Age of Consecutive Error (AoCE) in Remote Estimation

• Age of Consecutive Error (AoCE) is a semantics-aware metric that measures the duration a nonzero estimation error persists in remote estimation systems.
• It guides the design of transmission policies by triggering updates only when the current error’s persistence exceeds a calculated threshold, reducing semantic costs.
• AoCE is integrated within CMDP frameworks, enabling efficient, structure-aware algorithms to manage unbounded costs and optimize remote estimation performance.

Age of Consecutive Error (AoCE), denoted $\Delta_t$, is a semantics-aware metric for remote estimation that quantifies how long the same nonzero estimation error persists. In the finite-state Markov-source setting, AoCE counts how many consecutive slots the system has remained in the same nonzero estimation error; $\Delta_t=0$ indicates no error, a newly appearing error resets the metric to $1$, and persistence of the same error increments it by one (Luo et al., 24 Jul 2025). In related significance-aware formulations, AoCE is embedded into error-dependent distortion models and non-linear age functions so that different error types can incur different semantic costs (Luo et al., 2024).

1. Formal definition and state evolution

Let $X_t$ be the true source state at time $t$ and $\hat X_t$ the estimate at the receiver. AoCE is defined as

$$\Delta_t \;\coloneqq\; \begin{cases} \Delta_{t-1} + 1, & X_t\neq \hat X_t,\; (X_t,\hat X_t)=(X_{t-1},\hat X_{t-1}),\ 1, & X_t\neq \hat X_t,\; (X_t,\hat X_t)\neq(X_{t-1},\hat X_{t-1}),\ 0, & X_t=\hat X_t. \end{cases}$$

Accordingly, $\Delta_t=0$ means no error at time $t$; each time a new error begins AoCE resets to $1$; if the same error persists, AoCE increases by one (Luo et al., 24 Jul 2025).

In the significance-aware non-linear aging formulation, AoCE appears in the semantic cost

$\Delta_t=0$0

where $\Delta_t=0$1 if $\Delta_t=0$2, $\Delta_t=0$3, and $\Delta_t=0$4 otherwise, while $\Delta_t=0$5 is non-decreasing, possibly unbounded, with $\Delta_t=0$6. This formulation makes AoCE not only a counter of persistence but also an argument of an error-specific aging law (Luo et al., 2024).

2. Semantic interpretation and relation to other age metrics

AoCE captures the persistence, or lasting impact, of a wrong estimate: the longer the same error remains, the larger its semantic cost. In the finite-state Markov-source formulation, it is paired with the Age of Information (AoI), denoted $\Delta_t=0$7, which measures at the receiver how stale the last successfully received update is: $\Delta_t=0$8 AoI quantifies freshness of content, whereas AoCE quantifies severity of an uncorrected error. Together they form a semantics-aware metric pair: AoI governs the predictability of $\Delta_t=0$9 via the MAP estimator $1$0, whereas AoCE governs the penalty for leaving an error uncorrected (Luo et al., 24 Jul 2025).

Metric	Definition or reset behavior	Primary role
AoI	Measures stale last successfully received update; $1$1 ignores source-state	Freshness
AoII	Increments on any error; resets only on synchronization	Error-aging without error-type reset
AoCE	Resets whenever the type of error changes; allows per-error age functions $1$2	Persistence and semantic severity of the same error

AoCE is therefore a refinement relative to source-agnostic aging metrics. The significance-aware formulation states explicitly that AoCE resets whenever the type of error changes and allows different costs and non-linear age functions for different estimation errors to account for their relative importance to system performance. By choosing $1$3 or $1$4, one recovers traditional or exponential-age metrics, respectively (Luo et al., 2024).

3. Optimization formulations in remote estimation

In the semantics-aware remote estimation of a finite-state Markov chain with a MAP estimator, AoCE is embedded into both the state and the cost of a constrained Markov decision process (CMDP). The transmitter-side information state is

$1$5

and the one-step semantic cost under action $1$6 is

$1$7

The long-run objective is to minimize

$1$8

subject to the transmission frequency constraint

$1$9

Equivalently, one solves the unconstrained MDP

$X_t$0

The resulting optimization problem is a CMDP with unbounded costs (Luo et al., 24 Jul 2025).

A related model studies the semantics-aware remote state estimation of an asymmetric Markov chain with prioritized states. There the source is a finite-state Markov chain $X_t$1 with transition matrix $X_t$2, state $X_t$3 is “alarm,” the sensor action is $X_t$4, and the channel drop is $X_t$5. On success $X_t$6, the receiver sets $X_t$7; otherwise $X_t$8. The controlled Markov state is $X_t$9, the stage cost is

$t$0

and the long-run average-cost objective is

$t$1

For this model, there exists a stationary deterministic $t$2 and a bias function $t$3 satisfying the Bellman optimality equation (Luo et al., 2024).

4. Threshold structure of optimal transmission policies

A central structural result is that AoCE induces switching policies. In the finite-state Markov-source CMDP, for every triplet $t$4 with $t$5, there is a threshold $t$6 such that

$t$7

Hence transmission is triggered only when the current error has existed for at least $t$8 slots; otherwise the transmitter idles. The same work shows the existence of an optimal simple mixture policy, which randomly selects between two deterministic switching policies with a fixed probability (Luo et al., 24 Jul 2025).

Under additional symmetry, the switching structure simplifies further. For symmetric Markov chains

$t$9

together with Hamming distortion $\hat X_t$0, the MAP estimator reduces to zero-order hold and the optimal policy admits a single AoCE threshold $\hat X_t$1, independent of $\hat X_t$2. In that case,

$\hat X_t$3

This is the classic threshold policy (Luo et al., 24 Jul 2025).

The significance-aware non-linear aging formulation establishes an analogous switching theorem: for each error type $\hat X_t$4 with age $\hat X_t$5, there is a threshold $\hat X_t$6 such that

$\hat X_t$7

The stated intuition is that the sensor waits until the cost of consecutive error $\hat X_t$8 outweighs communication cost $\hat X_t$9. Under symmetric source dynamics $$\Delta_t \;\coloneqq\; \begin{cases} \Delta_{t-1} + 1, & X_t\neq \hat X_t,\; (X_t,\hat X_t)=(X_{t-1},\hat X_{t-1}),\ 1, & X_t\neq \hat X_t,\; (X_t,\hat X_t)\neq(X_{t-1},\hat X_{t-1}),\ 0, & X_t=\hat X_t. \end{cases}$$0 for $$\Delta_t \;\coloneqq\; \begin{cases} \Delta_{t-1} + 1, & X_t\neq \hat X_t,\; (X_t,\hat X_t)=(X_{t-1},\hat X_{t-1}),\ 1, & X_t\neq \hat X_t,\; (X_t,\hat X_t)\neq(X_{t-1},\hat X_{t-1}),\ 0, & X_t=\hat X_t. \end{cases}$$1 and uniform $$\Delta_t \;\coloneqq\; \begin{cases} \Delta_{t-1} + 1, & X_t\neq \hat X_t,\; (X_t,\hat X_t)=(X_{t-1},\hat X_{t-1}),\ 1, & X_t\neq \hat X_t,\; (X_t,\hat X_t)\neq(X_{t-1},\hat X_{t-1}),\ 0, & X_t=\hat X_t. \end{cases}$$2, $$\Delta_t \;\coloneqq\; \begin{cases} \Delta_{t-1} + 1, & X_t\neq \hat X_t,\; (X_t,\hat X_t)=(X_{t-1},\hat X_{t-1}),\ 1, & X_t\neq \hat X_t,\; (X_t,\hat X_t)\neq(X_{t-1},\hat X_{t-1}),\ 0, & X_t=\hat X_t. \end{cases}$$3, the optimal policy has a single threshold $$\Delta_t \;\coloneqq\; \begin{cases} \Delta_{t-1} + 1, & X_t\neq \hat X_t,\; (X_t,\hat X_t)=(X_{t-1},\hat X_{t-1}),\ 1, & X_t\neq \hat X_t,\; (X_t,\hat X_t)\neq(X_{t-1},\hat X_{t-1}),\ 0, & X_t=\hat X_t. \end{cases}$$4 for all errors; this recovers classical AoI/AoII-threshold policies as special cases (Luo et al., 2024).

5. Algorithmic computation and complexity reduction

The structural monotonicity in AoCE supports specialized algorithms. In the Markov-source CMDP, Algorithm 1, termed SPI in the summary, leverages monotonicity in $$\Delta_t \;\coloneqq\; \begin{cases} \Delta_{t-1} + 1, & X_t\neq \hat X_t,\; (X_t,\hat X_t)=(X_{t-1},\hat X_{t-1}),\ 1, & X_t\neq \hat X_t,\; (X_t,\hat X_t)\neq(X_{t-1},\hat X_{t-1}),\ 0, & X_t=\hat X_t. \end{cases}$$5 to search each $$\Delta_t \;\coloneqq\; \begin{cases} \Delta_{t-1} + 1, & X_t\neq \hat X_t,\; (X_t,\hat X_t)=(X_{t-1},\hat X_{t-1}),\ 1, & X_t\neq \hat X_t,\; (X_t,\hat X_t)\neq(X_{t-1},\hat X_{t-1}),\ 0, & X_t=\hat X_t. \end{cases}$$6 in ascending order, stopping once a transmit decision first becomes optimal. Building on the switching structure, the same work develops the efficient structure-aware algorithm Insec-SPI, which computes the optimal policy with reduced computation overhead (Luo et al., 24 Jul 2025).

In the significance-aware non-linear aging model, computation begins by truncating each error age to $$\Delta_t \;\coloneqq\; \begin{cases} \Delta_{t-1} + 1, & X_t\neq \hat X_t,\; (X_t,\hat X_t)=(X_{t-1},\hat X_{t-1}),\ 1, & X_t\neq \hat X_t,\; (X_t,\hat X_t)\neq(X_{t-1},\hat X_{t-1}),\ 0, & X_t=\hat X_t. \end{cases}$$7 via

$$\Delta_t \;\coloneqq\; \begin{cases} \Delta_{t-1} + 1, & X_t\neq \hat X_t,\; (X_t,\hat X_t)=(X_{t-1},\hat X_{t-1}),\ 1, & X_t\neq \hat X_t,\; (X_t,\hat X_t)\neq(X_{t-1},\hat X_{t-1}),\ 0, & X_t=\hat X_t. \end{cases}$$8

which yields a finite MDP state space $$\Delta_t \;\coloneqq\; \begin{cases} \Delta_{t-1} + 1, & X_t\neq \hat X_t,\; (X_t,\hat X_t)=(X_{t-1},\hat X_{t-1}),\ 1, & X_t\neq \hat X_t,\; (X_t,\hat X_t)\neq(X_{t-1},\hat X_{t-1}),\ 0, & X_t=\hat X_t. \end{cases}$$9. Theorem 4 states asymptotic optimality: $\Delta_t=0$0 exponentially fast in $\Delta_t=0$1, with rate $\Delta_t=0$2. Structured policy iteration then alternates between policy evaluation and policy improvement in increasing $\Delta_t=0$3 order; once $\Delta_t=0$4 is optimal at age $\Delta_t=0$5, the algorithm enforces $\Delta_t=0$6 for all larger ages. This reduces the search from $\Delta_t=0$7 policies to at most $\Delta_t=0$8 thresholds (Luo et al., 2024).

These algorithmic results are significant because AoCE leads naturally to countably infinite-state control problems with unbounded costs. The computational leverage comes from the threshold structure itself rather than from replacing the original objective.

6. Numerical behavior, performance, and related persistence-aware metrics

The numerical illustrations in the finite-state Markov-source study emphasize mixture and threshold effects. For $\Delta_t=0$9, $t$0, and $t$1, the two-threshold mixture policy shows that when the allowed frequency $t$2, no single threshold matches exactly, so the optimal policy mixes two policies whose thresholds differ by one. The reported piecewise-constant curves of $t$3 and $t$4 confirm that only switching policies and their mixtures are needed. Under the same $t$5, the minimum average cost achieved using the MAP estimator is significantly lower than that achieved using zero-order hold when transmission is scarce, indicating that AoI+AoCE yields lower cost than AoCE alone in that regime (Luo et al., 24 Jul 2025).

The significance-aware non-linear aging study reports that the AoCE-optimal policy strictly outperforms randomized, periodic, distortion-optimal, AoI/AoII-optimal, and naïve threshold policies on AoCE that ignore error type. In the described setup with $t$6, asymmetric $t$7, $t$8, $t$9, $1$0, exponential aging for missed alarms, and logarithmic aging for false alarms, thresholds increase first for false or normal errors as $1$1 increases or $1$2 decreases, whereas missed-alarm thresholds remain small. The same study states that with more semantic attributes—state-aware distortion and non-linear $1$3—the system saves up to $1$4–$1$5 of transmissions over distortion-only or AoI-only schemes at the same average cost (Luo et al., 2024).

A broader persistence-aware perspective appears in wireless status-update systems through the consecutive Age Violation Rate (C-AVR) vector, whose components quantify AoI-threshold violations over consecutive time windows of different lengths. That framework defines

$1$6

and aggregates them via a weighted objective $1$7. Although C-AVR is not an AoCE metric, it reflects the same methodological concern with temporal persistence rather than one-step or time-averaged freshness alone. This suggests a common research direction in which persistence-sensitive objectives are treated as first-class control variables in status-update and remote-estimation systems (Pan et al., 13 May 2026).