C-AoEI: Cross-layer Age of Error Information

• C-AoEI is a refined metric that integrates physical- and link-layer error effects, finite-blocklength coding, and HARQ to measure update freshness more accurately.
• It employs cross-layer optimization by coupling error detection outcomes with dynamic scheduling to quantify the trade-off between reliability and latency.
• The framework guides adaptive system designs in satellite IoT and next-gen networks, balancing resource allocation with stringent update freshness requirements.

The Cross-layer Age of Error Information (C-AoEI) framework generalizes classical Age of Information (AoI) by explicitly accounting for the impact of physical- and link-layer errors, code design, and decoding ambiguity in information freshness metrics. C-AoEI addresses the deficiencies of conventional AoI in environments with finite blocklength coding, hybrid ARQ (HARQ), or error detection mechanisms, enabling a precise cross-layer characterization of the freshness–efficiency trade-off in modern and next-generation communication systems, including satellite IoT and 6G/edge scenarios (Zhao et al., 28 Jan 2026, Raikar et al., 2023, Yao et al., 2020).

1. Formal Definition and Motivation

Classical instantaneous AoI at time $t$ is defined as $\Delta(t) = t - u(t)$, where $u(t)$ is the generation time of the most recently delivered update. However, in the presence of error detection (CRC), finite-blocklength error correction, or multi-round/layer HARQ protocols, the actual instant at which a packet at the receiver truly becomes error-free may deviate from application-layer ACK events, leading to ambiguity and underestimation of freshness loss.

C-AoEI refines the measurement of information staleness by coupling AoI increments or resets to the true moment an error-free update is established, as determined by the interplay of CRC indicators, decoding results, and potentially backtracking recovery. This metric thus bridges the physical-to-application-layer gap in age measurement.

For systems with packet-layer L-HARQ and backtracking, (Zhao et al., 28 Jan 2026) defines the average C-AoEI as

$$\Delta^E = \frac{\mathbb{E}[Y^2]}{2\,\mathbb{E}[Y]} + \mathbb{E}[B],$$

where $Y$ is the renewal interval (interdeparture time between successfully decoded packets) and $B$ the backtracking depth in time slots recovered upon HARQ success. In CRC-aided finite blocklength systems, "Reported AoI" (RAoI) increments if a CRC check fails and resets to a normalized blocklength value if it passes (Raikar et al., 2023).

2. System Models and C-AoEI-Driven Metrics

C-AoEI is tailored for cross-layer systems with:

• Time-slotted scheduling with packet transmission over noisy, finite blocklength channels;
• Layer-coded HARQ protocols with feedback and potential re-encoding/backtracking;
• Use of error detection codes (e.g., CRC), short blocklength error correction codes (cyclic or DL-based), and adaptive resource allocation (rate, power, code structure).

The relevant variables include:

• Blocklength $n$, message length $k$, CRC overhead $c$;
• Per-slot transmit power $P$, set by scheduling;
• Error detection (CRC pass/fail, $v(t)$), which determines whether RAoI resets or increments.

The RAoI evolution (Raikar et al., 2023) for user $i$ at slot $t$ is given by: $$\Delta_i(t) = \begin{cases} \Delta_i(t-1) + 1, &\text{if}\ u_i(t) v_i(t) = 0 \ n_i / N, &\text{if}\ u_i(t) v_i(t) = 1 \end{cases}$$ with $u_i(t)$ indicating if user $i$ is scheduled and $v_i(t)$ the CRC pass.

In L-HARQ-based systems (Zhao et al., 28 Jan 2026), C-AoEI incorporates both feedforward and backtracking success, thus integrating temporal effects of physical-layer diversity and packet-level mixing with AoI.

3. Optimization Formulations and Scheduling Policies

Minimizing C-AoEI generally involves optimizing over scheduling, coding, and resource allocation under system constraints.

In finite-blocklength, CRC-based systems (Raikar et al., 2023): $$\min_{\pi}\ A^{D,C} = \lim_{T\to\infty}\ \frac{1}{TM} \mathbb{E}\left[ \sum_{t=1}^T \sum_{i=1}^M w_i \Delta_i(t) \right]$$ subject to:

• Scheduling: $\sum_i u_i(t) \leq 1$, $u_i(t)\in\{0,1\}$
• Long-term average power: $$\lim_{T\to\infty} \frac{1}{T}\mathbb{E}\left[\sum_{t,i} u_i(t)P_i(t)\right] \leq \bar{P}$$
• Distortion: $$\lim_{T\to\infty} \frac{1}{T}\mathbb{E}\left[\sum_t u_i(t)v_i(t)d_i(k_i(t))\right] \leq \bar{d}_i$$

Key policy types include:

• Stationary Randomized Policy (SRP): Age-agnostic, selecting $(i,k,P)$ per slot using stationary probabilities, subject to power and distortion constraints. The achievable RAoI is bounded within a factor 2 of optimal.
• Drift-Plus-Penalty (DPP) Policy: Age-aware, incorporating virtual queues for constraints, using Lyapunov drift minimization to adaptively schedule transmissions, achieving tighter bounds with mean-rate stability.

For multi-rate, error-prone update channels (Yao et al., 2020), the update rate and error probability selection are optimized via threshold-based Markov decision processes, exploiting the quasi-convex structure of the cost function for efficient solution.

4. Cross-Layer Parameter Dependencies

C-AoEI links to lower-layer statistics in several key respects:

• Channel Dynamics: Channel error under fading (e.g., shadowed-Rician PDF) directly influences per-round packet error $\epsilon_s(z)$, altering $\mathbb{E}[Y]$ and $\mathbb{E}[B]$ (Zhao et al., 28 Jan 2026). Increased LoS power or bandwidth lowers $\epsilon_s$ and thus the C-AoEI.
• Protocol Parameters: Propagation and feedback delay, HARQ round count $K$, and code blocklengths impact both $Y$ and $B$, with more HARQ rounds generally reducing residual errors up to a point.
• Resource Control: Scheduling policies optimize over transmit SNR, coding rate, and packet-mixing ratios to jointly minimize C-AoEI while meeting throughput, power, and latency requirements.
• Error Detection Code Overhead: Longer CRCs improve detection but also reserve channel symbols, raising C-AoEI—a design trade-off between undetected error risk and update freshness (Raikar et al., 2023).

5. Algorithmic Implementations and Adaptivity

C-AoEI-aware algorithms have been proposed for diverse wireless contexts:

• Packet-Level Encoded L-HARQ: In multi-GBS satellite IoT, mixed retransmissions and backtracking decoding are executed, exploiting residual redundancy across time via explicit mixing and a priori information (Zhao et al., 28 Jan 2026).
• Adaptive Scheduling: Encoding decisions (packet mixes, weights) adapt to estimation of decoding probability. The adaptation is guided by closed-form C-AoEI sensitivities, balancing throughput and age performance using tunable thresholds and learning rates.
• Low-Complexity Solvers: In two-rate systems (Yao et al., 2020), threshold-based policies can be found via quasi-convex optimization and golden-section search, leveraging explicit cost structure.

6. Performance Metrics and Comparative Results

Empirical studies reveal several consistent insights:

Code Scheme / Policy	PRR	SRP	DPP
Cyclic (genie)	2.46	2.14	1.605
Cyclic (CRC-1)	1.92	2.07	1.555
DL-based (genie)	2.74	2.03	1.529
DL-based (CRC-1)	1.96	2.02	1.516
PPV bound	1.84	2.00	1.506

(PRR = Periodic Round Robin, SRP = Stationary Randomized, DPP = Drift-Plus-Penalty) (Raikar et al., 2023)

Key observed behaviors:

• DPP policy consistently outperforms age-agnostic strategies.
• DL-based short codes provide lower C-AoEI than classical cyclic codes at the same blocklength.
• Increasing average power or relaxing distortion constraints monotonically reduces C-AoEI.
• Lengthening CRC reliably closes the gap with genie-aided (perfect detection) limits but at the expense of slightly higher C-AoEI due to reduced payload efficiency.

In satellite IoT, the proposed C-AoEI-aware cross-layer approach achieves a 31.8% increase in transmission efficiency and 17.2% lower C-AoEI compared to stop-wait HARQ, and shows superior robustness to interference and channel dynamics (Zhao et al., 28 Jan 2026).

7. Main Insights and Implications

• Explicit error modeling is critical: Neglecting error detection (e.g., finite-blocklength CRC) leads to systematic underestimation of AoI and distorted resource trade-offs.
• Cross-layer design yields quantifiable benefits: By integrating channel, protocol, and application-layer metrics, C-AoEI-guided optimization achieves provable, near-optimal freshness under practical constraints.
• Adaptivity and learning are feasible: Structured policies—thresholds, weights, mixing ratios—can be efficiently learned or adapted based on closed-form C-AoEI sensitivities.
• Practical trade-off frontier: There exists a fundamental interplay among error detection strength, code blocklength, resource allocation, and update freshness, deterministically expressible via C-AoEI.

This framework supports a rigorous, analytically tractable approach for designing next-generation low-latency wireless protocols, especially under stringent reliability and efficiency requirements (Zhao et al., 28 Jan 2026, Raikar et al., 2023, Yao et al., 2020).