This paper investigates information freshness in a remote estimation system in which the remote information source is a continuous-time Markov chain (CTMC). For such systems, estimators have been mainly restricted to the class of martingale estimators in which the remote estimate at any time is equal to the value of the most recently received update. This is mainly due to the simplicity and ease of analysis of martingale estimators, which however are far from optimal, especially in query-based (i.e., pull-based) update systems. In such systems, maximum a-posteriori probability (MAP) estimators are optimal. However, MAP estimators can be challenging to analyze in continuous-time settings. In this paper, we introduce a new class of estimators, called structured estimators, which can seamlessly shift from a martingale estimator to a MAP estimator, enabling them to retain useful characteristics of the MAP estimate, while still being analytically tractable. Particularly, we introduce a new estimator termed as the $p$-MAP estimator which is a piecewise-constant approximation of the MAP estimator with finitely many discontinuities, bringing us closer to a full characterization of MAP estimators when modeling information freshness. In fact, we show that for time-reversible CTMCs, the MAP estimator reduces to a $p$-MAP estimator. Using the binary freshness (BF) process for the characterization of information freshness, we derive the freshness expressions and provide optimal state-dependent sampling policies (i.e., querying policies) for maximizing the mean BF (MBF) for pull-based remote estimation of a single CTMC information source, when structured estimators are used. Moreover, we provide optimal query rate allocation policies when a monitor pulls information from multiple heterogeneous CTMCs with a constraint on the overall query rate.

翻译：本文研究远程估计系统中的信息新鲜度问题，其中远程信息源为连续时间马尔可夫链（CTMC）。在此类系统中，估计器主要局限于鞅估计器类，即任意时刻的远程估计值等于最近接收到的更新值。这主要源于鞅估计器的简洁性与易分析性，但其性能远非最优，尤其在基于查询（即拉取式）的更新系统中。在此类系统中，最大后验概率（MAP）估计器是最优的。然而，在连续时间场景下分析MAP估计器具有挑战性。本文提出一类新型估计器——结构化估计器，其可在鞅估计器与MAP估计器间无缝切换，既能保留MAP估计的有用特性，又保持分析可处理性。特别地，我们提出一种称为$p$-MAP估计器的新估计器，它是对具有有限个间断点的MAP估计器的分段常数逼近，使我们在建模信息新鲜度时更接近完整刻画MAP估计器。事实上，我们证明对于时间可逆CTMC，MAP估计器可简化为$p$-MAP估计器。利用二元新鲜度（BF）过程刻画信息新鲜度，我们推导了新鲜度表达式，并为使用结构化估计器的单CTMC信息源拉取式远程估计提供了最大化平均BF（MBF）的最优状态依赖采样策略（即查询策略）。此外，我们针对监控器从多个异构CTMC拉取信息且总查询率受限的场景，给出了最优查询率分配策略。