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Structural Monotonicity in Transmission Scheduling for Remote State Estimation with Hidden Channel Mode

Hampei Sasahara

TL;DR

This work addresses transmission scheduling for remote state estimation over channels with a hidden binary mode by formulating the problem as a POMDP on the state $(\tau,\theta)$. To overcome the lack of order preservation due to partial observability, it introduces state-space folding, transforming the holding-time dynamics into a folded space where TP2 properties hold, enabling monotonicity analysis. The authors prove that the value function is increasing in the holding time $\tau$ and the belief $b$ of the unfavorable channel, and they establish a threshold structure for an associated optimal stopping policy with a monotone threshold $b_{\rm th}(\tau)$. This yields principled, low-complexity scheduling decisions and provides a foundation for efficient computation and robust remote estimation in partially observed channel environments.

Abstract

This study treats transmission scheduling for remote state estimation over unreliable channels with a hidden mode. A local Kalman estimator selects scheduling actions, such as power allocation and resource usage, and communicates with a remote estimator based on acknowledgement feedback, balancing estimation performance and communication cost. The resulting problem is naturally formulated as a partially observable Markov decision process (POMDP). In settings with observable channel modes, it is well known that monotonicity of the value function can be established via investigating order-preserving property of transition kernels. In contrast, under partial observability, the transition kernels generally lack this property, which prevents the direct application of standard monotonicity arguments. To overcome this difficulty, we introduce a novel technique, referred to as state-space folding, which induces transformed transition kernels recovering order preservation on the folded space. This transformation enables a rigorous monotonicity analysis in the partially observable setting. As a representative implication, we focus on an associated optimal stopping formulation and show that the resulting optimal scheduling policy admits a threshold structure.

Structural Monotonicity in Transmission Scheduling for Remote State Estimation with Hidden Channel Mode

TL;DR

This work addresses transmission scheduling for remote state estimation over channels with a hidden binary mode by formulating the problem as a POMDP on the state . To overcome the lack of order preservation due to partial observability, it introduces state-space folding, transforming the holding-time dynamics into a folded space where TP2 properties hold, enabling monotonicity analysis. The authors prove that the value function is increasing in the holding time and the belief of the unfavorable channel, and they establish a threshold structure for an associated optimal stopping policy with a monotone threshold . This yields principled, low-complexity scheduling decisions and provides a foundation for efficient computation and robust remote estimation in partially observed channel environments.

Abstract

This study treats transmission scheduling for remote state estimation over unreliable channels with a hidden mode. A local Kalman estimator selects scheduling actions, such as power allocation and resource usage, and communicates with a remote estimator based on acknowledgement feedback, balancing estimation performance and communication cost. The resulting problem is naturally formulated as a partially observable Markov decision process (POMDP). In settings with observable channel modes, it is well known that monotonicity of the value function can be established via investigating order-preserving property of transition kernels. In contrast, under partial observability, the transition kernels generally lack this property, which prevents the direct application of standard monotonicity arguments. To overcome this difficulty, we introduce a novel technique, referred to as state-space folding, which induces transformed transition kernels recovering order preservation on the folded space. This transformation enables a rigorous monotonicity analysis in the partially observable setting. As a representative implication, we focus on an associated optimal stopping formulation and show that the resulting optimal scheduling policy admits a threshold structure.
Paper Structure (12 sections, 14 theorems, 11 equations, 3 figures)

This paper contains 12 sections, 14 theorems, 11 equations, 3 figures.

Table of Contents

  1. INTRODUCTION
  2. Mathematical Preliminaries
  3. PROBLEM FORMULATION
  4. Remote State Estimation, Transmission Scheduling, Channel Model
  5. POMDP Formulation
  6. MONOTONICITY PROPERTIES
  7. Issue: Lack of Order Preservation in Transition Kernel
  8. Core Idea: State-Space Folding
  9. Monotonicity Properties
  10. Structural Threshold Property of Optimal Stopping Policy
  11. NUMERICAL EXAMPLE
  12. CONCLUSION

Key Result

Lemma 1

The FSD $p_1\leq_{\rm s}p_2$ holds iff $\sum_{x}v(x)p_1(x)\leq\sum_{x}v(x)p_2(x)$ for any increasing function $v:\mathcal{X}\to\mathbb{R}$.

Figures (3)

  • Figure 1: Overall system architecture.
  • Figure 2: State-space folding.
  • Figure 3: Threshold structure of optimal stopping policy.

Theorems & Definitions (25)

  • Definition 1
  • Lemma 1
  • Definition 2
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Definition 3
  • Lemma 5
  • Proposition 1
  • Proposition 2
  • ...and 15 more