Second Order Statistics of -Fisher-Snedecor Distribution and Their Application to Burst Error Rate Analysis of Multi-Hop Communications

An advantage of using the composite fading models (CFMs) is their ability to concurrently address the impact of multi-path and shadowing phenomena on the system performance in wireless communications. A Fisher-Snedecor (FS) <inline-formula> <tex-math notation="LaTeX">$\mathcal{F}$ </tex-math></inline-formula> CFM has been recently proposed as an experimentally verified and tractable fading model that can be efficiently applied for 5G and beyond 5G wireless communication systems. This paper provides second-order (s-order) performance analysis of the product of <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> independent but not identically distributed (i.n.i.d) FS <inline-formula> <tex-math notation="LaTeX">$\mathcal{F}$ </tex-math></inline-formula> random variables (RVs). In particular, accurate and closedform approximations for level crossing rate (LCR) and average fade duration (AFD) of the product of <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> i.n.i.d FS <inline-formula> <tex-math notation="LaTeX">$\mathcal{F}(N-{\mathrm {FS}}~ \mathcal{F})$ </tex-math></inline-formula> RVs are successfully derived by exploiting a general property of a Laplace approximation method for evaluation of the <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula>-folded integral-form LCR expression. Based on the obtained s-order statistical results, the burst error rate and maximum symbol rate of the <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula>-FS <inline-formula> <tex-math notation="LaTeX">$\mathcal{F}$ </tex-math></inline-formula> distribution are addressed and thoroughly examined. The numerical results of the considered performance measures are discussed in relation to the <inline-formula> <tex-math notation="LaTeX">$N-\mathrm{FS}~ \mathcal{F}$ </tex-math></inline-formula> multi-path and shadowing severity parameters. Moreover, the impact of the number of hops <inline-formula> <tex-math notation="LaTeX">$(N)$ </tex-math></inline-formula> of the <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula>-FS <inline-formula> <tex-math notation="LaTeX">$\mathcal{F}$ </tex-math></inline-formula> CFM on the s-order metrics, the burst error rate and maximum symbol rate are numerically evaluated and investigated. The derived s-order statistical results can be used to address the cooperative relay-assisted (RA) communications for vehicular systems. Monte-Carlo <inline-formula> <tex-math notation="LaTeX">$(\mathrm{M}-\mathrm{C})$ </tex-math></inline-formula> simulations for the addressed statistical measures are developed in order to confirm the provided theoretical results.

fading models (CFMs) is crucial due to their property to incorporate the simultaneous effect of the multi-path and shadowing on the system performances. The CFMs that are based on the mixture of log-normal random variables (RVs) have been reported in [2], [3]. Although the lognormal RVs are in accordance with the experiments for the shadowing effect, the performance analysis and mathematical manipulations with log-normal RVs can often lead to complex analytical expressions. In [4], [5], authors considered a Gamma-shadowed Nakagami-m distribution and provided theoretical verification for the Gamma RV as an adequate substitute for the log-normal RV. The communication systems over K and general-K composite fading channels have been addressed in [6] and [7], respectively. Moreover, the CFMs that can be applied in a variety of propagation scenarios are based on the mixture of some general and Gamma distributions [8], [9], [10].
The recently proposed composite Fisher-Snedecor (FS) F fading distribution is modeled with Nakagami-m and inverse Nakagami-m (I-Nakagami-m) RVs, where the average power of a Nakagami-m RV is subjected to the variations caused by an I-Nakagami-m RV [11]. It is important to note that the Nakagami-m distribution is capable of approximating the Rician distribution [12], which makes the FS F CFM attractive for application in line-of-sight (LOS) as well as non-LOS (N-LOS) multi-path/shadowing environments. The FS F CFM is confirmed by experiments for wireless communication between two nodes and has turned out to be mathematically more tractable than other CFMs. Moreover, FS F CFM has been efficiently applied for channel modeling and performance evaluation of re-configurable intelligent surface (RIS) assisted communications [13], [14], [15], [16], [17]. In [18], simultaneous wireless information and power transfer (SWIPT) networks in the presence of FS F fading are addressed. The diversity reception with selectioncombining (SC) and switched-and-stayed-combining (SSC) over the FS F fading channel have been explored in [19] and [20], respectively. Furthermore, statistics of the sum of FS F RVs with application to maximal-ratio-combining (MRC) are addressed in [21], [22]. Performance evaluation of dual-hop amplify-and-forward (AF) relay-assisted (RA) communications over the FS F fading channels is considered in [23]. Considerable efforts have been made in performance analysis of the channel capacity over FS F composite fading channels. In particular, a comprehensive study of the achievable channel capacity in the presence of FS F channels under different channel-state-information (CSI) availability assumptions has been explored in [24]. The ergodic channel capacity over the FS F fading under different power adaptation schemes are thoroughly addressed in [25], [26]. Furthermore, the effective channel capacity as well as the effective channel capacity in high SNR regime for single-input single-output (SISO) and multiple-input singleoutput (MISO) configurations are investigated in [27], [28]. Effective rate and achievable fixed rate capacity over composite F fading channels for emerging wireless systems such as vehicular communications are provided in [29], [30]. Physical layer security (PLS) over FS F fading channels are addressed in [31], [32]. Furthermore, the performance evaluation of wireless multiple access communication links over correlated FS F fading is investigated in [33]. The energy detection based spectrum sensing by applying FS F CFM is considered in [34]. In [35], authors introduced cascaded N-Fisher-Snedecor (N-FS) F distribution modeled as the product of N independent but not identically distributed (i.n.i.d) RVs and provided statistics such as probability density function (PDF), cumulative distribution function (CDF) and moment generating function (MGF). Capitalizing on the obtained results, [35] provided further outage probability and bit error rate as the important system performance measures of the cascaded N-FS F CFM. Moreover, the performance analysis of products and ratio of the products of FS F RVs and their applications to wireless communication scenarios are investigated in [36], [37]. Indeed, the products and the ratios of RVs play a significant role in wireless communications, since the products can be used to assess the performance of multi-hop relaying, keyhole MIMO communications, RIS assisted communications, RFID systems and multiple-scattered, multiple-shadowed fading environments [35], [36], [38], [39], [40], [41], [42], [43], while the ratio of RVs can be used in interference limited environments, spectrum sharing or full duplex communications [36], [37], [44], [45], [46]. However, the references mentioned above are only limited to the first-order (f-order) statistical performances of the various fading models. In order to get a better insight into the behaviour of time-variant channels, second-order (s-order) statistical analysis is important. The level crossing rate (LCR) and average fade duration (AFD) are widely considered as s-order statistical measures since they can be addressed through the integration of the first derivative of N-FS F random process (RP) and take into account the movement of the communicating objects [12], [47]. In particular, the LCR of N-FS F distribution is the time rate of change of the composite faded signal, while the AFD of N-FS F distribution is the mean time of the composite faded signal's being below a specified threshold. The s-order performance analysis can be practically used for optimal interleaver and channel coding design of wireless systems over fading channels. Furthermore, the s-order statistics and Doppler characterization provide useful knowledge of timevariant composite fading channels for a variety of 5G and beyond 5G scenarios [48], [49], [50], [51].
The s-order statistics of the composite F distribution are considered in [52] whereas the s-order statistics of a bivariate FS F CFM are considered in [53]. Furthermore, the LCR and AFD of FS F turbulence induced fading channels for N-hop free-space-optical (FSO) communications and dual-hop RIS assisted FSO communications are investigated in [54], [55], respectively. The s-order statistics of CFM has been reported in [56]. Additionally, the s-order metrics of cascaded and multi-hop communications have been extensively addressed in [57], [58], [59], [60], [61], [62] while s-order statistics of cooperative multi-hop communications with parallel links are addressed in [63], [64]. Namely, the available s-order statistical results of the N-hop cascaded fading channels for radio-frequency (RF) communications are only limited to N-Rayleigh and N-Nakagami-m channel models [57], [58], [59], [60], [61], [62], [65]. However, to the best of the authors' knowledge there is no reported results on the s-order statistics of the composite N-hop cascaded radio-frequency (RF) fading channels.
This paper provides the mathematical framework for calculation of novel, accurate and fast-to-compute closed-form s-order statistical measures such as LCR and AFD of N-FS F RF CFM. The general Laplace approximation (GLA) method has been exploited for derivation of closed-form approximations for LCR and AFD. The derived s-order statistics for the observed set of identically and non-identically distributed fading parameters are confirmed by Monte-Carlo (M-C) simulations. Moreover, capitalizing on the derived s-order statistical results and motivated by the fact that the available results for the burst error rate of cascaded fading channels are only limited to the dual-hop Nakagami-m and N-Nakagami-m fading channels [61], [66], we provide the burst error rate and maximum symbol rate for multi-hop communications over composite N-FS F fading channels which are based on the experimentally verified FS F CFM [11]. One need to have in mind that the N-FS F CFM can be used to capture propagation conditions more realistically if compared to N-Nakagami-m [61], since N-FS F CFM takes into account the simultaneous effect of the multi-path and shadowing on the multi-hop communication system. In the numerical results, novel remarks on the impact of N, multi-path and shadowing severity values of N-FS F CFM on the s-order measures, the burst error rate and maximum symbol rate have been drawn, graphically presented and analyzed in detail. Furthermore, the obtained s-order statistical measures of N-FS F distribution are compared to the previously published s-order statistical results for the special case when N = 1. Lastly, the obtained s-order results of N-FS F CFM are directly applied to cooperative communications established through independent direct link and independent multi-hop AF-RA links with selection-strategy (S-S) at reception.
The main contributions of this work are listed below: • Derivation of novel closed-form mathematical expressions for PDF and CDF of double FS F distribution, modeled as the product of two i.n.i.d FS F RVs and expressed in terms of hypergeometric functions. • Derivation of novel integral-form mathematical expressions for LCR and AFD of N-FS F distribution, modeled as the product of N i.n.i.d FS F RVs. • Derivation of the closed-form approximations for LCR and AFD of N-FS F distribution using the general Laplace approximation (GLA) formula and validation of the obtained closed-form formulas through simulations.
• Performance analysis of the burst error rate and maximum symbol rate of AF-RA multi-hop communications over FS F fading channels based on the s-order statistics. • S-order and burst error rate performance analysis of cooperative communications established through direct link and AF-RA multi-hop links over FS F channels with S-S at reception. • Analysis of the impact of N-FS F CFM system model parameters on the s-order statistics and burst error rate of the multi-hop and cooperative multi-hop AF-RA communications where the novel remarks have been provided. In particular, the obtained s-order statistical results can be useful in designing cooperative vehicular systems established through independent direct, multi-hop and cooperative multihop AF-RA links under LOS, N-LOS and N-LOS vehicle (N-LOSv) conditions with S-S at the destination. Indeed, vehicle-to-everything (V2X) 5G New Radio (NR) release 16 introduces the stochastic model for channel characterization in V2X communications where a path-loss formula is dependent on LOS, N-LOS and N-LOSv conditions [67]. The N-LOSv communication conditions are introduced to address the presence of blocking vehicles in V2X communications. Thus, the AF-RA vehicle-to-vehicle (V2V) communications can be used in vehicular networks in the absence of LOS V2V conditions [68], [69], [70], [71]. Moreover, the selection and switching strategies can be efficiently deployed in vehicular systems [49], [72], [73], [74], [75].
The paper is structured as follows. Sections II and III introduce FS F and double FS F CFMs, respectively. The s-order system performance analysis of N-FS F CFM modeled as the product of N i.n.i.d FS F RVs is presented in Section IV. In Section V, the s-order system performance analysis of multi-hop and cooperative multi-hop communications is considered. The numerical results and simulation results are given in Section VI. Lastly, Section VII concludes this paper.
The abbreviations that are used throughout the text are summarized in alphabetical order in Table 1.

II. FISHER-SNEDECOR F COMPOSITE FADING MODEL
The recently proposed Fisher-Snedecor (FS) F composite fading distribution is modeled with independent but not identically distributed (i.n.i.d) Nakagami-m and I-Nakagami-m RVs, where the average power of a Nakagami-m RV is subjected to the variations caused by an I-Nakagami-m RV [11]. The FS F RP, Z F (t) can be expressed as where X nm (t) and Y Inm (t) are Nakagami-m and I-Nakagami-m RPs, respectively (where the time t is omitted in the following text for simpler notation). Since the I-Nakagami-m RV can be expressed as Y Inm = 1 Y nm , the where (·) is the Gamma function [ where 2 F 1 (·, ·; ·; ·) is a hypergeometric function [76, eq. (9.100)]. Similarly, the F Z F (z F ) in (6) for m = and s = 1, reduces to the CDF of FS F distribution. Moreover, the composite FS F CFM reduces to the fading models without shadowing such as Nakagami-m (for m s →∞) and Rayleigh (for m s →∞ and m m = 1).

III. DOUBLE FISHER-SNEDECOR F COMPOSITE FADING MODEL
The product of two i.n.i.d FS F RPs can be written as The PDFs of X nm,i and Y nm,i = where m m i and m s i are the multi-path and shadowing severity parameters of double FS F CFM, respectively, whereas m i = i and s i = 1. The PDF of a double FS F RV After substituting (8) and (9) in (10) and using [76, eq. (3.259.3)], closed-form PDF expression of double FS F RV in terms of hypergeometric function is derived and presented at the bottom of the next page. Moreover, by transforming RVs, γ F d = z F d 2 , PDF of the SNR is obtained and provided as (12), shown at the bottom of the next page, whereγ i = i is the average SNR value of double FS F CFM. The CDF of double FS F distribution can be obtained from By applying the variable substitution,  (14) shown at the bottom of the page. Similarly, the closed-form SNR CDF expression whereγ i = i is given as (15), shown at the bottom of the page. In [35], authors introduced N-FS F CFM and provided the PDF and CDF of N-FS F distribution in terms of the Meijer's G functions. Although the Meijer's G functions are usually applied for obtaining useful and important analytical expressions they need to satisfy specific conditions [76, eq. (9.301)]. We provide useful and novel PDF and CDF expressions of double FS F RV given in terms of hypergeometric functions, where CDF is valid for integer values of m m 1 . Moreover, a similar analytical approach for derivation of the PDF and CDF of FSO RIS assisted communications over FS F turbulence channels has been performed in [55].

IV. N-FISHER-SNEDECOR F COMPOSITE FADING MODEL
The faded signal at the destination of cascaded N-FS F CFM can be modeled as the product of N i.n.i.d RPs where the PDFs of X nm,i and Y nm,i for i = 1, N are, respectively, The m m i and m s i are N-FS F multi-path and shadowing severity parameters of i-th hop, respectively, whereas m i = i is the N-FS F average power of i-th hop and s i = 1. Similarly, the composite N-FS F CFM becomes equivalent to the cascaded N-Nakagami-m (for m s i →∞) and N-Rayleigh (for m s i →∞ and m m i = 1) fading models.

A. LEVEL CROSSING RATE
Level crossing rate (LCR) of N-FS F CFM for a predetermined threshold z th can be obtained from the following expression [52, eq. (4)], [77] (16), X nm,1 can be expressed as can be written as 2N-1 folded, integral-form expression of the joint PDF of i.n.i.d RVs, . . , X nm,N and Y nm,N , as follows from [65, eq. (12)]. The equation is given at the bottom of the next page, where p Z F NŻ F N Y nm,1 X nm,2 ···Y nm,N for independent RVs can be written as [65, eq. (13)] The p Z F N |Y nm,1 X nm,2 ···Y nm,N (z F N |y nm,1 x nm,2 · · · y nm,N ) with a help of (20) can be expressed further as From ( (19)- (23)), the N Z F N (z th ) of N-FS F CFM can be written as expression (24), shown at the bottom of the page. From (16), the first derivative of Z F N can be expressed aṡ It is important to note that the first derivatives of 2N zero-mean (ZM) independent Gaussian RVs X nm,1 , Y nm,1 , X nm,2 , Y nm,2 , . . . , X nm,N and Y nm,N denoted as, respectivelyẊ nm,1 ,Ẏ nm,1 ,Ẋ nm,2 ,Ẏ nm,2 , . . . ,Ẋ nm,N andẎ nm,N are also 2N ZM independent Gaussian RVs. Since the linear transformation of independent Gaussian RVs is a Gaussian whereX nm,1 ,Ȳ nm,1 ,X nm,2 ,Ȳ nm,2 , . . . ,X nm,N andȲ nm,N are the mean values ofẊ nm,1 ,Ẏ nm,1 ,Ẋ nm,2 ,Ẏ nm,2 , . . . ,Ẋ nm,N anḋ Y nm,N , respectively. Based on (25) and with some additional mathematical manipulations, the σ 2 can be expressed as (27), given on the next page, where the variances σ 2 [78, eq. (4)] and [52], respectively. Moreover, in the case of N-FS F CFM, m i = i and s i = 1. It can be assumed further that the maximum Doppler frequencies are f m = f m x i = f m y i .
In accordance with (25)- (26), the integral expression (24) for the case when pŻ F N |Z F N ···Y nm,N (ż F N |z th · · · y nm,N ) has a conditional ZM Gaussian distribution can be evaluated as It is important to note that a similar analytical approach for derivation of (28) has been performed for the case of the s-order statistical evaluation of the Nakagami-m RV [79, eq. (16)], product of two independent Nakagami-m RVs [78, eq. (8)], the ratio of two independent Nakagami-m RVs [80, eq. (20)] and the product of N independent Rayleigh RVs [65, eq. (17)].
After introducing (17), (18) and (28) in (24), the LCR of N-FS F for a given threshold z th , yields where, L is 2N-1 folded, I-F expression provided as (30) shown at the bottom of the page.

1) EXACT LCR FOR N = 1
A closed-form LCR expression of FS F RV can be derived from the integral-form LCR expression given by (29). From (27)(28)(29)(30), the LCR for N = 1 can be written as where L 1 is one-folded integral-form expression given by Interestingly, the LCR of the N-FS F CFM provided in (31) The LCR expression of FS F can be useful for s-order performance analysis between two communication nodes.

2) LCR APPROXIMATION FOR N=1
The general Laplace approximation (GLA) formula has been initially proven by authors in [81], whereas in [82] authors have shown that the GLA formula fits well with exact expressions for the cases where the real-valued parameter γ takes small values. It has also been used in practice as a reliable and fast-computing mathematical tool for evaluation of many-folded integral-form expressions [54], [63], [65], [77], [78], [83], [84], [85] such as (29). In order to provide a detailed application of the GLA for evaluation of many-folded integral-form expressions, first we provide a closed-form approximate LCR expressions for N = 1.
Although the exact closed-form LCR expression is already provided in (33), here we present a derivation of an approximate LCR expression using the GLA formula for N = 1, as the simplest case since the GLA formula will be further utilized for derivation of an approximate closed-form LCR expression for general N. The GLA formula for the evaluation of one-folded integral expression in (32), can be written as [78, eq. (12)] where f 1 (y nm,1 ) and f 2 (y nm,1 ) are one-variate functions of y nm,1 , whereas γ is a real-valued parameter. After the following transformation in (32)  (34) is obtained as a real and positive value from ∂f 2 (y nm,1 )/∂y nm,1 = 0 forỹ nm,1 = y nm,1 as Finally, the LCR approximation for N = 1 is given by

B. AVERAGE FADE DURATION
The average fade duration (AFD) is the faded signal's average time below a threshold z th and for N-FS F CFM it can be calculated as where F Z F N is the CDF of N-FS F RP. Fortunately, the CDF of N-FS F RP in terms of the Meijer's G function has been recently provided in [35, eq. (13)] and is given by (52) shown at the bottom of the page. The approximate closedform expression for AFD of N-FS F CFM with respect to (50)-(52) is written as (53), shown at the bottom of the page.

C. BURST ERROR RATE
Deep fades can degrade the system performances of a wireless link. During a deep fade, the faded signal envelope is below a predetermined threshold, z F D < z th which causes symbol errors. The evaluation of the burst error rate can be useful for interleaver design of error correction codes that are usually applied as a preventive mechanism against the deep fades. The burst error rate is a process of correlated errors that are not like the f-order statistical analysis of uniformly spreading errors along time as addressed in [11] and [35] for FS F and N-FS F CFMs, respectively. Moreover, the 5G and 6G systems add the ultra-reliable low-latency communication (URLLC) which demand the performance analysis with respect to time. Indeed, the burst error rate analysis over time-varying fading channels represents useful performance metrics for various 5G and beyond 5G communication scenarios [86], [87], [88]. The burst error rate analysis based on s-order statistics of dual-hop and multi-hop relay communications over cascaded Nakagami-m and Log-normal fading channels has been considered in [61] and [66]. In this paper, the burst error rate analysis of the composite N-FS F distribution is addressed through time dependent s-order statistical measures N Z F N (z th ) and A Z F N (z th ) as a function of z th . The burst error rate is likely to occur when a time of fade duration A Z F N (z th ) increases. Moreover, the burst error rate has a similar behaviour to the outage probability F Z F N (z th ) since the burst error rate is expected to increase for higher z th values. In addition to the burst error rate analysis carried out in terms of number of hops and multi-path severities, provided in [61], [66], this paper provides the burst error rate analysis over N-FS F fading channels addressed in terms of shadowing severities. We evaluate the burst error rate of N-FS F CFM as [66, eq. (32)] The average number of lost symbols L Z F N (z th ) are determined as the ratio of the A Z F N (z th ) at a given threshold z th to the symbol transmission time as where  (54) can't exceed unity Based on (54)- (56) it can be concluded that the burst error rate presents an upper bound of the outage probability of N-FS F distribution for a given threshold z th that can be expressed as Moreover, the outage probability of N-FS F CFM is (52). Since the minimum value of L Z F N (z th ) in (55) is unity, than the minimum average number of lost symbols can be given as [61], [66, eq. (34)]. The maximum symbol rate of N-FS F CFM, which provides that one symbol on average is sent within each fade duration, can be determined by

V. PERFORMANCE OF MULTI-HOP AND COOPERATIVE MULTI-HOP COMMUNICATIONS A. MULTI-HOP AF-RA COMMUNICATION SYSTEM
The simplified scheme of the considered N-hop AF relaying system is presented on Fig. 1. The total fading gain of N-hop communication link assisted by fixed-gain (FG) AF relays over FS F channels can be modeled with the N-FS F CFM, as previously observed for N-hop AF-RA system over Rayleigh [65, eq. (39)] and Nakagami-m [61, eq. (4)] channels. We observed the case when the gain of each relay is constant and equal to one [90]. Thus, the total fading gain can be given by and D s are directly applied for performance assessment of N-hop FG AF-relaying system over FS F channels and numerically evaluated in the next section.

B. COOPERATIVE MULTI-HOP AF-RA COMMUNICATION SYSTEM IN NLOS CONDITIONS
Cooperative N-hop communications in N-LOS propagation conditions assisted by L parallel independent links with S-S at reception is presented in Fig. 2. The signal envelope at the output of the considered cooperative system can be given by is the signal envelope of j-th parallel link modeled with N-FS F distribution. The CDF at the output of cooperative system established through L independent N-hop AF-RA links with S-S at reception is [63, eqs. (8-9)] (52). The LCR at the output of the observed cooperative system for the threshold z th can be given by [63, eq. (23)] where the N Z F N (z th ) is provided in (29). The AFD is obtained from the previously derived expressions as Based on (54)-(56), the burst error rate for the considered case can be written as The provided s-order statistics for N Z F L (z th ), A Z F L (z th ) and BE L (z th ) are directly applied to the cooperative communications in N-LOS propagation conditions and numerically presented in the numerical and simulation results.

C. COOPERATIVE MULTI-HOP COMMUNICATIONS IN LOS CONDITIONS
The cooperative communications in LOS propagation conditions established through independent direct link and L independent N-hop RA links over FS F channels with S-S at reception is presented in Fig. 3. Similarly, the signal envelope at the output can be written as where Z F is the signal envelope given by (1). The CDF of the cooperative scenario in the presence of the LOS conditions can be written as where F z F is provided in (6). The LCR of the cooperative system under LOS conditions with S-S at reception over FS F fading channels is [63, eq. (22)] where N z F is provided in (33). The AFD of the considered cooperative system in LOS conditions can be given by Finally, the burst error rate for the considered case is The N Z F D,L (z th ), A Z F D,L (z th ) and BE D,L (z th ) are directly used for performance analysis of the cooperative communications in LOS propagation conditions and graphically presented in the following section.

VI. NUMERICAL AND SIMULATION RESULTS
In this section we provide the numerically evaluated integralform and approximate closed-form results for multi-hop FG AF-RA communications in terms of  [66].

A. LCR AND AFD OF MULTI-HOP COMMUNICATIONS OVER N-FS F CHANNELS
Numerical results for s-order statistical measures in terms of AFD and LCR of N-FS F distribution and their application to multi-hop AF-RA communications for identically and non-identically distributed fading parameters obtained from exact integral-form and approximate closed-form mathematical expressions are presented in Figs. 4-8. It can be noticed that the approximate expressions fit well with the exact expressions for the γ = 1 and observed system values.
Although the complexity of the GLA formula for obtaining the LCR approximations increases for higher N, it can be efficiently implemented for the considered N with the help of standardized software tools such as MATLAB or MATHEMATICA. Moreover, the presented s-order results are confirmed by M-C simulations. Fig. 4. shows the normalized LCR for different values of N, multi-path and shadowing severities of N-FS F CFM. As expected, more severe multi-path and shadowing fading conditions (e.g., when all m m i = m s i = 2) cause N Z F N (z th )/f m to take higher values in the low threshold area.   Fig. 5. It can be seen that by increasing the value of N, A Z F N (z th )f m increases for lower threshold values while it decreases for higher threshold values. It can be further observed that by increasing N-FS F CFM multi-path and shadowing severity parameters, A Z F N (z th )f m decreases for lower threshold values. Moreover, the impact of N on A Z F N (z th )f m for lower z th is stronger than N-FS F CFM multi-path and shadowing severities, while the impact of severities is more dominant for higher z th . It can be concluded that s-order system performance improvement for lower thresholds can be achieved for lower values of N (e.g., when N = 1) and higher values of multi-path and shadowing severities.

B. BURST ERROR RATE AND MAXIMUM SYMBOL RATE OF MULTI-HOP COMMUNICATIONS OVER N-FS F CHANNELS
The numerical results for the burst error rate and maximum data rate of N-FS F distribution for various sets of system model parameters evaluated from exact integral-form and approximate closed-form expressions are compared with M-C simulations on Figures 9-11, respectively. The burst error rate is evaluated for constant f m = 74.0741 Hz and D s = 10 5 symbols/s and for different N-FS F CFM parameters. It can be observed on Fig. 9. that the system performance improvement in relation to the burst error rate can be achieved by decreasing the number of hops and by increasing the values of multi-path and fading severity parameters of N-FS F CFM, since the burst error rate decreases. A similar observation for the burst error rate behaviour in relation to multi-path severity parameters has been noticed in [66] while the decrease of the burst error rate with increase of number of hops has been noticed in [61]. Fig. 10 shows the burst error rate as a function of for N = 2 and for different N-FS F CFM system model parameters. It is  obvious that the high average power region is influenced by N-FS F CFM severity conditions. The system performance improvement in relation to BE N in high average power region can be achieved in less severe composite fading conditions, as expected. The lower z th values cause decrease in the BE N . The maximum symbol rate over N-FS F CFM for a set of different system model parameters and constant f m = 74.0741 Hz is observed in Fig. 11. The maximum symbol rate is evaluated as D s = 1/A Z F N (z th ). The maximum D s is limited by the threshold value and is dependant on N-FS F CFM fading severity parameters and number of hops. As expected, the maximum D s increases with a decrease in the number of hops and increase in fading severity values.

C. S-ORDER STATISTICS AND BURST ERROR RATE OF COOPERATIVE MULTI-HOP COMMUNICATIONS OVER N-FS F CHANNELS
The N Z F ,L versus z th for N = 2, N-FS F CFM severities (m m 1 = 2, m s 1 = 2, m m 2 = 2, m s 2 = 2), average powers ( 1 = 2 = 1) and different L as well as N Z F ,D,L versus   Fig. 13. The burst error rate for the observed system model configuration can be decreased by increasing the number of available dual-hop parallel links. Moreover, the absence of direct LOS link on the burst error rate can be compensated by adding one or more dual-hop redundant links (e.g., BE D,L of a cooperative system with direct link assisted by one dual-hop AF-RA link has a similar behavior as BE L of a cooperative system established through three parallel dual-hop AF-RA links).
The s-order statistics and burst error rate of the observed dual-hop and cooperative dual-hop AF-RA communications (when N = 2) are derived and numerically evaluated by using novel analytical expressions of double FS F CFM (e.g., the CDF of double FS F CFM provided in (14) has been directly used for the derivation and numerical evaluation of the AFD expressions provided in (51), (61) and (65) as well as for the burst error rate expressions provided in (56), (62) and (66) for the special case when N = 2).
Although the derived s-order statistical results are developed for isotropic scattering conditions, as previously observed for different mobile and vehicular systems in [49], [61], [62], [63], [64], [65], [66], they can be useful for s-order and burst error rate performance analysis of cooperative multi-hop AF-RA V2X systems over composite fading channels in LOS, N-LOS and N-LOSv communication conditions.

VII. CONCLUSION
This paper provides accurate closed-form approximations for LCR and AFD of N-FS F distribution. The novel and fast-computing s-order statistical closed-form approximate results fit well with the exact integral-form results and M-C simulations. The mathematical proof that the LCR formula of N-FS F distribution for N = 1 reduces to the recently published results of composite F fading model has been provided. Capitalizing on the derived s-order statistical expressions, analytical and numerical results for the bust error rate and maximum data rate of N-FS F CFM are provided and applied to the multi-hop and cooperative multi-hop AF-RA communications established trough independent direct and L parallel links with S-S at destination. In general, the system performance improvement of the AF-RA multi-hop communication system in the lower threshold regime can be achieved by shifting from severe to less severe multi-path and shadowing fading conditions and by decreasing the number of hops, since AFD and burst error rate decrease. Numerical results for AF-RA multi-hop communications point out that the maximum symbol rate can be increased for lower thresholds in less severe multipath and shadowing fading conditions and for small number of hops. The cooperative AF-RA multi-hop communication performance improvement in terms of the burst error rate can be achieved by adding more redundant (parallel) multi-hop links. Moreover, the impact of the absence of direct LOS link on the burst error rate can be decreased by adding more redundant links. The provided s-order statistical results for cooperative multi-hop AF-RA communications assisted by L parallel links with and without direct LOS link over FS F CFM with S-S at destination can be useful for designing V2X systems in LOS, N-LOS and N-LOSv communication conditions. ANA GARCÍA ARMADA (Senior Member, IEEE) received the Ph.D. degree in electrical engineering from the Polytechnical University of Madrid in February 1998. She is currently a Professor with the University Carlos III of Madrid, Spain, where she has occupied a variety of management positions (the Head of Signal Theory and Communications Department, the Vice-Dean of Electrical Engineering, a Deputy Vice-Chancellor of International Relations, among others). She is leading the Communications Research Group with the University Carlos III of Madrid. She has been a Visiting Scholar with Stanford University, Bell Labs, and University of Southampton. She has participated (and coordinated most of them) in more than 30 national and 10 international research projects as well as 20 contracts with the industry, all of them related to wireless communications. She is the coauthor of eight book chapters on Wireless Communications and Signal Processing. She has published more than 150 papers in international journals and conference proceedings and holds four patents. She has contributed to International Standards Organizations, such as ITU and ETSI, is a member of the Expert Group of the European 5G PPP and a member of the Advisory Committee 5JAC of the ESA as expert appointed by Spain on 5G. Her main interests are multi-carrier and multi-antenna techniques and signal processing applied to wireless communications. She has received the Young Researchers Excellence Award from the University Carlos III of Madrid. She was awarded the third place Bell Labs Prize 2014 for shaping the future of Information and Communications Technology.