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Multi-Stage Structured Estimators for Information Freshness

Sahan Liyanaarachchi, Sennur Ulukus, Nail Akar

TL;DR

This work tackles information freshness in CTMC-driven systems by moving beyond simple martingale estimators to a multi-stage MAP framework. It introduces the $p$-MAP estimator, which models the MAP as a piecewise-constant function with finitely many stages, and derives closed-form MBF expressions for both $p$-MAP and $\tau$-MAP estimators under time-reversible CTMCs, along with a martingale baseline. The authors also formulate an optimal state-dependent sampling policy via a constrained SMDP to maximize freshness within a budget, and validate their approach with numerical experiments showing significant MBF gains over martingale policies, with the $p$-MAP estimator approaching MAP behavior as the number of stages increases. The results suggest practical gains in timeliness for pull-based update systems and provide a framework for extending MAP analysis beyond simple two-stage approximations.

Abstract

Most of the contemporary literature on information freshness solely focuses on the analysis of freshness for martingale estimators, which simply use the most recently received update as the current estimate. While martingale estimators are easier to analyze, they are far from optimal, especially in pull-based update systems, where maximum aposteriori probability (MAP) estimators are known to be optimal, but are analytically challenging. In this work, we introduce a new class of estimators called $p$-MAP estimators, which enable us to model the MAP estimator as a piecewise constant function with finitely many stages, bringing us closer to a full characterization of the MAP estimators when modeling information freshness.

Multi-Stage Structured Estimators for Information Freshness

TL;DR

This work tackles information freshness in CTMC-driven systems by moving beyond simple martingale estimators to a multi-stage MAP framework. It introduces the -MAP estimator, which models the MAP as a piecewise-constant function with finitely many stages, and derives closed-form MBF expressions for both -MAP and -MAP estimators under time-reversible CTMCs, along with a martingale baseline. The authors also formulate an optimal state-dependent sampling policy via a constrained SMDP to maximize freshness within a budget, and validate their approach with numerical experiments showing significant MBF gains over martingale policies, with the -MAP estimator approaching MAP behavior as the number of stages increases. The results suggest practical gains in timeliness for pull-based update systems and provide a framework for extending MAP analysis beyond simple two-stage approximations.

Abstract

Most of the contemporary literature on information freshness solely focuses on the analysis of freshness for martingale estimators, which simply use the most recently received update as the current estimate. While martingale estimators are easier to analyze, they are far from optimal, especially in pull-based update systems, where maximum aposteriori probability (MAP) estimators are known to be optimal, but are analytically challenging. In this work, we introduce a new class of estimators called -MAP estimators, which enable us to model the MAP estimator as a piecewise constant function with finitely many stages, bringing us closer to a full characterization of the MAP estimators when modeling information freshness.
Paper Structure (12 sections, 5 theorems, 12 equations, 4 figures)

This paper contains 12 sections, 5 theorems, 12 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. System Model
  4. Martingale Estimator
  5. $\tau$-MAP Estimator
  6. $p$-MAP Estimator
  7. Main Results
  8. Martingale Estimator
  9. Non-martingale Estimators
  10. Optimal State-Dependent Sampling Policy
  11. Numerical Results
  12. Conclusion

Key Result

Lemma 1

For time-reversible CTMCs, the elements of the transition matrix are given as follows, where $u_{ij}$ are the elements of matrix $U^T$ that diagonalizes $Q$.

Figures (4)

  • Figure 1: Query-based sampling of a Markov source.
  • Figure 2: Infinitely oscillating transition probabilities for a non time-reversible CTMC which has no unique stationary maximum. Here, $\bm\pi=\left\{\frac{1}{3},\frac{1}{3},\frac{1}{3}\right\}$. Since all $\pi_i$, $i=1, 2, 3$ are equal, there is no unique $\mathop{\mathrm{arg\,max}}\limits_i{\pi_i}$ here.
  • Figure 3: Variation of MBF with the sampling budget $\Omega$, under different state-dependent sampling policies, for a finite birth death chain (BDC) with $S=4$ states, with a unique stationary maximum.
  • Figure 4: Variation of MBF of the $p$-MAP estimator with the number of intermediate stages $K_i$ used to approximate an infinitely oscillating MAP estimator.

Theorems & Definitions (8)

  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Corollary 2
  • Remark 1
  • Remark 2
  • Remark 3