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Query-Based Sampling of Heterogeneous CTMCs: Modeling and Optimization with Binary Freshness

Nail Akar, Sennur Ulukus

TL;DR

This paper tackles remote monitoring of a heterogeneous collection of finite-state CTMC information sources via query-based sampling under a total sampling-rate budget. It introduces three freshness notions—fresh when equal (FWE), fresh when close (FWC), and fresh when sampled (FWS)—and derives closed-form mean freshness expressions using a martingale estimator. A water-filling optimization is developed to allocate sampling rates across sources to maximize a weighted sum of freshness, with quadratic worst-case complexity in the number of sources. Numerical results show the proposed policy outperforms several baselines, especially when sources are diverse, highlighting a practical, scalable approach for real-time heterogeneous CTMC monitoring.

Abstract

We study a remote monitoring system in which a mutually independent and heterogeneous collection of finite-state irreducible continuous time Markov chain (CTMC) based information sources is considered. In this system, a common remote monitor queries the instantaneous states of the individual CTMCs according to a Poisson process with possibly different intensities across the sources, in order to maintain accurate estimates of the original sources. \color{black}Three information freshness models are considered to quantify the accuracy of the remote estimates: fresh when equal (FWE), fresh when sampled (FWS) and fresh when close (FWC). For each of these freshness models, closed-form expressions are derived for mean information freshness for a given source. Using these expressions, optimum sampling rates for all sources are obtained so as to maximize the weighted sum freshness of the monitoring system, subject to an overall sampling rate constraint. This optimization problem leads to a water-filling solution with quadratic worst case computational complexity in the number of information sources. Numerical examples are provided to validate the effectiveness of the optimum sampling policy in comparison to several baseline sampling policies.

Query-Based Sampling of Heterogeneous CTMCs: Modeling and Optimization with Binary Freshness

TL;DR

This paper tackles remote monitoring of a heterogeneous collection of finite-state CTMC information sources via query-based sampling under a total sampling-rate budget. It introduces three freshness notions—fresh when equal (FWE), fresh when close (FWC), and fresh when sampled (FWS)—and derives closed-form mean freshness expressions using a martingale estimator. A water-filling optimization is developed to allocate sampling rates across sources to maximize a weighted sum of freshness, with quadratic worst-case complexity in the number of sources. Numerical results show the proposed policy outperforms several baselines, especially when sources are diverse, highlighting a practical, scalable approach for real-time heterogeneous CTMC monitoring.

Abstract

We study a remote monitoring system in which a mutually independent and heterogeneous collection of finite-state irreducible continuous time Markov chain (CTMC) based information sources is considered. In this system, a common remote monitor queries the instantaneous states of the individual CTMCs according to a Poisson process with possibly different intensities across the sources, in order to maintain accurate estimates of the original sources. \color{black}Three information freshness models are considered to quantify the accuracy of the remote estimates: fresh when equal (FWE), fresh when sampled (FWS) and fresh when close (FWC). For each of these freshness models, closed-form expressions are derived for mean information freshness for a given source. Using these expressions, optimum sampling rates for all sources are obtained so as to maximize the weighted sum freshness of the monitoring system, subject to an overall sampling rate constraint. This optimization problem leads to a water-filling solution with quadratic worst case computational complexity in the number of information sources. Numerical examples are provided to validate the effectiveness of the optimum sampling policy in comparison to several baseline sampling policies.
Paper Structure (11 sections, 4 theorems, 35 equations, 6 figures, 1 algorithm)

This paper contains 11 sections, 4 theorems, 35 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. System Model
  5. Analytical Expressions for Mean Freshness
  6. FWE Model
  7. FWC Model
  8. FWS Model
  9. Optimum Monitoring of Heterogeneous CTMCs
  10. Numerical Examples
  11. Conclusions

Key Result

Theorem 1

Let the irreducible CTMC $X(t) \in \{ 1,2,\ldots,K \}$ with generator $Q$ and steady-state vector $\pi$, be Poisson sampled with sampling rate $\lambda$. Then, for the FWE model, the mean freshness $f(\lambda) = \mathbb{E}[F_e]$ is given by, where $\bm{diag}[\cdot]$ represents a column vector composed of the diagonal entries of its matrix argument.

Figures (6)

  • Figure 1: Query-based status update system in which the monitor queries the CTMC $X_n(t)$ associated with source-$n$ with intensity $\lambda_n$, to maintain an estimate $\tilde{X}_n(t)$ of $X_n(t)$.
  • Figure 2: Sample paths of the processes $X(t)$, $\tilde{X}(t)$, $F_e(t)$, $F_c(t)$ and $F_s(t)$ for a source with states 1, 2 and 3, for an example scenario. Down arrows represent sampling operation.
  • Figure 3: The mean freshness as a function of the sampling rate $\lambda$ for the FWE, FWC, and FWS freshness models using the numerical (N) and analytical (A) methods.
  • Figure 4: The system freshness $F_S$ as a function of the sampling ratio $\kappa$ for the WF, UNIFORM, PROP, and INVPROP sampling policies: (a) FWE (b) FWS.
  • Figure 5: The optimum sampling rate $\lambda_n$ divided by $\kappa$ as a function of the source index $n$ for four values of the sampling ratio $\kappa$: (a) FWE (b) FWS.
  • ...and 1 more figures

Theorems & Definitions (9)

  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Corollary 2
  • proof
  • Remark
  • Theorem 2
  • proof