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Remote Estimation of Gauss-Markov Processes over Multiple Channels: A Whittle Index Policy

Tasmeen Zaman Ornee, Yin Sun

TL;DR

The paper studies scheduling sampling and transmission for N Gauss-Markov sources over L parallel channels with i.i.d. transmission times, aiming to minimize a weighted time-average MMSE. By formulating the problem as a continuous-time RMAB with continuous state space, it proves indexability and derives an exact Whittle index, unifying threshold-based sampling with Whittle-index scheduling. In the single-source, single-channel case, the optimal policy can be expressed both as a threshold-based sampler and a Whittle-index policy, with a zero index corresponding to idle channel and threshold crossing. Numerical results show the proposed signal-aware Whittle index policy achieving substantial gains over AoI-based, signal-agnostic, and MAF-ZW policies, especially when sources are highly unstable, highlighting practical impact for multi-source remote estimation over imperfect channels.

Abstract

We study a sampling and transmission scheduling problem for multi-source remote estimation, where a scheduler determines when to take samples from multiple continuous-time Gauss-Markov processes and send the samples over multiple channels to remote estimators. The sample transmission times are i.i.d. across samples and channels. The objective of the scheduler is to minimize the weighted sum of the time-average expected estimation errors of these Gauss-Markov sources. This problem is a continuous-time Restless Multi-armed Bandit (RMAB) problem with a continuous state space. We prove that the bandits are indexable and derive an exact expression of the Whittle index. To the extent of our knowledge, this is the first Whittle index policy for multi-source signal-aware remote estimation of Gauss-Markov processes. Our results unite two theoretical frameworks that were used for remote estimation and AoI minimization: threshold-based sampling and Whittle index-based scheduling. In the single-source, single-channel scenario, we demonstrate that the optimal solution to the sampling and scheduling problem can be equivalently expressed as both a threshold-based sampling strategy and a Whittle index-based scheduling policy. Notably, the Whittle index is equal to zero if and only if two conditions are satisfied: (i) the channel is idle, and (ii) the estimation error is precisely equal to the threshold in the threshold-based sampling strategy. Moreover, the methodology employed to derive threshold-based sampling strategies in the single-source, single-channel scenario plays a crucial role in establishing indexability and evaluating the Whittle index in the more intricate multi-source, multi-channel scenario. Our numerical results show that the proposed policy achieves high-performance gain over the existing policies when some of the Gauss-Markov processes are highly unstable.

Remote Estimation of Gauss-Markov Processes over Multiple Channels: A Whittle Index Policy

TL;DR

The paper studies scheduling sampling and transmission for N Gauss-Markov sources over L parallel channels with i.i.d. transmission times, aiming to minimize a weighted time-average MMSE. By formulating the problem as a continuous-time RMAB with continuous state space, it proves indexability and derives an exact Whittle index, unifying threshold-based sampling with Whittle-index scheduling. In the single-source, single-channel case, the optimal policy can be expressed both as a threshold-based sampler and a Whittle-index policy, with a zero index corresponding to idle channel and threshold crossing. Numerical results show the proposed signal-aware Whittle index policy achieving substantial gains over AoI-based, signal-agnostic, and MAF-ZW policies, especially when sources are highly unstable, highlighting practical impact for multi-source remote estimation over imperfect channels.

Abstract

We study a sampling and transmission scheduling problem for multi-source remote estimation, where a scheduler determines when to take samples from multiple continuous-time Gauss-Markov processes and send the samples over multiple channels to remote estimators. The sample transmission times are i.i.d. across samples and channels. The objective of the scheduler is to minimize the weighted sum of the time-average expected estimation errors of these Gauss-Markov sources. This problem is a continuous-time Restless Multi-armed Bandit (RMAB) problem with a continuous state space. We prove that the bandits are indexable and derive an exact expression of the Whittle index. To the extent of our knowledge, this is the first Whittle index policy for multi-source signal-aware remote estimation of Gauss-Markov processes. Our results unite two theoretical frameworks that were used for remote estimation and AoI minimization: threshold-based sampling and Whittle index-based scheduling. In the single-source, single-channel scenario, we demonstrate that the optimal solution to the sampling and scheduling problem can be equivalently expressed as both a threshold-based sampling strategy and a Whittle index-based scheduling policy. Notably, the Whittle index is equal to zero if and only if two conditions are satisfied: (i) the channel is idle, and (ii) the estimation error is precisely equal to the threshold in the threshold-based sampling strategy. Moreover, the methodology employed to derive threshold-based sampling strategies in the single-source, single-channel scenario plays a crucial role in establishing indexability and evaluating the Whittle index in the more intricate multi-source, multi-channel scenario. Our numerical results show that the proposed policy achieves high-performance gain over the existing policies when some of the Gauss-Markov processes are highly unstable.
Paper Structure (28 sections, 25 theorems, 220 equations, 4 figures, 4 algorithms)

This paper contains 28 sections, 25 theorems, 220 equations, 4 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Model and Formulation
  4. System Model
  5. MMSE Estimator
  6. Problem Formulation
  7. Main Results
  8. Signal-aware Scheduling
  9. Relaxation and Lagrangian Dual Decomposition
  10. Indexability
  11. Whittle Index Policy
  12. Unity of Whittle Index-based Scheduling and Threshold-based Sampling
  13. Signal-agnostic Scheduling
  14. Unity of Whittle Index-based Scheduling and Threshold-based Sampling
  15. Numerical Results
  16. ...and 13 more sections

Key Result

Proposition 1

If the $Y_{n,i}$'s are i.i.d. with $0<\mathbb{E}[Y_{n,i}] < \infty$, then $(S_{n,1} (\beta_n),S_{n,2} (\beta_n),\ldots)$ with a parameter $\beta_n$ is an optimal solution to per_arm_problem, where $D_{n,i} (\beta_n)= S_{n,i} (\beta_n)+ Y_{n,i}$, ${v_n}(\beta_n)$ is defined by $Q^{-1}(\cdot)$ and $K^{-1}(\cdot)$ are the inverse functions of $Q(x)$ in G and $K(x)$ in K, respectively, defined in th

Figures (4)

  • Figure 1: A multi-source, multi-channel remote estimation system.
  • Figure 2: Illustration of the Whittle index $\alpha_1(\varepsilon, \gamma)$ and the optimal threshold $v_1(\beta_1)$, where the parameters of the Gauss-Markov process are $\sigma_1 =1$ and $\theta_1 =0.1$ and the i.i.d. transmission times follow an exponential distribution with mean $\mathbb{E} [Y_{1,i}] = 2$.
  • Figure 3: Total time-average MSE vs the parameter $\sigma_1$ of the Gauss-Markov source 1, where the number of sources is $N=4$ and the number of channels is $L=2$. The transmission times are i.i.d, following a normalized log-normal distribution with parameter $\rho=1.5$, and $\mathbb{E} [Y_{n,i}] =1$. The other parameters of the Gauss-Markov sources are $\sigma_2 = 0.8, \sigma_3 = 0.9, \sigma_4=1$, and $\theta_1=-0.1, \theta_2 = \theta_3= \theta_4=0.1$.
  • Figure 4: Total time-average MSE vs the parameter $\theta_1$ of the Gauss-Markov source 1, where the number of sources is $N=4$ and the number of channels is $L=2$. The transmission times are i.i.d., following a normalized log-normal distribution with parameter $\rho=1.5$, and $\mathbb{E} [Y_{n,i}] =1$. The other parameters for the Gauss-Markov sources are $\sigma_1 = \sigma_2 = \sigma_3= \sigma_4=1$, and $\theta_2=0.2, \theta_3= 0.3, \theta_4= 0.1$.

Theorems & Definitions (40)

  • Definition 1
  • Proposition 1
  • proof
  • Theorem 1
  • Definition 2
  • Theorem 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 30 more