Throughput Analysis of Multipair Two-Way Replaying Networks With NOMA and Imperfect CSI

To improve the utilization of spectrum and system capacity, non-orthogonal multiple access (NOMA) is considered as a promising multiple access method for emerging communication technologies. Its advantageous benefits are implemented in two-way wireless networks. Power allocation factors assigned to multiple users play a key role in the successful deployment of fairness balance in NOMA. This paper considers fixed power allocation, and performance degradation is predicted in worse situation in which non-optimal power scheme and the imperfect channel state information (CSI) happen. More challenging issue happens in this system, and this paper addresses the situation of NOMA supporting multi-pair of users, in which group of users can communicate to other users to achieve acceptable throughout in two kinds of mode, i.e. delay-limited mode and delay-tolerant mode. Two performance metrics have been considered, namely, ergodic rate and outage probability. For both metrics, we derived closed-form analytical expressions as well as asymptotic performance. Numerical results show that the proposed system presents benefits to cellular networks by extending ability to serve more users who have different demands on data communication. Monte Carlo simulations are provided throughout to validate the accuracy of the derived analytical expressions.


I. INTRODUCTION
Recently, the introduction of emerging multimedia applications for wireless networks suffers from a significant massive users requirements and the dramatic rise data transmission [1]. Non-orthogonal multiple access (NOMA) has recently attracted a lot of research attention to meet such increasing needs and the main requirements such as low latency and massive connectivity. NOMA is proposed due to its advantages of achieving reduced access latency, enhanced user fairness, enlarged connections, and facilitated diverse The associate editor coordinating the review of this manuscript and approving it for publication was Ghufran Ahmed. quality of service (QoS) [2]. Different users in NOMA system occupy the same time, frequency and spreading code domain, different power levels. To guarantee that individual signals can be detected at receivers, different power allocation factors is exploited to serve users [3]. The most important aspect in NOMA is that signals from different users are superimposed at the transmit side and then transmitted over the same frequency resource at the same time slot. Successive interference cancellation (SIC) is necessary to detect signal at the receiver sides [4]. NOMA can also provide better system throughput in comparison to the conventional orthogonal multiple access (OMA) in term of the system throughput. Furthermore, NOMA provides improved scheme to reduce outage probability [5], [6]. Different from other kinds of multiple access techniques, NOMA users with weaker channel conditions need more allocated power and hence NOMA maintains the user fairness [7]- [9]. Furthermore, due to the ability to provide spatial diversity gain, cooperative communication has been introduced to improve system performance metrics [10], such as coverage and the communication reliability [11]. Therefore, cooperative networks and NOMA are required to form cooperative NOMA (CNOMA). CNOMA has been proposed as an efficient mean of improving the system efficiency and the spatial diversity, which has attract a great deal of research interests [12]- [15]. The authors in [15] examined performance degradation resulted from the problem of hardware impairment-aware design and such study valuated to implement cognitive radio (CR) in the context of combining CR and NOMA scheme in practice. Besides, nodes are able to communicate with each other in CNOMA systems. To provide additional transmission opportunities to the devices, relays are necessary helpers which improve the quality of the received signals to forward to distant devices [16], [17]. Two main schemes were considered in [16]. In particular, they introduced the first scenario in which the first device intends to communicate with the second device through the assistance of Amplify-and-Forward/ Decodeand-Forward (AF/DF)-assisted base station (BS) without energy harvesting. The second scenario in [16] presented the scenario as the wireless powered device was able to communicate with a non-energy harvesting device.
Regarding relaying scheme, the two-way relay (TWR) technique is introduced as a further advance. Due to its capability to boost the spectral efficiency, TWR has attracted a remarkable interests [18]. By exchanging information between two nodes with the help of a relay, TWR networks can provide improvement in term of spectrum efficiency [19]. On the other hand, full-duplex mode is applied in CNOMA and such a network benefits from combining the application of TWR into a relevant approach to enhance the spectral efficiency of systems. The authors in [20] proposed a two-way cooperative relay scheme employing NOMA in which two users operating in three phases benefits from bidirectional communication scheme. They showed that its performance is better than a conventional one way CNOMA system in terms of outage probability and ergodic rate. With characterizations of compressing data and high spectral efficiency in Network Coding (NC), the authors in [21] introduced a hybrid concept to employ a two-way relaying system (namely Hybrid-TWRS) which combines NOMA and NC.
Considering cognitive radio NOMA (CR-NOMA) networks, they introduced a two-way full-duplex (TW-FD) relaying to achieve better energy efficiency since the self-interference (SI) of FD can be regarded as a potential source related to harvest energy. Their main results indicated maximal security energy efficiency can be obtained for the secondary user (SU) system [21]. Other investigations as in [22] and [23] showed the advantages of FD TWR-CNOMA systems. In particular, the authors of [22] examined a successive group decoding scheme and a rate splitting factor in these FD TWR-CNOMA networks, in which theoretical performance evaluation was not performed. While secure transmission including base station (BS), a trusted relay, single and multiple eavesdroppers is explored in [23] to evaluate secure performance of a FD TWR-CNOMA network. However, their results are limited since only the achievable ergodic rate was considered.
However, most of the aforementioned studies assumed that perfect knowledge of the channel state information (CSI) known at the receivers. However, in practice, it is hard to achieve perfect CSI because of significant system overhead related to high accuracy of channel estimation [24]. Recently, the authors in [25] considered the channel estimation error as interference to examine the outage performance of a downlink NOMA system under impacts of imperfect CSI. NOMA-based downlink AF relay network in circumstance of imperfect CSI was studied in [26]. The authors of this work assumed the situation of the high signal to noise ratio (SNR) to find floor performance of the outage probabilities since the interference of the channel estimation error made degraded performance. However, in [25] and [26], they did not show optimal performance of uplink scenario in NOMA systems as the fixed-ordered decoding scheme is implemented based on the received signal strength in the long term. Returning to decoding order, the user with higher instantaneous received power should be detected first, and then the split from the received superposed signals to detect the remaining users. The most important point is that the order of instantaneous received powers of the superposed signals vary among different time blocks [27], and the decoding order should be varied following this change. As a result, critical problems in uplink NOMA system happens as considering the influence of the incorrect channel estimation.
Moreover, there still exist several existing implementation issues in practical scenarios of NOMA with the disadvantage of imperfect CSI [28]- [31]. It is noted that these unfavorable factors are the reasons of errors in the decoding process. In [20], [22], hard problem of multiple users in two-way NOMA are still not addressed, and open problem need be addressed in terms of ability of multiple pairs of users served by principle of NOMA and practical scenario of imperfect CSI. Motivated by these, to concentrate on the two-way NOMA and imperfect CSI, it is indicated that the degraded outage performance happens due to existence of channel estimation error in this considered NOMA system. In light of the above, the main contributions of this paper are summarized as follows: 1) We derive closed-form expressions of the ergodic capacity for the proposed multi-pairs system based on the instantaneous signal strength with imperfect CSI condition. Reporting from the simulation, this mechanism indicates that more pairs of users results in worse performance. However, the optimal throughput can be achieved by varying target rates. 2) We consider more insights related to the throughput performance in two modes, namely delay-limited transmission mode and delay-tolerant transmission mode. In addition, the impact of imperfect CSI on system performance is evaluated in the context of two-way transmission for NOMA system by analyzing diversity order. 3) Results show that under CSI imperfections, the considered TWR-NOMA scheme not only improves the spectral efficiency, but also increases throughput at several points related to power allocation factors and target rates. Moreover, it is shown that the throughput performance in delay-limited mode achieved with imperfect CSI is proven to be the best at case of 4 pairs of users. Furthermore, reducing levels of channel estimation errors achieves reasonable values of the throughput. The remainder of this paper is organized as follows. The system model of the proposed network is presented in Section II. We formulate in Section III the throughput in two cases including delay-limited transmission mode and delay-tolerant mode as well. Numerical examples and Monte Carlo results are presented in Section IV. Finally, we draw our conclusions in Section V.

II. SYSTEM MODEL AND COMPUTATION OF RECEIVED SIGNAL
A. SYSTEM MODEL In Fig. 1 our attentions put on the situation of two-way mode activated in NOMA network. In this system, the two-antenna relay R is main intermediate node is designed to serve N pairs of NOMA users. It is worth noting that these users are considered based on their distances to the relay node with regard to decoding order. Two groups G 1 (U 11 , U 11 , . . . , U N 1 ) and G 2 (U 12 , U 22 , . . . , U N 2 ) are facilitated under the support of a relay employing decode-and-forward (DF). It is noted that relay provides more advantages since two antennas are able to transmit and receive in the same time, namely A 1 and A 2 . To reduce hardware design at users, single antenna is equipped at users by utilizing superposition coding scheme. In practice, it has too high complexity to implement relaying mode related to decoding signal, i.e. DF protocol. However, DF protocol is assumed to ideal case to further examine more serious problem regarding CSI imperfection and ability of processing multiple pairs of users. In this model, two time slots are required to process signal at uplink and downlink in TWR-NOMA.
Additionally, to simple analysis, some assumptions need to be required. It is assumed that the direct links between two pairs of users do not exist due to the effects of serve shadowing [22]. Without loss of generality, all the wireless channels are modeled to be independent quasi-static block Rayleigh fading channels and disturbed by additive white Gaussian noise with mean power N 0 . Furthermore, we assumed that the feedback to the transmitter is instantaneous and error free, which means that CSI is also achievable at the transmitter whatever CSI the receiver has. Let the estimate for the channel h nk beĥ nk . By assuming MMSE estimation error, it holds that [25], [26], [28] h nk =ĥ nk +h nk , whereh nk stands for the channel estimation error and it has distribution of a complex Gaussian with zero mean and variance of δ nk , i.e.,h nk ∼ CN (0, δ nk ). Hence, the channel estimateĥ nk is also zero-mean complex Gaussian with vari- nk − δ nk , in which δ nk is denoted as the quality of channel estimation. In addition, the normalized channel estimation error factor is denoted as ζ nk = δ nk d η nk and then the variance can be known asˆ It is assumed thatˆ 1k >ˆ 2k > · · · >ˆ Nk , then it helps to further determine the order of the estimated channel gains as |ĥ 1k | 2 > |ĥ 2k | 2 > · · · > |ĥ Nk | 2 .

B. INFORMATION PROCESSING
In this paper, a dual time slot transmission provides two-way manner for TWR-NOMA network. In particular, the NOMA users in group G 1 , G 2 communicate in the first slot with R (uplink phase). Since two antennas in R provide higher bandwidth efficiency, but while the relay R receives the signals from the users in G 1 , interference signals from the pair of users in G 2 harm R. Now, at the antenna A k , the received signal at relay R can be expressed as where P as the transmit power at the user nodes, β nk specifies the uplink power control with β 1k > β 2k > · · · > β Nk , x nk denotes the unit energy signal transmitted by the user U nk and n k denotes the noise at the k-th antenna at R with N 0 being its variance. It is noted that (k,k) = (1, 2) or (k,k) = (2, 1)). Moreover, I k is the aggregated inter-group interference signal (IS) from Ak which can be given as where f By means of SIC, R successively decodes U nk 's information x nk (N ≥ n ≥ 1) while treating other x mk 's (N ≥ m ≥ n + 1), as interference. Hence, the received signal-tointerference-plus-noise ratio (SINR) at R to detect x nk then it is given by Pδ nk is the aggregated interference due to imperfect channel estimation. Each time x nk is correctly decoded, i.e., γ R→x nk ≥ τ nk where τ nk 2 2R nk − 1 withR nk being the data rate of x nk in bit-perchannel use (BCPU), the relay then utilizes SIC to alleviate interference term then decodes the information signal of the (n+1)-th user successively until x Nk is decoded. The received SINR at R to detect x Nk is given by In the second time slot, the information is exchanged between G 1 and G 2 by the assistance of the relay R. Therefore, just as downlink NOMA, R transmits the superposed signals √ α n2 x n2 to G 2 and G 1 by A 2 and A 1 , respectively, where α nk denotes the power allocation of U nk . In particular, to ensure the fairness between users in G k , a higher power is allocated to the distant user who has the worse channel condition. Hence, we assume that α Nk ≥ · · · ≥ α 2k ≥ α 1k . This paper limits research object by employing the fixed power allocation for two groups' NOMA users. 1 By exploiting channel reciprocal, the received signal at the nk-th user in the group G k is given by 1 Optimal power allocation factor will introduced to further improve the performance of systems, however, it is beyond the scope of this paper.
in which I nk specifies the aggregated IS at the k-th antenna 2 which can be given by where f (2) nk is modeled by CN (0, ξ 2 nk ). As normal principle of NOMA protocol, SIC is employed and the received SINR at U nk to detect x mk is given by where nk γ δ nk ,α mk m−1 t=1 α tk and ι nk γ |f (2) nk | 2 . From the above process, the exchange of information is achieved between the NOMA users who belong to groups, namely G 1 and G 2 . More specifically, the signal x nk of U nk is exchanged with the signal x nk of U nk .

III. SYSTEM PERFORMANCE ANALYSIS A. OUTAGE PROBABILITY OF x nk
To look at the system performance, the performance of TWR-NOMA is further characterized in terms of outage probabilities. Due to the channel's reciprocity, it is important to consider firstly on the outage probability of x nk as below.
In such a TWR-NOMA system, the outage event of x nk can be explained via its complementary events those are i) the event x nk is decoded correctly by the relay R and ii) the events all users U mk (n ≥ m ≥ 1) correctly decode x nk . The imperfect CSI situation certainly makes crucial affect on system performance. Particularly, the outage probability of x nk with imperfect CSI for TWR-NOMA system can be given by in which P 1 and P 2 denote the probabilities for the first and the second complementary events, respectively, which are formulated by To solve P ip,out nk , it needs more helps from simple computations. Therefore, the following lemma provides the outage probability of x nk for TWR-NOMA in an analytical manner.
Lemma 1: The closed-form expression for the outage probability of x nk of TWR-NOMA with imperfect CSI is given by (12), as shown at the bottom of the next page, in which 2 Unlike [32], where inter-group interference in the first and second time slots are dependent, the interference I k and I nk are independent. This assumption is certainly not only tractable analysis it is not object from the results in [32] with low IS levels, i.e., ξ 1 , ξ 2 ≈ 0.
and the term mk can be expressed in a recursive form as Proof: See Appendix A. Remark 1: From this interesting result, outage probability for the proposed TWR-NOMA is investigated. Considering that the channel gains information is prior known, outage probability depends on fixed power allocation factors. It is predicted that outage behavior may vary significantly due to impacts of power factors and CSI imperfection levels. Based on (12), the outage probability of x nk for TWR-NOMA with perfect CSI can be obtained by substituting δ nk = 0, ∀n, ∀k.

B. ASYMPTOTIC ANALYSIS
In order to gain deeper insights in considered system, the asymptotic analysis are presented in the high SNR regimes based on the derived outage probabilities. The diversity order is defined as [33] Based on the analytical result in (12), the asymptotic outage probability of x nk for imperfect CSI can be obtained by noticing that k + 1 → k in high SNR regime, i.e., when γ → ∞. As a result, the diversity orders of x nk with imperfect CSI are equal to zero.

C. DELAY-LIMITED THROUGHPUT
In delay-limited transmission scenario, throughput can be computed based on outage probability, and in this regard, the BS transmits message to users at a fixed rate, where system throughput will be subject to wireless fading channels. Under impacts of imperfect CSI, the corresponding throughput of TWR-NOMA is formulated as whereR n1 ,R n2 are the required transmission rates.

D. ERGODIC RATE OF x nk
To provide other metric regarding performance evaluation, the ergodic rate of TWR-NOMA is computed for considering the influence of signal's channel fading to target rate.
Since x nk can be detected at the relay as well as at U mk where m = n, n + 1, . . . , N , successfully. By the virtue of (4) and (8), the achievable rate of x nk for TWR-NOMA is written as where we have defined X nk min(γ R→x nk , γ 1k→1k , γ 2k→2k , . . . , γ nk→nk ) for brevity. Then, it can be calculated the ergodic rate of x nk , the corresponding CDF of X nk is presented in the following Lemma. Lemma 2: The CDF of the random variable X nk , denoted by F X nk , is presented by (18), as shown at the bottom of the next page, in which ε nk (γ ) Then, the corresponding ergodic rate of x nk is given by It can be seen that obtaining the closed-form expression of (19) for all values of n ∈ [1, N ] is intractable, not to mention impossible. However, with n = 1, we can obtain the exact expression of the above equation in the following proposition.
Proposition 3: The exact closed-form is examined for the ergodic rate of x 1k , ∀k, is given as , α tk α mk > τ tk , ∀t, (12) in which 1 = 1, 2 , and Proof: See Appendix C. To further obtain more insightful results, we consider the special cases of x nk with imperfect CSI for TWR-NOMA where there is no IS between the pair of antennas at the relay in the following part.

E. SLOPE ANALYSIS
By use of asymptotic results, we evaluate the high SNR slope in this subsection. The high SNR slope can capture the impact of various channel settings on the ergoic rate, which can be given by in which R ∞,erg nk (γ ) is the ergodic rate of x nk in high SNR regime, which can be obtained via yeilding nk + 1 → nk and n + 1 → n from F X nk (γ ). After that, substituting the achievable result into (19) to obtain the asymptotic Ergodic rate of x nk . Finally, our work showed that S nk = 0, ∀k, ∀n.
Remark 2: Due to zero slope, the ergodic rate of x nk reaches throughput ceilings in the high SNR regime even if perfect CSI is obtained at both antennas. Another notable reason is that the fixed decoding order in the first phase, i.e., uplink NOMA, and the influence of IS across user groups also contribute to the exist of the throughput limits.

F. DELAY-TOLERANT THROUGHPUT
In the delay-tolerant transmission scenario, the system throughput is determined by evaluating the ergodic rate. Based on the above derived results, the corresponding throughput of TWR-NOMA is given by

IV. NUMERICAL RESULTS
In this section, we evaluate the performance of our proposed TWR-NOMA by presenting some numerical examples of the derived expressions above. Monte Carlo simulations are presented throughout this section to validate the correctness of these expressions. In all our evaluations, we assume that all user nodes transmit with an identical power P while all relay nodes have the same power constraint. Without any further modification, the default settings are the SI levels as ξ 1 = ξ 2 = ξ = 0.01, the path-loss exponent as η = 2, the distance from user U nk to the relay as 1 + 9 n−1 N −1 measured in meters. Regarding estimation errors, the normalized values are ζ 1k = ζ 2k = · · · = ζ Nk ζ , thus the nearer the user located to the relay, the less severer that user is affected by imperfect CSI. For producing NOMA, the power allocation factors are β nk = 2 N −n+1 . For comparison purposes, TWR-OMA is considered in this section as a famous benchmark strategy. Particularly, the transmission takes a duration of 2 N + 1 time slots. In the first time slot, each user transmits the information signal x nk , ∀n, ∀k, to the relay R. In the next n time slot, in which n = 2, . . . , N , N + 1, the relay successively transmits the signal x n1 to the corresponding user U n2 . Finally, the relay, one-by-one, transmits the symbol x n2 to the user U n1 in the remaining time slots, i.e., the (N + 2) ÷ (2N + 1)-th time slot. The information transmission and the decoding at the relay in the first time slot of the conventional TWR-OMA network are similar to those of the proposed TWR-NOMA. Meanwhile, the relay transmits at full power in the remaining time slots for TWR-OMA. Note that the results of the dual-pair setting in this paper can be ultilized to characterize the TWR-NOMA system in [32]. Fig. 2 depicts the system throughput in delay-limited transmission mode versus SNR with different normalized chan-  nel estimation errors ζ = 0.5, 0.25, 0. It can be seen that the performance gaps among these cases are very small in low SNR ranges, i.e., smaller than 10 (dB). At high SNRs, ceiling throughput can be observed as SNR is greater than 30 (dB). The main reason is that the outage probability can be improved significantly at higher SNR, then corresponding throughput is enhanced accordingly. One can observe that TWR-NOMA exhibits explicit performance gaps as compared to these cases of ζ at high region of SNR. Such situation indicates that small impact of CSI can provide acceptable throughput value. The perfect CSI scenario (ζ = 0) is similar to Fig. 5 reported in [32] as the performance of the dual-pair TWR-OMA and the dual-pair TWR-NOMA are similar in the high SNR regime with N = 2. However, in general, the proposed TWR-NOMA outperforms the conventional TWR-OMA scheme under the imperfect CSI settings. It is shown in Fig. 3 that increasing SNR from 0 (dB) to 50 (dB), the ergodic capacity indicates upward trend. Especially, throughput increases very fast in range of SNR from 10 (dB) to 30 (dB). Moreover, in the high SNR region, the results show that TWR-NOMA converges to the throughput ceiling. Furthermore, it is worth noting that throughput in the delay-tolerant mode is higher than that of the delay-limited mode. For example, at SNR of 35 (dB), throughput under imperfect CSI in the delay-tolerant mode is 1.0 (BPCU) and such value is higher than the counterpart under perfect CSI settings. The main reason is that throughput in delay-limited mode is constrained by the threshold data rate. In addition, TWR-NOMA provides significant performance gain comparing to the considered TWR-OMA network.
The impact of the number of user pairs on throughput performance can be seen in Fig. 4. In the delay-limited mode, the throughput is somewhat improved when the number of user pairs is increased from N = 2 to N = 3, but then takes a dramatic turn beyond N = 3. This behavior is due to the event that the relay often insufficiently allocates power for some  users, which drastically reduces the performance at those users, and thus leads to a serious performance loss, especially with high values of N . Under the simulation settings, when N = 2, some users are over-allocated by the relay, and thus an additional pair can be deployed to make good uses of the wasted power and improve the sum-throughput. In addition, it is depicted that TWR-NOMA excellently outperforms TWR-OMA when the number of paired users is insignificant, but then underperforms TWR-OMA when N is considerably high. The reduction in the sum-throughput of TWR-OMA system is due to the additional time slots utilized for exchanging information between each group. In constrast, users in TWR-OMA systems are served with full power, regardless of the value of N , and thus perform more reliably than those in the TWR-NOMA system when there are too many users, i.e., N ≥ 6 for the delay-limited mode and N ≥ 7 for the delay-tolerant mode. In addition, the performance gap between the two throughput modes is driven to smaller gap by increasing the number of user pairs.
In Fig. 5, the maximum of throughput can be found as varying the threshold data rate sum rates. An exhausive search  algorithm is adopted to find the optimal power allocation (β 1k , β 2k , α 1k and α 2k ) for each value of the rate threshold R nk (BPCU), thus then obtains the optimal sum-thoughput. The outage performance is limited by higher data rate, and hence throughput is nearly zero atR nk = 1.5 (BPCU) for ζ = 0.1, atR nk = 2.25 (BPCU) for ζ = 0.01 and at R nk > 2.5 (BPCU) for the perfect CSI scenario. It is seen from the figure that throughput convergences to a very low value at some points of data rates and such situation results in different quality requirements. Interestingly, Fig. 6 shows that the maximum throughput in delay-limited mode can be found in numerical method, it happens as α 1k = 0.3 while it is not existence of optimal for throughput in delay-tolerant mode.
As a comparison in Fig. 7, the NOMA-based multi-pair network in imperfect CSI circumstance depends on the level of channel estimation errors. It is seen that the performance of the multi-pair TWR network with delay-tolerant mode significantly outperforms the counterpart under whole range of channel estimation error.

V. CONCLUSION
In this paper, a NOMA-based multi-pair TWR network has been studied in terms of throughput performance in two transmission modes. The metric of throughput in delay-limited mode and delay-tolerant mode are evaluated to find optimal performance. For each mode, we derived closed-form expressions of outage probability and the corresponding asymptotic performance is also investigated. Extensive simulation results in terms of ergodic rate and outage probability have been provided to validate the superiority of our proposed system under impact of imperfect CSI. Particularly, we have shown that the performance of our proposed system can be improved by reducing the levels of channel estimation error, selecting reasonable numbers of user pairs with appropriate power allocation factors. However, interesting topics such as power allocation optimization, MIMO, practical implementation with imperfect CSI, and novel multiple pair strategies still remain open for future investigations in the NOMA-based TWR network. Integration of these aspects will be studied in the future works.

APPENDIX A PROOF OF LEMMA 1
First, substituting (4) into P 1 , the probability P 1 can be obtained as Further, the PDF and the CDF of the random variable φ nk can be given as respectively. Due to the fact that ι k is the sum of independent and non-identically exponentially distributed random variables, i.e., β nk |f (1) nk | 2 , its PDF can be obtained as [33] It can be seen that the probability P 1 can be further derived via the following multifold integral f φ 1k (x 1k )dx 1k · · · dx nk · · · dx Nk dy, (A.4) VOLUME 8, 2020 within each fold, the lower bound of the integration, denoted by L mk with n ≥ m ≥ 1, is given by β tk x tk , (A. 5) and L mk = 0 with N ≥ m ≥ n + 1. For convenience, the integral P 1 can be rewritten as in which g v (y) denotes the integral up to the v-th fold of P 1 (1 ≤ v ≤ N ) and can be represented by in which g 0 (y) = 1. Subsequently, the first fold of P 1 can be solved with the help of (A.2) and (A.5) as Solving the second fold, we then obtain At this point, we then derive the probability P 2 , which is, from (14), is equivalent to Pr{φ mk ≥ θ nk (ι mk + mk + 1)}, (A.14) for α tk /α mk > τ tk , ∀t ∈ [n, N ]. In addition, the probability P 2 can be further expressed as where f ι mk (y) specifies the PDF of the aggregated interference power at the user D mk from the users in the group Gk , which can be given by Substituting f ι mk (y) in the above equation into (A.14) and solving the integral, we obtain Finally, by substituting (A.13) and (A.17) into P ip,out nk in (9), we then obtain Lemma 1. This is the end of the proof.

APPENDIX B PROOF OF LEMMA 2
The CDF of X nk is given by Denoting the first and second probabilities in (B.1) as P 1 (z) and P 2 (z), respectively. Substituting (4) into P 1 (z) and after some probabilistic approaches, P 1 (z) can then be derived as Using (A.3), the first integral part can be evaluated as Substituting (A.2) into I 2 , the second integral is derived as For the second probability at z < α nk /α nk , i.e., P 2 (z), with (8) and after some steps, we can obtain Substituting the complementary CDF φ mk , the PDF of ι mk and withλ mk /λ mk = 1 − ζ mk , (C.2) can be further derived as Substituting (B.5) and (B.7) into (B.1), we then obtain the result in Lemma 2. This is the end of the proof.

APPENDIX C PROOF OF PROPOSITION 3
Substituting (18) into (19) yields It is noted that the product term can be rewritten into the following finite sum via decomposition such that Substituting the above equation into (C.1) yields dγ .

(C.2)
Using partial fraction to decompose the product terms in the integral, we then obtain Applying the identity [34, Eq. (3.352.4)], into the above integrals, one then can obtain (20). This is the end of the proof.