Hybrid Protocols for Relay-Assisted Non-Orthogonal Multiple Access in Power Line Communications

We study the outage probability performance of hybrid protocols in relay-assisted, non-orthogonal multiple access (NOMA)-based power line communication. In particular, we consider two networks with source, relay and destination nodes, namely, cooperative relaying NOMA (CR NOMA) and single-stage NOMA (SS NOMA). We derive closed-form expressions for the outage probabilities of three hybrid relaying techniques, namely, hybrid decode- and amplify-and-forward (HDAF), incremental HDAF (IHADF) and IHDAF with maximal ratio combining (MRC) protocols. The HDAF protocol overcomes the limitations of decode-and-forward (DF) and amplify-and-forward (AF) protocols, by combining the advantages of both. In IHDAF protocol, the advantage of successful decoding at the receiver is exploited using the direct link between source and destination nodes, with the added benefit due to HDAF. Through an extensive simulation study, we (a) validate our analysis, (b) show that IHDAF-MRC outperforms IHDAF, HDAF, DF and AF, and (c) CR NOMA outperforms SS NOMA when IHDAF-MRC scheme is employed.


I. INTRODUCTION
P OWER line communication (PLC) is an attractive technology and has gained prominence, since it uses the existing wired infrastructure [1], which is useful in several applications, such as Internet-of-Things (IoT) and smart grids (SG). An SG enables bidirectional exchange of data, which helps in utilizing the ability to provide a wide range of services such as data transmission, data privacy, security, load management, reliability and smart metering [2], [3], [4]. To this end, the transmitter in PLC is incorporated with coupling circuits for the transmission of a modulated carrier signal over the power line, while the receiver is equipped with coupling circuit for decoding the signal [5], [6]. The key challenges in PLC channel for data transmission along with power include receiver noise, electromagnetic interference and multipath fading, which has detrimental effect on the system performance [2], [7], [8].
To combat these challenges, orthogonal frequency division multiplexing, multiple-input multiple-output, non-orthogonal multiple access (NOMA) and cooperative relaying (CR) have been proposed for PLC [9], [10], [11]. It is known that utilizing NOMA over PLC for decode-and-forward (DF) relay-assisted PLC networks provides several advantages including reduction in electromagnetic emissions, outage and enhancement of the average sum capacity and QoS across users, in contrast to orthogonal multiple access (OMA) schemes [12], [13], [14]. Cooperative communication in PLC finds several applications in fifth generation (5G) communication systems as well, since it enhances the reliability, QoS and provides wide range of coverage including last mile connectivity [4], [15], [16]. The well-known relaying schemes in PLC include amplify-and-forward (AF) and DF schemes [17], [18], [19], [20]. In the AF scheme, the signal received in a time slot is amplified and forwarded This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ VOLUME 4, 2023 1813 to the receiver by the relay. On the other hand, in the DF scheme, the symbols received in a time slot at the relay are decoded and forwarded to the receiver in the next consecutive time slot. It is well-known that the DF scheme has a higher processing cost, while the AF relay suffers from noise amplification. In recent years, NOMA has emerged as an innovative multiple access scheme for beyond 5G networks that can potentially reduce interference while increasing the device propagation performance [10], [11], [13]. In NOMA, several users are served simultaneously over the same time and frequency resources, by appropriately multiplexing the transmitted signals either in code-domain or power domain. In power-domain NOMA (PD-NOMA), the users are allocated different energy levels as per the channel gains. The far user with the least channel gain with more energy while the near user with best channel gain is assigned less energy. Signals are superimposed at the transmitter, and the receiver utilizes the successive interference cancellation (SIC) technique. Consequently, NOMA offers better spectral efficiency as compared to other OMA schemes [21]. More recently, several works on integrating NOMA with relay-assisted PLC have been proposed in the literature to exploit the advantages of NOMA. In [22], CR-based NOMA (CR NOMA) for PLC was proposed, where a source modem communicates to a destination modem through a DF relay. In the first time slot, a symbol is transmitted from the source to the two receiver modems. One of the receivers acts as a relay. In the next immediate time slot, the source sends another symbol to the receiver, and the relay forwards the symbol received in the previous time slot to the receiver. The symbol at the receiver is decoded by performing SIC. In [12], the authors proposed a single-stage NOMA (SS NOMA) for PLC, which is slightly different from CR NOMA. The operation of SS NOMA is similar to that of CR NOMA in the first time slot. However, in the immediate next time slot, the relay decodes the symbol and transmits it to the other node. The receiver decodes the symbols directly in both time slots. This idea was further investigated in [13], where a dual-stage (DS) setup was considered, which was shown to outperform the SS NOMA. More recently, the authors in [14] studied the outage probability performance of NOMA-PLC considering both SS and DS schemes. It was shown that DS-NOMA outperforms SS NOMA in terms of outage probability. Moreover, there are studies with different setups in a two-user NOMA for a hybrid wireless-PLC network [10], [23]. Additionally, it is known that the CR NOMA outperforms SS NOMA in terms of overall outage probability and average capacity [22].
In wireless communications, several hybrid relaying schemes have been proposed for performance enhancement in NOMA-based cooperative communication [24], [25], [26]. In [26], the authors demonstrated that the performance of DF-NOMA is superior to AF-NOMA. The authors in [27] proposed AF-NOMA with MRC at the destination. In [28], the authors proposed a hybrid decode-and AF (HDAF) relaying scheme for NOMA-based cooperative communication and showed that it offers better throughput than AF and DF schemes. The authors in [29] proposed a relay-assisted NOMA scheme with HDAF protocol, and showed that it achieves low latency and better system throughput in comparison with the existing multicast scheme. In [30], the authors proposed the incremental HDAF (IHDAF) scheme, where relay either chooses AF or DF to transmit data or remains silent, based on the channel gains. It was also demonstrated that the IHDAF scheme outperforms HDAF scheme, in terms of outage probability. In [31], the authors used the IHDAF scheme, based on [28], [30], and showed that it outperforms other relaying schemes, along with a study on trade-off in terms of the hardware implementation. In the incremental relaying scheme, a relay participates in the cooperative communication only when the quality of the channel from source to destination is below a given threshold, which helps in achieving a higher spectral efficiency [32], [33]. Note that the above-mentioned works are restricted to wireless networks, and do not consider PLC networks.
In this paper, motivated by the fact that the HDAF and IHDAF protocols offer a better outage performance than conventional AF and DF in NOMA-based cooperative wireless communication, we investigate the HDAF and IHDAF in NOMA-based, relay-assisted PLC networks. In particular, we study the outage probabilities of CR NOMA and SS NOMA. To the best of our knowledge, the performance of CR or SS schemes in NOMA-PLC networks for HDAF and IHDAF hybrid protocols, and their corresponding analysis and comparison of outage probabilities have not been reported in the literature so far. The major contributions of this work are listed as follows.
The remainder of the paper is organized is as follows. System models for CR NOMA and SS NOMA setups and details on the hybrid relaying schemes are discussed in Section II. Analytical expressions for the outage probabilities for HDAF, IHDAF and IHDAF-MRC at the relay and  destination nodes are derived in Section III. In Section IV, the simulation results are discussed. The concluding remarks of the paper are drawn in Section V. The notations used in this paper are given in Table 1.

II. SYSTEM MODEL
The system model, as illustrated in Fig. 1, consists of three PLC modems that operate in half-duplex mode, namely, one source (S) modem and two receiver modems. Out of the two receivers, one acts as a relay (R) and the other as the destination (D). For such a setup, two NOMA-based configurations have been proposed in the literature. These include the CR NOMA setup as shown in Fig. 1(a) [34], and the SS NOMA, which is shown in Fig. 1(b) [12]. It is important to note that in both setups, the transmission of information occurs over two-time slots. More details on these two techniques will be explained later. The channel gains for the PLC direct link S-R, S-D, and R-D are denoted by h SR , h SD , and h RD , respectively. Further, these random variables are modeled as independent lognormal distribution random variables, which are independent and identically distributed over time [6], [35], [36]. The PDFs of h SR , h SD and h RD are given by for i ∈ {SR,SD,RD}, where μ and σ 2 respectively denote the mean and the variance of 10 log 10 (h i ), i ∈ {SR,SD,RD}.
In the next two subsections, we provide details on the functionalities of CR NOMA and SS NOMA setups. Later in Section II-C, we explain the various relaying schemes considered.

A. CR NOMA SETUP
In the CR NOMA setup [34], S sends the symbol s 1 to R and D simultaneously in the first time slot. The signal at R and D are decoded directly and independently. Based on the relaying scheme chosen, R chooses to either DF or AF the received signal from the previous time slot to D in the second time slot. In addition, S sends symbol s 2 to D in the second time slot along with R. The symbol s 1 at D is decoded directly, while the symbol s 2 is decoded by performing SIC. The signals received in the first time slot at R and D are given as where n c denotes the additive white noise which is modeled as a Gaussian random variable with mean zero and variance σ 2 , and P S denotes the transmit power at S. Using (2), we calculate the signal-to-interference-plus-noise ratio (SINR) at D for decoding s 1 , which is given as where ρ S = P S n c denotes the transmit signal-to-noise ratio (SNR). Here, A 2 SD (d SD , f ) denotes the attenuation factor over the SD link, as a function of the distance between S-D, d SD , and the center frequency, f [12].

B. SS NOMA SETUP
In the SS NOMA setup [12], S transmits a superimposed signal comprising of symbols s 1 and s 2 to R and D simultaneously, in the first time slot. The symbol s 1 is decoded directly at R and D, while the symbol s 2 is decoded at R by performing SIC. Depending on the chosen relaying scheme, R operates either in DF or AF mode to transmit the received signal from the previous time slot to D, in the second time slot. The symbol s 1 is then decoded directly at D. The superimposed signal sent from S to R and D is given as where ζ 1 and ζ 2 denote the power allocation coefficients for symbols s 1 and s 2 respectively, such that ζ 1 + ζ 2 = 1. The signal received at R and D can be written as where the distance-dependent and frequency-dependent attenuations are defined as where d SR , d SD and d RD are the distances between S-R, S-D and R-D, and f is the frequency. The SINR at D for decoding s 1 can be expressed as Towards the end of the first time slot, R employs one of the forwarding protocols discussed next, and MRC is used at D for the performance enhancement.

C. RELAYING SCHEMES
In this section, we elaborate on the relay operations considered with CR NOMA and SS NOMA setups. In particular, we present the scenarios where AF, DF, HDAF, IHDAF and IHDAF-MRC are employed at the receiver. In each case, we describe the system functionalities of CR-and SS NOMA.

1) AMPLIFY-AND-FORWARD
CR NOMA: When R chooses to amplify the received signal in the previous time slot and forward it to D in the current slot, S sends the symbol s 2 to D simultaneously. Therefore the received signal at D can be written as where β is the amplification factor, given by The corresponding SINR for decoding s 1 at D can be written as (10), given at the bottom of this page. By employing the MRC at D, the total SINR for decoding s 1 can be expressed as SS NOMA: In SS NOMA, R forwards the signal received in the previous time slot to D, using the AF protocol in the next slot. Therefore the received signal at D is given as The corresponding SINR for decoding s 1 at D can be written as (13), given at bottom of this page. By employing the MRC at D, the total SINR for decoding s 1 can be expressed as 2) DECODE-AND-FORWARD CR NOMA: In this case, S sends the symbol s 2 to D, while R decodes s 1 directly by considering the symbol s 2 as noise.
In the same time slot, R forwards the decoded symbol s 1 to D. We assume perfect SIC at D for the ease of analysis. Hence, the symbol s 1 is decoded directly at D by considering s 2 as noise, while the symbol s 2 is decoded by performing SIC. Therefore, the signal received at D can be written as where P R denotes the transmit power at R. The corresponding SINR at R for decoding s 1 can be expressed as and the SINR at D for decoding the symbols s 1 and s 2 can be calculated as and respectively. The transmit SNR at R is given by ρ R = P R n c . Employing MRC at D, the total SINR for decoding s 1 can be expressed as SS NOMA: Even in this case, R decodes the symbol s 1 directly by considering the symbol s 2 as noise, and the symbol s 2 is decoded by performing SIC. Considering perfect SIC, the SINR at R for decoding symbols s 1 and s 2 are given by and respectively. If R successfully decodes s 1 and forwards it to D, the transmitted signal from R to D is given as The signal received at D from R is given by The SINR for decoding s 1 at D is expressed as Employing MRC at D, the total SINR for decoding s 1 can be expressed as

3) HYBRID DECODE-AND AMPLIFY-AND-FORWARD
The DF relaying scheme does not take account into the scenario when the signal forwarded to D is not correctly decoded at R [28], [30]. In the case of wireless links, when R does not decode s 1 correctly, it has been reported that employing the HDAF protocol enhances the network performance [29]. In principle, the idea behind HDAF is to combine the advantages of AF and DF relaying schemes. Here, if the SINR at D due to the signal s 1 falls below a threshold, i.e., when γ DF (SR,s 1 ) ≤ α 1 for some α 1 > 0, then AF protocol is employed. Otherwise, DF protocol is employed. A schematic representation of the working principle of HDAF protocol is described in Fig. 2.

4) INCREMENTAL HYBRID DECODE-AND AMPLIFY-AND-FORWARD
It has been established that the IHDAF protocol further improves the performance compared to the HDAF protocol, for wireless links. In the IHDAF protocol, if D fails to decode s 1 from the SD link in first time slot, i.e., when γ (SD,s 1 ) ≤ α 1 for some α 1 > 0, then R forwards it to D. Next, based on the feedback information obtained from D, S estimates whether the information has been successfully decoded by D and decides whether R needs to participate in the cooperative communication. In other words, R chooses DF protocol only when γ DF (SR,s 1 ) > α 1 and γ DF (RD,s 1 ) ≤ α 1 , else chooses the AF protocol. In this case, R is additionally equipped for gathering the decoding consequence of SD link from S [37]. Additionally, we assume that there is perfect reception of feedback information at S [38]. A schematic representation of the working of IHDAF protocol is given in Fig. 2.

5) INCREMENTAL HYBRID DECODE-AND AMPLIFY-AND-FORWARD WITH MRC
In the first time slot in IHDAF, if D is not able to decode the symbol s 1 from the direct link, then MRC is employed at the D. This is done by combining the signal received from SD link with the signal forwarded from R in the next time slot. The benefit of employing MRC along with IHDAF protocol allows the enhancement of outage performances of both CR NOMA and SS NOMA setups. This protocol is referred to as IHDAF-MRC.
In the next section, we present an analysis on the outage probabilities for the HDAF, IHDAF and IHDAF-MRF protocols for both CR NOMA and SS NOMA schemes.

III. OUTAGE PERFORMANCE ANALYSIS
We present the mathematical analysis for the outage probabilities for symbols s 1 and s 2 at R, for symbol s 1 at D. α 1 and α 2 are the threshold rate for symbols s 1 and s 2 respectively, at both R and D.

A. OUTAGE PROBABILITY AT R
In this subsection, we derive the outage probabilities at R for CR NOMA and SS NOMA setups. First, consider CR NOMA. The outage probability for symbol s 1 at R in this case happens when it fails to decode the symbol s 1 over SR link. Therefore, the outage probability for the s 1 at R is given as [22] Following the lognormal properties 1)-3) discussed in Appendix A, an expression for the approximate outage probability at R is given as where the complementary error function is represented by erfc(·). Next, we consider SS NOMA. The outage probability for symbols s 1 and s 2 at R for the SS NOMA occurs when it fails to decode either of the symbols over SR link. Therefore, the outage probability for both symbols at R, evaluated by using the properties 1)-3) of a lognormal distribution mentioned in Appendix A, can be written as [14] P SS where A = max(α 1 , α 2 ). In the next two subsections, we derive the overall outage probabilities at D for CR NOMA and SS NOMA setups, which is due to the symbol s 1 . For a given outage probability P out , recall that the achievable diversity order is defined as From (30), using high SNR approximation on the erfc(·) function, it is easy to show that the diversity order for P SS out is 1, similar to the discussion in [31].

B. CR NOMA SETUP
The following Propositions 1, 2 and 3 give the approximate expressions for the overall outage probabilities at D using HDAF, IHDAF and IHDAF-MRC protocols, respectively. Proposition 1: The approximate outage probability at D which uses the HDAF protocol due to the symbol s 1 can be written as (33), given in the bottom of this page.
Proof: See Appendix A.
In this case, observe that it is hard to analytically obtain the exact diversity order at the destination node under HDAF protocol, due to the complicated form of outage probability in Prop. 1.
Proposition 2: The approximate outage probability at D which uses the IHDAF protocol due to the symbol s 1 can be written as (35), given in the bottom of the page.
Proof: See Appendix B.
In the case of IHDAF protocol, even though analytical expression for diversity order at the destination node is hard to obtain, it is expected to be higher than 1 [31], which is later verified in Fig. 13.
Proposition 3: The approximate outage probability at D which uses the IHDAF-MRC protocol due to the symbol s 1 can be written as (37), given in the bottom of the next page.
Proof: The proof is similar to that of Proposition 1 given in Appendix A, and the corresponding mathematical details are skipped for brevity. Additionally, note that the CDF Pr{ (SD,s 1 ) ≤ α 1 } is calculated following the properties 1)-3) of a lognormal distribution mentioned in Appendix A.
The diversity order at the destination node for IHDAF-MRC protocol is known to be 2 [31].

C. SS NOMA SETUP
The following Propositions 4, 5 and 6 give the approximate expressions for the overall outage probabilities at D using HDAF, IHDAF and IHDAF-MRC protocols, respectively, for the SS NOMA setup. The proofs in this case are similar to the proofs in CR NOMA setup, and are skipped. Proposition 4: The approximate outage probability at D which uses the HDAF protocol due to the symbol s 1 can be written as (39), given in the bottom of the page.
Proposition 5: The approximate outage probability at D which uses the IHDAF protocol due to the symbol s 1 can be written as (41), given at the bottom of the page.
Proposition 6: The approximate outage probability at D which uses the IHDAF-MRC protocol due to the symbol s 1 can be written as (43), given at the bottom of the page.
Even for the SS NOMA case, it is hard to obtain the diversity order at the destination node for HDAF analytically, and is straightforward to verify that the diversity order at the destination for IHDAF and IHDAF-MRC protocols are 1 and 2, respectively. This will be later verified in Fig. 14.

IV. RESULTS AND DISCUSSION
In this section, we evaluate the outage performance of our considered relay-assisted NOMA networks and validate our expressions using Monte Carlo simulations. In particular, we first discuss the results corresponding to the CR NOMA setup, followed by the SS NOMA setup. Later, we present a performance comparison of all the hybrid protocols across both setups. Unless otherwise mentioned, the parameters considered for this study are chosen from Table 2 [12], [13], [33]. The attenuation model is chosen as . (41)

A. CR NOMA SETUP
We first consider the CR NOMA setup. Fig. 3 presents the variation in outage probability at R with respect to the transmit SNR, for different values of d SD . The markers denote the set of curves obtained due to Monte Carlo simulations. To note, as d SD increases, the outage probability increases due to the fact that the attenuation of the signal increases with an increase in distance. Further, a good agreement between analysis and Monte Carlo simulations can be observed. The outage at R is independent of the choice of the forwarding protocol. Fig. 4 presents the variation in outage probability at D for s 1 with respect to transmitting SNR, across all the hybrid forwarding protocols considered. In each case, it is observed that there exists a close match between our analysis and simulations. Further, it is noted that AF offers the least performance, while IHDAF-MRC performs the best. Fig. 5 presents the outage probability of D with varying threshold rates α 1 with respect to transmit SNR. The earlier set of conclusions hold even in this case. As expected, HDAF performs better than DF as the R switches between the AF and DF when the SINR at D is greater than α 1 , thereby the outage performance increases marginally as opposed to just DF or the AF protocols. Fig. 6 presents a comparison of the outage probabilities at D for OMA, DF-NOMA and IHDAF-MRC schemes for the CR NOMA setup. It is seen that the performance improvement of IHDAF-MRC over OMA is about 5 dB, for a given outage probability. Additionally, as expected, the IHDAF-MRC scheme outperforms the other schemes across all SNR values. In IHDAF-MRC, along with HDAF, D employs MRC for decoding the signal from the S-D link even with a failure from AF or DF forwarding protocol. Therefore, it is observed as IHDAF-MRC performs outperforms all others.

B. SS NOMA SETUP
Next, we consider the SS NOMA setup. Fig. 7 presents the variation in outage probability at the relay with respect to transmitting SNR. Fig. 8 presents the variation in outage probability at the destination for the symbol s 1 with respect to transmit SNR. The good fit between the analytical expressions and the Monte Carlo simulation establishes the accuracy of our analysis. Fig. 9 presents the outage probability of D with varying threshold rate α 1 with respect to transmit SNR. Fig. 10

C. PERFORMANCE COMPARISON OF CR NOMA AND SS NOMA
In this subsection, we compare the outage performances of both CR-and SS NOMA. Since IHDAF-MRC protocol was observed to offer the best performance among all the other hybrid protocols, we consider the variation in the outage probability of IHDAF-MRC at D for both setups in Fig. 11, in terms of d SD . It is interesting to note that the performance of both CR-and SS NOMA setups are almost similar, with CR NOMA slightly outperforming SS NOMA. This behaviour can be seen in contrast to the comparison observed in Fig. 12, which presents the variation in the overall outage probability of both setups with the S-D distance, for different transmit SNR values using the DF scheme. Note that the CR NOMA outperforms the SS NOMA considerably. Moreover, as the distance between R-D decreases, this performance improvement is significant. Therefore, it can be concluded that even though employing CR NOMA results in a significantly better overall outage with DF relaying in comparison with SS NOMA, the outage performance of the two are nearly equal when IHDAF-MRC protocol is employed. Figs. 13 and 14 present a comparison of the achievable diversity orders at D for HDAF, IHDAF and IHDAF-MRC schemes for the CR NOMA and SS NOMA setups, respectively. It is observed that as the transmit SNR (ρ) approaches ∞, the IHDAF-MRC scheme can achieve a diversity order of 2, since D uses the direct link to decode its own signal by treating IHDAF as a remedial solution.

V. CONCLUSION
We presented a study on outage probability performance of hybrid relaying protocols for NOMA-based PLC networks. We considered HDAF, IHDAF and IHDAF-MRC protocols, and derived expressions for their corresponding approximate  outage probabilities, using simple approximations on lognormal distribution. Two known PLC network configurations, namely, CR NOMA and SS NOMA were considered. Our Monte Carlo simulation study showed that the approximations used in our derivations are tight, and IHDAF-MRC outperforms IHDAF, HDAF, DF and AF schemes. Moreover, it was shown that CR NOMA setup marginally outperforms SS NOMA setup when IHDAF-MRC scheme is employed, while the performance improvement is significant under DF relaying.

APPENDIX A PROOF OF PROPOSITION 1
The outage for the symbol s 1 at D occurs when D fails to decode the symbol s 1 from either SD or RD links. Therefore, = Pr (SD, We start by considering D. The cumulative distribution function (CDF) of the received SINR at D is given by where, following the properties of a lognormal distribution, it is easy to see that h 2 SR ∼ LN (2μ CR (SD,s 1 ) , 4σ CR (SD,s 1 ) ). Now, recall that if H i ∼ LN (μ i , σ 2 i ) and H j ∼ LN (μ j , σ 2 j ), then 1) kH i ∼ LN (μ i + ln k, σ 2 i ), for k ∈ R + , 2) H i H j ∼ LN (μ l , σ 2 l ); μ l = μ i − μ j , and σ l = σ i + σ j , 3) H l = H i +H j is closely estimated as another lognormal distribution, with parameters μ l and σ 2 l , which can be calculated as [39], The moments E[H −1 l ] and E[H −2 l ] can be evaluated through simulations, or numerically as where f H 1 (h 1 ) and f H 2 (h 2 ) are the PDFs of H i and H j . Using the aforementioned properties, we obtain the CDF of (SD,s 1 ) using (45) as where μ CR (SD,s 1 ) and σ SS (SD,s 1 ) are evaluated using (46) and (47), respectively. Similarly considering D. The complementary CDF (CCDF) of the received SINR at D is given by = Q ln(y) − μ CR (SD,s 1 ) σ CR (SD,s 1 ) .
(51) Q(.) represents the Gaussian Q-function, which is given as Subsequently, the expressions for C and E are obtained in the similar manner.

APPENDIX B PROOF OF PROPOSITION 2
The outage for the symbol s 1 at D occurs when D fails to decode the symbol s 1 from the direct link or the cooperative link. That is given by equation (34), shown at the bottom of the p. 6 and we consider the first term in order to derive the analytical expressions. Since the random variables are i.i.d, (34) can be given as where F (SD,s 1 ) (α 1 ), F DF (RD,s 1 ) (α 1 ), F DF (SR,s 1 ) (α 1 ) and F AF (RD,s 1 ) (α 1 ) denote the CDF of random variables (SD,s 1 ) , DF (RD,s 1 ) , DF (SR,s 1 ) and AF (RD,s 1 ) , respectively, evaluated at a point α 1 ∈ R. Moreover,F DF (SR,s 1 ) (α 1 ) = 1 − F DF (SR,s 1 ) (α 1 ). Rest of the proof follows by calculating these CDFs, by approximating the random variables (SD,s 1 ) , DF (SR,s 1 ) , DF (RD,s 1 ) and AF (RD,s 1 ) as lognormal random variables with parameters obtained by following (46) and (47), as discussed in Appendix A. In particular, each of these lognormal parameters are calculated using the properties 1)-3) described in Appendix A. Substituting these parameters in the CDFs in (53) and further simplification gives the required result.