A Priority-Based Semi-Grant-Free NOMA: Protocol Design and Performance Analysis

This paper introduces a novel semi-grant-free non-orthogonal multiple access protocol that ensures reliable massive connectivity of both scheduled and random access devices of diverse data rate (DR) requirements while tripling the network’s capacity. In more detail, the protocol classifies the devices into two categories, namely primary and secondary. Primary devices (PDs) require high-data-rate scheduled or random access with high-transmission probability, and are one-third of the total number of devices, while, the rest, are classified as secondary devices (SDs) and have low-DR requirements as well as access the network in a random fashion. The base-station generates a number of radio resource blocks (RRBs) that is equal to the number of PDs and allocates a primary and two SDs in each RRB. The allocation is based on the requested DRs of the primary and SDs as well as on the transmission probability of the SDs. To prove the feasibility and efficiency of the presented protocol, we conduct a performance analysis that results in the extraction of novel and insightful closed-form expressions for the outage probability and achieved throughput for all devices.

The adoption of NOMA as one of the core enablers of nextgeneration wireless networks has been the focus of several research works, including [17], [18], [19], [20], [21], [22], [23], [24], [25], [26].In particular, the authors of [17] provided a thorough overview of the research challenges associated with the adoption of NOMA in wireless networks and outlined the future research directions.Similarly, in [18], the authors carried out a comprehensive survey with respect to NOMA schemes.They provided an overview of the main design principles and key principles and discussed the research challenges and future research directions.Finally, Elbayoumi et al. presented a thorough overview of the application of NOMA schemes towards supporting massive numbers of connected devices and evaluated the gains of adopting NOMA schemes in terms of network throughput [19].
The rest of the paper is organized as follows: Section II provides an overview of the relevant state-of-art regarding semi-GF NOMA schemes and describes this work's contributions.Furthermore, Section III introduces the system model and presents the priority-based SGF-NOMA protocol.The theoretical framework for the proposed protocol is presented in Section IV, while the analytical evaluation results are presented and discussed in Section V. Finally, a summary of the main findings and concluding remarks are provided in Section VI.
Notations: The absolute value, exponential and base-2 logarithm functions are respectively denoted by | • |, exp(•), and log 2 (•).Moreover, Pr(A) denotes the probability for event A to be valid, while X = {x 1 , x 2 , . . ., x L } defines the set X containing L elements (i.e., x 1 , x 2 , . . ., x L ).Also, the functions min(X ) and max(X ) respectively return the minimum and maximum value of X .Finally, f Y (•) and F Y (•) stand for the probability density function (PDF) and the cumulative distribution function (CDF) of random variable (RV) Y, respectively.Additionally, Table I summarizes the acronyms used throughout the paper.

II. RELATED WORKS AND CONTRIBUTIONS
From the performance analysis point of view, in [20], the outage probability (OP) of NOMA under partial channel information was extracted, while, in [21], the authors presented and analyzed the outage performance as well as the diversity order of cooperative NOMA.In [22], quantified the impact of phase noise on multi-carrier wireless systems that employ NOMA, by presenting closed-form expression for the corresponding OP, whereas, in [23], the authors evaluated the outage performance of NOMA-enabled systems in the presence of in-phase and quadrature imbalance.In addition, in [24], stochastic geometry tools were employed to derive the OP of NOMA-based downlink cloud radio access network, in which the remote radio heads were uniformly distributed and serve two paired users simultaneously.Meanwhile, a feasibility study for the application of NOMA in satellite communication networks was conducted in [25].Similarly, in [26], the authors characterized the sum capacity of NOMA-enabled underwater visible light communication networks.
The main drawback of NOMA is that, in the uplink, the base-station (BS) demands knowledge of both the set of active devices as well as their channel conditions.In several realistic scenarios of NG-IoT applications, this is not always feasible.As a response to this drawback, GF-NOMA was presented.The main idea behind GF-NOMA is to surpass the ALOHA approach in terms of reliability while achieving the same latency performance, by utilizing successive interference cancellation (SIC) and allowing multiple devices to use the same radio resource block (RRB) in a random fashion.The performance of GF-NOMA schemes has been the focus of several research works [27], [28].The authors of [27] proposed a dynamic compressive sensing (CS) multi-user-detection (MUD) scheme that exploits the temporal correlation of active devices and carried out an analysis in terms of performance.The respective performance analysis results show that the proposed scheme can achieve a higher performance in terms of bit-error-rate.In [29], the authors analyzed the performance of two uplink GF approaches over shared resources, namely a blind retransmission approach and a stop-and-wait (SAW) approach.The analysis reveals that the blind retransmission approach outperforms the SAW one in transmissions with low and medium frequencies, while the SAW approach is more efficient in high-frequency transmissions.Additionally, the authors of [15] developed a GF-NOMA framework that treats collisions as interference to the remaining received signals and derived simplified expressions for approximating the OP and network throughput.The analysis results show that SIC has a higher performance in terms of throughput compared to successive joint decoding and a similar performance in terms of OP.
In [30], Gharbieh et al. leveraged queuing theory and stochastic geometry to design a spatio-temporal analytical model for opportunistic uplink transmissions for wirelesspowered IoT environments.The model can jointly capture the characteristics of temporal traffic generation, transmission success probability, and energy harvesting.Similarly, the authors of [16] used stochastic geometry to model and analyze a GF NOMA system that employs CS for the MUD process.Based on the proposed model, they derived closed-form expressions for evaluating the user detection probability, channel estimation error, and total system DR.In [28], the authors proposed a priority-based GF access scheme that performs dynamic slot allocation to devices with different priority levels in a subframe-by-subframe manner.The proposed scheme can guarantee access to high-priority devices, while also offering a satisfactory performance to low-priority ones.In [31], the authors developed a tractable approach to derive the access delay violation probability in GF systems that utilize hybrid automatic repeat request schemes.Finally, the authors of [32] developed a beacon-aided slotted GF scheme for enabling massive device access and developed two algorithms that exploit the characteristic of sporadic uplink traffic to improve the device activity detection process.
Alternatively, semi-GF (SGF) access schemes have emerged to address the issues of GF access schemes, while also providing low-latency access to shared resources.The performance analysis of semi-GF-NOMA schemes has been the focus of several recent research works [33], [34], [35], [36], [37], [38], [39].In more detail, in [33], the authors proposed types of SGF communication schemes, in which a device uses grantbased (GB) access to a channel, while the others are admitted through contention control mechanisms.Also, Yang et al. [34] proposed an adaptive power allocation solution for uplink SGF-NOMA networks that guarantees the requirements of GB devices.Moreover, the authors derived closed-form expressions and asymptotic analytical results of the OP of both grant-based and GF devices.The authors of [35] developed a SGF-NOMA uplink approach for multiple grant-based and GF devices.Also, a power control mechanism was integrated in order to improve the performance of GF devices.Furthermore, in [36], the authors proposed a new SGF scheme that opportunistically combines the SGF schemes presented in [33].The scheme in [33] allows higher DRs and improved transmission robustness.In [37] the authors presented a performance analysis for a MIMO-assisted SGF scheme and derived closedform expressions for the preamble detection accuracy and access success probability.Zhang et al. in [38] investigated the ergodic rates of a SGF NOMA system by deriving closedform expressions for the ergodic rates of both grant-based and GF devices.In [39], stochastic geometry approaches are used to investigate the performance of SGF-NOMA networks, while a dynamic protocol is proposed that effectively reduces the GF interference experienced by GB devices.Additionally, Lu et al. in [40] evaluated the outage performance of two SGF schemes, where the resource allocation is performanceoriented and fairness-oriented, respectively.In the considered schemes, one GB device and one GF device are scheduled over the same radio resources.The authors of [41] conducted an outage performance analysis for a SGF system that employs rate-splitting multiple access.The radio resources are shared between a GB and a GF device.Finally, Bai and Gu [42] proposed a mathematical framework that leverages stochastic geometry to analyze the association success probability when two GF devices share radio resources with a single GB device.
Despite the paramount importance of GF and SGF NOMA, non of the aforementioned approaches enables the co-existence of scheduled and random access devices of diverse datarate requirements.Motivated by this, the current contribution is devoted to the presentation of a novel SGF-NOMA, that increases the network capacity while ensuring reliable connectivity of two both scheduled and random access devices.Particularly, in the proposed protocol, two device categories exist, namely primary and secondary devices (SDs).Due to their increased DR requirements, primary devices (PDs) require scheduled or highly-probable random access, whereas SDs have very low DR requirements and access the network in a random manner.Moreover, the BS creates RRBs based on the number of PDs, while primary and two SDs are scheduled in the same RRB.The scheduling of devices to a RRB is determined by their respective DR requirements and by the transmission probability of the SDs.To validate the feasibility and efficiency of the proposed protocol, we carry out a performance analysis to derive closed-form expressions for the respective OPs and throughput of all devices.Of note, a part of this paper has been presented in [1].

A. System Model
As illustrated in Fig. 1, we consider an uplink scenario, where a single BS provides access to N devices.We assume that M ≤ N PDs request high-DR continuous connectivity, while, the rest N − M SDs request low-DR event-driven connectivity.This scenario is considered realistic and common in NG-IoT environments, where video streaming, two and/or three-dimensional extended reality, and other high-DR applications, need to coexist with notification and alert-providing sensors, which require considerably lower DR [43], [44].
In what follows, we use to define the set of PDs.Note that in (1), P m with m ∈ [1, M ] stands for the index of the m-th PD.Similarly, and to define the sets of the DR requirements in bits/s/Hz for the PDs and SDs, respectively.In more detail, R Pm and R Sn with m ∈ [1, M ] and n ∈ [1, N − M ] respectively stand for the DR requirement of P m and S n .Notice, that in realworld deployments, the following inequality is expected to hold: Finally, we use P S = P S 1 , P S 2 , . . ., P S N −M (6) to define the set of SDs transmission probability.The BS creates the set of of M RRBs and in each R m , with m ∈ [1, M ], allocates a PD as well as a predetermined number of SDs, in a way that the DR requirements of all the allocated devices are ideally satisfied.For simplicity, we set the number of the maximum SDs that can be allocated in a single RRB to two.It is assumed that P m and S k and Thus, in the information transmission phase, the received signal from the R m at the BS can be expressed as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.The BS decodes s Pm first by treating the received signals by S k and S l as interference.Thus, the received signal-to-interference-plus-noise-ratio (SINR) for P m can be obtained as where E Pm , E S k , and E S l are the transmission powers of P m , S k , and S l , respectively, while N o stands for the noise power.
It is assumed that |h S k | > |h S l |; hence, according to the uplink NOMA principle [45], E S k > E S l .As a consequence, the BS will next decode the received by the S k device message.The SINR for S k can be expressed as Finally, the SNR for S l can be written as

B. Protocol Description
The protocol flow is depicted in Fig. 2, along with a mapping of the respective steps.The protocol consists of three phases, namely the discovery phase, the PDs GB access phase, and the SDs GF access phase.During the discovery phase, the BS broadcasts a synchronization beacon to notify all devices of its presence (Step 1).The beacon includes BS identification information and pilot symbols for channel estimation.Upon receiving the beacon, all devices carry out the channel estimation process.Then, each PD transmits an access request signal, along with its DR requirements (Step 2).Using that signal, the BS gains knowledge of the number of PDs, their DR requirements, and their respective channel state information (CSI).
In the PDs GB access phase, the BS generates a number of RRBs that is equal to the number of PDs (i.e., M) and allocates a single PD to each RRB in a way that the achievable DR is maximized and is greater than the required DR (Step 3).Next, the BS transmits an access grant signal to the PDs containing the PD-RRB allocation information (Step 4a).In addition, the BS broadcasts the available RRBs and the maximum interference that can be introduced to each RRB without violating the corresponding PD's required DR (Step 4b).
During the SDs GF access phase, the SDs receive the information about the available RRBs and broadcast their DR requirements their CSI, and their transmission probabilities through a secondary cognitive radio network that is formed among the SDs (Step 5).As a result, all SDs are aware of the DR requirements, the CSI, and transmission probabilities of the rest SDs, as well as the level of interference that each RRB can endure.Using this knowledge, each SD independently executes the same allocation policy algorithm in order to determine the RRB that can be used to transmit their data (Step 6).Specifically, the allocation policy considers the transmission probability of SDs as well as the DR requirements of both PDs and SDs, giving priority to the PDs' DR requirements.Different resource allocation algorithms can be utilized, based on the aim of the allocation policy (e.g., minimizing the total energy consumption, maximizing spectral and/or energy efficiency, maximizing system fairness, maximizing the total system throughput, etc.) [46], [47].
Of note, the BS performs successive interference cancellation (SIC) to distinguish the device signals.Also, since alternative channels are used for the SDs communications, in the SDs GF access phase, the signaling overhead between the BS and the SDs is minimized.In addition, the proposed protocol assumes that each device has been pre-configured as PD or SD.Consequently, based on its type, the device will be involved in different phases and steps.Specifically, PDs will be involved in Steps 2, 3, and 4a, while SDs will be involved in Steps 4b, 5, and 6.Finally, it is assumed that the DR requirements, SNR thresholds, and transmission probabilities remain steady during a particular communication cycle.

A. Outage Probability Analysis
In this section, the theoretical framework for the evaluation of the system OP is provided.Theorems 1-4 respectively return the OPs for P m , S k and S l for the following cases: i) Case 1: Only P m uses R n , ii) Case 2: Both P m and S k use R n , iii) Case 3: Both P m and S l use R n , and iv) Case 4: P m , S k , and S l use R n .
Theorem 1: In the case in which R n is used only by the primary user P m , the OP of P m can be evaluated as where γ Pm th is the SNR threshold of P m .Proof: For brevity, the proof of the Theorem is provided in Appendix A.
Theorem 2: In the case in which R n is used by both P m and S k , the OP of P m can be obtained as while, the OP of S k can be expressed as where γ S k th stands for the SNR threshold of S k .Proof: For brevity, the proof of the Theorem is provided in Appendix B.
Theorem 3: In the case in which R n is used by both P m and S l , the OP of P m can be obtained as while the OP of S l can be obtained as where γ S l th denotes the SNR threshold of S l .Proof: For brevity, the proof of the Theorem is provided in Appendix C.
Theorem 4: In the case in which R n is used by P m , S k , and S l , the OP of P m , S k and S l can be respectively obtained as where Pm can be expressed as Moreover, where can be written as Finally, Proof: For brevity, the proof of the Theorem is provided in Appendix D.
Proposition 1: The respective OPs of a PD (i.e., P m ) and two SDs (i.e., S k and S l ) sharing the m-th channel are derived as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Proof: Let C j with j ∈ [1,4] denote the event that case j is valid.Then, the OP of P m , S k and S l can be respectively obtained as and where and By substituting ( 29)-( 30) to ( 26)-( 28), we obtain ( 23)-( 25).This concludes the proof.

B. Throughput Analysis
In this subsection, an analysis of each device throughput is presented.For convenience, we assume that the maximum number of SDs that can be connected at each RRB are 2.In this case, the throughput of P m can be evaluated as where γ Pm th is the PD's signal-to-noise-ratio (SNR) threshold, while F γ Pm (γ Pm th ) is the CDF of the PD's instantaneous SNR and can be obtained as Similarly, the throughput of S k can be obtained as where is the CDF of the S k 's instantaneous SNR that can be obtained by Finally, the throughput of S l can be expressed as where F γ S l (γ S l th ) is the CDF of the S l 's instantaneous SNR which is obtained by

V. RESULTS & DISCUSSION
This section is focused on presenting the numerical results accompanied by insightful discussions that quantify the outage and throughput performance of the priority-based SGF-NOMA protocol and verify the theoretical framework as well as provide useful guidelines for the protocol deployment.The following scenario is considered: we assume that a single PD and two SDs have been assigned to the same RRB.Unless otherwise stated, we set γ Pm th to be equal to 1.5 dB and γ S k th = γ S l th = 0 dB. 1 For simplicity in what follows, we omit m and we set k = 1 and l = 2. Figure 3  Figure 6 shows the PD throughput as a function of the power-to-noise gain for different SD transmission probabilities.The power-to-noise gain of the PD is increased from 0 to 50 dB, while the power-to-noise gains of S 1 and S 2 are set to 0 and 10 dB, respectively.Also, the SNR threshold of both SDs is set to -5 dB.The PD throughput is evaluated for γ P th = 10dB and γ P th = 20dB, and for three cases of SD transmission probabilities, namely 0.25, 0.50, and 0.75.
As expected, the PD throughput increases as the PD powerto-noise gain is increased.For example, for γ P th = 10dB, P S 1 = P S 2 = 0.25, and E P /N o = 10dB, the PD throughput is 1 bps/Hz.When the power-to-noise gain is increased to E P /N o = 50dB, the PD throughput is 3.5 bps/Hz.On the other hand, for the case of γ P th = 20dB, the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Fig. 6.PD throughput vs power-to-noise gain for various SDs transmission probabilities.
respective PD throughput when E P /N o = 10dB is about 0 bps/Hz.However, when the power-to-noise gain is increased to E P /N o = 50dB, the corresponding PD throughput is 6.5 bps/Hz.This indicates that higher γ P th values lead to lower throughput when the E P /N o ratio is low since the PD will experience more outages.Nevertheless, for higher power-tonoise gains, the PD will achieve much higher throughput.
Concerning the transmission probabilities of the SDs, when the P S 1 and P S 2 are increased, the respective PD throughput will decrease as the SDs will introduce higher interference.For example, for γ P th = 20dB and E P /N o = 30dB, when P S 1 = P S 2 = 0.25, the PD throughput is about 4.5 bps/Hz, while when P S 1 = P S 2 = 0.75, the PD throughput is about 2 bps/Hz.In addition, for high values of E P /N o , the respective throughput of all cases converges, as the power-to-noise gain is high enough to minimize the experienced interference.
Figure 7 depicts the PD throughput as a function of the SNR threshold for various PD power-to-noise gains and SDs transmission probabilities.The SNR threshold of the PD is increased from 0 to 25 dB, while the PD power-to-noise gains are set to 10, 20, and 30 dB.Moreover, the SNR thresholds of SDs are set to -5 dB.Also, the PD throughput is evaluated for three cases of SDs transmission probabilities, namely 0.25, 0.50, and 0.75.
As γ P th is increased, the PD throughput increases up to a particular value and then it starts decreasing.This is expected since, for higher γ P th values, the device outage probability will have a higher impact on the throughput than the data rate.Considering the PD power-to-noise gain, higher values of gains lead to higher overall throughput.Also, when E P /N o = 10dB, the maximum PD throughput is achieved at γ P th = 8dB, while the corresponding maximum PD throughput when E P /N o = 30dB is achieved at γ th P = 24dB.This reveals that higher power-to-noise gains lead to decreased PD outage probability.Of note, for E P /N o = 10dB, when γ P th > 20dB, the PD throughput is almost zero since the power-to-noise gain is not enough to achieve the required SNR threshold.Moreover, when the P S 1 and P S 2 are increased, the respective PD throughput will decrease because of the increased interference caused by the SDs.For example, for E P /N o = 20dB and γ P th = 10dB, when P S 1 = P S 2 = 0.25, the PD throughput is about 1 bps/Hz, while when P S 1 = P S 2 = 0.75, the PD throughput is about 3 bps/Hz.Furthermore, for higher values of SNR threshold, the PD throughput in all cases is expected to converge to 0 bps/Hz as the outage probability of the device will continue to increase.
Finally, the respective throughput of S 1 and S 2 as functions of the SNR threshold for various transmission probabilities are illustrated in Fig. 8.The SNR threshold is increased from -5 to 15 dB, while the transmission probabilities of both SDs are set to 0.25, 0.50, and 0.75.Moreover, the power-to-noise gains of SD1 and SD2 are respectively set to 0 and 10 dB.
Similarly to the PD case, the SD throughput increases up to a particular value and then it starts decreasing, as the SNR threshold of the SDs is increased, due to the higher impact of the outage probability.As E S 1 /N o = 0dB, the throughput of S 1 for all cases of transmission probabilities will converge to 0 bps/Hz since the power-to-noise gain will not be able to achieve the required SNR threshold.On the other hand, since E S 2 /N o = 10dB, S 2 will achieve the required SNR threshold in the considered SNR range.
Finally, the respective S 1 and S 2 throughput decreases as the transmission probabilities of the SDs are increased.This happens because of the increased interference caused by the higher SD transmission frequency.For example, when γ SD th = 10dB, the S 1 throughput is 3, 2.5, and 2.2 bps/Hz for P S 1 = P S 2 = 0.25, P S 1 = P S 2 = 0.50, and P S 1 = P S 2 = 0.75, respectively.On the other hand, the corresponding S 2 throughput for the same cases is 2.5, 1.6, and 0.9 bps/Hz, respectively.
The aforementioned results are derived through a numerical analysis using the equations derived in Section IV.In this respect, the expected outage probabilities and device throughput can be determined by changing each of the functions' variables (e.g., required threshold, power budget, transmission probabilities).In real-world systems, the outage probabilities can be evaluated by configuring the BS to periodically record the received SNR and calculate how many times is below the required SNR threshold.Moreover, the achieved throughput of each device can be continuously monitored and recorded by the BS.The required threshold and transmission powers can either be pre-configured or defined during the BS-device association phase.

VI. CONCLUSION
In this paper, we introduced a novel SGF scheme that ensures connectivity for two device classes, namely the PDs and SDs.In more detail, by accounting for the sporadic transmission of the SDs, we reported a priority-based SFG NOMA scheme, and we presented a theoretical framework that quantifies its performance in terms of OP and throughput.We derived low-complexity and insightful closed-form expressions for the respective OP and throughput of each device, which are expected to serve as cornerstones of transmission parameters selection.The results revealed the capabilities of the presented scheme to triple the network's capacity while satisfying the SNR requirements.

APPENDIX A PROOF OF THEOREM 1
The OP of the PD in Case 1 is defined as By taking into account that in this case θ k = θ l = 0, and employing (10), (39), can be rewritten as or equivalently or where By applying (43) in (42), we obtain (13).This concludes the proof.

APPENDIX B PROOF OF THEOREM 2
The OP of the PD in Case 2 is defined as in (39).In this case θ k = 1 − θ l = 1; thus, after applying (10), the OP can be rewritten as which can be equivalently expressed as Note that |h Pm | 2 and |h S k | 2 are independent RVs.Hence, according to [48], (45) can be evaluated as where By applying ( 47) and ( 48) in (46) and after some algebraic manipulations, we obtain where With the aid of [49, eq.(3.321/4)], (50) can be evaluated as Finally, by applying (51) in ( 49), we obtain (14).
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Assuming that the decoding order in BS is (s Pm , s S k ), the OP of S k in case 2, can be defined as , θ l = 0 , (52) which, by taking into account (10) and (11), can be rewritten as or equivalently where and Next, we evaluate F 1 and F 2 .To evaluate (55), we exploit the fact that |h S k | 2 and |h Pm | 2 are independent RVs; thus, (55) can be calculated as where that can be obtained as while is the PDF of |h Pm | and can be expressed as By applying (58) and ( 59) in (57), we get or equivalently where and By applying [49, eq. (3.321/4)] in (62), we obtain Likewise, after some algebraic manipulations, (63) can be rewritten as By applying [49, eq.(3.321/4)] in (65), we obtain By employing (60) and ( 66), (61) can be expressed as Next, we evaluate F 2 .From (56), we can rewrite F 2 as which, by applying (58), concludes to Finally, by applying (67) and ( 69) in (54), we obtain After some algebraic manipulations (70) can be rewritten as (15).This concludes the proof.

APPENDIX C PROOF OF THEOREM 3
The OP of the PD in Case 3 is defined as in (39), assuming that θ k = 0 and θ l = 1.By applying (10) in (39) for θ k = 0 and θ l = 1, we can rewrite the OP of the PD, in this case, as or equivalently Notice that |h Pm | and |h S l | are independent RVs; thus, (72) can be evaluated as where By applying ( 47) and ( 74) in (73), we obtain where which can be equivalently written as or with and By applying [49, eq. (3.321/4)] in (79) and (80), we obtain and Next by employing (81) and ( 82) in (78), we get or, after some algebraic manipulations, Finally, by applying (84) in (75), we obtain (16).Assuming that, in this case, the decoding order in BS is (s Pm , s S l ), the OP of S k in case 2, can be defined as By taking into account ( 10) and ( 12), (85) can be rewritten as which, after some algebraic manipulations, can be expressed as or where and Next, we calculate F 3 and F 4 .By taking into account that |h S l | and |h Pm | are independent RVs, (89) can be rewritten as where that can be obtained as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

APPENDIX D PROOF OF THEOREM 4
The OP of the PD in Case 4 is defined as in (39), which by employing (10), for θ k = θ l = 1, can be expressed as where In order to evaluate the OP of the PD, we need first to characterize the distribution of I k ,l .By taking into account that |h S k | 2 and |h S l | 2 are independent RVs, the CDF of I k ,l can be calculated as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
which, by applying ( 47) and ( 74), and after some algebraic manipulations, can be as or which, after some algebraic manipulations, and by employing [49, eq.(3.321/4)], we obtain Moreover, the PDF of I kl can be obtained as or By accounting the fact that |h Pm | and I kl are independent RV, (107) can be evaluated as or, after employing (47) and (114), returns By applying [49, eq.(3.321/3)] in (116) after some algebraic manipulations, we obtain (18).Next, we need to evaluate the OP of S k in Case 4, which is defined as with the aid of ( 10) and (11), can be equivalently expressed as where Likewise, (118) can be rewritten as with and From (120), it becomes evident that in order to evaluate the OP of S k in Case 4, we need first to derive closed-form expressions for (121) and (122).In this direction, by taking into account that A and |h S k | are independent RVs, we can rewrite (121) as where f A (y) is the PDF of A. In other words, in order to evaluate F 5 , it is necessary to statistically characterize A. From (119), the CDF of A can be expressed as or equivalently Notice that |h Pm | and |h S l | 2 are independent RVs; thus, (125) can be analytically calculated as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

Fig. 3 .
Fig. 3. OP of the PD vs E P /No for different values of E S 1 , E S 2 , P S 1 , and P S 2 .

Fig. 7 .Fig. 8 .
Fig. 7. PD throughput vs SNR threshold for various values of E P /No and P S 1 = P S 2 .

TABLE I ACRONYMS
Pm , h S k , and h S l are the channel coefficients of the P m − BS, S k − BS, and S l − BS links, respectively.Moreover, s Pm , s S k , and s S l are the transmission signals by the P m , S k , and S l .Likewise, n m denotes the additive white Gaussian noise (AWGN) at R m .Finally, where h depicts the OP of the PD as a function of E P /N o for different values of E S 1 , E S 2 , P S 1 , and P S 2 .As expected, for fixed E S 1 , E S 2 , P S 1 and P S 2 , as E P /N o increases, the OP of the PD decreases.For instance, for E S 1 /N o = −20 dB and P S 1 = P S 2 = 0.3, the OP of the PD improves by approximately one order of magnitude, as E P /N o increases from 30 to 40 dB.Additionally, for given E P /N o , E S 1 /N o , Fig. 4. OP of the PD vs P S 2 for different values of E P , E S 1 , and E S 2 .P S 1 and P S 2 , as E S 2 /N o increases, the interference to the PD by S 2 increases; thus, the OP of the PD also increases.For instance, for E P /N o = 40 dB, E S 1 /N o = 0 dB, P S 1 = P S 2 = 0.3, the OP increases for about 10 times, as E S 2 /N o increases from 10 to 20 dB.This reveals the detrimental effect of S 2 interference level at the outage performance of PD.Similarly, for given E P /N o , E S 2 /N o , P S 1 and P S 2 , as E S 1 /N o increases, the interference to the PD by S 1 increases; as a consequence, an outage performance degradation is observed.For example, for E P /N o = 40 dB, E S 2 /N o = 10 dB, P S 1 = P S 2 = 0.3, the OP increases for about 5 times, as E S 1 /N o increases from −10 to 0 dB.From this example, it becomes evident that the same transmission SNR increase at both S 1 and S 2 results in different outage performance degradation.This is expected due to the fact that |h S 1 | > |h S 2 |.Moreover, from this figure, it becomes apparent that, for fixed E P /N o , E S 1 /N o , and E S 2 /N o , as P S 1 = P S 2 increases, the probability of interference to PD by S 1 and S 2 increases; hence, the OP for PD increases.For instance, E P /N o = 30 dB, E S 1 /N o = −20 dB, and E S 2 /N o = 10 dB, the OP of the PD increases by approximately 2.5 times, as P S 1 = P S 2 increases from 0.2 to 0.7.This highlights the importance of accurately capturing the transmission probabilities of SDs, when analyzing and deploying the priority-based SGF NOMA protocol.Figure 4 illustrates the OP of the PD as a function of P S 2 for different values of E P /N o , E S 1 /N o , and E S 2 /N o , assuming that P S 1 = 0.5.Notice that in the special case in which P S 2 = 0, no second SD uses the RRB.On the other hand, for P S 2 = 1, S 2 continuously transmits to the RRB.These two special cases respectively correspond to the best and worst scenarios for the OP of the PD.From this figure, we observe that, for given E P /N o , E S 1 /N o , and E S 2 /N o , as P S 2 increases, the probability of S 2 to cause interference to the PD increases; as a result, the OP also increases.Moreover, for fixed P S 2 , E S 1 /N o , and E S 2 /N o , as E P /N o increases, the received signal strength of the PD desired signal increases; thus, the average received SINR increases, which results to Fig. 5. OP of SDs vs E S 1 /No for different values of E S 2 , P S 1 and P S 2 .an outage performance enhancement.For example, for P S 2 = 0.2, E S 1 = 10dBm and E S 2 = 20dBm, the OP of the PD changes from 10 −0.5 to 10 −2 as E P increases from 30 to 60 dBm.Likewise, for fixed P S 2 , E P /N o , and E S 1 /N o , as E S 2 /N o increases, the interference caused by S 2 increases; as a consequence, the outage performance degrades.Finally, for given P S 2 , E P /N o , and E S 2 /N o , as E S 1 /N o increases, the interference caused by S 1 increases; as a consequence, the OP of the PD also increases.Figure 5 presents the OPs of the SDs as functions of E S 1 /N o for different values of E S 2 /N o , P S 1 , and P S 2 , assuming E P /N o =.For given E S 2 /N o , P S 1 and P S 2 , as E S 1 /N o increases, the OP of S 1 decreases, while the OP of S 2 increases.This is expected since, as the transmission SNR of S 1 increases, the average received SINR at S 1 increases, while the average received SINR at S 2 decreases.Moreover, for given E S 1 /N o , P S 1 and P S 2 , as E S 2 increases, the interference caused by S 2 to S 1 increases; thus the OP of S 1 increases.On the other hand, for given E S 1 /N o , P S 1 and P S 2 , as E S 1 /N o increases, the desired signal strength at S 2 increases; thus, the average SNR of s 2 at S 2 increases and in turn the OP of S 2 decreases.In addition, for fixed E S 1 /N o , P S 1 and P S 2 , as E S 2 /N o , the OP of S 2 decreases.Finally, for given E S 1 /N o , and E S 2 /N o , as P S 1 = P S 2 increases, both the OP of S 1 and S 2 increase.
|h Pm | 2 (•) stands for the CDF of |h Pm | 2 .Note that |h Pm | is a Rayleigh distributed RV with scale parameter equals to 1; thus, |h Pm | 2 follows an exponential distribution, with CDF that can be expressed as represents the PDF of |h S l |.Note that |h S l | is an ordered Rayleigh distributed RV with scale parameter that is equal to 1 and |h S k | < |h S l |; hence, the PDF of |h S l | can be written as 1 = Pm th E S l + E Pm γ Pm th E S l + 2E Pm Pm th E S l + E Pm γ Pm th E S l + 2E Pm Pm th E S l + E Pm γ Pm th E S l + 2E Pm Pm th E S l + E Pm γ Pm th E S l + 2E Pm Pm th E S l + E Pm γ Pm th E S k + E PmBy applying (138) and (140) in (120), we obtain (141), shown at the top of the next page.Finally, the OP of S l in Case 4 is defined as Pm th , γS k ≥ γ S k th , γ S l ≥ γ S l th |θ k = θ l = 1 , Pm |h Pm | 2 − γ Pm th E S k h S k 2 − γ Pm th N o γ Pm th E S l , E S k h S k 2 γ S k th − N o γ S kBy taking into account the basic assumption of this protocol, i.e.,E Pm |h Pm | 2 >> E S k |h S k | 2 , (143) can be simplified as Pm E S k γ Pm th exp − Pm th E S l + E Pm γ Pm th E S k + E Pm γ Pm th E S l + 2E Pm − E S k exp − By accounting the independency of |h S l | and |h S k | 2 , (146) can be evaluated as which can be written in a closed-form asF 5 = 1 − E Pm exp − − Pr γ Pm ≥ γ th E S l γ S k th (143)whereF 7 = Pr h S l 2 < E S k h S k 2 γ S k th − N o γ S k th E S l γ S k th (146)andF 8 = Pr h S l 2 < N o γ S l th E S l (147)Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
71 =By applying[49, eq.(3.321/4)] in (151) and (152), we obtainF 71 = E S l γ S k th 2E S k γ S k th + E S l γ S k S k γ S k th + E S l γ S k th E S l γ S k th + 1 y 2 dy (152) th E