Error-Aware Sampling & Transmission Policy

• Error-aware joint sampling and transmission policies are protocols that dynamically control sensor updates based on the current error and state, reducing reconstruction mismatches.
• They employ state-dependent, randomized decisions with constrained optimization to balance actuation cost and sampling rates.
• These methods adapt to various source models and resource limits, outperforming AoI and semantics-aware policies by significantly lowering error metrics.

An error-aware joint sampling and transmission policy is a class of protocols for networked real-time monitoring systems, wherein the sampling and forwarding of sensor measurements are dynamically controlled based on the instantaneous tracking error, source state, and actuation cost criteria, accounting for both communication unreliability and resource constraints. These policies are designed to minimize system-level performance metrics—such as real-time reconstruction error, actuation error, and error run-lengths—subject to constraints on sampling and transmission rates, with the operational objective of selectively allocating sampling/transmission opportunities to the most critical events or states. The central methodological shift is to make sampling depend explicitly on the current error or misalignment between the observed and reconstructed source state, sometimes further incorporating the semantic importance of different states.

1. Problem Formulation and Source Models

The canonical setting involves a finite-state Markov source $X(t)$ (often two-state, but generalizable to $N$ states) observed in discrete time by a sampler. The sampler may, at any slot, observe $X(t)$ and—according to its policy—choose to generate and transmit a sample. Transmission is typically over a memoryless erasure wireless channel: a sample, if sent, is successfully delivered with state-dependent probability $p_{s_i}$ (where $i$ indexes the current value of $X(t)$). The receiver/monitor reconstructs the last successfully received state as $\hat{X}(t)$, and employs $\hat{X}(t)$ to actuate a control or output (Salimnejad et al., 2023, Salimnejad et al., 2023, Salimnejad et al., 2023, Salimnejad et al., 21 Dec 2025).

Acknowledgment (ACK/NACK) feedback ensures that the transmitter maintains knowledge of the current reconstruction, essential for state-aware or error-aware decisions.

Key extensions include:

• Multiple correlated Markovian sources tracked by independent samplers over a shared channel (Salimnejad et al., 21 Dec 2025).
• Continuous-time Markov chains with general state spaces (Cosandal et al., 11 Jul 2024).
• Wiener or Ornstein–Uhlenbeck sources for mean-square estimation in continuous time (Pan et al., 2023, Çıtır et al., 30 Jul 2024, Pan et al., 2022).

2. Performance Metrics: Reconstruction, Actuation, and Error Runs

Error-aware policies directly target a broad set of metrics beyond raw Age of Information (AoI):

• Reconstruction error probability: $P_E = \mathbb{P}[X(t) \neq \hat{X}(t)]$, capturing the steady-state probability of mismatch (Salimnejad et al., 2023, Salimnejad et al., 2023, Salimnejad et al., 21 Dec 2025).
• Actuation error cost: $P_E^C = \sum_{i\neq j} C_{i,j} \pi_{i,j}$ quantifying penalties incurred when the reconstructed state leads to incorrect actuation, with $C_{i,j}$ state-dependent weights and $\pi_{i,j}$ the stationary joint probability (Salimnejad et al., 2023).
• Consecutive error (run-length): Distribution and expected length $\bar{C}_E$ of uninterrupted error periods, particularly sensitive in control or safety contexts (Salimnejad et al., 2023, Salimnejad et al., 2023).
• Importance-aware consecutive error: Run-length metrics focused on critical errors, e.g., only counting consecutive periods where $X=1,\hat{X}=0$ (Salimnejad et al., 2023).
• Other semantic or user-defined metrics: For instance, age of incorrect information (AoII), capturing time since the last correct actuation (Cosandal et al., 11 Jul 2024, Zakeri et al., 2023, Zakeri et al., 2023).

Closed-form expressions for all these metrics can be derived for randomized stationary policies and, in some models, for semantics-aware or multi-threshold rules.

3. State-Aware Randomized Stationary Policy (RS): Structure and Analysis

The principal error-aware scheduling paradigm is the state-aware randomized stationary (RS) policy (Salimnejad et al., 2023, Salimnejad et al., 2023, Salimnejad et al., 2023, Salimnejad et al., 21 Dec 2025). In this approach, the sampling-and-transmission decision is a Bernoulli trial whose probability $p_{\alpha^s_i}$ depends on the instantaneous source state $i$:

• At each slot $t$, if $X(t)=i$, the sampler draws $\alpha^s_i \sim \text{Bern}(p_{\alpha^s_i})$; if $\alpha^s_i=1$, a sample is taken and transmitted.
• If $p_{\alpha^s_0} \neq p_{\alpha^s_1}$, the policy is state-aware, assigning more (or less) frequent updates to critical states or to those with lower channel success rates $p_{s_i}$.

All stationary performance metrics can be written as analytical functions of the steady-state joint probability $\pi_{i,j}$ of source and reconstructed state, themselves algebraic expressions in the transition probabilities $p$, $q$, sampling probabilities $p_{\alpha^s_i}$, and channel parameters.

In the joint multi-source setting, each sampler operates a similar Bernoulli process with error-awareness: it only samples if the local process disagrees with the current reconstruction, and then only with prescribed probability $q_{\alpha_m}$ (Salimnejad et al., 21 Dec 2025).

4. Constrained Optimization and Optimal Parameter Selection

Error-aware policies are typically implemented under average sampling (or actuation) cost constraints:

• The mean sampling rate is $$\gamma = \mathbb{E}[\alpha^s(t)] = \pi_0 p_{\alpha^s_0} + \pi_1 p_{\alpha^s_1}$$, where $\pi_i$ is the stationary marginal of $X(t)$.
• A budget constraint $\delta\,\gamma \leq \delta_{\text{max}}$ (with $\delta$ the unit cost) imposes a linear constraint coupling the sampling probabilities.

The joint sampling/transmission scheduling problem is formulated as:

$$\begin{aligned} \min_{p_{\alpha^s_0},\,p_{\alpha^s_1}} \quad & P_E^C(p_{\alpha^s_0},\,p_{\alpha^s_1}) \ \text{subject to} \quad & \gamma(p_{\alpha^s_0},\,p_{\alpha^s_1}) \leq \eta \end{aligned}$$

where $\eta=\delta_{\max}/\delta$.

The Lagrangian $L$ for this convex/quasi-convex program admits explicit Karush-Kuhn-Tucker (KKT) solutions in several cases:

• If $p C_{1,0} \geq q C_{0,1}$, maximize $p_{\alpha^s_1}$ given budget, then solve a quadratic-in-ratio problem for $p_{\alpha^s_0}$ within admissible bounds.
• Conversely, the opposite for $q C_{0,1}>p C_{1,0}$.
• Boundary cases arise when cost or channel parameters are extreme; e.g., if only one state is critical, sample exclusively when in that state (Salimnejad et al., 2023, Salimnejad et al., 2023, Salimnejad et al., 21 Dec 2025).

For multiple sources, similar optimization is performed over the sampling probabilities $q_{\alpha_m}$ for each process, subject to a global sampling budget (Salimnejad et al., 21 Dec 2025).

5. Comparative Performance against Semantics-Aware and AoI-Based Policies

State-aware error-driven policies are benchmarked against semantics-aware and AoI-minimizing scheduling:

• Semantics-aware policies: Trigger updates only in response to meaningful source events (e.g., state changes impacting estimation or actuation). These are optimal under slow source evolution or when semantic mis-actions (e.g., acting on stale safety-critical information) dominate, but can be suboptimal under faster dynamics or when persistent error runs need breaking (Salimnejad et al., 2023, Salimnejad et al., 2023, Salimnejad et al., 2023).
• Uniform, periodic, or AoI-threshold policies: Ignore instantaneous error and transmit at regular intervals or when age exceeds a threshold. Such policies are asymptotically suboptimal, particularly in fast sources or under tight cost constraints.
• Error-aware policies: By selectively increasing sampling on critical states or when in persistent error, RS and error-aware variants strictly outperform semantics-aware and uniform policies in reconstruction error, actuation cost, and run-length metrics—reducing cost or error by 30–50% in typical regimes (Salimnejad et al., 2023, Salimnejad et al., 21 Dec 2025).

In continuous-state or mean-square estimation settings, the error-aware “threshold” policy—sampling when the instantaneous estimation error $|e(t)|$ exceeds an optimally tuned value—dominates age-based or zero-wait strategies, particularly under variable channel delays (Pan et al., 2023, Çıtır et al., 30 Jul 2024).

6. Extensions: Multi-Source, Semantic, and Continuous-Time Models

The error-aware policy framework generalizes to:

• Correlated multi-source Markov processes: Each sampler employs error-aware (or semantics-aware) logic, and the sampling intensities are optimized numerically, showing greatest gains when sources are strongly correlated and budgets tight (Salimnejad et al., 21 Dec 2025).
• Continuous-time Markov chains (CTMCs): The optimal error-aware policy (ESAT) is a multi-threshold rule, with thresholds tailored to both the current source state and the monitor's estimate. These thresholds are derived as solutions to a constrained semi-Markov decision process using multi-regime phase-type (MRPH) distributional analysis (Cosandal et al., 11 Jul 2024).
• Resource-harvesting and partially observable sources: POMDP-based error-aware policies adapt sampling and transmission to both belief state (inferred from feedback) and energy availability, yielding non-monotonic, multi-threshold structures (Zakeri et al., 2023, Zakeri et al., 2023).
• Mean-square estimation: Error-thresholding generalizes from two-state error indicators to continuous-valued estimation error, with sampling triggered by the first hitting of an error boundary optimized to minimize the Lagrange-relaxed MSE-plus-cost objective (Pan et al., 2023, Çıtır et al., 30 Jul 2024).

The table below summarizes representative policy structures and optimization regimes:

Policy	Sampling Trigger	Cost Constraint Handling
RS	State/error-aware randomized Bernoulli	KKT opt. on per-state sampling probs
Semantics	Only on meaningful/semantic state changes	Direct, budget via event frequency
Threshold		e(t)
Multi-source	Error-aware per source, feedback-based	Jointly optimize sampling vector
CTMC ESAT	Multi-threshold, estimation & state-aware	Policy iteration in CSMDP (MRPH)

7. Practical Insights and Design Guidelines

• Bias sampling to critical/difficult states: Assign higher update rates to states with high actuation error, poor channel delivery, or high mis-estimate penalties (Salimnejad et al., 2023).
• Tune sampling rate to cost budget: Aggressively increase sampling to the threshold permitted by constraints—optimal policies nearly always saturate the cost or rate budget (Salimnejad et al., 2023, Salimnejad et al., 2023).
• Error-aware triggering breaks long error runs: By incorporating error run-length into the policy, long out-of-sync periods are suppressed even under tight cost regimes, reducing the tail of error burst lengths and associated actuation risk (Salimnejad et al., 2023, Salimnejad et al., 2023).
• Parameter optimization is explicit or low-complexity: For discrete-state sources, polynomials, ratios and closed-form roots often suffice. For continuous-state estimation, one nonlinear equation defines the optimal error threshold (Pan et al., 2023, Çıtır et al., 30 Jul 2024).
• Non-monotonic and multi-threshold policies: In non-trivial partially observable or semantic metrics, optimal policies are not necessarily monotonic in age or error, requiring switching curves numerically derived from system dynamics (Zakeri et al., 2023, Cosandal et al., 11 Jul 2024).

In summary, error-aware joint sampling and transmission policies constitute the analytically and numerically optimal regime for resource-constrained networked monitoring and actuation in Markovian (and more general) sources. They combine analytical tractability, minimal performance metrics, and adaptability to system dynamics and constraints, outperforming a broad range of AoI, semantics-aware, and naive baselines across settings (Salimnejad et al., 2023, Salimnejad et al., 2023, Salimnejad et al., 21 Dec 2025).