Student Member, IEEE Mouktar Bello

Senior Member, IEEE, Inbar Fijalkow, Senior Member, IEEE Arsenia Chorti

Member, IEEE Wenjuan Yu

Senior Member, IEEE Leila Musavian

Asymptotic Performance Analysis of NOMA Uplink Networks Under Statistical QoS Delay Constraints

Keywords: Beyond 5G (B5G), effective capacity, low latency, non-orthogonal multiple access (NOMA), quality of service (QoS), user-pairing

In this paper, we study the performance of an uplink non-orthogonal multiple access (NOMA) network under statistical quality of service (QoS) delay constraints, captured through each user's effective capacity (EC). We first propose novel closed-form expressions for the EC in a two-user NOMA network and show that in the high signal-to-noise ratio (SNR) region, the "strong" NOMA user, referred to as U2, has a limited EC, assuming the same delay constraint as the "weak" user, referred to as U1. We demonstrate that for the weak user U1, OMA and NOMA have comparable performance at low transmit SNRs, while NOMA outperforms OMA in terms of EC at high SNRs. On the other hand, for the strong user U2, NOMA achieves higher EC than OMA at small SNRs, while OMA becomes more beneficial at high SNRs. Furthermore, we show that at high transmit SNRs, irrespective of whether the application is delay tolerant, or not, the performance gains of NOMA over OMA for U1, and OMA over NOMA for U2 remain unchanged. When the delay QoS of one user is fixed, the performance gap between NOMA and OMA in terms of total EC increases with decreasing statistical delay QoS constraints for the other user. Next, by introducing pairing, we show that NOMA with user-pairing outperforms OMA, in terms of total uplink EC. The best pairing strategies are given in the cases of four and six users NOMA, raising once again the importance of power allocation in the optimization of NOMA's performance.

I. INTRODUCTION

N On-orthogonal multiple access (NOMA) schemes have attracted a lot of attention recently, allowing multiple users to be served simultaneously with enhanced spectral efficiency; it is known that the boundary of achievable rate pairs using NOMA is outside the capacity region achievable with orthogonal multiple access (OMA) techniques [START_REF] Islam | Power-domain non-orthogonal multiple access (NOMA) in 5G systems: Potentials and challenges[END_REF]- [START_REF] Ding | On the performance of non-orthogonal multiple access in 5G systems with randomly deployed users[END_REF]. Superior achievable rates are attainable through the use of superposition coding at the transmitter and of successive interference cancellation (SIC) at the receiver [START_REF] Saito | Non-orthogonal multiple access (NOMA) for cellular future radio access[END_REF]. The SIC receiver decodes multi-user signals with descending received signal power and subtracts the decoded signal(s) from the received superimposed signal, so as to improve the signalto-interference ratio. The process is repeated until the signal of interest is decoded [START_REF] Higuchi | Non-orthogonal multiple access (NOMA) with successive interference cancellation for future radio access[END_REF]. The interest in NOMA is linked to the multiple possibilities it offers, for example, in massive machine type communications (mMTC) systems where a large number of smart Internet of things (IoT) devices try to access the shared resources simultaneously.

In uplink NOMA networks, the strongest user's signal is decoded first (reverse order with respect to the downlink). However the use of SIC limits the promised performance gain brought by NOMA due to error propagation [START_REF] He | Successive interference cancellation and fractional frequency reuse for LTE uplink communications[END_REF]- [START_REF] Manglayev | NOMA with imperfect SIC implementation[END_REF]. The authors in [START_REF] Chen | Multiuser detection for uplink non-orthogonal multiple access system[END_REF] introduce an iterative interference cancellation (IIC) detection scheme for uplink NOMA, and proposed a new detection scheme based on IIC, which is called advanced IIC M. Bello, A. Chorti and I. Fijalkow are with ETIS UMR8051, CY Cergy Paris University, ENSEA, CNRS, F-95000, Cergy, France, ({mouktar.bello, arsenia.chorti, inbar.fijalkow}@ensea.fr) W. Yu is with the EEE Dep. of the Univ. of Lancaster (w.yu8@lancaster.ac.uk) and L. Musavian is with the CSEE, Univ. of Essex (leila.musavian@essex.ac.uk). M. Bello (AIIC). Its shown that the bit error rate performance of AIIC is much better than that of SIC.

Similarly, the combination of NOMA with other emerging techniques and technologies such as new modulation techniques, user pairing, resource allocation algorithms (power and channel), MIMO, etc., improves its performance [START_REF] Cai | Modulation and multiple access for 5G networks[END_REF]- [START_REF] Zeng | Capacity comparison between MIMO-NOMA and mimo-oma with multiple users in a cluster[END_REF]. The authors in [START_REF] Zhu | Optimal user pairing for downlink non-orthogonal multiple access (noma)[END_REF] investigate the optimal power allocation in a NOMA system with two users, analyze user pairing in a NOMA system with four users and propose a closed-form globally optimal power allocation solution for a general NOMA downlink networks. In [START_REF] Kara | Optimal power allocation for dl noma systems[END_REF], an iterative gradient ascent-based power allocation method is proposed for downlink NOMA that achieves better performance compared to fixed and fractional power allocation strategies.

Furthermore, the power allocation strategy for energy efficient improvement in a downlink NOMA system is discussed in [START_REF] Glei | Power allocation for energy-efficient downlink noma systems[END_REF]. A novel power allocation algorithm based on particle swarm optimization is presented in [START_REF] Diao | Joint computing resource, power, and channel allocations for d2d-assisted and noma-based mobile edge computing[END_REF]. To tackle the power allocation problem in downlink multi-carrier NOMA networks, a dynamic power allocation algorithm is proposed in [START_REF] Xie | Power allocation scheme for downlink and uplink noma networks[END_REF], while a joint subchannel and power allocation is proposed based on the Dinkelbach algorithm in [START_REF] Huang | Energy efficient resource allocation algorithm in multi-carrier noma systems[END_REF]. In the same framework, the joint power allocation and time switching control for energy efficiency optimization is investigated in [START_REF] Tang | Energy efficiency optimization for noma with swipt[END_REF], with the aim to optimize the energy efficiency of the system under maximum transmit power budget among others.

In [START_REF] Alkhawatrah | Buffer-aided relay selection for cooperative noma in the internet of things[END_REF], a novel prioritization-based buffer-aided relay selection scheme which is able to combine NOMA and OMA transmission in a relay network is proposed and an analytical expression for the average throughput of the proposed scheme is derived. In [START_REF] Xu | Performance analysis of buffer-aided hybrid noma/oma in cooperative uplink system[END_REF], dynamic and fixed power controls at users are discussed in the case of a cooperative uplink system with a buffer-aided NOMA and OMA transmission and an efficient buffer-aided hybrid NOMA/OMA based mode selection scheme is proposed. A hybrid OMA-NOMA scheme is presented in [START_REF] Pischella | Performance analysis of uplink noma-relevant strategy under statistical delay qos constraints[END_REF], in which NOMA is only employed if all users gain in terms of effective rates. Finally, NOMA offers a natural scenario for physical layer security as one user's signal is naturally degraded with respect to the other's [START_REF] Yu | Effective secrecy rate for a downlink NOMA network[END_REF] and constitutes the equivalent of a helping interferer [START_REF] Chorti | Achievable secrecy rates in physical layer secure systems with a helping interferer[END_REF].

Besides, in a number of emerging applications, delay constraints become increasingly important, e.g., ultra reliable low latency communication (URLLC) systems such as autonomous vehicles and enhanced reality. Furthermore, in future wireless networks, users are expected to necessitate flexible delay guarantees for achieving different service requirements. In order to satisfy diverse delay requirements, a simple and flexible delay quality of service (QoS) model is imperative to be applied and investigated. In this respect, the effective capacity (EC) theory can be employed [START_REF] Yu | Tradeoff analysis and joint optimization of link-layer energy efficiency and effective capacity toward green communications[END_REF]- [START_REF] Tang | Cross-layer modeling for quality of service guarantees over wireless links[END_REF]. The EC denotes the average maximum constant arrival rate which can be served by a given service process, while guaranteeing the required statistical delay provisioning [START_REF] Musavian | Effective capacity maximization with statistical delay and effective energy efficiency requirements[END_REF].

The delay-constrained communications for a downlink NOMA network was studied in [START_REF] Yu | Link-layer capacity of NOMA under statistical delay QoS guarantees[END_REF], where the EC theory was utilized. The present analysis on uplink complements [START_REF] Yu | Link-layer capacity of NOMA under statistical delay QoS guarantees[END_REF] which focused on downlink transmissions. NOMA, as a more spectrum-efficient technique, is considered to be promising for supporting a massive number of devices in the uplink connections.

The present work extends our recent publication [START_REF] Bello | Performance analysis of noma uplink networks under statistical qos delay constraints[END_REF] in which novel closed form expressions for the effective rate in a two user uplink NOMA network were presented. Our contributions in this works are articulated around six lemmas and four propositions; the main contributions of this paper are listed below:

• First, using the theory of order statistics, we derive closed-form expressions for the effective capacity of each user in a two-user uplink NOMA network; The expressions are validated through Monte-Carlo simulations. • We then provide an asymptotic analysis of the individual and sum ECs for both OMA and NOMA, in the case of delay-constrained and delay-tolerant applications. A detailed comparison between NOMA and OMA is provided; through an extensive set of simulation results, we show that NOMA does not always perform better than OMA in the presence of delay constraints. For illustration purposes, we depict the regions of the transmit signal-tonoise ratio (SNR) where the earlier outperforms the latter for generic values of the system parameters. • With respect to the strong NOMA user, we prove that, its EC reaches a plateau in the high SNRs, in contrast to OMA, a consequence of the fact that the strong user is interference limited. • The impact of the delay QoS exponent and of the transmit SNR on the individual and sum ECs is investigated as well. Specifically, we show that, through an extensive set of simulation results, the individual ECs decrease as the delay constraints becomes more stringent, and consequently so does the sum EC. ically, to be the pairing of users with the maximum channel gains gaps, which is in agreement to previous results in systems without delay constraints [START_REF] Shahab | User pairing schemes for capacity maximization in non-orthogonal multiple access systems[END_REF]. The rest of the paper is organized as follows. In Section II, we introduce the system model and define the notion of EC in an uplink NOMA system under delay QoS constraints. In Section III, an asymptotic analysis of the EC is provided for a two-user system. In Section IV, the EC of multiple pairs is studied to investigate the impact of pairing. Simulation results are given in Section V, followed by conclusions in Section VI.

II. EFFECTIVE CAPACITY IN UPLINK NOMA

The notation used throughout the rest of the paper is given in Table I for convenience.

A. General Case: M -User NOMA Assume a M -user NOMA uplink network with users U 1 , U 2 , ...U M in Rayleigh block-fading propagation channels [START_REF] Biglieri | Limiting performance of blockfading channels with multiple antennas[END_REF], with respective channel gains during a transmission block denoted by |h i | 2 , i = 1, . . . , M , that without loss of generality are ordered as

|h 1 | 2 < • • • < |h M | 2 .
The users transmit corresponding unit power symbols s 1 , . . . , s M respectively, with E[|s i | 2 ] = 1, i = 1, . . . , M with a total transmit power constraint P T = M i=1 P i = 1. We note in passing that the total power constraint does not capture the individual user's budgets, but rather regulatory requirements imposing that the transmit power in any given resource block cannot exceed a maximum value [START_REF] Yang | A general power allocation scheme to guarantee quality of service in downlink and uplink NOMA systems[END_REF]. The received superimposed signal can be expressed as [START_REF] Zhang | Uplink nonorthogonal multiple access in 5G systems[END_REF]:

z = M i=1 P i h i s i + w, (1)
where w denotes a zero mean circularly symmetric complex Gaussian random variable with variance σ 2 , i.e., w ∼ CN (0, σ 2 ).

The receiver first decodes the symbols of the strongest user treating the transmission of the weaker users as interference. After decoding it, the receiver suppresses it from z and decodes the signal of the second strongest user, and so on until the decoding of the weakest user's signal. Following the SIC principle and denoting the transmit SNR ρ = 1 σ 2 , the achievable rate, in b/s/Hz, for user U i , i = 1, . . . , M , assuming no error propagation, is expressed as [START_REF] Fan | Uplink achievable rate for massive MIMO systems with low-resolution ADC[END_REF]:

R i = log 2 1 + ρP i |h i | 2 1 + ρ i-1 l=1 P l |h l | 2 . ( 2
)
Notice that in the present work we don't consider the impact of path loss, but it could be easily taken into account by inserting a multiplicative coefficient on the user received SNR or SINR, accounting for the loss in received power due to distance. For the sake of simplicity, we do not account for this effect in the present, as common in related published work [START_REF] Islam | Power-domain non-orthogonal multiple access (NOMA) in 5G systems: Potentials and challenges[END_REF], [START_REF] Makki | A survey of noma: Current status and open research challenges[END_REF], [START_REF] Yu | Link-layer capacity of NOMA under statistical delay QoS guarantees[END_REF], [START_REF] Zhang | Uplink nonorthogonal multiple access in 5G systems[END_REF].

Next, let θ i be the statistical delay exponent of the i-th user, i.e., θ i captures how strict the delay constraint of the user i is, and assume that the service process satisfies the Gärtner-Ellis theorem [START_REF] Wu | Effective capacity: a wireless link model for support of quality of service[END_REF]. A slower decay rate can be represented by a smaller θ i , which indicates that the system is more delay tolerant, while a larger θ i corresponds to a system with more stringent QoS requirements. Applying the EC theory in an uplink NOMA with M users, the i-th user's EC over a blockfading channel is defined as:

E i c = - 1 θ i T f B ln E e -θiT f BRi (in b/s/Hz) , (3)
where T f is the duration of each fading-block, B is the system bandwidth and E [•] denotes expectation over the channel gains. By inserting R i into (3), we obtain the following expression for the EC of the i-th user

E i c = 1 β i log 2   E   1 + ρP i |h i | 2 1 + ρ i-1 l=1 P l |h l | 2 βi     (4)
where

β i = - θiT f B
ln 2 , i = 1, . . . M , is the normalized (negative) QoS exponent. Developing (4), we have that:

E i c = 1 β i log 2 ∞ 0 ∞ x1 ∞ x2 ... ∞ xi-1 1 + ρP i x i 1 + i-1 l=1 ρP l x l βi f X(1) , X(2) ,..., X(i) (x 1 , x 2 , ..., x i ) d xi d xi-1 ...d x1 , (5)
where f X(1) , X(2) ,...,

X(i) (x 1 , x 2 , ..., x i ) is the joint distribution of x i = |h i | 2 , i = 1, . . . , M .
To evaluate the joint distribution of the channel gains, we make use of the theory of order statistics [START_REF] Yang | Order statistics in wireless communications: diversity, adaptation, and scheduling in MIMO and OFDM systems[END_REF]. The probability density function (PDF) of the i-th ordered random variable in a population of M is given by:

f X(i) (x) = ψ i f (x)(1 -F (x)) M -i F (x) i-1 , (6)
where

ψ i = 1 B(i,M -i+1) ,
(x) = 1 -e -x .
The joint distribution of M order statistics is given by [START_REF] Yang | Order statistics in wireless communications: diversity, adaptation, and scheduling in MIMO and OFDM systems[END_REF]:

f X(1)...X(M ) (x 1 , x 2 , . . . , x M ) = M ! f X(1) (x 1 ) . . . f X(M ) (x M ), (7)
where x 1 ≤ x 2 ≤ . . . ≤ x M , while for any two order statistics, we have that:

f X(l) , X(k) (x l , x k ) = M ! (l -1)! (k -l -1)! (M -k)! × (1 -F (x)) l-1 f (x)(F (x) -F (y)) k-l-1 f (y)(F (y)) M -k . (8)
Closed-form expressions for multi-users uplink NOMA can be obtained by inserting ( 7) into ( 5), but considering the complexity of the analytical development of these integrals, we will just consider in this study, the simple case of two users. Furthermore, there is also a question of practical limitation. In fact, the execution of several SICs in series at the base station can lead to additional processing delay; thus, an increase in terms of latency, especially for the last decoded user. Moreover, if we have imperfect SIC, additional errors due to error propagation can lead to decoding failure for weaker users, as they are the last to be decoded [START_REF] Sun | Non-orthogonal multiple access with sic error propagation in downlink wireless mimo networks[END_REF]. Due to the above reasons, in the following we focus on deriving closed-form expressions only for the two-user case.

B. Case of Two-User NOMA Uplink Network (M =2)

Using ( 6), we obtain

f X(1) (x 1 ) = 2e -2x1 . (9)
Furthermore, by setting M = 2, l = 1 and k = 2 in (8), we get:

f X(1) , X(2) (x 1 , x 2 ) = 2f (x 1 )f (x 2 ) = 2e -x1 e -x2 . (10)
As a result, the EC of U 1 , denoted by E 1 c , is expressed as

E 1 c = 1 β 1 log 2 E[(1 + ρP 1 x 1 ) β1 ] = 1 β 1 log 2 ∞ 0 (1 + ρP 1 x 1 ) β1 f X(1) (x 1 )dx 1 = 1 β 1 log 2 2 P 1 ρ × U 1, 2 + β 1 , 2 ρP 1 , (11)
where U (•, •, •) denotes the confluent hypergeometric function [START_REF] Yu | Link-layer capacity of NOMA under statistical delay QoS guarantees[END_REF].

On the other hand, the EC of U 2 is evaluated as

E 2 c = 1 β 2 log 2 E 1 + ρP 2 x 2 1 + ρP 1 x 1 β2 = 1 β 2 log 2 ∞ 0 ∞ x1 1 + ρP 2 x 2 1 + ρP 1 x 1 β2 f X(1) , X(2) (x 1 , x 2 )dx 2 dx 1 = 1 β 2 log 2 2P 1-β2 2 (ρP 2 ) β2 e 1 ρP 2 e -(P 1 -P 2 ) ρP 2 + 1 β 2 log 2 -β2 j=0 -β 2 j (ρP 1 ) j × ∞ k=0 (-1) k (P 2 -P 1 ) k k! (1 + j + k) × Γ 2 + β 2 + j + k, 1 ρP 2 -(ρP 2 ) -1-j-k Γ 1 + β 2 , 1 ρP 2 (12)
with Γ(•, •) denoting the incomplete Gamma function [START_REF] Yu | Link-layer capacity of NOMA under statistical delay QoS guarantees[END_REF].

Proof: The proof is provided in Appendix I. The closed-form expression of the sum EC in the case of a two-user NOMA uplink network can be easily obtained by summing up the individual ECs.

C. Case of a Two-User OMA Network

Similarly, using time division multiple access (TDMA), the achievable data rate of the i-th user in a two-user OMA network, denoted by R i , i = 1, 2, is given by

R i = 1 2 log 2 1 + 2ρP i |h i | 2 , i = 1, 2. (13)
Note that 1 2 is due to the equal allocation of resources to both users. Furthermore, it is important to note that the power of each OMA user is double that of NOMA, for the sake of fairness [START_REF] Yu | Link-layer capacity of NOMA under statistical delay QoS guarantees[END_REF]. The corresponding ECs of both users in an OMA network are denoted by E i c :

E i c = 1 β i log 2 E (1 + 2ρP i |h i | 2 ) β i 2 . (14)
A general expression of the ECs of M TDMA OMA users is given in [START_REF] Yu | Link-layer capacity of NOMA under statistical delay QoS guarantees[END_REF]; applying this to a two-user network we can be easily obtain:

E 1 c = 1 β 1 log 2 1 ρP 1 × U 1, 2 + β 1 2 , 1 ρP 1 , (15)
E 2 c = 1 β 2 log 2 1 ρP 2 1 k=0 1 k (-1) k × U 1, 2 + β 2 2 , 1 + k 2ρP 2 . ( 16
)
The difference in these expressions is due to the different PDFs of ordered channel gains.

III. ASYMPTOTIC ANALYSIS

In this Section, an asymptotic analysis with respect to the transmit SNR ρ is presented. This analysis consists in describing the limiting behavior of individual and total ECs, and how they evolve with the transmit SNR ρ. Our results are summarized in the following Propositions and Lemmas.

A. Case 1: Delay-Constrained Users Proposition 1: reaches an upper limit, allowing OMA to outperform NOMA after some SNR value (which depends on the system parameters). Proposition 1 is the synthesis of Lemmas 1, 2 and 3, discussed in detail next.

1) At low transmit SNR, ρ → 0, E 1 c , E 1 c , E
Lemma 1: In the low and high SNR regimes, respectively, the following conclusions hold:

1) When ρ → 0, then, E 1 c → 0, E 2 c → 0, E 1 c → 0, E 2 c → 0, E 1 c -E 1 c → 0, E 2 c -E 2 c → 0; 2) When ρ → +∞, then E 1 c → +∞, E 2 c → 1 β2 log 2 E 1 + P2|h2| 2 P1|h1| 2 β2 , E 1 c → +∞, E 2 c → +∞, E 1 c -E 1 c → +∞, E 2 c -E 2 c → -∞. Proof: The proof is provided in Appendix II.
To further analyze the impact of ρ on the individual EC, the partial derivatives with the respect of ρ are investigated [START_REF] Yu | Link-layer capacity of NOMA under statistical delay QoS guarantees[END_REF].

Lemma 2: For the EC of the U 1 , in a two-user uplink network the following hold:

1) ∂E 1 c ∂ρ ≥ 0 and ∂ E 1 c ∂ρ ≥ 0, ∀ρ; 2) When ρ → 0, then lim ρ→0 ( ∂(E 1 c -E 1 c ) ∂ρ ) = 0; 3) When ρ >> 1, then ∂(E 1 c -E 1 c ) ∂ρ ≈ 1 2ρ ln 2 ≥ 0 and it approaches 0 when ρ → ∞. Proof:
The proof is provided in Appendix III. Lemma 3: For the EC of the U 2 , in a two-user uplink network the following hold:

1) ∂E 2 c ∂ρ ≥ 0 and ∂ E 2 c ∂ρ ≥ 0, ∀ρ; 2) When ρ → 0, then lim ρ→0 ( ∂(E 2 c -E 2 c ) ∂ρ ) = 0 3) When ρ >> 1, then ∂(E 2 c -E 2 c ) ∂ρ ≈ -1 2 ln 2 1
ρ < 0 and it approaches 0 when ρ → ∞. Proof: The proof is provided in Appendix IV. Finally, we investigate the sum ECs when using OMA and NOMA, denoted by V N and V O , respectively, i.e.,

V N = E 1 c + E 2 c , (17)
V O = E 1 c + E 2 c . ( 18
)
Proposition 2: 1) At low transmit SNR ρ, V N and V O increase at a constant rate that depends on the average of the channel power gains and the allocated power coefficients. 2) When ρ >> 1, V N and V O tend to ∞, and reach a plateau when the transmit SNR ρ → ∞.

Proposition 2 is the consequence of the Lemma 4. Lemma 4: For the sum EC with NOMA, denoted by V N , and with OMA, denoted by V O , in a two-user uplink network, the following hold:

1) ∂V N ∂ρ ≥ 0 and ∂V O ∂ρ ≥ 0, ∀ρ; 2) When ρ → 0, V N → 0, lim ρ→0 ( ∂V N ∂ρ ) = P1 ln 2 E[|h 1 | 2 ] + P2 ln 2 E[|h 2 | 2 ] ≥ 0, and V O → 0, lim ρ→0 ( ∂V O ∂ρ ) = P1 ln 2 E[|h 1 | 2 ] + P2 ln 2 E[|h 2 | 2 ] ≥ 0; 3) When ρ >> 1, V N → ∞, lim ρ→∞ ( ∂V N ∂ρ ) = 0, and V O → ∞, lim ρ→∞ ( ∂V O ∂ρ ) = 0. Proof: The proof is provided in Appendix V.

B. Case 2: Delay-Tolerant Applications

A case of particular interest is presented when the users' applications are delay tolerant, i.e., when the delay exponent becomes negligible. In this case, investigation of the ECs of the two-user, uplink NOMA and OMA networks, is performed without delay constraints. The impact of the transmit SNR ρ in this case is also investigated.

Proposition 3: 1) For both OMA and NOMA, when there is no delay constraint (θ = 0), the individual ECs of both users are equal to their ergodic capacities. 2) At high transmit SNRs, irrespective of whether there's a tolerance for delay or not, the conclusions on the performance gain of NOMA over OMA for U 1 , and OMA over NOMA for U 2 remain the same. Proposition 3 is the consequence of the Lemma 5.

Lemma 5: Considering the EC for the weaker user with θ 1 → 0, in NOMA and OMA, the following hold:

a) When θ 1 → 0, lim θ1→0 E 1 c = E[R 1 ], lim θ1→0 E 1 c = E[ R 1 ], lim θ1→0 (E 1 c -E 1 c ) = E[R 1 ] -E[ R 1 ], b) When θ 1 → 0, ρ → ∞, lim θ1→0 ρ→∞ E 1 c = ∞, lim θ1→0 ρ→∞ E 1 c = ∞, lim θ1→0 ρ→∞ (E 1 c -E 1 c ) = ∞.
Considering the EC for the stronger user with θ 2 → 0, in NOMA and OMA, we prove that:

c) When θ 2 → 0, lim θ2→0 E 2 c = E[R 2 ], lim θ2→0 E 2 c = E[ R 2 ], lim θ2→0 (E 2 c -E 2 c ) = E[R 2 ] -E[ R 2 ], d) When θ 2 → 0, ρ → ∞, lim θ2→0 ρ→∞ E 2 c = E log 2 1 + P2|h2| 2 P1|h1| 2 , lim θ2→0 ρ→∞ E 2 c = ∞, lim θ2→0 ρ→∞ (E 2 c -E 2 c ) = -∞.
Proof: The proof is provided in Appendix VI.

IV. EFFECTIVE CAPACITY OF MULTIPLE NOMA PAIRS

The M NOMA users scenario assumes that the resource block is shared among M users. For large values of M , stronger users are penalized due to high interference level from weaker users since they are decoded first. Pairing allows us to mitigate interference from weaker users on stronger ones. A popular approach for alleviating this effect in an M user network, is to form M 2 groups with indices i = 1, . . . , M 2 , where each group involves only 2 users. Inside each group, NOMA is implemented, while across different groups TDMA is applied.

The achievable data rate of the two users, U 1 and U 2 of the i th group, where |h 1i | 2 ≤ |h 2i | 2 , can be formulated as follow:

R 1i = 2 M log 2 1 + ρP 1i |h 1i | 2 , (19)
R 2i = 2 M log 2 1 + ρP 2i |h 2i | 2 1 + ρP 1i |h 1i | 2 , ( 20
)
with 2 M the fraction of resources at the disposal of the two users inside a NOMA group.

On the other hand, if all users utilize TDMA, their achievable data rates are given as follows:

R j = 1 M log 2 1 + 2P j ρ|h j | 2 , j ∈ {1 i , 2 i } . (21)
The factor 1 M is to indicate that each user has only one time slot to transmit.

By replacing ( 19) and ( 20) in (3), we get respectively the following ECs for U 1 and U 2 in the i th group:

E 1i c = 1 β 1i log 2 E (1 + ρP 1i |h 1i | 2 ) 2β 1 i M , (22)
E 2i c = 1 β 2i log 2   E 1 + ρP 2i |h 2i | 2 1 + ρP 1i |h 1i | 2 2β 2 i M   . (23)
On the other hand, replacing [START_REF] Xie | Power allocation scheme for downlink and uplink noma networks[END_REF] in (3) we get the expressions for both users while using TDMA:

E 1i c = 1 β 1i log 2 E (1 + 2ρP 1,i |h 1i | 2 ) β 1 i M , (24)
E 2i c = 1 β 2i log 2 E (1 + 2ρP 2,i |h 2i | 2 ) β 2 i M . (25)
Next, we analyze the total sum EC of multiple NOMA pairs, denoted by E tot c , in comparison with the total sum EC for the M OMA users, E c tot defined as:

E tot c = M 2 i=1 (E 1i c + E 2i c ), (26)
E c tot = M 2 i=1 ( E 1i c + E 2i c ). (27)
To investigate the performance of the user-pairing, the following Proposition and Lemma are provided.

Proposition 4: 1) NOMA user-pairing outperforms OMA at low transmit SNRs and this performance gain carries on at very high transmit SNRs, with the possibility to be improved by optimizing the power allocation. Proposition 4 is the consequence of Lemma 6.

∂(E tot c -Ec tot ) ∂ρ = 0. b) When ρ → ∞, E tot c -E c tot → constant, given in (28),
and

lim ρ→∞ ∂(E tot c -Ec tot ) ∂ρ = 0. lim ρ→∞ (E tot c -E c tot )= M 2 i=1 1 β 1,i log 2 2 -β 1,i M E (P 1,i |h 1,i | 2 ) β 1,i M + 1 β 2,i log 2     E 1 + P2,i|h2,i| 2 P1,i|h1,i| 2 2β 2,i M E (2P 2,i |h 2,i | 2 ) β 2,i M         . (28)
Proof: The proof is provided in Appendix VII. From Lemma 6, we can conclude that E tot c -E c tot initially starts at 0, first increases at low transmit SNRs ρ, and finally approaches the constant value given in (28) that depends on the power allocation, at high transmit SNRs, i.e., this performance gain of NOMA with user-pairing over OMA can be optimized by finding the best pairing strategy.

V. NUMERICAL RESULTS

In this Section, the Propositions and Lemmas presented in previous sections are validated through Monte-Carlo simulations. We first consider a two-user uplink NOMA system, with the following parameters: normalized transmission power for both users, P 1 = 0.2, P 2 = 0.8, normalized delay exponent β 1 = β 2 = -1 for both users, unless otherwise stated. Fixed power allocation is used for the sake of simplicity.

Fig. 1 provides validation of the proposed closed-form expressions of E 1 c and E 2 c respectively in [START_REF] Chen | Multiuser detection for uplink non-orthogonal multiple access system[END_REF] and [START_REF] Cai | Modulation and multiple access for 5G networks[END_REF]. The analytical expressions of these individual ECs ( , ) and the corresponding Monte-Carlo simulations ( , ) are indistinguishable, showcasing the accuracy of the proposed closed-from expressions. In Fig. 2, the ECs of the two-user uplink NOMA and OMA networks are depicted versus the transmit SNR. We note that for U 1 , NOMA and OMA perform equally well at very low transmit SNRs, and NOMA is advantageous compared to OMA at high transmit SNRs. In contrast, for U 2 , NOMA is better at low SNRs and OMA is advantageous at high transmit SNRs. We notice also that the EC of U 2 reaches a plateau at high SNRs, validating Lemma 1. Moreover, we note that, at low transmit SNRs, E 2 c is higher than E 1 c , despite the interference that U 2 experiences; but with the transmit SNR increasing, E 1 c increases without bound and therefore at some point surpasses E 2 c which is capped to an upper value. Fig. 3 and Fig. 4 show, respectively, the EC of U 1 and U 2 , versus the transmit SNR, for different values of the delay exponent. When the delay constraints become more stringent, i.e., β decreases (equivalently, θ increases), both E 1 c and E 2 c decrease.

In Fig. 5, the ECs of the strong and weak users are depicted in the high SNR regime (ρ = 30 dB) as functions of the (negative) normalized delay exponent, for NOMA and OMA. We noticed that the EC curves are identical. On the other hand, in Fig. 6, where E 1 c and E 2 c are depicted across different SNR values, ∈ {1, 10, 30, 40, 50} dB, as functions of the (negative) normalized delay exponent, the EC of both users increase with the transmit SNR ρ increasing. Fig. 7 shows E 1 c -E 1 c versus the transmit SNR. This curve initially starts at zero, increases at the high transmit SNRs. Also, we can note that this gap decreases with delay constraints becoming more stringent (β decreasing). This confirms Lemma 2.

Fig. 8 shows E 2 c -E 2 c versus the transmit SNR. This curve initially starts at zero, increases to a certain maximum and starts decreasing without bound at high values of the transmit SNR. This confirms Lemma 3. We note that the maximum of these curves decreases when the delay becomes more stringent. Furthermore, as the negative delay exponent decreases, the zero crossing point moves to higher SNRs; this implies that as the delay QoS constraints become more stringent, the region of SNRs over which NOMA outperforms OMA increases.

To investigate the impact of ρ on the performance of the total EC for the two-user system, in Fig. 9, the plots for V N in NOMA and V O in OMA, versus the transmit SNR are depicted for various delay exponents. The curves demonstrate that for both NOMA and OMA, the total EC for the two users starts at the initial value of 0 and then increases with the transmit SNR, as outlined in Lemma 4. When ρ is very small, the total EC for the two users in NOMA, V N , increases faster than V O in OMA. On the contrary, with the increase of the transmit SNR, V O becomes gradually higher than V N . At very high values of the transmit SNR, the gap between V N and V O increases further. Finally, when the delay becomes more stringent, both V N and V O decrease.

V N ( =-1) V N ( =-2) V O ( =-1) V O ( =-2)
Fig. 9. V N and V O versus ρ, for various values of normalized delay exponent.

Fig. 10 and Fig. 11 depict V N -V O versus ρ, for several values of the (negative) normalized delay exponent. In Fig. 10, the delay of U 2 is fixed, while the delay exponent of U 1 varies. It is shown that in that case, the smallest delay QoS (i.e., the highest negative normalized delay exponent) of U 1 corresponds to the highest gap in V N -V O . With more stringent delay constraints for U 1 , NOMA outperforms OMA in an increasing region of SNRs. On the other hand, when the delay of U 1 is fixed, Fig. 11 shows that the smallest delay QoS (i.e., the highest negative normalized delay exponent) for U 2 corresponds to the largest gap in V N -V O . The curve of V N -V O , initially starts at zero, increases maximum, returns to negative values. In the regions in which it is positive, NOMA outperforms OMA in terms of the total EC; the opposite is true in the regions in which it is negative.

Next, we focus on the comparison of multiple NOMA pairs and OMA, i.e., E tot c and E c tot . Fig. 12-( versus the transmit SNR, for various settings of user-pairing. Initially these start at zero at low transmit SNRs, increasing to a maximum at high values of ρ. This confirms Lemma 6. Specifically, we set the total number of users M = 4; the normalized delay of all users are assumed to be equal

β 1,i = β 2,i = -1, i = 1, . . . , M 2 .
The best pairing policy in the case of M = 4 is (1,4)- [START_REF] Makki | A survey of noma: Current status and open research challenges[END_REF][START_REF] Ding | A survey on non-orthogonal multiple access for 5G networks: Research challenges and future trends[END_REF]. We noticed that even the worst pairing strategy outperforms OMA in terms of the total EC. Fig. 13 depicts the result of the exhaustive search, done in order to find the pairing strategy which gives the highest total EC in the case of M = 6. The curves are normalized to the worst pairing. It appears that when these six users are divided in three groups of two users, the pairing strategy: (1,6)-(2,5)- (3,4) gives the highest total sum EC. We believe that this is due to the fact that coupling the strongest user and the weakest user produces the lowest interference at decoding. Fig. 14, on the other hand, depicts the result of the exhaustive search, of valid pairs, when all six users are divided in two groups of three users. It appears that the best pairing policy in terms of total sum EC is : (1,2,6)- [START_REF] Ding | A survey on non-orthogonal multiple access for 5G networks: Research challenges and future trends[END_REF][START_REF] Saito | System-level performance evaluation of downlink non-orthogonal multiple access (noma)[END_REF][START_REF] Ding | On the performance of non-orthogonal multiple access in 5G systems with randomly deployed users[END_REF]. The results for the best pairing are aligned with the literature on NOMA pairing without delay constraints.

Fig. 15 depicts a comparison between full NOMA, i.e., when all users transmit in the same resource block, NOMA user-pairing, NOMA user-grouping (groups of 3 users) and OMA, for M = 6 users. Considering the best power allocation policies in the case of user-pairing and user-grouping, it appears that full NOMA outperforms all of them in terms of the total EC, followed by NOMA with user-grouping, assuming absence of error propagation due to decoding errors.

In the simulation results given above, the user pairing is presented for a specific values of the system parameters and can be further improved by the power allocation optimization, which in the present was ignored to simplify the analysis. We

Fig. 14. Sum EC versus ρ for various grouping settings, normalized to the worst grouping which is (1,2,3)(4,5,6). M = 6.

note in passing that when the number of users M is not a multiple of two or three, a hybrid pairing can be used, i.e., using both clusters of two and three users.

VI. CONCLUSIONS AND FUTURE WORK

The concept of EC enabled us to study the performance gain of NOMA over OMA in systems with statistical delay QoS constraints. First, we investigated the EC of the uplink of a two-user NOMA network, assuming a Rayleigh block fading channel. We derived novel closed-form expressions for the ECs of the two users and provided a comparison between NOMA and OMA. The results show that, the EC of U 1 can surpass the EC of U 2 , as the latter is limited due to interference. Furthermore, we showed that the ECs of both users decrease as the delay constraints become more stringent. For both users, when the delay QoS of one of them is fixed, the smallest values of the other's delay QoS give the highest performance gap between NOMA and OMA in terms of total EC. On the other hand, we investigated NOMA with user pairing and found the optimal pairing strategy that gave the highest EC, for M = 4 and M = 6. It turns out that NOMA grouping and NOMA pairing does not better than full NOMA, but one can get close to it when users transmit with optimal power. NOMA with user pairing is interesting as it can be an alternative to mitigate interference on stronger users and reduce the impact of error propagation. These results raise questions on the possibility of switching between NOMA and OMA according to the individual users' delay constraints and transmit power.

APPENDIX I

E 1 c = 1 β 1 log 2 ∞ 0 (1 + ρP 1 x 1 ) β1 e -2x1 dx 1 . (29)
Set t = ρP 1 x 1 i.e., x 1 = t ρP1 and since

x 1 : 0 → ∞ =⇒ t : 0 → ∞, dx 1 = 1
ρP1 dt, we can get that:

E 1 c = 1 β 1 log 2 2 P 1 ρ ∞ 0 (1 + t) β1 e -2t P 1 ρ )dt . (30)
Also, by setting a

= 1, (b-a-1) = β 1 , =⇒ b = β 2 +2, z = 2
P1ρ and denoting by U (., ., .) the confluent hypergeometric function:

U (a, b, z) = 1 Γ(a)
∞ 0 e -zt t a-1 (1 + t) b-a-1 dt, we have that:

∞ 0 (1 + t) β1 e -2t P 1 ρ dt = U 1, 2 + β 1 , 2 ρP1
, which means that:

E 1 c = 1 β 1 log 2 2 P 1 ρ × U 1, 2 + β 1 , 2 ρP 1 . (31)
For the U 2 , we have that:

E 2 c = 1 β 2 log 2 E 1 + ρP 2 x 2 1 + ρP 1 x 1 β2 = 1 β 2 log 2 2 ∞ 0 ρP 2 1 + ρP 1 x 1 β2 e -x1 ∞ x1 1 + ρP 1 x 1 ρP 2 + x 2 β2 e -x2 dx 2 dx 1 . (32)
We set z = 1+ρP1x1 ρP2 +x 2 , i.e., we have that:

x 2 = z -1+ρP1x1 ρP2
and dx 2 = dz, so that

x 2 → x 1 , =⇒ z → 1+ρP1x1 ρP2 + x 1 = 1+ρx1 ρP2 and x 2 → ∞ =⇒ z → ∞. E 2 c = 1 β 2 log 2 2 ∞ 0 ρP 2 1 + ρP 1 x 1 β2 e -x1 ∞ 1+ρx 1 ρP 2 z β2 e -z- 1+ρP 1 x 1 ρP 2 dzdx 1 = 1 β 2 log 2 2e 1 ρP 2 ∞ 0 ρP 2 1 + ρP 1 x 1 β2 e -x1 e P 1 x 1 P 2 ∞ 1+ρx 1 ρP 2 z β2 e -z dzdx 1 . (33)
We note that:

∞ a e -x x b dx = a -b 2 e -a 2 W -b 2 , 1-b 2 (a)
where W is the Whittaker W function. Hence, we get that:

E 2 c = 1 β 2 log 2 2e 1 ρP 2 ∞ 0 ρP 2 1 + ρP 1 x 1 β2 e -x1 e P 1 x 1 P 2 1 + ρx 1 ρP 2 β 2 2 e -1+ρx 1 2ρP 2 W β 2 2 , 1+β 2 2 1 + ρx 1 ρP 2 dx 1 = 1 β 2 log 2 2 (ρP 2 ) β 2 2 e 1 2ρP 2 ∞ 0 (1 + ρP 1 x 1 ) -β2 (1 + ρx 1 ) β 2 2 e (2P 1 -2P 2 -1)x 1 2P 2 W β 2 2 , 1+β 2 2 1 + ρx 1 ρP 2 dx 1 . ( 34
) Note that W u-1 2 ,u (z) = e 1 2 z z 1 2 -u Γ(2u, z), so that we have W β 2 2 , 1+β 2 2 1+ρx1 ρP2 = e 1+ρx 1 2ρP 2 1+ρx1 ρP2 - β 2 2 Γ 1 + β 2 , 1+ρx1 ρP2
.

By substituting it in E 2 c , we have that:

E 2 c = 1 β 2 log 2 2(ρP 2 ) β2 e 1 ρP 2 ∞ 0 (1 + ρP 1 x 1 ) -β2 e (P 1 -P 2 )x 1 P 2 × Γ 1 + β 2 , 1 + ρx 1 ρP 2 dx 1 . (35)
To continue we set 1+ρx1 ρP2 = y, i.e., x 1 = P 2 y -1 ρ , and dx 1 = P 2 dy. x 1 → 0 =⇒ y → 1 ρP2 and x 1 → ∞ =⇒ y → ∞. Recall that without loss of generality we have set P 1 + P 2 = 1. Then we get that

E 2 c = 1 β 2 log 2 2(ρP 2 ) β2 e 1 ρP 2 ∞ 0 (1 + ρP 1 x 1 ) -β2 × e (P 1 -P 2 )x 1 P 2 Γ 1 + β , 1 + ρx 1 ρP 2 dx 1 = 1 β 2 log 2 2P 2 (ρP 2 ) β2 e 1 ρP 2 e -(P 1 -P 2 ) ρP 2 × ∞ 1 ρP 2 P -β2 2 (1 + ρP 1 y) -β2 e (P1-P2)y Γ(1 + β 2 , y)dy . (36)
Using binomial expansion we have (1 + 1 y) -β2 = -β2 j=0 -β2 j (ρP 1 y) j when β 2 is integer, otherwise we use β 2 . And, using Taylor series expansion we have that e (P1-P2)y = e -(P2-P1)y = ∞ k=0

(-1) k (P2-P1) k k!
y k , which converges.

E 2 c = 1 β 2 log 2 2P 1-β2 2 (ρP 2 ) β2 e 1 ρP 2 e -(P 1 -P 2 ) ρP 2 × ∞ 1 ρP 2 (1 + ρP 1 y) -β2 e (P1-P2)y Γ(1 + β 2 , y)dy = 1 β 2 log 2 2P 1-β2 2 (ρP 2 ) β2 e 1 ρP 2 e -(P 1 -P 2 ) ρP 2 × -β2 j=0 -β 2 j (ρP 1 ) j × ∞ k=0 (-1) k (P 2 -P 1 ) k k! × ∞ 1 ρP 2 y j+k Γ (1 + β 2 , y) dy . (37)
Note that

∞ c z b Γ (A, z) dz= 1 1 + b -c 1+b Γ (A, c)+Γ (1 + A + b, c) i.e., ∞ 1 ρP 2 y j+k Γ (1 + β 2 , y) dy = 1 1 + j + k × -(ρP 2 ) -1-j-k Γ(1 + β 2 , 1 ρP 2 )+Γ(2 + β 2 + j + k, 1 ρP 2 ) . (38)
Finally, by inserting [START_REF] Zhang | Uplink nonorthogonal multiple access in 5G systems[END_REF] in [START_REF] Yang | A general power allocation scheme to guarantee quality of service in downlink and uplink NOMA systems[END_REF] we obtain [START_REF] Cai | Modulation and multiple access for 5G networks[END_REF].

APPENDIX II

By inserting ρ → 0 into (11) and ( 12), we get 1) of Lemma 1, i.e.,

lim ρ→0 (E 1 c -E 1 c ) = 1 β 1 log 2    E 1 + ρP 1 |h 1 | 2 β2 E (1 + 2ρP 1 |h 1 | 2 ) β 2 2    = 0, lim ρ→0 (E 2 c -E 2 c ) = 1 β 2 log 2    E 1 + ρP2|h2| 2 1+ρP1|h1| 2 β2 E (1 + 2ρP 2 |h 1 | 2 ) β 2 2    = 0.
In the same way, by inserting ρ → ∞ into ( 11) and ( 12), we get 2) in Lemma 1, given below.

lim ρ→∞ E 2 c = 1 β 2 log 2 E 1 + P 2 |h 2 | 2 P 1 |h 1 | 2 β2 , lim ρ→∞ (E 1 c -E 1 c ) = 1 β 1 log 2   (ρP 1 ) β 1 2 E ( 1 ρP1 + |h 1 | 2 ) β2 E ( 1 ρP1 + 2|h 1 | 2 ) β 2 2   = ∞, lim ρ→∞ (E 2 c -E 2 c ) = 1 β 2 log 2       E 1 ρ +P1|h1| 2 +P2|h2| 2 1 ρ +P1|h1| 2 β2 ρ β 2 2 E 1 ρ + 2P 2 |h 2 | 2 β 2 2       = -∞.

APPENDIX III

To analyze the trends of E 1 c and E 1 c with respect to ρ, we start with

∂E 1 c ∂ρ = 1 β 1 ln 2 E[(1 + ρP 1 |h 1 | 2 ) β1 ] E[(1 + ρP 1 |h 1 | 2 ) β1 ] = P 1 ln 2 E[|h 1 | 2 (1 + ρP 1 |h 1 | 2 ) β1-1 ] E[(1 + ρP 1 |h 1 | 2 ) β1 ] ≥ 0. (39)
Similarly, for U 1 in OMA we have that

∂ E 1 c ∂ρ = 1 β 1 ln 2 E[(1 + 2ρP 1 |h 1 | 2 ) β 1 2 ] E[(1 + 2ρP 1 |h 1 | 2 ) β 1 2 ] = P 1 ln 2 E[|h 1 | 2 (1 + 2ρP 1 |h 1 | 2 ) β 1 2 -1 ] E[(1 + 2ρP 1 |h 1 | 2 ) β 1 2 ] ≥ 0. (40)
Then, we get that

∂(E 1 c -E 1 c ) ∂ρ = P 1 ln 2 E[|h 1 | 2 (1 + ρP 1 |h 1 | 2 ) β1-1 ] E[(1 + ρP 1 |h 1 | 2 ) β1 ] - P 1 ln 2 E[|h 1 | 2 (1 + 2ρP 1 |h 1 | 2 ) β 1 2 -1 ] E[(1 + 2ρP 1 |h 1 | 2 ) β 1 2 ]
.

and lim ρ→0 (

∂(E 1 c -E 1 c ) ∂ρ ) = (P1-P1) ln 2 E[|h 1 | 2 ] = 0. When ρ >> 1, we have that ∂(E 1 c -E 1 c ) ∂ρ ) = P 1 ρ ln 2 E[|h 1 | 2 (P 1 |h 1 | 2 ) β1-1 ] E[(P 1 |h 1 | 2 ) β1 ] - P 1 ρ ln 2 E[|h 1 | 2 (2P 1 |h 1 | 2 ) β 1 2 -1 ] E[(2P 1 |h 1 | 2 ) β 1 2 ] = 1 2ρ ln 2 ≥ 0. (42)
When ρ → ∞, this term approaches 0.

APPENDIX IV

E 2 c = 1 β 2 log 2 E 1 + ρP 2 |h 2 | 2 1 + ρP 1 |h 1 | 2 β2 . (43)
And

∂E 2 c ∂ρ = 1 β 2 ln 2 E 1 + ρP2|h2| 2 1+ρP1|h1| 2 β2 E 1 + ρP2|h2| 2 1+ρP1|h1| 2 β2 = 1 ln 2 E P2|h2| 2 2 ) 2 1 + ρP2|h2| 2 1+ρP1|h1| 2 β2-1 E 1 + ρP2|h2| 2 1+ρP1|h1| 2 β2 ≥ 0. (44)
In the same way, for the U 2 in OMA, we have that:

∂ E 2 c ∂ρ = 1 β 2 ln 2 E[(1 + 2ρP 2 |h 2 | 2 ) β 2 2 ] E[(1 + 2ρP 2 |h 2 | 2 ) β 2 2 ] = P 2 ln 2 E[|h 2 | 2 (1 + 2ρP 2 |h 2 | 2 ) β 2 2 -1 ] E[(1 + 2ρP 2 |h 2 | 2 ) β 2 2 ] ≥ 0, (45)
and

∂(E 2 c -E 2 c ) ∂ρ = 1 ln 2 E P2|h2| 2 (1+ρP1|h1| 2 ) 2 1 + ρP2|h2| 2 1+ρP1|h1| 2 β2-1 E 1 + ρP2|h2| 2 1+ρP1|h1| 2 β2 - P 2 ln 2 E[|h 2 | 2 (1 + 2ρP 2 |h 2 | 2 ) β 2 2 -1 ] E[(1 + 2ρP 2 |h 2 | 2 ) β 2 2 ] . (46)
When ρ → 0, we have that lim

ρ→0 ( ∂(E 2 c -E 2 c ) ∂ρ ) = 0. When ρ is very large, ∂(E 2 c -E 2 c ) ∂ρ = E P2|h2| 2 ρ 2 ( 1 ρ +P1|h1| 2 ) 2 (1 + ρ ρ (P2|h2| 2 ) ( 1 ρ +P1|h1| 2 ) ) β2-1 ln 2E (1 + ρ ρ P2|h2| 2 ( 1 ρ +P1|h1| 2 ) ) β2 - P 2 ln 2 1 ρ E[|h 2 | 2 ( 1 ρ + 2P 2 |h 2 | 2 ) β 2 2 -1 ] E[( 1 ρ + 2P 2 |h 2 | 2 ) β 2 2 ] = P 2 ρ 2 P 2 1 ln 2 E |h2| 2 (|h1| 2 ) 2 1 + P2|h2| 2 P1|h1| 2 β2-1 E 1 + P2|h2| 2 P1|h1| 2 β2 - 1 2 ln 2 1 ρ = P2 P 2 1 ln 2 A -1 2 ln 2 ρ ρ 2 , ( 47
)
where

A = E |h 2 | 2 (|h 1 | 2 ) 2 1+ P 2 |h 2 | 2 P 1 |h 1 | 2 β 2 -1 E 1+ P 2 |h 2 | 2 P 1 |h 1 | 2 β 2
, unrelated to ρ. And it gradually approaches 0 when ρ → ∞.

APPENDIX V Note that

V N = E 1 c + E 2 c
. By using Lemma 1, we have that lim ρ→0 (V N ) = 0 and lim ρ→∞ (V N ) = ∞. Then, we get that,

∂V N ∂ρ = ∂(E 1 c + E 2 c ) ∂ρ = P 1 ln 2 E[|h 1 | 2 (1 + ρP 1 |h 1 | 2 ) β1-1 ] E[(1 + ρP 1 |h 1 | 2 ) β1 ] + 1 ln 2 E P2|h2| 2 (1+ρP1|h1| 2 ) 2 1 + ρP2|h2| 2 1+ρP1|h1| 2 β2-1 E 1 + ρP2|h2| 2 1+ρP1|h1| 2 β2 ≥ 0. (48) When ρ → 0, we have that lim ρ→0 ( ∂V N ∂ρ ) = P1 ln 2 E[|h 1 | 2 ] + P2 ln 2 E[|h 2 | 2 ]. When ρ → ∞, we get that lim ρ→∞ ∂V N ∂ρ = 1 ρ ln 2 + E P2|h2| 2 (P1|h1| 2 ) 2 1 + P2|h2| 2 P1|h1| 2 β2-1 ρ 2 ln 2E 1 + P2|h2| 2 P1|h1| 2 β2 =0.
For V O in the case of OMA, we note that

V O = E 1 c + E 2 c . By using Lemma 1, we have lim ρ→0 (V 0 ) = 0 and lim ρ→∞ (V 0 ) = ∞. Then, ∂V 0 ∂ρ = ∂( E 1 c + E 2 c ) ∂ρ = P 1 ln 2 E[|h 1 | 2 (1 + 2ρP 1 |h 1 | 2 ) β 1 2 -1 ] E[(1 + 2ρP 1 |h 1 | 2 ) β 1 2 ] + P 2 ln 2 E[|h 2 | 2 (1 + 2ρP 2 |h 2 | 2 ) β 2 2 -1 ] E[(1 + 2ρP 2 |h 2 | 2 ) β 2 2 ] ≥ 0. (49)
When ρ → 0, we have that lim ρ→0

( ∂V O ∂ρ ) = P1 ln 2 E[|h 1 | 2 ] + P2 ln 2 E[|h 2 | 2 ]. When ρ → ∞, we have that lim ρ→∞ ( ∂V O ∂ρ ) = lim ρ→∞ ( 1 2ρ ln 2 + 1 2ρ ln 2 ) = lim ρ→∞ ( 1 ρ ln 2
), which equals to 0.

APPENDIX VI We have

E 2 c = 1 β 2 log 2 E 1 + ρP 2 |h 2 | 2 1 + ρP 1 |h 1 | 2 β2 = - 1 θ 2 T f B E - θ 2 T f B ln 2 ln 1 + ρP 2 |h 2 | 2 1 + ρP 1 |h 1 | 2 . ( 50
)
When θ 2 → 0, we get an indeterminate form. By applying the L'Hopital's rule one can get

E 2 c = - 1 T f B E - T f B ln 2 ln 1 + ρP 2 |h 2 | 2 1 + ρP 1 |h 1 | 2 = E log 2 1 + ρP 2 |h 2 | 2 1 + ρP 1 |h 1 | 2 . ( 51
)
Hence, we get that

lim θ2→0 E 2 c = E log 2 1 + ρP 2 |h 2 | 2 1 + ρP 1 |h 1 | 2 ,
which equals to E[R 2 ], the ergodic capacity. Proceeding in the same way, one can find

lim θ1→0 E 1 c = E log 2 1 + ρP 1 |h 1 | 2 = E[R 1 ], lim θ1→0 E 1 c = E 1 2 log 2 1 + 2ρP 1 |h 1 | 2 = E[ R 1 ], lim θ2→0 E 2 c = E 1 2 log 2 1 + 2ρP 2 |h 2 | 2 = E[ R 2 ], lim θ1→0 (E 1 c -E 1 c ) = E[R 1 ] -E[ R 1 ], lim θ2→0 (E 2 c -E 2 c ) = E[R 2 ] -E[ R 2 ].
To look further the impact of the transmit SNR ρ on the EC considering delay-unconstrained user:

lim θ1→0 ρ→∞ E 1 c = lim ρ→∞ E log 2 1 + ρP 1 |h 1 | 2 = ∞,
We also have that

lim θ2→0 ρ→∞ E 2 c = lim ρ→∞ E log 2 1 + ρP 2 |h 2 | 2 1 + ρP 1 |h 1 | 2 = E log 2 1 + P 2 |h 2 | 2 P 1 |h 1 | 2
Similarly, we have for OMA Therefore, we have that

lim θ1→0 ρ→∞ E 1 c -E 1 c = lim ρ→∞ E log 2 1 + ρP 1 |h 1 | 2 (1 + 2ρP 1 |h 1 | 2 ) 1 2 = lim ρ→∞ E log 2 ρP 1 |h 1 | 2 2 = ∞. lim θ2→0 ρ→∞ E 2 c -E 2 c = -∞.

APPENDIX VII

Using the Lemma 1, when ρ → 0, we can show that E Similarly,

∂ E c tot ∂ρ = M 2 i=1 ∂ E 1 c ∂ρ + ∂ E 2 c ∂ρ , = M 2 i=1 1 M ln 2 E 2P 1,i |h 1,i | 2 (1 + 2ρP 1,i |h 1,i | 2 ) β 1,i M -1 E (1 + 2ρP 1,i |h 1,i | 2 ) β 1,i M + 1 M ln 2 E 2P 2,i |h 2,i | 2 (1 + 2ρP 2,i |h 2,i | 2 ) β 2,i M -1 E (1 + 2ρP 2,i |h 2,i | 2 )

  is supported by the CY Tech Doctoral Training School, A. Chorti and I. Fijalkow are supported by the INEX project eNiGMA.

  and, B(a, b) is the beta function B(a, b) = Γ(a)Γ(b) Γ(a+b) , with Γ(a) = (a -1)!. Assuming a Rayleigh wireless environment, the channel gains, denoted by x i = |h i | 2 , are exponentially distributed with PDF and cumulative density function (CDF) respectively given by f (x) = e -x , and F

Lemma 6 :

 6 Considering E tot c -E c tot , we prove that: a) When ρ → 0, E tot c -E c tot → 0, and lim ρ→0

2 Fig. 1 .

 21 Fig. 1. Validation of the closed-form expressions in uplink two-user NOMA system.

Fig. 3 .

 3 Fig. 2. E 1 c , E 2 c , and Ec 1 , Ec 2 , versus the transmit SNR ρ

Fig. 4 .Fig. 5 . E 1 c , E 2 c , and Ec 1 and Ec 2 ,

 451212 Fig. 4. E 2 c versus the transmit SNR ρ for different delay requirements.

Fig. 6 .

 6 Fig. 6. E 1 c , E 2 c versus normalized delay β, for different values of ρ.

5 Fig. 7 .

 57 Fig. 7. E 1 c -E 1 c versus ρ, for several values of the normalized delay exponent.

5 Fig. 8 .

 58 Fig. 8. E 2 c -E 2 c versus ρ, for various values of normalized delay exponent.

Fig. 10 .

 10 Fig. 10. V N -V O versus ρ for various values of normalized delay exponent.

Fig. 11 .

 11 Fig. 11. V N -V O versus ρ for various values of normalized delay exponent.

Fig

  Fig. 12. (a): E tot c and Ec tot ; (b): E tot c -Ec tot versus ρ for various pairing settings. M = 4.

Fig. 13 .

 13 Fig. 13. E tot c versus ρ for various pairing settings, normalized to the worst pairing which is (1,2)(3,4)(5,6). M = 6.

Fig. 15 .

 15 Fig. 15. Sum EC and ∆Sum EC for various setting versus ρ. M = 6.

2 1 +

 1 2ρP 2 |h 2 | 2 = ∞.

  2 c and E 2 c start at zero and then increase at the same rate for any user. 2) At high values of the transmit SNR, ρ >> 1, E 1 c increases faster than E 1 c and NOMA becomes more advantageous than OMA, for U 1 . While for U 2 , E 2 At very high values of the transmit SNR, ρ → ∞, the performance gain of NOMA over OMA increases at gradually reducing rate, for U 1 . Albeit, for U 2 , E 2

	c c , although NOMA is outperform-increases faster than E 2
	ing OMA.
	3) c

  On the other side, when ρ → ∞, ρP1,i |h 1,i | 2 ) 2β 1,i M E (1 + 2ρP 1,i |h 1,i | 2 ) E (1 + 2ρP 2,i |h 2,i | 2 ) E ( 1 ρ + 2P 1,i |h 1,i | 2 ) E ( 1 ρ + 2P 2,i |h 2,i | 2 ) -β 1,i M E (P 1,i |h 1,i | 2 ) E |h 1,i | 2 (1 + ρP 1,i |h 1,i | 2 )

	lim ρ→∞	∂E tot c ∂ρ	= lim ρ→∞	M 2 i=1	2 M ln 2ρ					E tot c -E c	tot =	M 2 i=1	1 β 1,i	log 2	 	E (1 + β 1,i M	 
	+	2P 2,i M ln 2ρ 2	E	P1,i|h1,i| 2 P2,i|h2,i| 2 P2,i|h2,i| 2 M (P1,i|h1,i| 2 ) 2 1 + E 1 + |h2,i| 2 P1,i|h1,i| 2 2β 2,i	2β 2,i M -1	= 0.	+			1 β 2,i	log 2	   	E 1 +	ρP2,i|h2,i| 2 1+ρP1,i|h1,i| 2	2β 2,i M β 2,i M	   
												=	M 2 i=1	1 β 1,i	log 2	  ρ	β 1,i M	E ( 1 ρ + P 1,i |h 1,i | 2 ) E ( 1 ρ + 2P 1,i |h 1,i | 2 ) 2β 1,i M M β 1,i	 
												+	1 β 2,i	log 2	    ρ -β 2,i M	P2,i|h2,i| 2 ρ +P1,i|h1,i| 2 1 ρ + 2P 2,i |h 2,i | 2 ) E 1 + E ( 1	2β 2,i M β 2,i M	    . (52)
												Then,
												E tot c -E c	tot	=
															M 2 i=1	1 β 1,i	log 2	 	E ( 1 ρ + P 1,i |h 1,i | 2 )	M 2β 1,i M β 1,i	 
															+	1 β 2,i	log 2	  	E (1 +	P2,i|h2,i| 2 ρ +P1,i|h1,i| 2 ) 1	2β 2,i M β 2,i M	   .
															(53)
															lim ρ→∞	(E tot c -E c	tot	) =
															M
															2 i=1	1 β 1,i	log 2 2 β 1,i M
															+	1 β 2,i	log 2	   	P2,i|h2,i| 2 P1,i|h1,i| 2 E (2P 2,i |h 2,i | 2 ) E 1 +	2β 2,i M β 2,i M	    ,
												which is a constant with respect to ρ. Furthermore, to analyze lim ρ→0 (	∂(E tot c -Ec ∂ρ	tot )	) and
												lim ρ→∞	(	∂(E tot c -Ec ∂ρ	tot )	), we start with	∂E tot c ∂ρ	and ∂ Ec tot ∂ρ .
															M
												∂E tot c ∂ρ	=	2 i=1	∂E 1,i c ∂ρ	+	∂E 2,i c ∂ρ
												=	M 2 i=1	2P 1,i M ln 2	E (1 + ρP 1,i |h 1,i | 2 )	2β 1,i M -1 2β 1,i M
	E 1,i c → 0 and E 2,i c -E 2,i c → 0. Then E tot c -E c E tot c -E c tot M 2 =	tot	1,i c -→ 0, since	+		2P 2,i M ln 2	E	|h2,i| 2 (1+ρP1,i|h1,i| 2 ) 2 (1 + E (1 + ρP2,i|h2,i| 2 ρP2,i|h2,i| 2 1+ρP1,i|h1,i| 2 ) 2β 2,i M 1+ρP1,i|h1,i| 2 )	2β 2,i M -1	, (54)

i=1 (E 1,i c + E 2,i c -E 1,i c -E 2,i c ), we get lim ρ→0 E tot c -E c tot = 0.

where (.)' a first derivative with respect to ρ. Then,

lim ρ→0 ∂E tot c ∂ρ = M 2 i=1 2P1,i M ln 2 E[|h 1,i | 2 ] + 2P2,i M ln 2 E[|h 2,i | 2 .