Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Structured Estimators: A New Perspective on Information Freshness

Sahan Liyanaarachchi, Sennur Ulukus, Nail Akar

TL;DR

This work tackles remote estimation of a CTMC under binary freshness by challenging the prevailing martingale approach in pull-based update systems. It introduces a family of structured estimators that interpolate between martingale and MAP, namely Exponential, Erlang, and $τ$-MAP, and provides closed-form (or readily computable) expressions for the binary freshness metric $E[Δ]$. Theoretical results include exact formulas (and time-reversible simplifications) for these estimators, a bound showing martingale performance is dominated by a suitable $τ^*$-MAP, and convergence of the Erlang family to $τ$-MAP as its parameter grows. Numerical experiments demonstrate substantial freshness gains from the structured estimators, especially the $τ$-MAP, while highlighting tunable trade-offs with sampling rate and Erlang parameters; these estimators also pave the way for rate-allocation in multi-CTMC scenarios.

Abstract

In recent literature, when modeling for information freshness in remote estimation settings, estimators have been mainly restricted to the class of martingale estimators, meaning the remote estimate at any time is equal to the most recently received update. This is mainly due to its simplicity and ease of analysis. However, these martingale estimators are far from optimal in some cases, especially in pull-based update systems. For such systems, maximum aposteriori probability (MAP) estimators are optimum, but can be challenging to analyze. Here, we introduce a new class of estimators, called structured estimators, which retain useful characteristics from a MAP estimate while still being analytically tractable. Our proposed estimators move seamlessly from a martingale estimator to a MAP estimator.

Structured Estimators: A New Perspective on Information Freshness

TL;DR

This work tackles remote estimation of a CTMC under binary freshness by challenging the prevailing martingale approach in pull-based update systems. It introduces a family of structured estimators that interpolate between martingale and MAP, namely Exponential, Erlang, and -MAP, and provides closed-form (or readily computable) expressions for the binary freshness metric . Theoretical results include exact formulas (and time-reversible simplifications) for these estimators, a bound showing martingale performance is dominated by a suitable -MAP, and convergence of the Erlang family to -MAP as its parameter grows. Numerical experiments demonstrate substantial freshness gains from the structured estimators, especially the -MAP, while highlighting tunable trade-offs with sampling rate and Erlang parameters; these estimators also pave the way for rate-allocation in multi-CTMC scenarios.

Abstract

In recent literature, when modeling for information freshness in remote estimation settings, estimators have been mainly restricted to the class of martingale estimators, meaning the remote estimate at any time is equal to the most recently received update. This is mainly due to its simplicity and ease of analysis. However, these martingale estimators are far from optimal in some cases, especially in pull-based update systems. For such systems, maximum aposteriori probability (MAP) estimators are optimum, but can be challenging to analyze. Here, we introduce a new class of estimators, called structured estimators, which retain useful characteristics from a MAP estimate while still being analytically tractable. Our proposed estimators move seamlessly from a martingale estimator to a MAP estimator.
Paper Structure (15 sections, 7 theorems, 22 equations, 5 figures)

This paper contains 15 sections, 7 theorems, 22 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Structured Estimators
  5. Martingale Estimator
  6. Exponential Estimator
  7. Erlang Estimator
  8. $\tau$-MAP Estimator
  9. Main Results
  10. Numerical Results
  11. Conclusion
  12. Proof of Lemma \ref{['lem:T']}
  13. Proof of Theorem \ref{['thrm:erlang_BF']}
  14. Proof of Corollary \ref{['cor:erl_tr']}
  15. Proof of Theorem \ref{['thrm:mart_v_map']}

Key Result

Lemma 1

If the $i^*=\mathop{\mathrm{arg\,max}}\limits_{i\in S}\pi_i$ is unique, then $\exists~\tau^*<\infty$, such that the $\mathop{\mathrm{arg\,max}}\limits_{i \in S} \bm{v}_j^TP(t)=i^*$ for all $t>\tau^*$ and $\forall j\in S$, where $\bm{v}_j$ is a vector of all zeros except for an one at the $j$th index

Figures (5)

  • Figure 1: Query-based (pull-based) sampling of a Markovian source.
  • Figure 2: State transition diagram for $Y(t)$.
  • Figure 3: Variation of binary freshness with the sampling rate for $\Gamma=10$ and $\lambda=\frac{1}{\tau^*}$ for a generic 4-state CTMC.
  • Figure 4: Variation of binary freshness with the sampling rate for different $\Gamma$ values with $\lambda=\frac{1}{\tau^*}$ and $\tau=\tau^*$ for a time reversible 5-state CTMC.
  • Figure 5: Variation of binary freshness with the sampling rate for different $\tau$ under $\tau$-MAP estimator for a 2-state CTMC.

Theorems & Definitions (10)

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