NOMA Versus OMA: Scheduling to Minimize the Age of Information

In this work, we consider multiple internet-of-things devices (IoTDs) sensing and reporting the status of a time-varying signal to a central node. We analytically compare the performance of orthogonal multiple access (OMA) and non-orthogonal multiple access (NOMA) scheduling schemes for the expected system age of information (AoI) and throughput for generic channel distribution. Further, an IoTD pairing scheme is proposed that adaptively decides to pair IoTDs to minimize AoI depending on the operating transmission signal-to-noise ratio while only requiring the knowledge of channel distribution. Lastly, the proposed results are verified through numerical evaluation.


I. INTRODUCTION
Information freshness is a critical requirement in the emerging real-time applications of the internet-of-things (IoT) such as smart homes, autonomous driving, etc., where the destination takes the control decisions depending on the input received from the IoT devices (IoTDs) measuring the underlying time-varying signal.A new metric, namely age of information (AoI), has been introduced and widely accepted to characterize the information freshness [1].The instantaneous AoI is measured at the destination as the time elapsed since the time of generation of the most recently received packet, and hence, it depends on the scheduling scheme for the IoTDs.Only one IoTD can be scheduled for transmission at a time instant in the traditional orthogonal multiple access (OMA).However, the recently proposed non-orthogonal multiple access (NOMA) has paved the way for simultaneous transmission by multiple IoTDs at a time instant which could be beneficial for minimizing the AoI.Motivated by this, AoI-aware adaptive OMA/NOMA schemes have been proposed in [2] and [3].AoI-independent OMA and NOMA scheduling schemes based on the instantaneous channel gain have been proposed in [4].Physical layer network coding (PNC) enabled NOMA has been proposed in [5] to minimize AoI.In contrast to these works, [6] has proposed an AoI-independent scheduling scheme for NOMA in the decentralized setting of grant-free non-orthogonal massive access in IoT networks.
The AoI-aware schemes proposed in [2], [3] adaptively select between OMA and NOMA in each time slot, whereas the scheme proposed in [4] requires the knowledge of the instantaneous channel state information (CSI) of all the IoTDs.High computational complexity in [2], [3] and additional signaling overhead in [4] limit the applicability of these schemes for real-time IoT systems.Thus, one of the motivations for this work is the requirement of lightweight and The key contributions of this work are as follows.
1) Through analytical investigation and for a generic channel distribution, we highlight the scenarios wherein OMA outperforms NOMA and vice-versa in terms of AoI and throughput as throughput is the conventional metric to quantify another important system requirement, i.e., reliability.2) A pairing scheme for IoTDs for minimizing the AoI is proposed which adaptively decides to pair IoTDs based on the transmission signal-to-noise ratio (SNR) for a generic channel distribution.Further, the pairing scheme is also applicable to imperfect successive interference cancellation (SIC) and imperfect CSI scenarios.The key differences between our work and the existing literature are as follows.
1) The proposed scheme is AoI-agnostic and selects between the IID OMA and (N)OMA schemes before the system starts and then probabilistically schedules the IoTDs for transmission in each slot.This reduces the computational complexity as compared to the AoI-aware schemes proposed in [2], [3] which select between the OMA and NOMA schemes in each slot based on the Markov decision process framework.2) Further, the proposed scheme only requires the instantaneous CSI of only the scheduled IoTDs in the slot, and hence, reduces the signaling overhead as compared to [4] which requires instantaneous CSI of all the IoTDs in each slot.The proposed scheme uses the traditional SIC decoder which has lower complexity than the PNC-based decoder used in [5].The lower computational and signaling overhead makes the proposed scheme more scalable to a large number of IoTDs as compared to the existing schemes.3) The proposed scheme is centralized as compared to the decentralized setting considered in [6], and hence, can prevent the events of unsuccessful transmissions due to the transmission conflicts of the IoTDs.

II. SYSTEM MODEL
In this work, we consider an uplink IoT scenario.There are K IoTDs, K time-varying signals, and a central node (CN) in the system, as depicted in Fig. 1.Each of the K IoTDs senses and updates the status of the corresponding time-varying signal to the CN when it is scheduled for transmission.The time axis is slotted.The IoTDs transmit their 0018-9545 © 2023 IEEE.Personal use is permitted, but republication/redistribution requires IEEE permission.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
packet to the CN using a single channel, and hence, interference occurs when multiple IoTDs are scheduled for transmission in a time slot.A fresh packet arrives for each IoTD at the beginning of each slot and can be transmitted within the slot if the IoTD is scheduled for transmission.Further, at the end of each time slot, each IoTD discards the packet which could not be transmitted in order to maintain the information freshness.In this work, we focus on two scheduling schemes: 1) Independently and identically distributed (IID) OMA Scheme, wherein, the CN schedules the IoTD i for transmission with probability r i in each slot such that K i=1 r i = 1.IID schemes have been considered in the literature for analytical tractability and ease of implementation [1].
2) IID (N)OMA Scheme, wherein, the CN schedules either an IoTD or a pair of two IoTDs for transmission in each slot.
Let j be the index for an IoTD or a pair of IoTDs such that j ∈ {1, 2, . . ., K} denote the K single IoTD each and pairs of two IoTDs.Let ψ j denote the set of two IoTDs paired together for the index } and B j,m = ψ j \ ψ j,m denote the set of interfering IoTDs for the m th IoTD in ψ j .Further, we use τ j to denote the probability that an IoTD or a pair of two IoTDs indexed by j is scheduled for transmission such that i denote the probability of successful transmission by the IoTD i when scheduled for transmission in the IID OMA scheme.Further, the probability that the IoTD i is scheduled for transmission and its transmission is successful in the IID OMA scheme is denoted by p O i and Similarly, in the IID (N)OMA scheme, p N i→B i denotes the probability of successful transmission by IoTD i when it is scheduled for transmission paired with the IoTDs contained in the set B i .In this work, we restrict the sets B i to singletons ∀i.In addition to the randomness of the wireless channel, the probability of successful transmission of IoTD i in NOMA also depends on its transmission power as well as the transmission powers of the IoTDs contained in B i ∀i.The transmission power control factor of IoTD i is denoted by α i .For the remaining text, the power control factors are absorbed in the probability of successful transmission and are explicitly mentioned only when required.Please refer to Appendix A for the discussion on the optimization of the transmission power control factors.We use ρ = P T σ 2 to denote the transmission SNR where, P T and σ 2 denote the transmission and noise power, respectively.In this work, we consider both throughput and AoI as performance metrics of interest.Throughput is defined as the number of packets successfully received in a slot.The expected system throughput for the IID OMA and IID (N)OMA schemes, denoted by E[S O ] and E[S N ], respectively, are given by The instantaneous AoI of the IoTD i for the IID OMA scheme at the beginning of time slot t, denoted by A similar definition holds for the instantaneous system AoI for the IID (N)OMA scheme, denoted by A i,N (t).Then, the expected system AoI for the IID OMA and (N)OMA schemes, denoted by E[A O ] and E[A N ], respectively, are given by A detailed derivation for (2) is provided in Appendix B of [7].The basic assumptions which underlie the utility of this manuscript are as follows.
2) The delay incurred in the transmission of scheduling decisions from the CN to the IoTD is considered to be nominal for analytical convenience.3) Perfect CSI and SIC are assumed at the CN while decoding the received signals [2], [6].Please note that the CN selects between the IID OMA and IID (N)OMA scheme before the beginning of the system and then exploits it for the entire time horizon.In this context, we next present the analytical results which aid the CN in the selection of a suitable scheduling scheme for both K = 2 as well as K > 2.

III. ANALYTICAL INSIGHTS FOR K = 2
In the IID (N)OMA scheme for K = 2, we consider that both the IoTDs are simultaneously scheduled for transmission and p N i→j denotes the probability of successful transmission of IoTD i when it is paired with IoTD j.We now present the following theorem which provides analytical insights into the performance of IID OMA and IID (N)OMA schemes for K = 2.
Theorem 1: For a system with two IoTDs, i.e., for 2) Please recall that q O i denotes the probability of successful transmission by IoTD i when scheduled for transmission in the IID OMA scheme.Further, p N i→j denotes the probability of successful transmission of IoTD i in the IID (N)OMA scheme when it is paired with the IoTD j.
Theorem 1 does not make any assumption on the distribution of the channel, and hence, given the knowledge of the probability of successful transmission in both IID OMA and IID (N)OMA schemes, the system designer can use Theorem 1 to adaptively choose between IID OMA and IID (N)OMA schemes for each value of transmission SNR.Although the theorem is presented for K = 2, it can be extended to K > 2 when all the K IoTDs simultaneously transmit in each slot in the IID (N)OMA scheme.It can also be observed from Theorem 1 that the optimum operating points may not be the same for both AoI and throughput. Assumption Please recall that ρ denotes the transmission SNR.Further, p N i→j and p N j→i for Rayleigh channel, statistical CSI-based decoding order, perfect SIC, and where, θ = Please recall that α i denotes the transmission power control factor of IoTD i in the IID (N)OMA scheme.We now present the following lemma which extends the Theorem 1 for the Assumption 1.
Lemma 1: Let K = 2, β 1 ≤ β 2 , and the fading channel is Rayleigh distributed.Without loss of generality, consider and hence, p O i , p N 1→2 , and p N 2→1 are given by ( 5) and ( 6), respectively.Then, where, and where The following corollary extends Theorem 1 and Lemma 1 for the asymptotic case of high SNR, i.e., ρ → ∞.
Corollary 1: Considering the same setting as in Lemma 1, in the asymptotic case of high SNR, i.e., when ρ → ∞, Remark 1: Please note that the optimum values for α i which minimizes the expected system AoI are a function of ρ and are non-trivial to obtain in closed form due to the involvement of exponential terms.However, in Corollary 1, we consider that the α i are selected such that the condition λ 2 > λ 1 β 1 α 2 α 1 is satisfied ∀ρ.Thus, the system designer can use the Corollary 1 to determine the constant values for α i such that the IID (N)OMA scheme performs better than IID OMA scheme in high SNR regime.
Let ρ S th (respectively, ρ A th ) be the sufficiently high value of ρ such that IID (N)OMA scheme performs better than the IID OMA scheme in expected system throughput (respectively, AoI) ∀ρ > ρ S th (respectively, ρ A th ).We now present the following proposition which provides the exact value for ρ S th for the case of no power control.Proposition 1: For the special case of no power control, i.e., th , where, ρ S th is given by The exact expressions of ρ S th and ρ A th for the general case when the power control is applied are non-trivial to obtain due to the exponential terms in Γ S and Γ A , respectively.Therefore, the following proposition presents the approximated expressions of ρ S th and ρ A th for the general case.
Proposition 2: If θ > 1 2 , the approximated expression of ρ S th for the general case is given by Similarly, if θ > 1 2 , the approximated expression of ρ A th for the general case is obtained by solving Please note that the approximated values for ρ S th and ρ A th , obtained from ( 11) and ( 12), respectively, are only valid when θ > 1  2 which is in line with the Corollary 1. Further, α i are considered as constant for Proposition 2 as described in the Remark 1 for analytical tractability.

IV. ANALYTICAL INSIGHTS FOR K > 2
In this section, we aim to analytically investigate the performance of IID OMA and (N)OMA schemes for the generic case of K > 2. The IID OMA scheme for K > 2 is the same as that for K = 2, i.e., the CN schedules the IoTD i with probability r i .Analogous to K = 2, all the IoTDs can be simultaneously scheduled for transmission in the IID (N)OMA scheme for K > 2 for which the performance analysis can be obtained by extending the Theorem 1.However, the pairing of multiple IoTDs complicates the decoding process at the receiver and increases the likelihood of unsuccessful reception, and hence, following [3], we only consider the pairing of two IoTDs.Therefore, in the IID (N)OMA scheme, we consider that the CN either selects a cluster of two IoTDs, which we hereafter refer to as a super IoTD, or a single IoTD which has not been paired with any other IoTD.Then, the following corollary presents the analytical insight for K > 2 with one super IoTD pair in the IID (N)OMA scheme.
Corollary 2: Consider K IoTDs which have been arranged in the descending order of the channel gains, i.e., E[|h In the IID (N)OMA scheme, we consider that IoTDs u and v are paired as a super IoTD such that and IID (N)OMA scheme outperforms IID OMA scheme at high SNR if λ v > λuβuαv αu for the special case of Rayleigh fading channel.Based on the Corollary 2, we present the adaptive IoTD pairing scheme in Algorithm 1 which only pairs the IoTDs if the expected system AoI achieved after optimizing over transmission power control factors in IID (N)OMA scheme is less than the expected system AoI achieved in IID OMA scheme.The adaptive IoTD pairing can be achieved by constructing a conflict graph and obtaining the maximum weighted independent set (MWIS) [9].Each vertex of the conflict graph is either a single IoTD or a possible pair of 2 IoTDs which is obtained in lines 3-10.Lines 11-13 check for the existence of an edge between the two vertices and lines 14-16 compute the weight for each vertex.Please note that the weight associated with each vertex is negative so that the IoTD pairs obtained from MWIS in line 17 minimize the expected system AoI.Lines 18-20 compute the optimum probability of selecting an IoTD (or a super IoTD) i, denoted by τ * i , and line 21 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Sort the IoTDs i ∈ {1, 2, . . ., |K|} in the descending order of their channel gains, i.e., E[|h i | 2 ] 3: for each i ∈ K do 4: V ← V ∪ {i} 5: end 6: for each i, j ∈ {1, 2, . . ., |K|} such that i < j do 7: E i,j = (V i ∩ V j = ∅) 13: end 14: for each i ∈ {1, 2, . . ., |V|} do 15: computes the minimum expected AoI for the IID (N)OMA scheme, denoted by E[A N ] min .We now present the following corollary which provides analytical insights on the performance of IID OMA and IID (N)OMA schemes for the expected system throughput.Corollary 3: Consider K IoTDs which have been arranged in the descending order of the channel gains, i.e., E[|h IID (N)OMA scheme can achieve higher system throughput than IID OMA scheme if a super IoTD pair is formed by pairing an IoTD i > 1 with IoTD 1 which simultaneously satisfies i = arg max j>1 (max α 1 ,α j (p N 1→j + p N j→1 )) and max α 1 ,α j (p N 1→j + p N j→1 ) > q O 1 .Remark 2: Note that the Algorithm 1 can output more than one super-IoTDs.Further, we assume the perfect CSI and SIC in this work.However, note that Theorem 1, Corollary 2, and Algorithm 1 hold for the imperfect CSI and SIC scenarios also.

V. NUMERICAL RESULTS
In this section, we present some numerical results averaged over 10 6 slots.Two cases are considered -1) Case A: power control is applied, i.e., α i ∈ (0, 1) ∀i such that the α i of the IoTDs paired together sum up to 1, and 2) Case B: power control is not applied, i.e., α i = 1 ∀i.   is less than its exact value, and hence, the approximated value of ρ A th for Case A, given by (12), is less than its exact value.A similar reason holds for the approximated value of ρ S th for Case A, given by (11), being larger than the exact value.The relatively larger and smaller difference between the exact and approximated values of ρ S th and ρ A th for Case A is due to the first and second-order approximation, respectively.In Case B, as E[|h 2 | 2 ] increases, the interference from the weak IoTD to the strong IoTD increases which in turn reduces the probability of successful transmission for the strong IoTD.Hence, ρ S th increases for larger values of E[|h 2 | 2 ] in Case B, as observed from Fig. 3.However, in Case A, the increase in E[|h 2 | 2 ] does not significantly increase the interference of weak IoTD to the strong IoTD due to the power control but instead boosts the probability of successful transmission for the weak IoTD resulting in a reduction of ρ S th .Fig. 4 presents the variation of the E[A] for the IID OMA scheme, given by ( 14), and the adaptive scheme, computed using Algorithm 1, against the different values of ρ for both Case A and B and K = 5 [2].The simulation parameters are Ri = 1 ∀i and h i = d −τ i g i , where, d i = (K + 1) − i, τ = 2, and g i ∼ CN(0, 1) [2].The greedy algorithm (Algorithm 2 in [10]) is employed in line 17 of Algorithm 1 to obtain the MWIS which has the computational complexity of O(|V| 2 ) and |V| ∼ O(K 2 ) [10].We observe from Fig. 4 that the adaptive scheme achieves less E[A] than the IID OMA scheme for both Case A and B. Although Case B is not energy-efficient, it has certain advantages.First, the formation of super IoTD is possible at a lower value of ρ as the probability of successful transmission of weaker IoTD is higher which is reflected in less E[A] for Case B than Case A. Second, Case B eliminates the procedure of determining the optimum power control fraction leading to reduced computational complexity than Case A. Please note that E[A] can be further reduced by AoI-aware policies as proposed in [2] which requires the determination of optimal transmission policy for each time slot leading to high computational complexity, especially for large values of K. Whereas, the adaptive scheme proposed in this work needs to be run only once for the operating value of ρ followed by the probabilistic selection of an IoTD (or, a super IoTD) independently in each time slot.
Let denote the factor of imperfection of SIC in the IID (N)OMA scheme such that 0 ≤ ≤ 1, where, = 0 implies perfect SIC and = 1 implies no SIC [11].Fig. 5 presents the variation of E[A] for the IID OMA scheme and Case A of the adaptive pairing scheme against different values of ρ and .The analytical expression for E[A] for imperfect SIC is given in Appendix C of [7].It can be observed from Fig. 5 that the adaptive IoTD pairing scheme converges to the IID OMA scheme for larger as larger leads to larger imperfection in SIC and in turn less advantage of IID (N)OMA scheme over IID OMA scheme.Thus, the proposed pairing scheme is adaptive to imperfection in SIC.A discussion of imperfect CSI is provided in Appendix C of [7].

VI. PROOF OUTLINES
Due to space constraints, we present an outline of the proofs.Please refer to Appendix B of [7] for the detailed proofs.For Theorem 1, using (2), the expected system AoI for IID OMA and (N)OMA schemes are given by E[A O ] = 1

Figs. 2
and 3 present the variation of the approximated and exact values of ρ A th and ρ S th against different values of E[|h 2 | 2 ], respectively, for K = 2.The simulation parameters are E[|h 1 | 2 ] = 1, Ri = 1 ∀i, α 1 = 0.8 and α 2 = 0.2 for Case A and α i = 1 ∀i for Case B. The assumption for constant values of α 1 and α 2 for Case A is in line with Proposition 2 and Remark 1.The expected AoI computed by using the second-order Taylor series approximation of the exponential terms

Fig. 2 .Fig. 3 .
Fig. 2. Variation of exact and approximated values of ρ A th against different values of E[|h 2 | 2 ] for Case A and K = 2.

Fig. 4 .
Fig. 4. Variation of E[A] against different values of ρ for OMA scheme and Case A and B of the Adaptive scheme for K = 5.

Fig. 5 .
Fig. 5. Variation of E[A] against different values of ρ and for IID OMA scheme and Case A of the Adaptive scheme for K = 5.
Manuscript received 27 March 2023; revised 26 May 2023 and 22 July 2023; accepted 7 August 2023.Date of publication 11 August 2023; date of current version 17 January 2024.This work was supported by SERB Grant on Versatile Scheduling Policies with Provable Guarantees for Latency Sensitive Systems.
1: Let h i and E[|h i | 2 ] denote the fading channel coefficient and expected channel gain between IoTD i and CN.Then, |h i | follows independent and non-identical Rayleigh distribution, and consequently, |h i | 2 follows exponential distribution with rate parameter λ i such that E[|h i | 2 ] = 1 λ i .Let Ri denote the minimum data rate required for successful reception of the packet of IoTD i and β i = 2 Ri − 1.Then, p O i for the Rayleigh channel is given by