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Optimal Sampling and Scheduling for Remote Fusion Estimation of Correlated Wiener Processes

Aimin Li, Elif Uysal

TL;DR

This work addresses remote fusion estimation of a pair of positively correlated Wiener processes observed by distributed sensors over a shared, delay-prone channel. It establishes a separation principle: the MMSE fusion estimator, the Maximum Age First (MAF) scheduler, and a design-friendly sampling policy can be optimized in stages, with an important result that, in the infinite-horizon setting, AoI optimization is equivalent to minimizing the MSE under pull-based communication. The authors formulate the problem via a fractional objective, convert it to a tractable average-cost MDP using a state-space transformation, and prove that the MAFF scheduler is optimal; this leads to a practical age-based sampling policy, namely a threshold-triggered water-filling scheme that achieves near-optimal AoI. Simulations demonstrate that the proposed WF+MAF policy outperforms benchmarks and validates the theoretical AoI–MSE equivalence, offering a scalable approach for joint design in correlated multi-source remote estimation.

Abstract

In distributed sensor networks, sensors often observe a dynamic process within overlapping regions. Due to random delays, these correlated observations arrive at the fusion center asynchronously, raising a central question: How can one fuse asynchronous yet correlated information for accurate remote fusion estimation? This paper addresses this challenge by studying the joint design of sampling, scheduling, and estimation policies for monitoring a correlated Wiener process. Though this problem is coupled, we establish a separation principle and identify the joint optimal policy: the optimal fusion estimator is a weighted-sum fusion estimator conditioned on Age of Information (AoI), the optimal scheduler is a Maximum Age First (MAF) scheduler that prioritizes the most stale source, and the optimal sampling can be designed given the optimal estimator and the MAF scheduler. To design the optimal sampling, we show that, under the infinite-horizon average-cost criterion, optimizing AoI is equivalent to optimizing MSE under pull-based communications, despite the presence of strong inter-sensor correlations. This structural equivalence allows us to identify the MSE-optimal sampler as one that is AoI-optimal. This result underscores an insight: information freshness can serve as a design surrogate for optimal estimation in correlated sensing environments.

Optimal Sampling and Scheduling for Remote Fusion Estimation of Correlated Wiener Processes

TL;DR

This work addresses remote fusion estimation of a pair of positively correlated Wiener processes observed by distributed sensors over a shared, delay-prone channel. It establishes a separation principle: the MMSE fusion estimator, the Maximum Age First (MAF) scheduler, and a design-friendly sampling policy can be optimized in stages, with an important result that, in the infinite-horizon setting, AoI optimization is equivalent to minimizing the MSE under pull-based communication. The authors formulate the problem via a fractional objective, convert it to a tractable average-cost MDP using a state-space transformation, and prove that the MAFF scheduler is optimal; this leads to a practical age-based sampling policy, namely a threshold-triggered water-filling scheme that achieves near-optimal AoI. Simulations demonstrate that the proposed WF+MAF policy outperforms benchmarks and validates the theoretical AoI–MSE equivalence, offering a scalable approach for joint design in correlated multi-source remote estimation.

Abstract

In distributed sensor networks, sensors often observe a dynamic process within overlapping regions. Due to random delays, these correlated observations arrive at the fusion center asynchronously, raising a central question: How can one fuse asynchronous yet correlated information for accurate remote fusion estimation? This paper addresses this challenge by studying the joint design of sampling, scheduling, and estimation policies for monitoring a correlated Wiener process. Though this problem is coupled, we establish a separation principle and identify the joint optimal policy: the optimal fusion estimator is a weighted-sum fusion estimator conditioned on Age of Information (AoI), the optimal scheduler is a Maximum Age First (MAF) scheduler that prioritizes the most stale source, and the optimal sampling can be designed given the optimal estimator and the MAF scheduler. To design the optimal sampling, we show that, under the infinite-horizon average-cost criterion, optimizing AoI is equivalent to optimizing MSE under pull-based communications, despite the presence of strong inter-sensor correlations. This structural equivalence allows us to identify the MSE-optimal sampler as one that is AoI-optimal. This result underscores an insight: information freshness can serve as a design surrogate for optimal estimation in correlated sensing environments.
Paper Structure (19 sections, 6 theorems, 41 equations, 4 figures)

This paper contains 19 sections, 6 theorems, 41 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Related Work
  4. Contributions of This Work
  5. System Model and Problem Formulation
  6. Coupled Correlated Sources
  7. Multi-source Network Model
  8. Sampling, Scheduling, and Estimating Policies
  9. Problem Formulation
  10. Optimal Estimating, Sampling, and Scheduling
  11. Separation Principle and Optimal Estimator
  12. Maximum Age First Scheduling is All You Need
  13. Optimal Sampling Policy
  14. Age-Optimal Sampling is All you Need
  15. Simulation Results
  16. ...and 4 more sections

Key Result

Theorem 1

Let $\rho \in [0,1]$ denote the correlation coefficient between $W_t^{(1)}$ and $W_t^{(2)}$. Let $\Delta_{m}(t)$ denote the AoI of source $m$. Given any scheduling policy $\pi$ and sampling policy $f$, the MMSE estimate of $W_t^{(1)}$ is: The MMSE estimation of $W_t^{(2)}$ is: where the functions $G_{t,\rho}(x,y)$ and $Q_{t,\rho}(x,y)$ are defined by: and we refer to them as fusion weights.

Figures (4)

  • Figure 1: System Model.
  • Figure 2: Relationship between different MDP formulations.
  • Figure 3: Finite-horizon time-average MSE and AoI under different scheduling and sampling policies. Here $Y_{\max}=20$ and $p=0.95$.
  • Figure 4: Long-term average MSE and AoI under different scheduling and sampling policies. We set $d=1$ for constant-wait sampling. (a) $p=0.95$; (b) $Y_{\max}=25$.

Theorems & Definitions (11)

  • Theorem 1: MMSE Fusion Estimator for Correlated Wiener Processes
  • proof
  • Lemma 1: Equivalence via Dinkelbach
  • proof
  • Lemma 2: Monotonicity in the age gap $\Gamma$
  • proof
  • Theorem 2: Separation principle and optimality of MAF scheduling
  • proof
  • Theorem 3: Average-cost MDP Equivalence
  • proof
  • ...and 1 more