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

Abstract

The Version Age of Information (VAoI) quantifies information freshness by measuring the number of versions the receiver lags behind. This paper studies VAoI minimization in an $M$-user uplink non-orthogonal multiple access (NOMA) system where users maintain single-packet buffers and transmissions are constrained by average power and information-quality constraints, modeled by a general distortion function. A fundamental trade-off arises: transmitting more bits per update improves information quality but increases power consumption, reducing transmission opportunities and increasing VAoI, while transmitting fewer bits has the opposite effect. We formulate a weighted-sum VAoI minimization problem as a convex optimization problem. However, users' power allocations are coupled through multiple-access capacity constraints per channel state, leading to exponential complexity. To address this, we develop a VAoI-agnostic stationary randomized policy that jointly optimizes scheduling, bit allocation, and power control without tracking instantaneous VAoI, and achieves a provable 2-approximation to the globally optimal average VAoI. Leveraging Lagrangian dual decomposition, we derive closed-form expressions for the scheduling probabilities and power allocations, and efficiently determine the optimal successive interference cancellation decoding order, avoiding exhaustive search Numerical results show that NOMA significantly outperforms time-division multiple access (TDMA): at high power budgets, NOMA achieves near-zero VAoI, whereas TDMA saturates at a non-zero value, consistent with the analysis. The proposed general distortion framework accommodates diverse bit-priority structures by assigning unequal importance to different bits within an update.

Figures (4)

• Figure 1: Long-term expected average VAoI vs. (a) power bound $\bar{P}_i = \bar{P}$, for $\bar{D}_i = 0.05$, $\lambda_i = 0.5$; (b) distortion bound $\bar{D}_i = \bar{D}$, for $\bar{P}_i = 2$, $\lambda_i = 0.4, \;\forall i$. System parameters: $M=3$, $w_i = 1/3$, $\rho_i \in \{0,1,2\}$, $\mathcal{H}_i = \{0.1, 1\}$ with equal probability, $\forall i \in \{1,2,3\}$.
• Figure 2: Long-term expected average VAoI vs. probability of packet arrival, $\lambda_i = \lambda$, for different scheduling policies. System parameters: $M=3$, $\bar{P}_i = 5$, $\bar{D}_i = 0.05$, $w_i = 1/3$, $\rho_i \in \{0,1,2\}$, $\mathcal{H}_i = \{0.1, 1\}$ with equal probability, $\forall i \in \{1,2,3\}$.
• Figure 3: Achievable long-term expected average VAoI regions for NOMA and TDMA schemes under VA-SRP w/o PA by varying user weights $(w_1, w_2)$ subject to $w_1 + w_2 = 1$. System parameters: $M=2$, $\bar{P}_i = 5$, $\bar{D}_i = 0.05$, $\lambda_i = 0.9$, $\rho_i \in \{0,1,2\}$, $\mathcal{H}_i = \{0.1, 1\}$ with equal probability, $\forall i \in \{1,2\}$.
• Figure 4: Variation of the scheduling probabilities $\mu(h,\rho)$ (shown as table entries) and the long-term expected average VAoI under the VA-SRP, $\bar{\Delta} = \lim_{T \to \infty} \frac{1}{T} \sum_{t=1}^{T} \mathbb{E}[\Delta(t)] = \lambda ((\sum_{\rho}\mathbb{E}_h[\mu(h, \rho)] \mathbb{I}\{\rho > 0\} )^{-1} -1)$, for different distortion functions, $\delta(\rho)$ and bounds on the average distortion, $\bar{D}$. The considered distortion functions are $\delta_1(\rho)=e^{-\rho}$, $\delta_2(\rho)=1$ for $\rho< (r_{\rm max} -1)$, $\delta_2(r_{\rm max} - 1)=0.05$, and $\delta_2(\rho)=0$ otherwise, $\delta_3(\rho) = 1-\rho/r_{\rm max}$ and $\delta_4(\rho)=(\cos(\pi \rho/2r_{\rm \max})^{0.3}$. System parameters: $M = 1, \bar{P} = 10, \lambda = 0.9, r_{\rm max} = 5$, $\rho \in \{\rho_0=0,\rho_1=1,\rho_2=2,\rho_3=3,\rho_4=4,\rho_5=5\}$, $\mathcal{H}=\{h_0=0.1,\,h_1=1\}$ with equal probability.

Theorems & Definitions (9)

• Theorem 1
• proof
• Lemma 2
• proof
• Remark 1
• Remark 2: Dual Variable Convergence and Averaging
• Remark 3
• Theorem 3
• proof