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Optimal Sampling for Uncertainty-of-Information Minimization in a Remote Monitoring System

Xiaomeng Chen, Aimin Li, Shaohua Wu

TL;DR

This work studies uncertainty of information (UoI) minimization in a remote monitoring system with a binary Markov source and random transmission delays. It models sampling decisions as a POMDP and reduces it to a semi-Markov decision process via a belief state, then solves the optimal long-run average UoI using a two-layer bisec-RVI algorithm, providing a globally optimal policy. To reduce computation, it derives a sub-optimal index-based threshold policy that is provably near-optimal under large expected delays and demonstrates, through simulations, that both proposed policies outperform traditional zero-wait and AoI-optimal policies, with the sub-optimal policy closely tracking the optimal one under heavy delays. The results underscore the benefit of state-aware UoI metrics for sampling in the presence of random delays, with practical implications for remote monitoring and IoT systems where channel delays are variable and sizable.

Abstract

In this paper, we study a remote monitoring system where a receiver observes a remote binary Markov source and decides whether to sample and transmit the state through a randomly delayed channel. We adopt uncertainty of information (UoI), defined as the entropy conditional on past observations at the receiver, as a metric of value of information, in contrast to the traditional state-agnostic nonlinear age of information (AoI) penalty functions. To address the limitations of prior UoI research that assumes one-time-slot delays, we extend our analysis to scenarios with random delays. We model the problem as a partially observable Markov decision process (POMDP) problem and simplify it to a semi-Markov decision process (SMDP) by introducing the belief state. We propose two algorithms: A globally optimal bisection relative value iteration (bisec-RVI) algorithm and a computationally efficient sub-optimal index-based threshold algorithm to solve the long-term average UoI minimization problem. Numerical simulations demonstrate that our sampling policies surpass traditional zero wait and AoI-optimal policies, particularly under conditions of large delay, with the sub-optimal policy nearly matching the performance of the optimal one.

Optimal Sampling for Uncertainty-of-Information Minimization in a Remote Monitoring System

TL;DR

This work studies uncertainty of information (UoI) minimization in a remote monitoring system with a binary Markov source and random transmission delays. It models sampling decisions as a POMDP and reduces it to a semi-Markov decision process via a belief state, then solves the optimal long-run average UoI using a two-layer bisec-RVI algorithm, providing a globally optimal policy. To reduce computation, it derives a sub-optimal index-based threshold policy that is provably near-optimal under large expected delays and demonstrates, through simulations, that both proposed policies outperform traditional zero-wait and AoI-optimal policies, with the sub-optimal policy closely tracking the optimal one under heavy delays. The results underscore the benefit of state-aware UoI metrics for sampling in the presence of random delays, with practical implications for remote monitoring and IoT systems where channel delays are variable and sizable.

Abstract

In this paper, we study a remote monitoring system where a receiver observes a remote binary Markov source and decides whether to sample and transmit the state through a randomly delayed channel. We adopt uncertainty of information (UoI), defined as the entropy conditional on past observations at the receiver, as a metric of value of information, in contrast to the traditional state-agnostic nonlinear age of information (AoI) penalty functions. To address the limitations of prior UoI research that assumes one-time-slot delays, we extend our analysis to scenarios with random delays. We model the problem as a partially observable Markov decision process (POMDP) problem and simplify it to a semi-Markov decision process (SMDP) by introducing the belief state. We propose two algorithms: A globally optimal bisection relative value iteration (bisec-RVI) algorithm and a computationally efficient sub-optimal index-based threshold algorithm to solve the long-term average UoI minimization problem. Numerical simulations demonstrate that our sampling policies surpass traditional zero wait and AoI-optimal policies, particularly under conditions of large delay, with the sub-optimal policy nearly matching the performance of the optimal one.
Paper Structure (16 sections, 5 theorems, 67 equations, 4 figures, 2 algorithms)

This paper contains 16 sections, 5 theorems, 67 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. System Model and Problem Formulation
  3. System Model
  4. Problem Formmulation
  5. Optimal Sampling policy
  6. Belief State
  7. An Optimal Sampling Policy
  8. Sub-optimal Index-based Policy
  9. Numerical Results
  10. Conclusion
  11. The Proof of lemma \ref{['lemmabs']}
  12. The proof of Theorem \ref{['theorem1']}
  13. The proof of Theorem \ref{['theorem2']}
  14. Proof of Lemma \ref{['lemma2']}
  15. Proof of Part ($i$)
  16. ...and 1 more sections

Key Result

Lemma 1

Given $\Omega(t)=\omega$, $\Omega(t+k)$ can be explicitly calculated by where $\omega \in \{ p^{(n)},\, 1-q^{(n)} \}$, $n,k \in \mathbb{N}$. For short-hand notations, we leverage $\tau^k(\omega)$ to denote the right-hand side of eq3-3.

Figures (4)

  • Figure 1: UoI vs. AoI and the latest observed state $S_0$.
  • Figure 2: System model of the considered remote monitoring system.
  • Figure 3: Average UoI and Average AoI v.s. $y$ with i.d.d random delay, where $P[Y_i = 1] = 0.8$ and $P[Y_i = y] = 0.2$, the dynamics of the Markov source depicted as $p = 0.05$ and $q = 0.2$.
  • Figure 4: Average UoI and Average AoI v.s. $y$ with i.d.d random delay, where $P[Y_i = 1] = 0.8$ and $P[Y_i = y] = 0.2$, the dynamics of the Markov source depicted as $p = 0.7$ and $q = 0.95$.

Theorems & Definitions (10)

  • Lemma 1
  • proof
  • Corollary 1
  • proof
  • Theorem 1
  • proof : Proof sketch
  • Theorem 2
  • proof : Proof sketch
  • Lemma 2
  • proof