Minimizing Functions of Age of Incorrect Information for Remote Estimation

TL;DR

This work tackles remote estimation with a general AoII-based cost by modeling a push-based, threshold-driven transmission policy as a discrete-time semi-Markov decision process. It introduces dual-regime absorbing Markov chains (DR-AMC) and dual-regime phase-type (DR-PH) distributions to accurately characterize the time-to-absorption and AoII evolution under a policy, enabling closed-form or efficiently computable expressions for SMDP parameters. The authors derive explicit parameter calculations for polynomial AoII penalties, implement a policy-iteration solution, and demonstrate that the resulting multi-threshold policy significantly outperforms benchmarks and matches exhaustive-search optimality. The framework provides a scalable and flexible approach to optimizing AoII-costs with arbitrary penalties in remote estimation systems.

Abstract

The age of incorrect information (AoII) process which keeps track of the time since the source and monitor processes are in sync, has been extensively used in remote estimation problems. In this paper, we consider a push-based remote estimation system with a discrete-time Markov chain (DTMC) information source transmitting status update packets towards the monitor once the AoII process exceeds a certain estimation-based threshold. In this paper, the time average of an arbitrary function of AoII is taken as the AoII cost, as opposed to using the average AoII as the mismatch metric, whereas this function is also allowed to depend on the estimation value. In this very general setting, our goal is to minimize a weighted sum of AoII and transmission costs. For this purpose, we formulate a discrete-time semi-Markov decision process (SMDP) regarding the multi-threshold status update policy. We propose a novel tool in discrete-time called 'dual-regime absorbing Markov chain' (DR-AMC) and its corresponding absorption time distribution named as 'dual-regime phase-type' (DR-PH) distribution, to obtain the characterizing parameters of the SMDP, which allows us to obtain the distribution of the AoII process for a given policy, and hence the average of any function of AoII. The proposed method is validated with numerical results by which we compare our proposed method against other policies obtained by exhaustive-search, and also various benchmark policies.

Figures (5)

• Figure 1: A remote estimation system with the source process $X_t$ and the monitor process $\hat{X}_t$. At each time slot, the source can sample the process $X_t$, and initiate a transmission. If the process changes before the end of the time slot, the transmission is preempted, otherwise, it reaches the monitor with probability $\sigma$, i.e., packet transmissions are geometrically distributed with parameter $\sigma$. The monitor updates its estimation with received updates (marked with dashed circles).
• Figure 2: A sample path of processes $X_t$, $\hat{X}_t$, and AoII$_t$ for a general policy, and AoII penalty functions $f_1(x)=\sqrt{x}$, $f_2(x)=x^2$, and $f_3(x)=x$, where $x = \text{AoII}_t$. Successful transmissions are shown with blue arrows, and crossed red and purple arrows indicate failed and preempted transmissions, respectively. AoII is reset upon synchronization (indicated with star). Otherwise, it increases by one at every time slot as long as the mismatch condition between $X_t$ and $\hat{X}_t$ stays.
• Figure 3: A sample path with two complete cycles for a threshold policy.
• Figure 4: Each point shows the average AoII cost when using the threshold pair $(\tau_1,\tau_2)$ with a color map. Circle and cross markers are used to show the threshold values that minimize the average cost using exhaustive search and the proposed SMDP algorithm, respectively, for a given weight $\lambda$, the values of which are given along with the markers. As an example, when $\lambda \in \{ 68,\ldots,75 \}$, the threshold pair $(5,10)$ is shown to be the optimum multi-threshold policy using both methods.
• Figure 5: Comprasion of benchmark policies with proposed SMDP policy for $\bm{Q}_2$$(N=3)$, and $\bm{Q}_3$$(N=10)$ with varying $\lambda$

Theorems & Definitions (6)

• Definition 1: Embedded Points
• Definition 2: Dual-regime Absorbing Markov Chain
• Definition 3: Dual-regime Phase Type Distribution
• Lemma 1
• Lemma 2: Polynomial AoII Penalty Functions
• Definition 4: Factorial moment