On the Role of Age and Semantics of Information in Remote Estimation of Markov Sources

Abstract

This paper studies semantics-aware remote estimation of Markov sources. We leverage two complementary information attributes: the urgency of lasting impact, which quantifies the significance of consecutive estimation error at the transmitter, and the age of information (AoI), which captures the predictability of outdated information at the receiver. The objective is to minimize the long-run average lasting impact subject to a transmission frequency constraint. The problem is formulated as a constrained Markov decision process (CMDP) with potentially unbounded costs. We show the existence of an optimal simple mixture policy, which randomizes between two neighboring switching policies at a common regeneration state. A closed-form expression for the optimal mixture coefficient is derived. Each switching policy triggers transmission only when the error holding time exceeds a threshold that depends on both the instantaneous estimation error and the AoI. We further derive sufficient conditions under which the thresholds are independent of the instantaneous error and the AoI. Finally, we propose a structure-aware algorithm, Insec-SPI, that computes the optimal policy with reduced computation overhead. Numerical results demonstrate that incorporating both the age and semantics of information significantly improves estimation performance compared to using either attribute alone.

Figures (11)

• Figure 1: Remote state estimation of a Markov source.
• Figure 2: Time ordering of the relevant variables.
• Figure 3: Evolution of the belief value of a binary Markov chain. (a) The stationary distribution of the chain is $\mu = (0.6, 0.4)$. When the source was last observed in state $1$, i.e., $Z_t = 1$, the MAP estimate is $\hat{X}_t = 1$ if $0 \leq \Theta_t \leq 3$, and $\hat{X}_t = 0$ otherwise. When $Z_t = 0$, the MAP estimate remains $\hat{X}_t = Z_t = 0$ for all $\Theta_t \geq 0$. In (b), the MAP estimator enters a steady state when $Z_t = 0$ and $\Theta_t \geq 3$, or when $Z_t = 1$ and $\Theta_t \geq 4$.
• Figure 4: Schematic representation of the Insec-SPI algorithm.
• Figure 5: Illustration of the bisection and intersection search methods. The function $F^\lambda$ is piecewise constant and non-increasing, while $\mathcal{L}^\lambda$ is piecewise linear and concave. Crosses in (a) indicate bisection points, and squares in (b) denote intersection points. The optimal Lagrange multiplier $\lambda^*$ corresponds to a breakpoint of $F^\lambda$ and a corner point of $\mathcal{L}^\lambda$.

Theorems & Definitions (18)

• Definition 1
• Definition 2
• Definition 3
• Proposition 1
• Proposition 2
• Theorem 1
• Lemma 1
• Proposition 3
• Definition 4
• Definition 5