Version AoI Optimization under Power and General Distortion Constraints in Uplink NOMA

1 V ersion AoI Optimization under Po wer and General Distortion Constraints in Uplink NOMA Gangadhar Kare vv anav ar , Rajshekhar V . Bhat, and Nikolaos Pappas, Senior Member , IEEE Abstract —The V ersion Age of Information (V AoI) quantifies information freshness by measuring the number of versions the r eceiver lags behind. This paper studies V AoI minimization in an M -user uplink non-orthogonal multiple access (NOMA) system where users maintain single-packet buffers and trans- missions are constrained by a verage power and information- quality constraints, modeled by a general distortion function. A fundamental trade-off arises: transmitting more bits per update impro ves information quality but increases power consumption, reducing transmission opportunities and increasing V AoI, while transmitting fewer bits has the opposite effect. W e formulate a weighted-sum V AoI minimization problem as a con vex opti- mization problem. However , users’ power allocations are coupled through multiple-access capacity constraints per channel state, leading to exponential complexity . T o address this, we develop a V AoI-agnostic stationary randomized policy that jointly optimizes scheduling, bit allocation, and power control without tracking instantaneous V AoI, and achieves a pro vable 2-approximation to the globally optimal av erage V AoI. Leveraging Lagrangian dual decomposition, we derive closed-form expr essions f or the schedul- ing probabilities and power allocations, and efficiently determine the optimal successive interference cancellation decoding order , av oiding exhaustive search Numerical results show that NOMA significantly outperforms time-division multiple access (TDMA): at high power budgets, NOMA achieves near -zero V AoI, whereas TDMA saturates at a non-zero value, consistent with the anal- ysis. The proposed general distortion framework accommodates diverse bit-priority structures by assigning unequal importance to different bits within an update. Index T erms —V ersion age of inf ormation, non-orthogonal multiple access, successive interference cancellation, stationary randomized policy , information quality . I . I N T RO D U C T I O N Real-time monitoring and control applications, such as vehicular networks and industrial Internet of Things (IoT), require not only timely updates but also sufficient information quality for ef fective decision-making. In such systems, a central node collects information from multiple distributed sources. Traditional metrics such as throughput and latency do not capture information freshness, which is quantified by the age of information (AoI), defined as the time elapsed since the generation of the freshest recei ved update at the destination [1]–[4]. While AoI captures timeliness, it treats Gangadhar Karevv anav ar and Rajshekhar V . Bhat are with the Indian Institute of T echnology Dharwad, Dharwad, India (e-mail: 212021007@iitdh.ac.in; rajshekhar.bhat@iitdh.ac.in). Nikolaos Pappas is with the Dept. of Computer and Information Science, Link ¨ oping University , Link ¨ oping, Sweden (e-mail: nikolaos.pappas@liu.se). The work of Rajshekhar V . Bhat was supported by the TTDF , DoT , Government of India, under Proposal ID TTDF/6G/492 through TCOE India. The work of Nikolaos Pappas was supported by ELLIIT and the European Union (6G-LEADER, 101192080). all successfully received updates equally , irrespecti ve of their completeness or informational significance. V ersion age of information (V AoI) refines this notion by measuring version staleness rather than time elapsed, i.e., how many versions the receiv er lags behind [5]–[10]. Howe ver , neither AoI nor V AoI accounts for update quality , motiv ating the incorporation of distortion metrics. In practice, not all bits within an update carry equal impor- tance. For instance, in sensor networks, initial bits may encode critical metadata (e.g., anomaly flags), while later bits provide finer-grained measurements. Similarly , in video streaming, header information and the essential coarse components of intra-coded (I-) frames are typically prioritized over fine- grained refinement data. T o capture such heterogeneous bit priorities, we model information quality using a general dis- tortion metric δ ( ρ ) that depends on the number of successfully transmitted bits ρ ∈ { 0 , 1 , . . . , r max } , without restricting δ ( · ) to be conv ex or monotonically decreasing, as is commonly assumed [11], [12]. This framew ork captures scenarios where initial bits con ve y task-critical information, and later bits provide refinement, with δ ( · ) assigning higher penalties to incomplete transmission of high-priority bits. In this work, we aim to minimize the long-term average V AoI in M -user uplink multiple access channel (MA C) subject to av erage power and distortion constraints. A fundamental trade-off arises between timeliness and information quality: transmitting fe wer bits per update conserves po wer and enables more frequent updates, reducing V AoI at the cost of higher distortion, while transmitting more bits improves information quality but consumes more po wer and reduces transmission opportunities, potentially increasing V AoI. This trade-off be- comes more sev ere in uplink non-orthogonal multiple access (NOMA) systems, where users’ power allocations are coupled through interference, and the decoding strategy , such as the successiv e interference cancellation (SIC), directly impacts both achiev able rates and required transmit powers. W e revie w related work to position our contributions. The AoI framew ork, studied in [1]–[4], characterized the trade-off between update frequency and information freshness and was later extended to V AoI to capture version-based staleness [5]. These concepts ha ve been e xtended to multiple sources, energy harvesting, and advanced multiple access [13], [14]. Minimiz- ing AoI and its variants under practical constraints, such as power , distortion, and energy harvesting, yields challenging joint optimization problems. Approximation algorithms and stationary randomized policies hav e emerged as effecti ve low-comple xity solutions with prov able performance guaran- tees [15]–[21]. In this work, we develop a low-complexity 2 stationary randomized policy and derive a performance bound. T o support simultaneous uplink transmissions in the MAC, we adopt NOMA, which enables users to share resources via superposition coding and receiver -side SIC. Several recent works have inv estigated NOMA for AoI optimization. In [22], a time-di vision multiple access (TDMA) network is augmented with two-user cognitive-radio NOMA, where the secondary user is decoded first with a limited rate to ensure primary- user priority , showing that NOMA improves AoI by increasing transmission opportunities and reducing access delay . The up- link coexistence of mission-critical (MC) and enhanced mobile broadband (eMBB) services has been studied by comparing puncturing and NOMA: puncturing yields fresher MC updates, whereas NOMA can increase eMBB rates by up to five times at the cost of some freshness loss [23], [24]. NOMA-assisted grant-free uplink schemes with randomly arri ving packets hav e also been sho wn to reduce AoI compared to OMA- based approaches [24] significantly . In contrast to some of these works, which focus on two-user systems, we consider a general M -user setting and jointly optimize transmitting bits and transmit powers under av erage po wer and distortion constraints, as identified as a future direction in [22]. Sev eral works study AoI optimization with distortion, high- lighting the trade-of f between timeliness and update quality . AoI minimization under constant and age-dependent distortion constraints shows that longer processing improves quality at the cost of increased AoI [25]. Online scheduling formulations balancing AoI, energy , and quality hav e been proposed, with greedy algorithms achieving provable competitiveness [12]. Related metrics, such as the age of incorrect estimates and age of incorrect information (AoII), jointly capture freshness and accuracy , leading to threshold-type optimal policies [26], [27]. Joint sampling and compression strategies hav e been shown to achiev e asymptotically optimal age–distortion trade- offs [28]. In discrete-time settings, optimal age–distortion poli- cies have been characterized via dynamic programming [29], and deep reinforcement learning has been applied to semantic- empowered NOMA systems for AoII minimization under power and bandwidth constraints [30], [31]. Collectively , these works underscore the importance of distortion-aware policies for ensuring timely , high-quality updates. Our work is most closely related to [6], [11], [17]. In [17], AoI minimization in fading uplink MA Cs under TDMA and NOMA is studied. While both works consider NOMA-based uplink MA Cs, our approach differs in key aspects. Fir st , we optimize V AoI [5], [7], which measures version staleness and increases only upon ne w arri vals, capturing both timeliness and content staleness and inducing a different Markov structure than AoI. Second , we incorporate general distortion metric δ ( ρ ) enabling flexible bit-priority modeling (e.g., critical meta- data in initial bits [2], [32] and refinement in later bits), leading to joint optimization of whether to transmit and how many bits to transmit per user—a trade-off absent in [17]. Compared to [11], which studies AoI minimization with power–distortion constraints under TDMA-only MA C, we consider NOMA, introducing 2 M − 1 MAC constraints for all feasible user rate vectors; moreover , while [11] assumes a fixed distortion function, our general δ ( ρ ) captures a broad class of bit-priority structures. Relativ e to [6], which studies V AoI minimization in a broadcast channel, the uplink MAC considered here exhibits different interference coupling, requiring coordinated power allocation to satisfy MAC constraints across rate vec- tors, whereas [6] assumes base-station-controlled transmis- sions with fixed bit allocation. Finally , distortion constraints require ne w analytical tools for jointly optimizing scheduling, bit allocation, and power control. Sev eral other works address related but distinct problems. W orks such as [12], [26], [27] incorporate information quality via AoII or estimation-error metrics but do not study the joint optimization of V AoI and distortion under NOMA. Recent studies [22]–[24] analyze AoI in NOMA systems but are restricted to two-user settings and do not incorporate distortion constraints. W orks on AoI–distortion trade-of fs [25], [28], [29] focus on single-user or TDMA systems, thereby av oiding the complexity of NOMA-MA C constraints. Although po wer allocation in MA C is well studied in the capacity litera- ture [33], extending these techniques to V AoI optimization under distortion constraints requires new formulations and methods. The main contributions of this work are as follows: • W e formulate V AoI minimization in uplink NOMA sys- tems under av erage power and distortion constraints as a con vex optimization problem. Unlike prior NOMA-based age optimization works [17], [22], we explicitly incorporate information-quality constraints that capture the trade-off be- tween update frequency and distortion. A flexible distortion metric captures div erse bit-priority structures, from uniform importance to metadata-critical updates. • W e propose a V AoI-agnostic, stationary , randomized policy (V A-SRP) that operates without tracking instantaneous V AoI and achiev es a prov able 2-approximation to the globally optimal V AoI across all policies. Le veraging the structure of the MA C capacity region, we employ Lagrangian dual decomposition and deri ve closed-form expressions for the scheduling probabilities and power allocations via SIC, ef fi- ciently handling the 2 M − 1 MAC constraints. The optimal SIC decoding order is obtained via sorting in O ( M log M ) time, av oiding exhausti ve O ( M !) search. • Numerical results demonstrate substantial V AoI gains of NOMA ov er TDMA. At high power budgets, NOMA achiev es near-zero V AoI by enabling simultaneous multi- user transmissions with status update delivery probabilities approaching unity . In contrast, TDMA saturates at a non- zero V AoI due to its single-user-per -slot limitation. W e compare V A-SRP with heuristic policies, demonstrating per- formance and computational advantages, and illustrate the impact of distortion structures, including uniform (linear), metadata-critical (step), and refinement-oriented (con vex) models. I I . S Y S T E M M O D E L W e consider a system with M sources transmitting status updates to a base station (BS) in slotted time. W e describe the packet arriv al, channel, distortion, power , and V AoI evolution models, and formulate a long-term average V AoI minimization problem subject to av erage power and distortion constraints. 3 A. P acket Arrival Model Each user i ∈ M ≜ { 1 , 2 , . . . , M } receiv es a new version update packet at the beginning of each slot with probability λ i . Each user maintains a single-pack et queue that stores only the most recent packet, rendering previous packets obsolete. Let A i ( t ) be an indicator variable such that A i ( t ) = 1 if the i th user recei ves a new packet at time t , and A i ( t ) = 0 otherwise. The sequence { A i ( t ) } t ≥ 1 is an independent and identically distributed (i.i.d.) Bernoulli process with P ( A i ( t ) = 1) = λ i , for all i ∈ M and t ∈ { 1 , 2 , . . . } . Each update packet contains r max bits. In slot t , User i transmits ρ i ( t ) ∈ { 0 , 1 , . . . , r max } bits, where ρ i ( t ) > 0 corresponds to a valid update. The quality of the received update is captured by the distortion metric δ ( ρ i ( t )) defined in Section II-C and depends on the number of successfully transmitted bits. For instance, when initial bits encode crit- ical information, and later bits provide refinement, δ ( · ) can reflect this priority structure. If a packet is not transmitted, is considered for transmission in the next slot unless replaced by a new arriv al. If ρ i ( t ) > 0 bits are transmitted in a slot, the remaining r max − ρ i ( t ) bits are discarded at the end of the slot, and the queue is deemed empty . W e assume unsent bits are dropped rather than buf fered to avoid state-space ex- plosion and to reflect grant-free NOMA and ultra-reliable lo w latency communication (URLLC) operation, finite-blocklength coding, and AoI/V AoI packet management [1], [5], [34]–[36]. This yields a tractable V AoI model that depends only on current decisions and admits stationary randomized policies with prov able guarantees, while ignoring cross-slot distortion accumulation, which is left for future work. B. Channel Model W e model the wireless channel as block fading, remaining constant within a slot and changing independently across slots. Let H i ( t ) denote the channel power gain between User i and the BS in slot t , where { H i ( t ) } t ≥ 1 are i.i.d. random v ariables taking values in a finite set H i ⊂ R + . The channel realization h i ( t ) is perfectly estimated at the beginning of each slot using pilot signals. The channel state vector h ( t ) ≜ ( h 1 ( t ) , . . . , h M ( t )) is known at the start of each slot. W e denote H as the the set of all channel states. C. Distortion Model Let Q i ( t ) ∈ { 0 , 1 } denote whether User i has a pending packet at the beginning of slot t , i.e., Q i ( t ) = 1 if User i has a packet in the queue, and Q i ( t ) = 0 otherwise. Per-slot distortion is defined as d i ( t ) =    δ ( ρ i ( t )) I { ρ i ( t ) > 0 } , if Q i ( t ) = 1 , 0 , if Q i ( t ) = 0 , where δ : [0 , r max ] → R + is a function of transmitted bits ρ i ( t ) that captures bit importance, e.g., higher penalties when high-priority bits are missing and diminishing returns for extra bits. The distortion is zero when ρ i ( t ) = 0 or when Q i ( t ) = 0 , corresponding to no transmission or empty queue. An alternativ e model could incur distortion whenever a packet is pending b ut not transmitted, but this would cause V AoI and distortion to e volv e similarly , coupling timeliness and quality . In our formulation, distortion captures trans- mission quality , while V AoI captures staleness , enabling in- dependent control of frequent low-quality versus infrequent high-quality updates. Models where distortion accrues during deferral are better suited to settings with intrinsic information decay and are beyond the scope of this work. D. Rate-P ower Relationship In an uplink MA C using the NOMA scheme, users trans- mit simultaneously to the BS. In slot t , User i ∈ M transmits ρ i ( t ) ∈ { 0 , 1 , . . . , r max } bits using power f i ( t ) under channel power gain h i ( t ) . All users encode and transmit ( ρ 1 ( t ) , . . . , ρ M ( t )) bits simultaneously using powers ( f 1 ( t ) , . . . , f M ( t )) ; if ρ i ( t ) = 0 , then f i ( t ) = 0 . The BS receiv es a superimposed signal comprising the simultaneous transmissions from all users and decodes all users’ transmis- sions. For successful decoding under channel state h ( t ) , the rate–power pairs ( ρ i ( t ) , f i ( t )) i ∈M must satisfy X i ∈S ρ i ( t ) ≤ g X i ∈S f i ( t ) h i ( t ) ! , ∀S ⊆ M , ∀ t. (1) Here g ( · ) is a concav e, non-decreasing function with g (0) = 0 . The MAC capacity region induces 2 M − 1 constraints when users transmit simultaneously using the NOMA scheme. E. P erformance Metric and Pr oblem formulation Let z i ( t ) denote the version index of the packet in User i ’ s queue at the be ginning of slot t , gi ven by z i ( t ) = P t τ =1 A i ( τ ) , which increments only upon new arriv als. Let y i ( t ) denote the version of the most recently receiv ed packet from User i at the BS, which updates only when ρ i ( t ) > 0 . The instantaneous V AoI is defined as ∆ i ( t ) = z i ( t ) − y i ( t ) , measured at the end of each time slot, and ev olves as follows: ∆ i ( t ) =  ∆ i ( t − 1) + A i ( t )  1 − I { ρ i ( t ) > 0 }  , (2) for all i ∈ M and t ∈ { 1 , 2 , . . . } , with ∆ i (0) = 0 , ∀ i . The long-term expected average V AoI is lim T →∞ 1 T P T t =1 P M i =1 w i E [∆ i ( t )] , where ( w 1 , w 2 , . . . , w M ) are user weights. Our goal is to design a scheduling policy that jointly selects transmitting bits and power per user in each slot to minimize the long-term expected average V AoI subject to average po wer and distortion constraints. Specifically , we solve the following optimization problem. V opt = min ϕ lim T →∞ 1 T T X t =1 M X i =1 w i E [∆ i ( t )] , (3a) subject to lim T →∞ 1 T T X t =1 E  f i ( t )  ≤ ¯ P i , ∀ i ∈ M , (3b) lim T →∞ 1 T T X t =1 E  d i ( t )  ≤ ¯ D i , ∀ i ∈ M , (3c) 4 (1) , (2) . Here, ϕ denotes the scheduling policy . The parameters ¯ P i and ¯ D i denote the bounds on the av erage transmit power and the av erage distortion, respectiv ely . All expectations are taken ov er the randomness in the system under policy ϕ . W e also consider a TDMA benchmark (which allows at most one user to transmit in each slot), obtained by solving (3) with the following additional constraint: M X i =1 I { ρ i ( t ) > 0 } ≤ 1 , ∀ t ∈ { 1 , 2 , . . . , T } . I I I . O P T I M I Z A T I O N A NA LY S I S W e study V AoI-agnostic stationary randomized policies (V A-SRPs) to solve (3), deri ve the optimal policy , and obtain a performance bound relativ e to V opt . A. V AoI-Agnostic Stationary Randomized P olices (V A-SRP) W e define the class of V A-SRPs, reformulate (3) under this class, and solve the resulting problem. 1) Pr oblem Reformulation: Define R ≜ { ( ρ 1 , ρ 2 , . . . , ρ M ) | ρ i ∈ { 0 , 1 , . . . , r max } , ∀ i ∈ M} = { 0 , 1 , . . . , r max } M . Policy (V A-SRP): In each slot, when channel state is h , the policy selects a rate vector ρ ≜ ( ρ 1 , . . . , ρ M ) ∈ R with pr obability µ ( h , ρ ) , where ρ i ∈ { 0 , 1 , . . . , r max } specifies the number of bits transmitted by User i . The corresponding transmit-powers vector f ( h , ρ ) ≜ ( f i ( h , ρ )) i ∈M satisfies the MA C constraints. This policy does not require kno wledge of instantaneous V AoI, simplifying implementation. W e next deriv e expressions for the long-term av erage V AoI, power , and distortion. a) Long-term Expected A verag e V AoI under V A-SRP: Under V A-SRP , the conditional probability of a successful status update deli very from User i giv en channel state h is P ρ ∈ R µ ( h , ρ ) I { ρ i > 0 } . By (2), ρ i > 0 implies a successful update and V AoI reset. Thus, the per-slot success probability for User i is p i ( µ ) = P ρ ∈ R E h [ µ ( h , ρ ) I { ρ i > 0 } ] , where µ ≜ ( µ ( h , ρ )) h , ρ . Under V A-SRP , the V AoI process forms a Markov chain since the policy depends only on the channel state and arriv als and channels are i.i.d. Although [6] considers a broadcast channel, the per-user V AoI e v olution under an SRP has the same Marko v structure (see Fig. 3 in [6]). Consequently , the objectiv e function in (3) under V A-SRP can be obtained from the stationary distribution of this Markov chain. Applying [6, Theorem 1], the long-term average V AoI for User i is lim T →∞ 1 T T X t =1 E [∆ i ( t )] = λ i  1 p i ( µ ) − 1  , ∀ i ∈ M . (4) b) Long-term Expected A verage T ransmit P ower under V A-SRP: Let P i ( µ , F ) denote the expected av erage trans- mit po wer of User i , which under NOMA depends on all users’ transmission decisions. It is giv en by P i ( µ , F ) = P ρ ∈ R E h [ µ ( h , ρ ) f i ( h , ρ )] , where f i ( h , ρ ) is the power re- quired to transmit ρ i bits under joint transmission ρ and channel state h . Here, F ≜ ( f i ( h , ρ )) i, h , ρ denotes the collection of power allocations. The average power constraints under V A-SRP are P i ( µ , F ) ≤ ¯ P i , for all i ∈ M . c) Long-term Expected A verage Distortion under V A- SRP: By the law of total expectation, the per-slot ex- pected distortion for User i is E [ d i ( t )] = E [ d i ( t ) | Q i ( t ) = 1] P ( Q i ( t ) = 1) , since d i ( t ) = 0 when Q i ( t ) = 0 . Conditioned on a pending packet, the expected per-slot distortion under V A-SRP is E [ d i ( t ) | Q i ( t ) = 1] = P ρ ∈ R E h [ µ ( h , ρ ) δ ( ρ i ) I { ρ i > 0 } ] . Thus, E [ d i ( t )] = P ( Q i ( t ) = 1) P ρ ∈ R E h [ µ ( h , ρ ) δ ( ρ i ) I { ρ i > 0 } ] . The steady-state probability of a pending packet for User i follows from [6, Theorem 1] as P ( Q i ( t ) = 1) = λ i λ i (1 − p i ( µ )) + p i ( µ ) , ∀ i ∈ M , (5) where λ i is the packet arriv al probability and p i ( µ ) is the status update deliv ery probability . This occupancy probability depends only on arriv als and transmissions, and is independent of power and distortion constraints. Substituting yields the long-term av erage distortion for User i D i ( µ ) = λ i P ρ ∈ R E h  µ ( h , ρ ) δ ( ρ i ) I { ρ i > 0 }  λ i (1 − p i ( µ )) + p i ( µ ) , ∀ i ∈ M . Thus, the distortion constraints under V A-SRP , D i ( µ ) ≤ ¯ D i , are equiv alently written as D ′ i ( µ ) ≜ λ i X ρ ∈ R E h  µ ( h , ρ ) δ ( ρ i ) I { ρ i > 0 }  ≤ ¯ D ′ i ( µ ) ≜ ¯ D i  λ i − λ i p i ( µ ) + p i ( µ )  , ∀ i ∈ M . d) Reformulation of (3) under V A-SRP: Using (4) and the power and distortion constraints, (3) is reformulated un- der V A-SRPs. T o expose its con ve x structure, we introduce auxiliary variables π i ( h , ρ ) ≜ µ ( h , ρ ) · f i ( h , ρ ) , (6) for all i ∈ M , h ∈ H , ρ ∈ R , representing the e xpected power contrib ution of User i under channel state h and rate vector ρ . The reformulated problem is V SRP = min µ , π M X i =1 w i λ i  1 p i ( µ ) − 1  , (7a) subject to X ρ ∈ R E h [ π i ( h , ρ )] ≤ ¯ P i , ∀ i ∈ M , (7b) D ′ i ( µ ) ≤ ¯ D ′ i ( µ ) , ∀ i ∈ M , (7c) µ ( h , ρ ) · g − 1 X i ∈S ρ i ! ≤ X i ∈S π i ( h , ρ ) h i , ∀S ⊆ M , h ∈ H , ρ ∈ R , (7d) 0 ≤ µ ( h , ρ ) ≤ 1 , ∀ h ∈ H , ρ ∈ R , (7e) X ρ ∈ R µ ( h , ρ ) = 1 , ∀ h ∈ H , (7f) where π ≜ ( π i ( h , ρ )) i, h , ρ . The MA C constraint (7d) fol- lows from (1) since g ( · ) is increasing: g − 1  P i ∈S ρ i  ≤ P i ∈S f i ( h , ρ ) h i , which, after multiplying by µ ( h , ρ ) ≥ 0 and using (6), yields (7d). 5 Giv en the optimal ( µ ∗ , π ∗ ) for (7), we have f ∗ i ( h , ρ ) =    π i ( h , ρ ) µ ( h , ρ ) , µ ( h , ρ ) > 0 , 0 , µ ( h , ρ ) = 0 , (8) for all i ∈ M and ( h , ρ ) . Theorem 1. (7) is a con vex optimization pr oblem. Pr oof. See Appendix A. Despite con vexity , solving (7) directly using standard solvers is computationally challenging due to the exponential number of MA C constraints. For each ( h , ρ ) pair , there are 2 M − 1 constraints corresponding to all non-empty subsets S ⊆ M , yielding a total of | H | | R | (2 M − 1) constraints. This exponential scaling renders direct methods infeasible even for moderate M . T o ov ercome this, we employ a Lagrangian dual decomposition that exploits the problem structure and significantly reduces complexity . 2) Lagr angian Dual F ormulation: Due to the con vexity of (7), we solve the problem via Lagrangian duality . Slater’ s condition holds provided the problem is feasible. In particular , there exists ε > 0 such that a randomized policy with strictly positiv e probabilities µ ( h , ρ ) > 0 for all ( h , ρ ) and suffi- ciently small π i ( h , ρ ) = ε strictly satisfies the power , distor- tion, and MAC constraints, while maintaining 0 < µ ( h , ρ ) < 1 and P ρ µ ( h , ρ ) = 1 . Hence, Slater’ s condition is satisfied and strong duality holds [37]. T o enable decomposition, we reformulate the problem so that the Lagrangian is linear in µ . W e introduce auxiliary variables η ≜ ( η i ) i ∈M to linearize the objectiv e by replacing 1 /p i ( µ ) , with linking constraints p i ( µ ) ≥ 1 /η i . The resulting equiv alent formulation is V SRP = min µ , π , η M X i =1 w i λ i ( η i − 1) , (9a) subject to p i ( µ ) ≥ 1 η i , ∀ i ∈ M , (9b) D ′ i ( µ ) ≤ ¯ D ′ i ( η i ) , ∀ i ∈ M , (9c) (7b) , (7d) , (7e) , (7f) , where ¯ D ′ i ( η i ) = ¯ D i ( λ i − λ i /η i + 1 /η i ) , ∀ i ∈ M . Since w i λ i > 0 , constraint (9b) binds at optimality , yielding η i = 1 /p i ( µ ) and ensuring equiv alence between (9) and (7). W e dualize the power , distortion, and linking constraints us- ing Lagrange multipliers β ≜ ( β ) i ∈M ≥ 0 , α ≜ ( α ) i ∈M ≥ 0 , and ν ≜ ( ν ) i ∈M ≥ 0 . By conv exity and Slater’ s condition, strong duality holds, and the primal solutions ( µ ∗ , π ∗ , η ∗ ) are optimal if and only if they minimizes the Lagrangian for op- timal multipliers ( β ∗ , α ∗ , ν ∗ ) . The corresponding Lagrangian is giv en by L ( β , α , ν , µ , π , η ) = M X i =1 w i λ i ( η i − 1) − ν i  p i ( µ ) − 1 η i  + β i  X ρ ∈ R E h [ π i ( h , ρ )] − ¯ P i  + α i  D ′ i ( µ ) − ¯ D ′ i ( η i )  ! . (10) The dual function is defined as z ( β , α , ν ) = min µ , π , η L ( β , α , ν , µ , π , η ) , subject to (7d) , (7e) , (7f) . The dual problem is ( β ∗ , α ∗ , ν ∗ ) = arg max β ≥ 0 , α ≥ 0 , ν ≥ 0 z ( β , α , ν ) . (11) Once dualized, the po wer , distortion, and linking constraints are enforced implicitly through the optimal multipliers. Solv- ing the dual problem (11) yields the optimal multipliers ( β ∗ , α ∗ , ν ∗ ) and the optimal dual objectiv e z ( β ∗ , α ∗ , ν ∗ ) . By strong duality , V SRP = z ( β ∗ , α ∗ , ν ∗ ) , and any ( µ ∗ , π ∗ , η ∗ ) minimizing the Lagrangian for optimal multipliers are the primal optimal solutions. For fixed ( β , α , ν ) , the dual function is minimized by first optimizing π for fixed ( µ , η ) , followed by optimization over ( µ , η ) . a) P ower Allocation Optimization for F ixed µ and η : For fixed ( µ , η ) , we optimize over the power variables F , with π recovered via (6). The optimal F ∗ solves minimize F M X i =1 X h ∈ H P ( h ) X ρ ∈ R β i µ ( h , ρ ) f i ( h , ρ ) , subject to (7d) , where P ( h ) denotes the probability that the channel state is h . Since the objectiv e is separable across channel states, the problem decomposes into independent subproblems for each ( h , ρ ) , given by minimize f ( h , ρ ) M X i =1 β i f i ( h , ρ ) , (12) subject to g − 1 X i ∈S ρ i ! ≤ X i ∈S f i ( h , ρ ) h i , ∀S ⊆ M . Problem (12) is independent of µ and η ; hence, f ∗ ( h , ρ ) is identical for all feasible ( µ , η ) , and states with µ ( h , ρ ) = 0 are immaterial, since such states do not contribute to either the scheduling probability or the average power constraints. Problem (12) minimizes a weighted sum power over the MA C capacity region to achiev e rate vector ρ . Since the MA C region is a polymatroid and we minimizing a linear objectiv e (weighted sum power) o ver this region, the optimum is attained at a vertex, corresponding to a specific SIC decoding order . NOMA-SIC P ower Allocation: Under SIC, the BS de- codes users sequentially according to a decoding order θ ≜ ( θ 1 , . . . , θ M ) , where θ j is index of the user decoded at stage j . The required transmit powers for users, which satisfies the MA C constraints, gi ven by f θ j ( h , ρ , θ ) = g − 1 ( ρ θ j ) h θ j M Y k = j +1  1 + g − 1 ( ρ θ k )  , ∀ j ∈ M . (13) with the conv ention that g − 1 (0) = 0 and the empty product equals 1. 6 T o solve (12) and eliminate the MA C constraints, we consider the transmit po wer under NOMA–SIC and determine the decoding order that minimizes the weighted sum po wer θ ∗ ( h , ρ , β ) = arg min θ M X j =1 β θ j f θ j ( h , ρ , θ ) . (14) A brute-force search ov er all M ! decoding orders is pro- hibitiv e; howe ver , the optimal order can be found ef ficiently . Lemma 2. Given Lagrang e multipliers β , transmission rate vector ρ , and channel state h , the optimal decoding order θ ∗ ( h , ρ , β ) = ( θ ∗ 1 , . . . , θ ∗ M ) that solves (14) satisfies h θ ∗ 1 I { ρ θ ∗ 1 > 0 } β θ ∗ 1 ≥ h θ ∗ 2 I { ρ θ ∗ 2 > 0 } β θ ∗ 2 ≥ · · · ≥ h θ ∗ M I { ρ θ ∗ M > 0 } β θ ∗ M . Pr oof. See Appendix B. Remark 1. By Lemma 2, the optimal decoding order is ob- tained by sorting the ratios h i I { ρ i > 0 } /β i in non-incr easing or der . This sorting operation has computational complexity O ( M log M ) for each ( h , ρ ) pair , which is significantly lower than the O ( M !) complexity of brute-force sear ch. Let f ∗ ( h , ρ , β ) denote the optimal NOMA-SIC power allocation, where the optimal decoding order depends on β . Since (12) is independent of ( µ , η ) , the optimal decoding order and power allocation are identical for all feasible ( µ , η ) . Giv en any µ and β , we obtain the corresponding π ∗ ( µ , β ) via π ∗ i ( h , ρ ) = µ ( h , ρ ) · f ∗ i ( h , ρ , β ) , for all i ∈ M and ( h , ρ ) . b) Sc heduling Pr obability and Auxiliary V ariable Op- timization: Giv en the optimal power allocations π ∗ ( µ , β ) , we optimize ov er µ and η . Substituting π ∗ i ( h , ρ ) = µ ( h , ρ ) f ∗ i ( h , ρ , β ) into (10), and using the definitions of p i ( µ ) , D ′ i ( µ ) , and ¯ D ′ i ( η i ) yields L ( β , α , ν , µ , π ∗ ( µ ) , η ) = M X i =1  w i λ i η i + ν i η i − α i ¯ D i (1 − λ i ) η i  + X h ∈ H X ρ ∈ R P ( h ) µ ( h , ρ ) " M X i =1  β i f ∗ i ( h , ρ , β ) + α i λ i δ ( ρ i ) I { ρ i > 0 } − ν i I { ρ i > 0 }  # + C, (15) where C = − P M i =1 ( w i λ i + β i ¯ P i + α i ¯ D i λ i ) is a constant independent of µ and η . The dual function is z ( β , α , ν ) = min µ , η L ( β , α , ν , µ , π ∗ ( µ , β ) , η ) , subject to (7e) , (7f) . As (15) is separable in µ and η , the minimization decouples. Optimization over µ : The terms in volving µ ( h , ρ ) in (15) further decompose across channel states h , since the con- straints 0 ≤ µ ( h , ρ ) ≤ 1 , ∀ h , ρ and P ρ µ ( h , ρ ) = 1 , ∀ h are imposed independently for each h . Hence, for each ( h , ρ ) , µ ∗ ( h , ρ ) is optimal if and only if it solution to the following subproblem minimize µ ( h , ρ ) , ∀ ρ X ρ ∈ R µ ( h , ρ ) A ( h , ρ ) , (16) subject to X ρ ∈ R µ ( h , ρ ) = 1 , 0 ≤ µ ( h , ρ ) ≤ 1 , ∀ ρ . Here A ( h , ρ ) = P M i =1  β i f ∗ i ( h , ρ , β ) + α i λ i δ ( ρ i ) I { ρ i > 0 } − ν i I { ρ i > 0 }  . Problem (16) is a linear program with a closed-form solution. Let J ∗ ( h ) = { ρ ∈ R : A ( h , ρ ) = min ρ ′ A ( h , ρ ′ ) } denote the set of minimizers. Then, for each ( h , ρ ) pair, the optimal µ ∗ ( h , ρ ) is µ ∗ ( h , ρ ) =      1 , if | J ∗ ( h ) | = 1 and ρ ∈ J ∗ ( h ) , 1 / | J ∗ ( h ) | , if | J ∗ ( h ) | > 1 and ρ ∈ J ∗ ( h ) , 0 , otherwise . (17) When multiple minimizers exist, µ ∗ ( h , ρ ) is chosen as the uniform distribution over J ∗ ( h ) . Optimization over η : The terms in (15) inv olving η i are separable across i . Minimizing with respect to η i yields ∂ L ∂ η i = w i λ i − ν i − α i ¯ D i (1 − λ i ) η 2 i = 0 . Solving for η i giv es η ∗ i = s ν i − α i ¯ D i (1 − λ i ) w i λ i . For given ( β , α , ν ) , the optimal primal variables ( µ ∗ , π ∗ , η ∗ ) are obtained; arguments are omitted for brevity . The dual problem is therefore ( β ∗ , α ∗ , ν ∗ ) = arg max β ≥ 0 , α ≥ 0 , ν ≥ 0 z ( β , α , ν ) , (18) where z ( β , α , ν ) = L ( β , α , ν , µ ∗ , π ∗ , η ∗ ) . By strong duality , V SRP = z ( β ∗ , α ∗ , ν ∗ ) , and the corresponding Lagrangian minimizers solve (7). Next, we describe a subgradient ascent algorithm to solve (18). c) Subgr adient Ascent Algorithm for Solving the Dual Pr oblem: The dual function z ( β , α , ν ) is concave but gen- erally nonsmooth [37]; it is not clear that it is dif feren- tiable e verywhere. Hence, we employ a subgradient ascent method [38], [39]. The subgradients of the dual function with respect to the dual variables are giv en by ζ β i =   X ρ ∈ R E h [ π i ( h , ρ )] − ¯ P i   ∈ ∂ β i z ( β , α , ν ) , ζ α i =  D ′ i ( µ ) − ¯ D ′ i ( η i )  ∈ ∂ α i z ( β , α , ν ) , ζ ν i =  1 η i − p i ( µ )  ∈ ∂ ν i z ( β , α , ν ) , for all i ∈ M , where ∂ z ( · ) denotes the set of subgradients (i.e., the subdifferential) of the dual function z ( · ) at the dual variables. These subgradients correspond to violations of the 7 power , distortion, and linking constraints, respecti vely . Starting from any nonnegati ve initial dual variables ( β 0 , α 0 , ν 0 ) , the dual variables are updated via subgradient ascent iterations β i,k +1 =   β i,k + s k   X ρ ∈ R E h [ π ∗ i,k ( h , ρ )] − ¯ P i     + , (19a) α i,k +1 =  α i,k + s k  D ′ i ( µ ∗ k ) − ¯ D ′ i ( η ∗ i,k )  + , (19b) ν i,k +1 = " ν i,k + s k 1 η ∗ i,k − p i ( µ ∗ k ) !# + , (19c) for all i ∈ M and k . Here, k is the iteration index, [ · ] + = max { 0 , ·} , and s k > 0 is a diminishing step size satisfying P k s k = ∞ and P k s 2 k < ∞ (e.g., s k = 1 /k ). At each iteration k , the primal variables ( µ ∗ k , π ∗ k , η ∗ k ) min- imize the Lagrangian for ( β k , α k , ν k ) . Under the above step- size conditions, the iterates con ver ge to ( β ∗ , α ∗ , ν ∗ ) , and by strong duality the corresponding Lagrangian minimizers solve (7) [37]–[39]. T o ev aluate the terms 1 /η ∗ i,k appearing in (19c) and (19b) (through ¯ D ′ i ( η ∗ i,k ) ), it is necessary to ensure that η ∗ i,k > 0 for all i . T o this end, we employ an ϵ r -regularized primal update to avoid degenerac y of η ∗ i,k , where ϵ r > 0 is a small regularization parameter . Specifically , η ∗ i,k is computed as η ∗ i,k = s max  ν i,k − α i,k ¯ D i (1 − λ i ) , ϵ r  w i λ i , ∀ i, k . (20) For any fixed ϵ r > 0 , the regularized dual problem remains con vex and con verges under standard diminishing step sizes, with an optimality gap that vanishes as ϵ r → 0 . Alternatively , a vanishing regularization ϵ r,k ∝ s k preserves con ver gence to the unregularized optimum. Algorithm 1 summarizes the procedure for solving (7). Remark 2 (Dual V ariable Conv ergence and A veraging) . When the dual pr oblem (18) admits multiple optimal solutions, the subgradient method may not con ver ge to a unique dual point; instead, the dual iterates may oscillate among optimal solutions while attaining the optimal dual value. In such cases, the primal solutions corr esponding to individual dual iterates may violate some primal constraints, no single dual solution necessarily enfor ces all primal constraints simultaneously; rather , appr opriate con vex combinations of the associated solutions ( µ ∗ , π ∗ ) are r equir ed. This behavior is well known in dual decomposition and subgradient-based methods [40]. Nevertheless, under standar d diminishing step-size condi- tions, the ergodic (time-averaged) primal iterates ¯ µ L = 1 L L X l =1 µ ∗ ( β ∗ l , α ∗ l , ν ∗ l ) , ¯ π L = 1 L L X l =1 π ∗ ( β ∗ l , α ∗ l , ν ∗ l ) , ar e guaranteed to con ver ge to the optimal primal solutions and satisfy all primal constraints, including MAC constraints [40]. Her e L is the averaging window (e.g ., L = 500 ). Specifically , once the dual iterates ar e deemed to have con verg ed in the sense that | z ( β k , α k , ν k ) − z ( β k +1 , α k +1 , ν k +1 ) | < ϵ Algorithm 1 Subgradient-based solution of (7) Require: M , r max , ( λ i , w i , ¯ P i , ¯ D i ) i ∈M , H , P ( h ) for all h ∈ H , tolerance ϵ > 0 , ϵ r > 0 1: Initialize iteration index k = 0 , dual variables β k ≥ 0 , α k ≥ 0 , ν k ≥ 0 2: r epeat 3: For each ( h , ρ ) , compute optimal decoding order θ ∗ ( h , ρ , β k ) by sorting ( h i I { ρ i > 0 } /β i,k ) i ∈M in non-increasing order , and obtain the optimal po wer allocation f ∗ ( h , ρ , β k ) using (13) . 4: For each ( h , ρ ) , compute µ ∗ k ( h , ρ ) using (17). 5: For each i ∈ M , compute p i ( µ ∗ k ) , D ′ i ( µ ∗ k ) , η ∗ i,k using (20), and ¯ D ′ i ( η ∗ i,k ) . 6: For each i ∈ M , update β i,k +1 , α i,k +1 and ν i,k +1 using (19). 7: k ← k + 1 8: until | z ( β k , α k , ν k ) − z ( β k +1 , α k +1 , ν k +1 ) | < ϵ 9: At con ver gence, set β ∗ ← β k , α ∗ ← α k , ν ∗ ← ν k , and obtain µ ∗ ( β ∗ , α ∗ , ν ∗ ) using (17). 10: F or each i, ( h , ρ ) , compute π ∗ i ( h , ρ ) = µ ∗ ( h , ρ ) · f ∗ i ( h , ρ , β ∗ ) , and thereby obtain π ∗ ( β ∗ , α ∗ , ν ∗ ) . Ensure: Optimal solutions for (7): µ ∗ ( β ∗ , α ∗ , ν ∗ ) and π ∗ ( β ∗ , α ∗ , ν ∗ ) . Obtain F ∗ via (8). the subgradient ascent algorithm is run for an additional L iterations, and the ergodic primal solutions are computed. The a veraged power allocations obtained from ( ¯ µ L , ¯ π L ) satisfy the MA C constraints (1) but may not correspond to a single SIC decoding order . T o retain SIC-based decoding, we record the empirical frequency of each decoding order over the av eraging window and construct a randomized policy by defining scheduling probabilities according to these frequen- cies. The polic y then selects a rate v ector and a decoding order , for which the corresponding power allocation is computed via SIC using (13). Remark 3. The pr oposed dual decomposition approac h offers substantial computational advantag es over directly solving (7) , which would requir e handling | H | × | R | × (2 M − 1) MAC con- straints, exponential in the number of users. By exploiting the SIC structur e, the power allocation is obtained analytically , completely eliminating these constraints. Moreover , Lemma 2 r educes the decoding order optimization fr om factorial com- plexity O ( M !) to O ( M log M ) per ( h , ρ ) via sorting. The algorithm computes µ and η in closed form using (17) and (20) , avoiding iterative con vex optimization. F or each h , evaluating A ( h , ρ ) over ρ ∈ R incurs complexity O ( | R | M ) , while computing η ∗ r equires O ( M ) . 3) V A-SRP under TDMA Scheme: W e also solve (7) under a TDMA constraint, where at most one user transmits ρ > 0 bits in any slot. This is enforced by restricting the rate vector ρ to the set S ≜ { 0 } ∪ M [ i =1 { ρ e i | ρ ∈ { 1 , 2 , . . . , r max }} , 8 instead of R . Here, e i denotes the i th standard basis vector in R M , and 0 is the all-zero vector . Thus, S includes either no transmission or exactly one active user transmitting between 1 and r max bits, ensuring the TDMA constraint. B. P erformance Bound on V A-SRP W e no w establish a provable approximation guarantee for the V A-SRP solution. Theorem 3. The optimal objective value of (7) , V SRP , is at most twice the optimal objective value of (3) , V opt , i.e., V SRP ≤ 2 V opt . Pr oof. W e follo w a standard lower -bounding and policy- construction argument, adapted from [6]. See Appendix C for details. C. V A-SRP with P ower Adjustment (V A-SRP w/ P A) The proposed V A-SRP is both V AoI- and queue-agnostic, and may therefore schedule a user ev en when its queue is empty . While this does not af fect V AoI (which is zero when the queue is empty), it leads to unnecessary power accounting in the SRP formulation. In practice, no power is consumed when a scheduled user has an empty queue. Consequently , the actual av erage power usage is lo wer than that imposed by the original SRP constraints. W e therefore refer to the policy obtained from (7) as V A-SRP without P ower Adjustment (V A- SRP w/o P A) . Next, we introduce refined power constraints and denote the corresponding policy as V A-SRP with P ower Adjustment (V A-SRP w/ P A) . W e define the instantaneous transmit power of User i in slot t as P i ( t ) = f i ( h ( t ) , ρ ( t )) if Q i ( t ) = 1 , and P i ( t ) = 0 otherwise. The expected per-slot transmit power is E [ P i ( t )] = E [ f i ( h ( t ) , ρ ( t )) | Q i ( t ) = 1] P ( Q i ( t ) = 1) . Conditioned on Q i ( t ) = 1 , the expected transmit power under V A-SRP is E [ f i ( h ( t ) , ρ ( t )) | Q i ( t ) = 1] = X ρ ∈ R E h [ µ ( h , ρ ) f i ( h , ρ )] . Using the queue occupancy probability , P ( Q i ( t ) = 1) , from (5), the unconditional expected per-slot transmit power , E [ P i ( t )] = λ i P ρ ∈ R E h [ µ ( h , ρ ) f i ( h , ρ )] λ i (1 − p i ( µ )) + p i ( µ ) , ∀ i ∈ M . Hence, the long-term av erage po wer under V A-SRP w/ P A is P i ( µ , F ) = E [ P i ( t )] and the av erage power constraints are giv en by P i ( µ , F ) ≤ ¯ P i , ∀ i ∈ M , which can be equiv alently expressed as P ′ i ( µ , F ) = λ i X ρ ∈ R E h [ µ ( h , ρ ) f i ( h , ρ )] ≤ ¯ P ′ i ( µ ) = ¯ P i ( λ i − λ i p i ( µ ) + p i ( µ )) , ∀ i ∈ M . (21) Replacing the power constraint (7b) with (21) in (7) yields the V A-SRP w/ P A. D. Heuristic P olicies In this section, we describe several heuristic policies. 1) V AoI-A war e Greedy P olicy: In this policy , all transmis- sion vectors ρ consistent with the current queue state are considered (i.e., ρ i = 0 if Q i = 0 ). For each ρ , transmit powers are obtained by solving instantaneous po wer minimiza- tion problem subject to the MAC constraints. Among all rate vectors, only those that satisfy the running average power and distortion constraints are retained. The optimal transmission vector ρ ∗ is chosen to minimize the instantaneous total V AoI while maximizing the aggregate number of transmitted bits. 2) Max-V AoI-F irst P olicy (TDMA): This polic y is a TDMA-restricted variant of the V AoI-aware Greedy policy . In each slot, at most one user is allo wed to transmit, and the user with the largest instantaneous V AoI is selected for transmission. The transmission rate is then chosen to satisfy the running average power and distortion constraints while maximizing the number of transmitted bits. 3) Round-Robin P olicy: This policy is V AoI-agnostic and selects users in a round-robin manner, irrespecti ve of V AoI and queue states. Each user is given an equal opportunity to be served in a cyclic order , and only the scheduled user is allowed to transmit in a giv en time slot. For the selected user , ρ ∗ ∈ { 0 , 1 , . . . , r max } is chosen to satisfy the running average power and distortion constraints while maximizing the number of transmitted bits. I V . N U M E R I C A L R E S U LT S This section presents numerical results. Unless otherwise stated, we use g ( x ) = log 2 (1 + x ) and δ ( x ) = (1 − x/r max ) 2 . In Fig. 1, we present the v ariation of the average V AoI under different scheduling policies. Fig. 1a illustrates how the long-term expected av erage V AoI varies with the bound on the average transmit power for users, ¯ P i , for both NOMA and TDMA schemes under v arious policies, including the proposed V A-SRP and sev eral heuristic baselines. As ¯ P i increases, the average V AoI decreases and ev entually saturates, since higher power budgets enable more frequent status update deli veries. Under NOMA, V A-SRP w/ P A consis- tently outperforms V A-SRP w/o P A by eliminating redundant power usage when queues are empty , enabling more frequent transmissions. V A-SRP w/ P A also outperforms Max-V AoI- First and Round-Robin policies and achiev es performance comparable to the V AoI-A ware Greedy policy . Importantly , V A-SRP is simpler , as it is V AoI-agnostic and avoids per-slot V AoI tracking. Now , we compare the TDMA and NOMA schemes. At lower values of ¯ P i , both TDMA and NOMA schemes exhibit similar performance under the V A-SRP and Greedy policies. As the av ailable transmit power increases, NOMA begins to outperform TDMA. At low power levels, NOMA effecti vely behav es like TDMA by serving a single user , whereas at higher power levels it enables simultaneous multi-user transmissions. Under V A-SRP , as ¯ P i increases, the average V AoI under NOMA approaches zero, whereas under TDMA it saturates at P M i =1 w i λ i ( M − 1) for an M -user system. This behavior follows directly from the status update deli very probability constraints. The long-term expected average V AoI under V A- SRP is given by P M i =1 w i λ i ( p − 1 i − 1) , where p i ∈ [0 , 1] 9 0 2 4 6 8 10 12 14 0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 ¯ P , Bound on the average transmit power lim T →∞ 1 T P T t =1 P M i =1 w i E [∆ i ( t )] Long-term expected average V AoI NOMA, V A-SRP w/ P A NOMA, V A-SRP w/o P A TDMA, V A-SRP w/ P A TDMA, V A-SRP w/o P A NOMA, V AoI-Aw are Greedy Max-V AoI-First Round-Robin (a) 0 0 . 02 0 . 04 0 . 06 0 . 08 0 . 1 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 ¯ D , Bound on the average distortion lim T →∞ 1 T P T t =1 P M i =1 w i E [∆ i ( t )] Long-term expected average V AoI NOMA, V A-SRP w/ P A NOMA, V A-SRP w/o P A TDMA, V A-SRP w/ P A TDMA, V A-SRP w/o P A NOMA, V AoI-Aw are Greey Max-V AoI-First Round-Robin (b) Fig. 1: Long-term expected average V AoI vs. (a) power bound ¯ P i = ¯ P , for ¯ D i = 0 . 05 , λ i = 0 . 5 ; (b) distortion bound ¯ D i = ¯ D , for ¯ P i = 2 , λ i = 0 . 4 , ∀ i . System parameters: M = 3 , w i = 1 / 3 , ρ i ∈ { 0 , 1 , 2 } , H i = { 0 . 1 , 1 } with equal probability , ∀ i ∈ { 1 , 2 , 3 } . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 λ , Probability of packet arriv al lim T →∞ 1 T P T t =1 P M i =1 w i E [∆ i ( t )] Long-term expected average V AoI NOMA, V A-SRP w/ P A NOMA, V A-SRP w/o P A TDMA, V A-SR w/ P A TDMA, V A-SRP w/o P A NOMA, V AoI-Aw are Greedy Max-V AoI-First Round-Robin Fig. 2: Long-term expected average V AoI vs. probability of packet arrival, λ i = λ , for different scheduling policies. System parameters: M = 3 , ¯ P i = 5 , ¯ D i = 0 . 05 , w i = 1 / 3 , ρ i ∈ { 0 , 1 , 2 } , H i = { 0 . 1 , 1 } with equal probability , ∀ i ∈ { 1 , 2 , 3 } . denotes the probability of successful status update deli very from User i . In the TDMA scheme, even with infinite power , only one user can transmit a status update at any given time. Hence, with infinite ¯ P i , the optimal status update deliv ery probability becomes p ∗ i = 1 / M , ∀ i , leading to a minimum achiev able av erage V AoI of P M i =1 w i λ i ( M − 1) . In contrast, with infinite ¯ P i , NOMA enables simultaneous transmissions to multiple users, allowing p ∗ i = 1 , ∀ i , which results in a minimum achiev able average V AoI of zero. In Fig. 1b, we study the variation of the av erage V AoI with respect to the bound on the average distortion for users, ¯ D i . Similar trends as discussed previously are observed from the figure: as ¯ D i increases, the average V AoI decreases and ev entually saturates. For a given ¯ P i , at lower values of ¯ D i , the 0 0 . 5 1 1 . 5 2 2 . 5 0 0 . 5 1 1 . 5 2 2 . 5 lim T →∞ 1 T P T t =1 E [∆ 1 ( t )] , Long-term av erage V AoI of User 1 lim T →∞ 1 T P T t =1 E [∆ 2 ( t )] , Long-term av erage V AoI of User 2 NOMA TDMA Fig. 3: Achievable long-term expected average V AoI regions for NOMA and TDMA schemes under V A-SRP w/o P A by varying user weights ( w 1 , w 2 ) subject to w 1 + w 2 = 1 . System parameters: M = 2 , ¯ P i = 5 , ¯ D i = 0 . 05 , λ i = 0 . 9 , ρ i ∈ { 0 , 1 , 2 } , H i = { 0 . 1 , 1 } with equal probability , ∀ i ∈ { 1 , 2 } . Greedy policy outperforms all other policies. Howe ver , as ¯ D i increases, the V A-SRP w/ P A with the NOMA scheme begins to outperform the Greedy policy . In Fig. 2, we illustrate how the a verage V AoI varies with the packet arriv al probability , λ i . As λ i increases, the average V AoI also rises. Nev ertheless, the relative performance trends among the different policies remain consistent with those observed in the previous results. It is observed that at λ i = 1 , with the NOMA scheme, both V A-SRP w/ P A and V A-SRP w/o P A achiev e identical performance, since the constraint in (7b) and (21) becomes equal when λ i = 1 . Howe ver , for λ i < 1 , the V A-SRP w/ P A outperforms the V A-SRP w/o P A. 10 T ABLE I: Comparison of average online run-time (per time slot) of different scheduling policies. Each v alue is obtained by averaging 10 sample paths, each consisting of 500k time slots. All the simulations are conducted on a 12 th Generation Intel Core i 5 - 12600 Processor with 16 GB RAM, using Python 3 . 10 . System parameters: M = 3 , ¯ P i = 2 , ¯ D i = 0 . 06 , λ i = 0 . 5 , w i = 1 / 3 , ρ i ∈ { 0 , 1 , 2 } , H i = { 0 . 1 , 1 } with equal probability , ∀ i ∈ { 1 , 2 , 3 } . NOMA, V A-SRP w/ P A TDMA, V A-SRP w/ P A NOMA, V AoI- A ware Greedy Max- V AoI- First Round- Robin Online Run-time (milli seconds) 0 . 026 0 . 021 176 . 26 51 . 48 50 . 05 Offline Run-time (seconds) 85 . 7 16 . 3 - - - A verage V AoI 0 . 2762 1 . 00 0 . 3067 0 . 4185 0 . 4349 In Fig. 3, we illustrate the achiev able long-term a verage V AoI regions for the TDMA and NOMA schemes under the V A-SRP w/o P A for the two-user case. The regions are obtained by v arying the user weights w 1 and w 2 between 0 and 1 , subject to w 1 + w 2 = 1 . The figure shows that the achie vable region of the NOMA scheme entirely encompasses that of the TDMA scheme. In TDMA, achieving a lower average V AoI for one user leads to a significantly higher V AoI for the other user , since it can serve only one user at a time. In T able I, we report the offline and online computational times of the different policies. The V A-SRP takes a higher offline computational time than the V AoI-A ware Greedy , Max- V AoI-First, and Round-Robin policies, as these require no offline optimization. This is because the V A-SRP is computed using a subgradient ascent algorithm with several hundred iter - ations (until con vergence), where each iteration in volv es sort- ing operations, maximization over a set, and the computation of primal variables using closed-form expressions. Con versely , for online execution, the V AoI-A ware Greedy , Max-V AoI-First and Round-Robin policies require significantly more time than V A-SRP , since they track the V AoI, average power , and distor- tion of all users and must solve power -minimization problems under the MA C constraints. Overall, V A-SRP achie ves the lowest av erage V AoI among the considered policies at the cost of higher offline computation, while maintaining lo w online complexity due to its stationary and V AoI-agnostic structure. As discussed in the introduction, not all bits within an update are equally important in practice. Our framew ork captures such bit-priority structures through a distortion metric δ ( ρ ) , which depends on the number of transmitted bits, ρ . This allows modeling scenarios where the first few bits carry task-critical information, while the remaining bits provide refinement. T o illustrate this flexibility , Fig. 4 presents dif ferent distortion functions corresponding to different application- specific bit-priority structures. F or different distortion func- tions, we sho w the v ariation of the long-term av erage V AoI ( ¯ ∆ ) and scheduling probabilities µ ( h, ρ ) under the V A-SRP , for fixed bound on a verage transmit power , ¯ P . From Fig. 4, we observe that for a fixed distortion function (e.g., δ 1 ( ρ ) ), relaxing the distortion constraint (i.e., increasing ¯ D ) leads to a reduction in the average V AoI. This is achiev ed ρ 0 ρ 1 ρ 2 ρ 3 ρ 4 ρ 5 ρ 0 ρ 1 ρ 2 ρ 3 ρ 4 ρ 5 ρ 0 ρ 1 ρ 2 ρ 3 ρ 4 ρ 5 ρ 0 ρ 1 ρ 2 ρ 3 ρ 4 ρ 5 ρ 0 ρ 1 ρ 2 ρ 3 ρ 4 ρ 5 ρ 0 ρ 1 ρ 2 ρ 3 ρ 4 ρ 5 ρ 0 ρ 1 ρ 2 ρ 3 ρ 4 ρ 5 ρ 0 ρ 1 ρ 2 ρ 3 ρ 4 ρ 5 h 0 h 0 h 0 h 0 h 1 h 1 h 1 h 1 h 0 h 0 h 0 h 0 h 1 h 1 h 1 h 1 δ 1 ( ρ ) , ¯ D = 0 . 07 , ¯ ∆ = 0 . 2498 δ 2 ( ρ ) , ¯ D = 0 . 07 , ¯ ∆ = 0 . 7115 δ 3 ( ρ ) , ¯ D = 0 . 07 , ¯ ∆ = 0 . 8531 δ 4 ( ρ ) , ¯ D = 0 . 07 , ¯ ∆ = 1 . 2998 δ 1 ( ρ ) , ¯ D = 0 . 01 , ¯ ∆ = 0 . 7771 δ 1 ( ρ ) , ¯ D = 0 . 05 , ¯ ∆ = 0 . 3353 δ 1 ( ρ ) , ¯ D = 0 . 1 , ¯ ∆ = 0 . 1544 δ 1 ( ρ ) , ¯ D = 0 . 2 , ¯ ∆ = 0 1 2 3 4 5 0 0 . 2 0 . 4 0 . 6 0 . 8 1 ρ δ ( ρ ) δ 1 ( ρ ) : Conve x δ 2 ( ρ ) : Step δ 3 ( ρ ) : Linear δ 4 ( ρ ) : Concave Fig. 4: V ariation of the scheduling probabilities µ ( h, ρ ) (shown as table en- tries) and the long-term expected average V AoI under the V A-SRP , ¯ ∆ = lim T →∞ 1 T P T t =1 E [∆( t )] = λ (( P ρ E h [ µ ( h, ρ )] I { ρ > 0 } ) − 1 − 1) , for different distortion functions, δ ( ρ ) and bounds on the average distortion, ¯ D . The considered distortion functions are δ 1 ( ρ ) = e − ρ , δ 2 ( ρ ) = 1 for ρ < ( r max − 1) , δ 2 ( r max − 1) = 0 . 05 , and δ 2 ( ρ ) = 0 otherwise, δ 3 ( ρ ) = 1 − ρ/r max and δ 4 ( ρ ) = (cos( π ρ/ 2 r max ) 0 . 3 . System parameters: M = 1 , ¯ P = 10 , λ = 0 . 9 , r max = 5 , ρ ∈ { ρ 0 = 0 , ρ 1 = 1 , ρ 2 = 2 , ρ 3 = 3 , ρ 4 = 4 , ρ 5 = 5 } , H = { h 0 = 0 . 1 , h 1 = 1 } with equal probability . by transmitting fe wer bits more frequently , as reflected by increased scheduling probabilities for smaller ρ and reduced probabilities for larger ρ . Conv ersely , tighter distortion con- straints fav or transmitting more bits per update b ut less fre- quently , as required by applications demanding near-complete packet reconstruction. Importantly , the choice of the distortion function significantly affects the scheduling policy even for a fixed ¯ D . For example, with a conv ex distortion function δ 1 ( ρ ) , V A-SRP tends to transmit fewer bits with higher probability , resulting in a lower average V AoI. In contrast, step and linear distortion functions δ 2 ( ρ ) and δ 3 ( ρ ) fa vor transmitting more bits less frequently , resulting in higher average V AoI. This sho ws that distortion functions can be selected based on application requirements regarding the preferred number of transmitting bits. Overall, by appropriately selecting δ ( ρ ) , 11 the proposed framework enables a flexible trade-off between av erage V AoI and the frequency of transmitting a larger number of bits. This allows the V A-SRP to adapt across applications ranging from semantic or flag-based updates to full-packet reconstruction scenarios. V . C O N C L U S I O N This paper studied V AoI minimization in uplink NOMA systems under av erage power and information-quality con- straints captured through a general distortion metric. W e dev eloped a V AoI-agnostic stationary randomized policy that jointly optimizes scheduling, bit allocation, and power control, and achiev es a prov able 2-approximation to the globally opti- mal V AoI. By lev eraging Lagrangian dual decomposition, we reduce the computational complexity of obtaining the policy . The algorithm has polynomial-time complexity per iteration, dominated by a sorting step to determine the decoding order for SIC. Numerical results demonstrate that NOMA substan- tially outperforms TDMA, achieving near-zero V AoI at high power budgets through simultaneous multi-user transmissions. At the same time, TDMA remains limited by its single- user-per -slot constraint. Our distortion framework accommo- dates div erse bit-priority structures, from uniform importance to metadata-critical scenarios, enabling application-specific trade-offs between timeliness and information quality . 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Since f ( x ) = 1 /x is con vex and decreasing for x > 0 , the composition 1 /p i ( µ ) is con vex [37]. Hence, (7a), being a nonnegati ve weighted sum of con ve x functions, is con ve x. P ower constraint (7b) : This constraint is linear in π and independent of µ , and therefore defines a conv ex set. Distortion constraint (7c) : The left-hand side D ′ i ( µ ) = λ i P ρ E h [ µ ( h , ρ ) δ ( ρ i ) I { ρ i > 0 } ] is affine in µ . The right- hand side ¯ D ′ i ( µ ) = ¯ D i  λ i − λ i p i ( µ ) + p i ( µ )  is also affine since p i ( µ ) is affine. Thus, the constraint defines a con ve x set. MA C constraint (7d) : For fixed ( S , h , ρ ) , the constraint µ ( h , ρ ) g − 1  X i ∈S ρ i  ≤ X i ∈S π i ( h , ρ ) h i is linear in ( µ , π ) , since g − 1 ( P i ∈S ρ i ) is constant. Hence, each constraint defines a halfspace, whose intersection is con vex. Constraints (7e) and (7f) : These are affine constraints in µ and therefore con ve x. Since the objecti ve function is con ve x and all constraints define conv ex sets, problem (7) is a conv ex optimization problem. B. Pr oof of Lemma 2 W e first prove the lemma under the assumption that ρ θ j > 0 for all j ∈ M , and then extend the result to the case where ρ θ j = 0 for some users. a) Case 1 ( ρ θ j > 0 for all j ∈ M ): For a giv en decoding order θ , the objective function in (14) can be written as P ( θ ) = M X j =1 β θ j g − 1 ( ρ θ j ) h θ j M Y k = j +1  1 + g − 1 ( ρ θ k )  , where all terms are strictly positiv e under the assumption ρ θ j > 0 . Consider any two adjacent users x = θ t and y = θ t +1 in the decoding order , and let θ ′ denote the order obtained by swapping x and y , with all other user positions unchanged. All terms in P ( θ ) and P ( θ ′ ) are identical except those inv olving users x and y . A direct comparison yields P ( θ ) − P ( θ ′ ) = G g − 1 ( ρ x ) g − 1 ( ρ y )  β x h x − β y h y  , where G > 0 is a common positiv e factor independent of x and y . Hence, P ( θ ) ≤ P ( θ ′ ) if and only if β x /h x ≤ β y /h y . Therefore, any decoding order containing an adjacent pair violating this condition can be improved by swapping the pair . By repeated adjacent exchanges, the optimal decoding order must satisfy β θ ∗ 1 h θ ∗ 1 ≤ β θ ∗ 2 h θ ∗ 2 ≤ · · · ≤ β θ ∗ M h θ ∗ M , or equiv alently , h θ ∗ 1 /β θ ∗ 1 ≥ · · · ≥ h θ ∗ M /β θ ∗ M . b) Case 2 ( ρ θ j = 0 for some users): If ρ i = 0 , then g − 1 ( ρ i ) = 0 , and User i contributes zero po wer and zero cost to P ( θ ) , regardless of its position in the decoding order . Howe ver , placing such users early in the SIC order is undesirable, since earlier decoding stages correspond to higher effecti ve interference lev els and require higher transmit powers. This is resolv ed by modifying the sorting metric to h i I { ρ i > 0 } /β i , which preserves the optimal ordering among transmit- ting users while forcing all non-transmitting users to the end of the decoding order . Their relativ e order is immaterial since they incur zero power . Accordingly , the optimal decoding order satisfies h θ ∗ 1 I { ρ θ ∗ 1 > 0 } β θ ∗ 1 ≥ · · · ≥ h θ ∗ M I { ρ θ ∗ M > 0 } β θ ∗ M , which completes the proof. C. Pr oof of Theor em 3 W e provide a complete proof of Theorem 3, following a three-step argument. Step 1: Lower bound on optimal V AoI. F or a single user , the expected sum of V AoI o ver an inter -deliv ery interval of I slots is E [∆ I ] = λ ( I 2 − I ) / 2 [6], where λ is the packet arriv al probability . Consider a time horizon of T slots. Let D ( T ) = P T t =1 I { ρ ( t ) > 0 } denote the number of successful updates, I [1] , . . . , I [ D ( T )] denotes inter-deliv ery interv als, and R denote the remaining slots after the last deliv ery . The time-av erage expected V AoI is expressed as 1 T T X t =1 E [∆( t )] = λ 2   D ( T ) X k =1 I 2 [ k ] − I [ k ] T + R 2 − R T   . By Jensen’ s inequality ( ¯ M [ I 2 ] ≥ ( ¯ M [ I ]) 2 , where ¯ M [ · ] denotes the sample mean) and minimizing with respect to R yields lower bound to the long-term average V AoI as lim T →∞ 1 T T X t =1 E [∆( t )] ≥ λ 2  1 q − 1  , where q = lim T →∞ (1 /T ) E [ D ( t )] = lim T →∞ (1 /T ) E [ P T t =1 I { ρ ( t ) > 0 } ] is the probability of status update deliv ery . Recall that ρ > 0 corresponds to a successful update. For the M -user system, we have V ϕ = lim T →∞ 1 T T X t =1 M X i =1 w i E [∆ i ( t )] ≥ 1 2 M X i =1 w i λ i 1 q ϕ i − 1 ! , 13 for any policy ϕ , where q ϕ i is the probability of status update deliv ery for User i under policy ϕ . Minimizing ov er all admissible policies yields V opt ≥ L B , where V opt = min ϕ V ϕ , subject to (1) , (2) , (3b) , (3c) , and L B = min ϕ 1 2 M X i =1 w i λ i 1 q ϕ i − 1 ! , (22) subject to (1) , (2) , (3b) , (3c) . Step 2: Construction of a feasible V A-SRP . Let ϕ LB denote the optimal policy achieving L B , which specifies transmission decisions ( ρ LB i ( t ) , f LB i ( t )) i ∈M for each time slot t . W e con- struct a V A-SRP by defining scheduling probabilities based on the empirical frequencies under ϕ LB : ˆ µ ( h , ρ ) = lim T →∞ 1 |T T ( h ) | E   X t ∈T T ( h ) I { ρ LB ( t ) = ρ }   , for all h ∈ H , ρ ∈ R , where T T ( h ) = { t ∈ { 1 , . . . , T } : h ( t ) = h } , and ρ LB ( t ) ≜ ( ρ LB i ( t )) i ∈M . The expectation is taken over the randomness in the system under the lower - bound policy ϕ LB . W e hav e ˆ µ ≜ ( ˆ µ ( h , ρ )) h ∈ H , ρ ∈ R , and the probability of status update deliv ery for User i under constructed V A-SRP , p i ( ˆ µ ) = P ρ ∈ R E h [ ˆ µ ( h , ρ ) I { ρ i > 0 } ] . Let ˆ V SRP denote the long-term expected a verage V AoI achie ved by the constructed V A-SRP . Using (4), we obtain ˆ V SRP = P M i =1 w i λ i (1 /p i ( ˆ µ ) − 1) , with ˆ µ satisfying average power and distortion constraints. W e define the probability of status update deliv ery under ϕ LB as q LB i = lim T →∞ 1 T E " T X t =1 I { ρ LB i ( t ) > 0 } # = X h ∈ H X ρ ∈ R P ( h ) ˆ µ ( h , ρ ) I { ρ i > 0 } , ∀ i ∈ M . Substituting q LB i into the lower -bound expression yields L B = 1 2 M X i =1 w i λ i  1 q LB i − 1  . (23) Since the constructed V A-SRP employs the same schedul- ing probabilities ˆ µ ( h , ρ ) for all ( h , ρ ) , the resulting sta- tus update probability for User i satisfies p i ( ˆ µ ) = q LB i , and the corresponding av erage V AoI is giv en by ˆ V SRP = P M i =1 w i λ i  1 /q LB i − 1  . Since the constructed V A-SRP inherits the scheduling fre- quencies of ϕ LB , it satisfies the average power and distortion constraints. Comparing with (23) yields ˆ V SRP = 2 L B . P ower constraint satisfaction: Power constraint satisfaction follows directly from [6]: the constructed V A-SRP inherits feasibility since ˆ µ ( h , ρ ) is derived from the lower -bound policy . Distortion constraint satisfaction: Since the lower-bound problem (22) includes distortion constraints, ϕ LB satisfies the av erage distortion constraints: lim T →∞ 1 T E " T X t =1 d i ( t ) # ≤ ¯ D i , ∀ i ∈ M . (24) W e can decompose the left-hand side by conditioning on the queue state. Under any policy , the per-slot expected distortion is: E [ d i ( t )] = E [ d i ( t ) | Q i ( t ) = 1] · P ( Q i ( t ) = 1) , since d i ( t ) = 0 when Q i ( t ) = 0 . For the lo wer-bound policy , the conditional expected distortion is E ϕ LB [ d i ( t ) | Q i ( t ) = 1] = lim T →∞ 1 T E " T X t =1 δ ( ρ LB i ( t )) I { ρ LB i ( t ) > 0 } # = X h ∈ H X ρ ∈ R P ( h ) ˆ µ ( h , ρ ) δ ( ρ i ) I { ρ i > 0 } . The constructed SRP uses the same scheduling probabilities ˆ µ ( h , ρ ) for all ( h , ρ ) . Therefore, the conditional expected distortion under the constructed V A-SRP is E SRP [ d i ( t ) | Q i ( t ) = 1] = E ϕ LB [ d i ( t ) | Q i ( t ) = 1] . From (5), the steady-state queue occupancy probability de- pends only on λ i and the status update probability p i . Since p i ( ˆ µ ) = q LB i , we hav e P SRP ( Q i ( t ) = 1) = λ i λ i (1 − q LB i ) + q LB i = P ϕ LB ( Q i ( t ) = 1) . Therefore, the expected average distortion under the con- structed SRP is E SRP [ d i ( t )] = E SRP [ d i ( t ) | Q i ( t ) = 1] · P SRP ( Q i ( t ) = 1) = E ϕ LB [ d i ( t ) | Q i ( t ) = 1] · P ϕ LB ( Q i ( t ) = 1) = E ϕ LB [ d i ( t )] ≤ ¯ D i , where the last inequality follows from (24). Hence, the con- structed V A-SRP satisfies the distortion constraints. Step 3: Establishing the 2-appr oximation. Since the schedul- ing probabilities ˆ µ of the constructed V A-SRP constitute a feasible solution to problem (7), the corresponding objectiv e value satisfies ˆ V SRP ≥ V SRP . Combining all the established bounds, we obtain L B ≤ V opt ≤ V SRP ≤ ˆ V SRP = 2 L B ≤ 2 V opt . Consequently , the V A-SRP achie ves a 2 -approximation of the optimal V AoI, i.e., V SRP ≤ 2 V opt .