
(38)

4.1.2 Second hop link

From Eq. 07.34.06.006.01 in [36], as \(\bar {\gamma }_{\textsf {r}_{1},\textsf {r}_{2}} \rightarrow \infty \) or, equivalently, as z→0, the Meijer G-function can be approximated using the following series representation:
$$ \begin{aligned} &\mathrm{G}^{m,n}_{p,q} \left[ \begin{array}{c|c} & a_{1},\ldots,a_{p}\\[-4pt] z &\\[-4pt] & b_{1},\ldots,b_{q} \end{array}\right]\\ &=\sum_{k=1}^{m}\frac{\prod^{m}_{j=1, j\neq k}\Gamma(b_{j}-b_{k})\prod^{n}_{j=1}\Gamma(1-a_{j}+b_{k})}{\prod^{p}_{j=n+1}\Gamma(a_{j}-b_{k}) \prod^{q}_{j=m+1}\Gamma(1-b_{j}+b_{k})}z^{b_{k}}(1+o(z)), \end{aligned} $$
(39)
where pq is required. Here, we use the same approach that was used in [43] in writing the outage probability for this case. Defining ν=min{ζ 2,α,β}, then we have
$$\begin{array}{*{20}l} F_{\gamma_{\textsf{R}_{1},\textsf{R}_{2}}}(\gamma)\simeq\Upsilon \left(\frac{\gamma}{\bar{\gamma}_{\textsf{r}_{1},\textsf{r}_{2}}}\right)^{\frac{\nu}{r}}, \end{array} $$
(40)
where Υ is constant. In order to find the value of Υ, we first rewrite the CDF \(F_{\gamma _{\textsf {R}_{1},\textsf {R}_{2}}}(\gamma)\) provided in (20) using the asymptotic expression of the Meijer provided in (39). By matching the Meijer function with its asymptotic representation, we find that m=3r,n=1,p=r+1,q=3r+1. Now, the asymptotic CDF \(F_{\gamma _{\textsf {R}_{1},\textsf {R}_{2}}}(\gamma)\) can be written with ignoring the high order terms as
$$\begin{array}{*{20}l} F_{\gamma_{\textsf{R}_{1},\textsf{R}_{2}}}(\gamma)&=A\sum_{k=1}^{3r}\frac{\prod^{3r}_{j=1, j\neq k}\Gamma(b_{j}-b_{k})\Gamma(1-a_{1}+b_{k})}{\prod^{r+1}_{j=2}\Gamma(a_{j}-b_{k}) \Gamma(1-b_{3r+1}+b_{k})}\\&\quad \times B^{b_{k}/r}\left(\frac{\gamma}{\bar{\gamma}_{\textsf{r}_{1},\textsf{r}_{2}}}\right)^{\frac{b_{k}}{r}}, \end{array} $$
(41)
where b j =χ 2(j), j=1,…,3r, b 3r+1=0, b k =min{ζ 2,α,β}, a j =χ 1(j), for j=1,…,r+1, a 1=1, and b 3r+1=0. For the case of IM/DD receiver (r=2), (41) simplifies to
$$\begin{array}{*{20}l} F_{\gamma_{\textsf{R}_{1},\textsf{R}_{2}}}(\gamma)&=A\sum_{k=1}^{6}\frac{\prod^{6}_{j=1, j\neq k}\Gamma(b_{j}-b_{k})\Gamma(b_{k})}{\prod^{3}_{j=2}\Gamma(a_{j}-b_{k}) \Gamma(1+b_{k})}\\&\quad\times B^{b_{k}/2}\left(\frac{\gamma}{\bar{\gamma}_{\textsf{r}_{1},\textsf{r}_{2}}}\right)^{\frac{b_{k}}{2}}, \end{array} $$
(42)
where b j =χ 2(j), j=1,…,6, b 7=0, a j =χ 1(j), for j=1,…,3, a 1=1, and b 7=0. Comparing (42) with the second term of (40) shows that the constant Υ equals
$$\begin{array}{*{20}l} \Upsilon=A\sum_{k=1}^{6}\frac{\prod^{6}_{j=1, j\neq k}\Gamma(b_{j}-b_{k})\Gamma(b_{k})}{\prod^{3}_{j=2}\Gamma(a_{j}-b_{k}) \Gamma(1+b_{k})}B^{b_{k}/2}, \end{array} $$
(43)

where b k =ν.

4.1.3 Third hop link

Similar to first hop link analysis, as \(\bar {\gamma }_{\textsf {r}_{2},u} \rightarrow \infty \), the CDF and PDF of the third hop SNR \(F_{\gamma _{\textsf {R}_{2},\textsf {D}}}(\gamma)\) and \(F_{\gamma _{\textsf {R}_{2},\textsf {D}}}(\gamma)\) simplify to \(\phantom {\dot {i}\!}\lambda _{\textsf {r}_{2},u}\gamma \) and \(\phantom {\dot {i}\!}\lambda _{\textsf {r}_{2},u}\), respectively. Upon substituting these statistics in (18) and integrating the result using \(\int _{0}^{\gamma }f_{\gamma _{\textsf {R}_{2},\textsf {D}_{\text {Sel}}}}(t)dt\), we get
$$\begin{array}{*{20}l} F_{\gamma_{\textsf{R}_{2},\textsf{D}_{\text{Sel}}}}(\gamma)\simeq&{K_{2}-1 \choose N_{2}-1}K_{2}(\lambda_{\textsf{r}_{2},u})^{K_{2}-N_{2}+1}\\&\times\sum_{j=0}^{N_{2}-1}\frac{{N_{2}-1 \choose j}(-1)^{j}(\lambda_{\textsf{r}_{2},u})^{j}}{(j+K_{2}-N_{2}+1)}\gamma^{j+K_{2}-N_{2}+1}. \end{array} $$
(44)
The CDF in (44) is still dominant for the first term of the summation and hence, it can be further simplified to
$$\begin{array}{*{20}l} F_{\gamma_{\textsf{R}_{2},\textsf{D}_{\text{Sel}}}}(\gamma)\simeq&{K_{2}-1 \choose N_{2}-1}K_{2}(\lambda_{\textsf{r}_{2},u})^{K_{2}-N_{2}+1}\frac{\gamma^{K_{2}-N_{2}+1}}{(K_{2}-N_{2}+1)}. \end{array} $$
(45)
With the aim of obtaining the diversity order and coding gain of the system, the CDF in (16) can be simplified at high SNR values to be
$$\begin{array}{*{20}l} F_{\gamma_{\textsf{D}}}(\gamma)\simeq&F_{\gamma_{\textsf{U}_{\text{Sel}},\textsf{R}_{1}}} (\gamma)+F_{\gamma_{\textsf{R}_{1},\textsf{R}_{2}}}(\gamma)+F_{\gamma_{\textsf{R}_{2},\textsf{D}_{\text{Sel}}}}(\gamma), \end{array} $$
(46)

where the remaining terms in (16) are omitted and this is accurate for high SNRs.

Substituting (38), (39), and (45) in (46), the outage probability can be written at high SNR values as
$$ \begin{aligned} P_{\textsf{out}}^{\infty}=&{K_{1}-1 \choose N_{1}-1}K_{1}(\bar{\gamma}_{u,\textsf{r}_{1}})^{-(K_{1}-N_{1}+1)}\frac{(\gamma_{\textsf{ out}})^{K_{1}-N_{1}+1}}{(K_{1}-N_{1}+1)}\\ &+\left(\frac{\Upsilon^{-\frac{r}{\nu}}}{\gamma_{\textsf{ out}}}\bar{\gamma}_{\textsf{r}_{1},\textsf{r}_{2}}\right)^{-\frac{\nu}{r}}+{K_{2}-1 \choose N_{2}-1}K_{2}(\bar{\gamma}_{\textsf{r}_{2},u})^{-(K_{2}-N_{2}+1)}\\ &\times\frac{(\gamma_{\textsf{out}})^{K_{2}-N_{2}+1}}{(K_{2}-N_{2}+1)}. \end{aligned} $$
(47)

It is clear from (47) that the performance of the considered relay network will be dominated by the worst link among the available three links, the first RF link, the FSO link, and the third RF link. This domination depends on the parameters of these links. Therefore, the diversity order (G d) of the triple-hop mixed RF/FSO/RF relay network with generalized order user scheduling is equal to min(K 1N 1+1,ν/r,K 2N 2+1) and based on the value of the diversity order, one of the following three cases could represent the overall system performance:

Case 1 (One hop is dominant).

In this case, the coding gain (G c) can be written as
$$ \begin{aligned} G_{\textsf{c}}=\left\{ \begin{array}{ll} \left[{K_{1}-1 \choose N_{1}-1}K_{1}\frac{(\gamma_{\textsf{out}})^{K_{1}-N_{1}+1}}{(K_{1}-N_{1}+1)}\right]^{-\frac{1}{K_{1}-N_{1}+1}}, & G_{\text{\textsf{d}}}=K_{1}-N_{1}+1, \\ \frac{\Upsilon^{-\frac{r}{\nu}}}{\gamma_{\text{\textsf{out}}}}, & G_{\text{\textsf{d}}}=\frac{\nu}{r}, \\ \left[{K_{2}-1 \choose N_{2}-1}K_{2}\frac{(\gamma_{\text{\textsf{out}}})^{K_{2}-N_{2}+1}}{(K_{2}-N_{2}+1)}\right]^{-\frac{1}{K_{2}-N_{2}+1}}, & G_{\text{\textsf{d}}}=K_{2}-N_{2}+1.\\ \end{array} \right. \end{aligned} $$
(48)

Case 2 (Two hops are dominant).

In this case, the coding gain can be written as
$$\begin{array}{@{}rcl@{}} G_{\textsf{c}}=\left\{ \begin{array}{ll} \frac{1}{2}\bigg\{\left[{K_{1}-1 \choose N_{1}-1}K_{1}\frac{(\gamma_{\textsf{out}})^{K_{1}-N_{1}+1}}{(K_{1}-N_{1}+1)}\right]^{-\frac{1}{K_{1}-N_{1}+1}}\\+\left[{K_{2}-1 \choose N_{2}-1}K_{2}\frac{(\gamma_{\textsf{out}})^{K_{2}-N_{2}+1}}{(K_{2}-N_{2}+1)}\right]^{-\frac{1}{K_{2}-N_{2}+1}}\bigg\}, & G_{\text{\textsf{d}}}=K_{1}-N_{1}+1=K_{2}-N_{2}+1,\\ \frac{1}{2}\bigg\{\left[{K_{1}-1 \choose N_{1}-1}K_{1}\frac{(\gamma_{\textsf{out}})^{K_{1}-N_{1}+1}}{(K_{1}-N_{1}+1)}\right]^{-\frac{1}{K_{1}-N_{1}+1}}+\frac{\Upsilon^{-\frac{r}{\nu}}}{\gamma_{\textsf{ out}}}\bigg\}, & G_{\text{\textsf{d}}}=K_{1}-N_{1}+1=\frac{\nu}{r},\\ \frac{1}{2}\bigg\{\left[{K_{2}-1 \choose N_{2}-1}K_{2}\frac{(\gamma_{\textsf{out}})^{K_{2}-N_{2}+1}}{(K_{2}-N_{2}+1)}\right]^{-\frac{1}{K_{2}-N_{2}+1}}+\frac{\Upsilon^{-\frac{r}{\nu}}}{\gamma_{\textsf{ out}}}\bigg\}, & G_{\text{\textsf{d}}}=K_{2}-N_{2}+1=\frac{\nu}{r}.\\ \end{array} \right. \end{array} $$
(49)

Case 3 (Three hops have the same diversity order).

In this case, the coding gain can be written as
$$\begin{array}{@{}rcl@{}} G_{\textsf{c}}=\left\{ \begin{array}{ll} \frac{1}{3}\left\{\left[{K_{1}-1 \choose N_{1}-1}K_{1}\frac{(\gamma_{\textsf{out}})^{K_{1}-N_{1}+1}}{(K_{1}-N_{1}+1)}\right]^{-\frac{1}{K_{1}-N_{1}+1}} +\frac{\Upsilon^{-\frac{r}{\nu}}}{\gamma_{\textsf{sout}}}\right.\\ +\left.\left[{K_{2}-1 \choose N_{2}-1}K_{2}\frac{(\gamma_{\textsf{out}})^{K_{2}-N_{2}+1}}{-(K_{2}-N_{2}+1)}{\vphantom{\sum_{0}^{0}}}\right] ^{-\frac{1}{K_{2}-N_{2}+1}}\right\}, & G_{\text{\textsf{d}}}=K_{1}-N_{1}+1=K_{2}-N_{2}+1=\frac{\nu}{r}.\\ \end{array} \right. \end{array} $$
(50)

In summary, the system performance could be dominated by the following: (1) the first hop link (i.e., K 1 and N 1) when it is the worst link among the three links; (2) the second hop link (i.e., ζ 2, α, and β) when it is the worst link among the three links; and (3) the third hop link (i.e., K 2 and N 2) when it is the worst link among the three links. It is very important to mention here that if the diversity orders of two hops are equal and they are the minimum compared to that of the third hop, the coding gain of the system in this case equals the summation of the coding gains of the two hops which dominate the system performance divided by 2. Similarly, if the diversity orders of the three hops are equal, the coding gain of the system in this case equals the summation of the coding gains of the three hops divided by 3.

4.2 Power allocation

In this section, we aim to derive the optimum adaptive power allocation for the transmitting nodes in the system. In the proposed power allocation protocol, the powers of all transmitting nodes (selected source, first relay, and second relay) in the system are considered to be variables. Assuming that we may have two different cases in regard to the transmission power of the first relay: the allocated power for the first relay is less than the peak/maximum allowed power and in this case, the power allocation protocol is optimum and the second case where the allocated power for the first relay is larger than the peak power and here the extra allocated power will be saved.

We denote the distance between the first hop K 1 sources and relay R1 by \(\phantom {\dot {i}\!}d_{\textsf {s},\textsf {r}_{1}}\), the distance between the relays R1 and R2 by \(d_{\textsf {r}_{1},\textsf {r}_{2}}\phantom {\dot {i}\!}\), while the distance between the relay R2 and third hop K 2 destinations by \(d_{\textsf {r}_{2},\textsf {d}}\phantom {\dot {i}\!}\). We consider a total distance of D tot between the first hop sources and third hop destinations. The total distance D tot can be written as \(\phantom {\dot {i}\!}D_{\text {tot}}=d_{\textsf {s},\textsf {r}_{1}}+d_{\textsf {r}_{1},\textsf {r}_{2}}+d_{\textsf {r}_{2},\textsf {d}}\). Under the scenario where the received power decays with the distance, we can express the average value of SNR in the hop between the K 1 sources and relay R1 as \(\bar {\gamma }_{\textsf {s},\textsf {r}_{1}}={P_{\textsf {s},\textsf {r}_{1}}}{d^{-\mu }_{\textsf {s},\textsf {r}_{1}}}\), where \(P_{\textsf {s},\textsf {r}_{1}}=\frac {P_{u,\textsf {r}_{1}}}{K_{1}}= \frac {E_{\textsf {s},\textsf {r}_{1}}}{N_{0}}\), μ is the path loss exponent and is equal for all hops to a value greater than 1, and N 0 is AWGN power which is assumed equal for the three hops. Similarly, we can express the average value of SNR in the second hop as \(\bar {\gamma }_{\textsf {r}_{1},\textsf {r}_{2}}={P_{\textsf {r}_{1},\textsf {r}_{2}}}{d^{-\mu }_{\textsf {r}_{1}, \textsf {r}_{2}}}\), where \(P_{\textsf {r}_{1},\textsf {r}_{2}}=\frac {E_{\textsf {r}_{1},\textsf {r}_{2}}}{N_{0}}\). The average value of SNR in the hop between the relay R2 and K 2 destinations can be expressed as \(\bar {\gamma }_{\textsf {r}_{2},\textsf {d}}={P_{\textsf {r}_{2},\textsf {d}}} {d^{-\mu }_{\textsf {r}_{2},\textsf {d}}}\), where \(P_{\textsf {r}_{2},\textsf {d}}=\frac {P_{\textsf {r}_{2},\textsf {d}}}{K_{1}}= \frac {E_{\textsf {r}_{2},\textsf {d}}}{N_{0}}\phantom {\dot {i}\!}\). Finally, the power constraint can be written as \(\phantom {\dot {i}\!}P_{\text {tot}}=P_{\textsf {s},\textsf {r}_{1}}+P_{\textsf {r}_{1},\textsf {r}_{2}}+P_{\textsf {r}_{2},\textsf {d}}\).

We derive the optimal power allocation that minimizes the outage probability subject to sum power constraint as below
$$\begin{array}{*{20}l} & \left(P^{*}_{\textsf{s},\textsf{r}_{1}},P^{*}_{\textsf{r}_{1},\textsf{r}_{2}}, P^{*}_{\textsf{r}_{2},\textsf{d}}\right) =\text{arg}\underset{\left(P_{\textsf{s},\textsf{r}_{1}}, P_{\textsf{r}_{1},\textsf{r}_{2}},P_{\textsf{r}_{2},\textsf{d}}\right)}{\text{min}} F_{\gamma_{\textsf{D}}}(\gamma_{\textsf{out}}),\\ &\text{subject to}\ P_{\text{tot}}=P_{\textsf{s},\textsf{r}_{1}}+P_{\textsf{r}_{1},\textsf{r}_{2}}+P_{\textsf{r}_{2},\textsf{d}}. \end{array} $$
(51)
The asymptotic expression for \(F_{\gamma _{\textsf {D}}}(\gamma _{\textsf {out}})\) can be rewritten as follows
$$ { \begin{aligned} F_{\gamma_{\textsf{D}}}(\gamma_{\textsf{out}})\simeq&{K_{1}-1 \choose N_{1}-1}K_{1}\left(\frac{d_{\textsf{s},\textsf{r}_{1}}^{\mu}}{P_{\textsf{s}, \textsf{r}_{1}}}\right)^{K_{1}-N_{1}+1}\frac{(\gamma_{\textsf{out}})^{K_{1}-N_{1}+1}}{(K_{1}-N_{1}+1)}\\&+\frac{\gamma_{\textsf{out}} AB {d^{\mu}_{\textsf{r}_{1},\textsf{r}_{2}}}}{P_{\textsf{r}_{1}, \textsf{r}_{2}}}+{K_{2}-1 \choose N_{2}-1}K_{2}\\ &\times\left(\frac{d_{\textsf{r}_{2},\textsf{d}}^{\mu}}{P_{\textsf{r}_{2},\textsf{d}}}\right)^{K_{2}-N_{2}+1} \frac{(\gamma_{\textsf{out}})^{K_{2}-N_{2}+1}}{(K_{2}-N_{2}+1)}. \end{aligned}} $$
(52)

To simplify the understanding of the next steps of optimization, we introduce here the Lagrangian multipliers method which is used in deriving of the optimal transmission powers of the transmitting nodes of the system. This method is a very common strategy used for finding the local maxima and minima of a function subject to equality constraints. The method depends on defining a Lagrange multiplier, Lagrange function, and a constraint. The Lagrange function is then defined as a summation of the constraint multiplied by the Lagrange multiplier and the original constrained problem or function. Note that the original constrained function and constraint are functions of the unknowns which we need to find their optimum values. Then, the Lagrange function is differentiated with respect to each unknown and equated by zero trying to find a solution to that unknown in terms of the Lagrange multiplier. Then, the unknowns are written again (in terms of the Lagrange multiplier) to formulate the constraint. After that, the constraint is solved in order to obtain the Lagrange multiplier. Finally, the obtained Lagrange multiplier is used in finding the optimal values of the unknowns.

Now, using the Lagrangian multipliers method for our problem, we define the Lagrange function as
$$\begin{array}{*{20}l} \mathcal{F}(P_{\textsf{s},\textsf{r}_{1}},P_{\textsf{r}_{1},\textsf{r}_{2}},P_{\textsf{r}_{2}, \textsf{d}},\lambda)&=F_{\gamma_{\textsf{D}}}(P_{\textsf{s},\textsf{r}_{1}},P_{\textsf{r}_{1}, \textsf{r}_{2}},P_{\textsf{r}_{2},\textsf{d}})\\&\quad+\lambda g(P_{\textsf{s},\textsf{r}_{1}},P_{\textsf{r}_{1},\textsf{r}_{2}},P_{\textsf{r}_{2},\textsf{d}}), \end{array} $$
(53)
where λ is the Lagrange multiplier and g(.) is the power constraint. The function in (53) can be rewritten as
$$ {\begin{aligned} \mathcal{F}(P_{\textsf{s},\textsf{r}_{1}},P_{\textsf{r}_{1},\textsf{r}_{2}},P_{\textsf{r}_{2},\textsf{d}},\lambda)=&{K_{1}-1 \choose N_{1}-1}K_{1}\left(\frac{d_{\textsf{s},\textsf{r}_{1}}^{\mu}}{P_{\textsf{s},\textsf{ r}_{1}}}\right)^{K_{1}-N_{1}+1}\\ &\times\frac{(\gamma_{\textsf{out}})^{K_{1}-N_{1}+1}}{(K_{1}-N_{1}+1)} +\frac{\gamma_{\textsf{out}} AB {d^{\mu}_{\textsf{r}_{1},\textsf{r}_{2}}}}{P_{\textsf{r}_{1},\textsf{r}_{2}}}\\ &+\!{K_{2}-1 \choose N_{2}-1}\!\!\times\! K_{2}\left(\frac{d_{\textsf{r}_{2},\textsf{d}}^{\mu}}{P_{\textsf{r}_{2}, \textsf{d}}}\right)^{K_{2}-N_{2}+1}\\ &\times\frac{(\gamma_{\textsf{out}})^{K_{2}-N_{2}+1}}{(K_{2}-N_{2}+1)} +\lambda(P_{\textsf{s},\textsf{r}_{1}}\\&+P_{\textsf{r}_{1},\textsf{r}_{2}}+P_{\textsf{r}_{2},\textsf{d}}-P_{\text{tot}}). \end{aligned}} $$
(54)
Taking the first derivative with respect to \(P_{\textsf {s},\textsf {r}_{1}}\phantom {\dot {i}\!}\) and equate it by zero, we get
$$\begin{array}{*{20}l} \frac{\partial \mathcal{F}}{\partial P_{\textsf{s},\textsf{r}_{1}}}= &-{K_{1}-1 \choose N_{1}-1}K_{1}\left(d_{\textsf{s},\textsf{r}_{1}}^{\mu}\gamma_{\textsf{out}}\right)^{K_{1}-N_{1}+1}\\ &\times\left(P^{*}_{\textsf{s},\textsf{r}_{1}}\right)^{-(K_{1}-N_{1}+2)}+\lambda=0. \end{array} $$
(55)
Solving for \(P^{*}_{\textsf {s},\textsf {r}_{1}}\), we get
$$\begin{array}{*{20}l} P^{*}_{\textsf{s},\textsf{r}_{1}}=\left[{K_{1}-1 \choose N_{1}-1}K_{1}\right]^{K_{1}-N_{1}+3}d_{\textsf{s},\textsf{r}_{1}}^{\mu}\gamma_{\textsf{out}}\lambda^{\frac{-1}{(K_{1}-N_{1}+2)}}. \end{array} $$
(56)
Similarly, differentiating \(\mathcal {F}\) with respect to \(\phantom {\dot {i}\!}P_{\textsf {r}_{1},\textsf {r}_{2}}\) and equating the result by zero and solving for \(P_{\textsf {r}_{1},\textsf {r}_{2}}\phantom {\dot {i}\!}\), we get
$$\begin{array}{*{20}l} P^{*}_{\textsf{r}_{1},\textsf{r}_{2}}=\left[ABd_{\textsf{r}_{1},\textsf{r}_{2}}^{\mu}\gamma_{\textsf{out}}\right]^{\frac{1}{2}} \lambda^{\frac{-1}{2}}. \end{array} $$
(57)
Following the same method of the first hop with the third hop, we get
$$\begin{array}{*{20}l} P^{*}_{\textsf{r}_{2},\textsf{d}}=\left[{K_{2}-1 \choose N_{2}-1}K_{2}\right]^{K_{2}-N_{2}+3}d_{\textsf{r}_{2},\textsf{d}}^{\mu}\gamma_{\textsf{out}}\lambda^{\frac{-1}{(K_{2}-N_{2}+2)}}. \end{array} $$
(58)
Now, summing the three individual powers’ results in the total power as
$$\begin{array}{*{20}l} &\left[{K_{1}-1 \choose N_{1}-1}K_{1}\right]^{K_{1}-N_{1}+3}d_{\textsf{s},\textsf{r}_{1}}^{\mu}\gamma_{\textsf{ out}}\lambda^{\frac{-1}{(K_{1}-N_{1}+2)}}\\&+\left[ABd_{\textsf{r}_{1},\textsf{r}_{2}}^{\mu}\gamma_{\textsf{out}}\right]^{\frac{1}{2}} \lambda^{\frac{-1}{2}}+\left[{K_{2}-1 \choose N_{2}-1}K_{2}\right]^{K_{2}-N_{2}+3}d_{\textsf{r}_{2},\textsf{d}}^{\mu}\\ &\times\gamma_{\textsf{out}}\lambda^{\frac{-1}{(K_{2}-N_{2}+2)}}=P_{\text{tot}}. \end{array} $$
(59)
It is clear from (59) that finding a closed-form expression for λ would be very difficult. However, a numerical solution can be found by standard iterative root-finding algorithms, such as the Bisection’s method and Newton’s method. It is worthwhile to mention that the closed-form expressions for some special cases can be found such as the case where K 1=K 2=1 and N 1=N 2=1. For this case, (59) reduces to
$$\begin{array}{*{20}l} d_{\textsf{s},\textsf{r}_{1}}^{\mu}\gamma_{\textsf{out}}\lambda^{\frac{-1}{2}}+\left[ABd_{\textsf{r}_{1}, \textsf{r}_{2}}^{\mu}\gamma_{\textsf{out}}\right]^{\frac{1}{2}}\lambda^{\frac{-1}{2}}+d_{\textsf{r}_{2}, \textsf{d}}^{\mu}\lambda^{\frac{-1}{2}}=P_{\text{tot}}. \end{array} $$
(60)
Solving for λ gives
$$\begin{array}{*{20}l} \lambda=\left\{\frac{P_{\text{tot}}}{\gamma_{\textsf{out}}(d_{\textsf{s},\textsf{r}_{1}}^{\mu}+d_{\textsf{r}_{2}, \textsf{d}}^{\mu})+\left[ABd_{\textsf{r}_{1},\textsf{r}_{2}}^{\mu}\gamma_{\textsf{out}}\right]^{\frac{1}{2}}}\right\}^{-2}. \end{array} $$
(61)
Upon substituting (61) in (56), (57), and (58), we get the optimum transmission powers as
$$\begin{array}{*{20}l} P^{*}_{\textsf{s},\textsf{r}_{1}}=\frac{d_{\textsf{s},\textsf{r}_{1}}^{\mu}\gamma_{\textsf{out}}P_{\text{tot}}} {\gamma_{\textsf{out}}(d_{\textsf{s},\textsf{r}_{1}}^{\mu}+d_{\textsf{r}_{2},\textsf{d}}^{\mu})+\left[ABd_{\textsf{r}_{1}, \textsf{r}_{2}}^{\mu}\gamma_{\textsf{out}}\right]^{\frac{1}{2}}}, \end{array} $$
(62)
$$\begin{array}{*{20}l} P^{*}_{\textsf{r}_{1},\textsf{r}_{2}}=\frac{\left[ABd_{\textsf{r}_{1},\textsf{r}_{2}}^{\mu}\gamma_{\textsf{out}}\right]^{\frac{1}{2}} P_{\text{tot}}}{\gamma_{\textsf{out}}(d_{\textsf{s},\textsf{r}_{1}}^{\mu}+d_{\textsf{r}_{2},\textsf{d}}^{\mu})+\left[ABd_{\textsf{r}_{1}, \textsf{r}_{2}}^{\mu}\gamma_{\textsf{out}}\right]^{\frac{1}{2}}}, \end{array} $$
(63)
$$\begin{array}{*{20}l} P^{*}_{\textsf{r}_{2},\textsf{d}}=\frac{d_{\textsf{r}_{2},\textsf{d}}^{\mu}\gamma_{\textsf{out}}P_{\text{tot}}} {\gamma_{\textsf{out}}(d_{\textsf{s},\textsf{r}_{1}}^{\mu}+d_{\textsf{r}_{2},\textsf{d}}^{\mu})+ \left[ABd_{\textsf{r}_{2},\textsf{d}}^{\mu}\gamma_{\textsf{out}}\right]^{\frac{1}{2}}}. \end{array} $$
(64)

The effectiveness of the derived optimal power allocation solutions will be verified in the following section where a numerical example which compares the system performance with and without power allocation is provided and discussed.

5 Simulation and numerical results

In this section, we validate the accuracy of the achieved analytical and asymptotic expressions via comparison with Monte Carlo simulations. Additionally, some numerical examples are provided and discussed to illustrate the impact of the number of available sources and order of selected source at the first hop, number of available destinations and order of selected destination at the third hop, and turbulence fading parameters and pointing errors on the system performance. The effectiveness of the proposed power allocation algorithm is also shown in this section. A total number of 2×105 samples/SNR value have been used in generating the simulation results. Also, the BPSK modulation scheme has been used in the results of ASEP.

The effect of the order of selected source at the first hop (N 1) and order of selected destination at the third hop (N 2) is illustrated in Fig. 2 for the case where N 1=N 2. Excellent matching between the analytical and asymptotic results with Monte Carlo simulations can be seen in this figure. Also, it is clear that under weak turbulence conditions (α=9.708 and β=8.198), as N 1=N 2 decreases or, equivalently, as the quality of the selected source and destination enhances, better the achieved performance. This is because, for weak turbulence conditions, the system performance is dominated by both the first and third RF hops and hence, the diversity order of the system (G d) equals to K 1N 1+1=K 2N 2+1. For fixed number of sources and destinations (K 1=K 2), reducing N 1=N 2 increases the diversity order of the system and enhances the system performance. For this case, the coding gain of the system (G c) is expressed by the first case in (49).
Fig. 2

P out vs SNR of multiuser mixed RF/FSO/RF relay network with generalized order user scheduling for different values of N 1=N 2

Figure 3 portrays the outage behavior of the system under weak turbulence conditions for different numbers of sources and destinations when they are equal (K 1=K 2). Again, it can be seen from this figure that the exact and asymptotic results perfectly match with the Monte Carlo simulations. Additionally, it is clear that under weak turbulence conditions (α=9.708 and β=8.198), as K 1=K 2 increases or, equivalently, the more the number of available sources and destinations, the better the achieved performance. This is because, for weak turbulence conditions, the system performance is dominated by both the first and third RF hops and hence, the diversity order of the system (G d) equals to K 1N 1+1=K 2N 2+1. For fixed order of selected source and destination (N 1=N 2), increasing K 1=K 2 increases the diversity order of the system and enhances the system performance. Again, the coding gain (G c) for this case is expressed by the first case in (49). It is important to mention here that for both Figs. 2 and 3, the diversity order of the system is linearly proportional with the number of available sources and destinations and order of selected source and destination.
Fig. 3

P out vs SNR of multiuser mixed RF/FSO/RF relay network with generalized order user scheduling for different values of K 1=K 2

It is worth mentioning here that the issue of achieving the best performance and satisfying fairness among users is indeed a trade-off. The scheduling schemes which exist in literature can be divided according to two criteria: sum-rate capacity and fairness among users. Maximum rate or conventional scheduling maximizes the sum capacity of the system at the expense of fairness among users, whereas proportional fair user selection scheme satisfies fairness among users at the expense of system sum-rate [44, 45]. Therefore, the selection of the scheduling scheme depends on the system requirements and nature of the system. As an example on the suitability of the scheme to be used, although the proportional fair user scheduling could be helpful for users of weak channels, the loss that occurs in throughput when this scheduling scheme is used can be large in situations where users are scattered across the cell [46]. In summary, the opportunistic and even the generalized order user selection schemes are suitable for systems where the system overall sum-rate capacity or the overall performance is the main requirement of the system; conversely, the proportional fair user scheduling scheme is more desirable in systems where fairness among users is the first priority.

The outage performance vs SNR is portrayed in Fig. 4 under weak turbulence conditions (α=8.650 and β=7.142) for different values of outage threshold γ out. Two main cases are illustrated in this figure: the case where all links’ average SNRs are increasing with increasing the x-axis value and the case where one of these SNRs is fixed. In the case where all SNRs are varying, the performance keeps enhancing as SNR increases and no noise floor appears in the results, whereas in the case where one SNR is kept fixed, a noise floor appears in the results, and a zero diversity order is achieved by the system. This behavior is expected as the system performance is dominated by the worst link among the three links. It is clear also from this figure that in both cases, the outage threshold γ out affects the system performance by only affecting its coding gain.
Fig. 4

P out vs SNR of multiuser mixed RF/FSO/RF relay network with generalized order user scheduling for different values of γ out with fixed average SNRs and varying average SNRs

Figure 5 studies the outage performance of the system vs order of selected source/destination (N 1=N 2) under weak turbulence conditions for different values of average SNR/hop. As expected, increasing N 1=N 2 or, equivalently, decreasing the quality of the selected source and destination increases the outage probability and degrades the system performance. Also, it is obvious from this figure that the best performance is achieved at the largest value of SNR.
Fig. 5

P out vs order of selected user of multiuser mixed RF/FSO/FSO relay network with generalized order user scheduling for different values of average SNR

The effectiveness of the proposed power allocation algorithm is illustrated in Fig. 6 under weak turbulence conditions for different values of outage threshold γ out. Clearly, the system with optimum power allocation gives better performance compared to the case with no power allocation. Also, this figure shows that the outage threshold γ out degrades the system performance through reducing the coding of the system and not the diversity order. This is in a full match with the asymptotic results which show that the outage threshold affects only the coding gain of the system. Note that in plotting this figure, the total distance (D tot) between the sources and destinations was assumed to be 1 and divided between the three hops as follows: \(\phantom {\dot {i}\!}D_{\textsf {s},\textsf {r}_{1}}=0.3\), \(\phantom {\dot {i}\!}D_{\textsf {r}_{1},\textsf {r}_{2}}=0.3\), and \(\phantom {\dot {i}\!}D_{\textsf {r}_{2},\textsf {d}}=0.4\).
Fig. 6

P out vs SNR of multiuser mixed RF/FSO/RF relay network with generalized order user scheduling for different values of γ out without and with power optimization

Figure 7 studies the impact of pointing error, represented by ζ on the error probability performance of the system under severe atmospheric turbulence conditions (α=4.341 and α=1.309). As the figure is generated for severe atmospheric turbulence conditions, the system performance will be dominated by the FSO link parameters (α, β and ζ 2). Based on the values of these three parameters, the curves in this figure could be divided into two sets: the set of curves where the diversity order (G d) is affected by changing ζ and the set of curves where the coding gain (G c) is affected by changing ζ. In the first case, the diversity order of the system is affected/determined by ζ 2 as it is the minimum parameter among α, β, and ζ 2. On the other hand, when the minimum value of these three parameters becomes equal β, changing ζ affects the coding gain of the system and not the diversity order which is in this case determined by β. This result is in a full match with the asymptotic result provided in the second part of (48). Note that in this figure, the type of the detector, represented by r, should also affect the diversity order of the system. It is assumed to be a heterodyne receiver with r=1.
Fig. 7

ASEP vs SNR of multiuser mixed RF/FSO/RF relay network with generalized order user scheduling for different values of ζ

The two types of detection (heterodyne and IM/DD) are studied in Fig. 8, where the error probability performance is portrayed versus SNR under various atmospheric turbulence conditions. As expected, due to its ability to better overcome the thermal noise effect in the FSO systems, the heterodyne detection mode (i.e., r=1) gives better results compared to the IM/DD detection mode (i.e., r=2). This gain in the system performance is achieved at the expense of system complexity. For one type of detection, say r=1 or r=2, it is clear that under strong and medium atmospheric turbulence conditions, the system performance is dominated by the FSO link parameters and as the minimum value among the parameters (α, β, and ζ 2) which is β is almost constant, moving from strong to medium turbulence conditions is only affecting the coding gain of the system and not the diversity order. Whereas, moving from medium turbulence conditions to weak turbulence conditions makes the RF links dominate the system performance, where the diversity order equals K 1N 1+1=K 2N 2+1. This new value for the diversity order is clearly larger than its value in the severe and medium turbulence conditions which explains why the system performance gets better under weak turbulence conditions compared to the other two cases.
Fig. 8

ASEP vs SNR of multiuser mixed RF/FSO/RF relay network with generalized order user scheduling for different values of α, β, and r

Figure 9 illustrates more on the effect of the number of sources/destinations (K 1 and K 2) and order of selected sources/destinations (N 1 and N 2) on the error probability performance of the considered system. Two main cases are shown in this figure: severe atmospheric turbulence conditions (α=4.341 and β=1.309) and weak atmospheric turbulence conditions (α=6.993 and β=5.460). It is clear that under severe turbulence conditions (α=4.341 and β=1.309), trying to increase K 1=K 2 is not beneficial for the diversity order and coding gain of the system as they are determined by the FSO parameters that dominate the overall system performance. This case is represented by the brown and black curves on the figure. Whereas under weak turbulence conditions (α=6.993 and β=5.460), it is obvious that the diversity order and coding gain of the system are dominated by the RF links parameters (K 1, N 1, K 2, and N 2). We can see that the diversity order of the system in this case is determined by the minimum value among the terms K 1N 1+1 and K 2N 2+1. Keeping one term constant and increasing the other one enhances the coding gain of the system but not the diversity order. Finally, increasing both terms affects the minimum value among them and hence, increases the system diversity order as can be seen in the last curve in this figure (green-star curve).
Fig. 9

ASEP vs SNR of multiuser mixed RF/FSO/RF relay network with generalized order user scheduling for different values of (K 1, N 1), and (K 2, N 2)

The ergodic channel capacity versus SNR is plotted in Fig. 10 under weak atmospheric turbulence conditions for different values of N 1=N 2. The gain achieved in the system capacity due to decreasing N 1=N 2 or, equivalently, due to selecting a better source and destination is shown in this figure. This gain in the system performance is expected as enhancing the quality of the RF parts of the system is beneficial for the system performance under weak turbulence conditions. It is clear also from this figure that the analytical expressions are in excellent match with the simulation results.
Fig. 10

Capacity vs SNR of multiuser mixed RF/FSO/RF relay network with generalized order user scheduling for different values of N 1=N 2

6 Conclusions

In this paper, the performance of a new scenario of triple-hop multiuser mixed RF/FSO/RF relay network with generalized order user scheduling was evaluated. Also, a power allocation algorithm to calculate the optimum transmission power was proposed. Closed-form expressions were derived for the outage probability, average symbol error probability, and channel capacity assuming Rayleigh and Gamma-Gamma fading models for the RF and FSO links, respectively. The effect of pointing errors was also considered in the derivations. Furthermore, the system performance was studied at the high-SNR regime where an approximate expression for the outage probability, in addition to the diversity order and coding gain were provided. Monte Carlo simulations were provided to illustrate the validity of the achieved exact and asymptotic results. The main results illustrated that the system performance is dominated by the worst hop among the three hops. The diversity order and coding gain are determined by the parameters of the link/s which dominate the system performance. These parameters are: number of users and order of selected users in the RF links and atmospheric turbulence parameters, pointing error, and type of detector in the FSO link. Finally, the results showed that the proposed power allocation algorithm clearly enhances the system performance compared to the case with no power allocation.

Declarations

Acknowledgements

This work was funded by the National Plan for Science, Technology and Innovation (Maarifah)—King Abdulaziz City for Science and Technology—through the Science and Technology Unit at the King Fahd University of Petroleum and Minerals (KFUPM)—the Kingdom of Saudi Arabia, under grant number 15-ELE4157-04.

Competing interests

The author declares that he has no competing interests.

Authors’ information

Anas M. Salhab is a member of IEEE.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Electrical Engineering Department, King Fahd University of Petroleum and Minerals

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