---
title: Age of Consecutive Error (AoCE) in Remote Estimation
url: https://www.emergentmind.com/topics/age-of-consecutive-error-aoce
type: topic
---

# Age of Consecutive Error (AoCE) in Remote Estimation

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 [2507.18514]. 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 [2410.03637].

## 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 [2507.18514].

In the significance-aware non-linear aging formulation, AoCE appears in the semantic cost
\[
c(S_t)=\bar d(X_t,\hat X_t)\;g_{X_t,\hat X_t}(\Delta_t),
\]
where \(\bar d(X,\hat X)=D_{i,j}\) if \((X,\hat X)=(i,j)\), \(i\neq j\), and \(0\) otherwise, while \(g_{i,j}(\delta)\ge 0\) is non-decreasing, possibly unbounded, with \(g_{i,j}(0)=g_{i,j}(1)=1\). This formulation makes AoCE not only a counter of persistence but also an argument of an error-specific aging law [2410.03637].

## 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 \(\Theta_t\), which measures at the receiver how stale the last successfully received update is:
\[
\Theta_t = t - \max\{\tau\le t: Y_\tau\neq\mathcal E\}.
\]
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 \(X_t\) via the MAP estimator \(g(Z_t,\Theta_t)\), whereas AoCE governs the penalty for leaving an error uncorrected [2507.18514].

| Metric | Definition or reset behavior | Primary role |
|---|---|---|
| AoI | Measures stale last successfully received update; \(AoI(t)=t-U_t\) 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 \(g_{i,j}(\cdot)\) | 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 \(g_{i,j}(\delta)=\delta\) or \(g_{i,j}(\delta)=e^{\beta\delta}\), one recovers traditional or exponential-age metrics, respectively [2410.03637].

## 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
\[
S_t=(\,X_t,\;Z_{t-1},\;\Theta_{t-1},\;\Delta_{t-1}\,),
\]
and the one-step semantic cost under action \(U_t\in\{0,1\}\) is
\[
l(S_t,U_t)=\E\bigl[d(X_t,\hat X_t)\,\rho(\Delta_t)\mid S_t,U_t\bigr]+\lambda\,U_t.
\]
The long-run objective is to minimize
\[
\mathcal J(\pi)=\limsup_{T\to\infty}\frac1T
\E^\pi\Bigl[\sum_{t=0}^{T-1}d(X_t,\hat X_t)\,\rho(\Delta_t)\Bigr]
\]
subject to the transmission frequency constraint
\[
\limsup_{T}T^{-1}\sum_t\{U_t=1\}\le F_{\max}.
\]
Equivalently, one solves the unconstrained MDP
\[
\inf_\pi\bigl\{\mathcal J(\pi)+\lambda\,F(\pi)\bigr\}.
\]
The resulting optimization problem is a CMDP with unbounded costs [2507.18514].

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 \(X_t\in\{1,\dots,M\}\) with transition matrix \(Q_{i,j}=\Pr\{X_{t+1}=j|X_t=i\}\), state \(1\) is “alarm,” the sensor action is \(A_t\in\{0,1\}\), and the channel drop is \(H_t\sim\mathrm{Bernoulli}(p_s)\). On success \((H_t=1,A_t=1)\), the receiver sets \(\hat X_{t+1}=X_t\); otherwise \(\hat X_{t+1}=\hat X_t\). The controlled Markov state is \(S_t=(X_t,\hat X_t,\Delta_t)\), the stage cost is
\[
l(S_t,A_t)=c(S_t)+\lambda\cdot 1\{A_t=1\},
\]
and the long-run average-cost objective is
\[
\mathcal L(\pi)=\limsup_{T\to\infty}\tfrac1T E^\pi\bigl[\sum_{t=1}^T l(S_t,A_t)\bigr].
\]
For this model, there exists a stationary deterministic \(\pi^*\) and a bias function \(h(s)\) satisfying the Bellman optimality equation [2410.03637].

## 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 \((x,z,\theta)\) with \(x\ne g(z,\theta)\), there is a threshold \(\delta^*_{x,z,\theta}\) such that
\[
\pi(x,z,\theta,\delta)
=
\begin{cases}
1, & x\neq g(z,\theta)\quad\text{and}\quad \delta\ge \delta^*_{x,z,\theta},\\
0, & \text{otherwise.}
\end{cases}
\]
Hence transmission is triggered only when the current error has existed for at least \(\delta^*_{x,z,\theta}\) 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 [2507.18514].

Under additional symmetry, the switching structure simplifies further. For symmetric Markov chains
\[
Q_{i,j}=
\begin{cases}
\sigma,&i\neq j,\\
1-(|\mathcal X|-1)\sigma,&i=j,
\end{cases}
\qquad
\sigma\le\frac1{|\mathcal X|},
\]
together with Hamming distortion \(d(x,\hat x)=1_{x\neq \hat x}\), the MAP estimator reduces to zero-order hold and the optimal policy admits a single AoCE threshold \(\delta^*\), independent of \((x,z,\theta)\). In that case,
\[
\pi(x,z,\theta,\delta)=1\iff\delta\ge\delta^*.
\]
This is the classic threshold policy [2507.18514].

The significance-aware non-linear aging formulation establishes an analogous switching theorem: for each error type \((i,j)\) with age \(\delta\ge 1\), there is a threshold \(\tau^*_{i,j}\) such that
\[
\pi^*(i,j,\delta)=
\begin{cases}
1,& \delta\ge \tau^*_{i,j},\\
0,& \text{otherwise.}
\end{cases}
\]
The stated intuition is that the sensor waits until the cost of consecutive error \(g_{i,j}(\delta)\cdot D_{i,j}\) outweighs communication cost \(\lambda\). Under symmetric source dynamics \(Q_{i,j}=p\) for \(i\neq j\) and uniform \(D_{i,j}=D\), \(g_{i,j}=g\), the optimal policy has a single threshold \(\tau^*\) for all errors; this recovers classical AoI/AoII-threshold policies as special cases [2410.03637].

## 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 \(\delta\) to search each \(\delta^*_{x,z,\theta}\) 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 [2507.18514].

In the significance-aware non-linear aging model, computation begins by truncating each error age to \(N\) via
\[
\Delta_t(N)=\min\{\Delta_t,N\},
\]
which yields a finite MDP state space \(S^N\). Theorem 4 states asymptotic optimality:
\[
L^*(N)\to L^*
\]
exponentially fast in \(N\), with rate \(\sim (\max_i Q_{i,i}p_f)^N\). Structured policy iteration then alternates between policy evaluation and policy improvement in increasing \(\Delta\) order; once \(a=1\) is optimal at age \(\delta\), the algorithm enforces \(a=1\) for all larger ages. This reduces the search from \(2^{|S^N|}\) policies to at most \(\prod_{i\neq j}\tau^*_{i,j}\) thresholds [2410.03637].

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 \(|\mathcal X|=3\), \(\sigma=0.1\), and \(\rho(\delta)=1.2e^{0.55\delta}+0.3\), the two-threshold mixture policy shows that when the allowed frequency \(F_{\max}<0.25\), no single threshold matches exactly, so the optimal policy mixes two policies whose thresholds differ by one. The reported piecewise-constant curves of \(F^\lambda\) and \(\mathcal J^\lambda\) confirm that only switching policies and their mixtures are needed. Under the same \(F_{\max}\), 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 [2507.18514].

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 \(M=4\), asymmetric \(Q\), \(\lambda\in\{0,\dots,5\}\), \(p_s\in[0.4,1]\), \(D_{i,j}=1\), exponential aging for missed alarms, and logarithmic aging for false alarms, thresholds increase first for false or normal errors as \(\lambda\) increases or \(p_s\) decreases, whereas missed-alarm thresholds remain small. The same study states that with more semantic attributes—state-aware distortion and non-linear \(g_{i,j}\)—the system saves up to \(30\)–\(50\%\) of transmissions over distortion-only or AoI-only schemes at the same average cost [2410.03637].

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
\[
\mathcal V_t^k=\bigcap_{i=1}^k\{\Delta_r(t-k+i)>\zeta\},
\qquad
\Psi_r^k=\limsup_{T\to\infty}\frac{1}{T-k+1}\sum_{t=k}^T\mathbb E[\mathbf 1\{\mathcal V_t^k\}],
\]
and aggregates them via a weighted objective \(\bar\Psi=\sum_{k=1}^{k_{\max}} w_k\Psi_r^k\). 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 [2605.13002].

Source: https://www.emergentmind.com/topics/age-of-consecutive-error-aoce