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AoII-Optimum Sampling of CTMC Information Sources Under Sampling Rate Constraints

Ismail Cosandal, Nail Akar, Sennur Ulukus

TL;DR

This work tackles remote tracking of an $N$-state CTMC under a sampling-rate constraint by optimizing the Age of Incorrect Information (AoII). It establishes that the optimal transmission policy is a threshold rule on AoII, with thresholds dependent on the source-monitor pair, and introduces a relaxed problem where thresholds depend only on the monitor state. Using a novel CSMDP formulation based on absorbing Markov chains, solved via policy iteration and a Lagrangian multiplier with a bisection step, the authors achieve an $\mathcal{O}(N^4)$ algorithm that yields optimum thresholds for CTMCs with hundreds of states. Numerical results validate substantial MAoII improvements over baselines at low budgets and confirm scalability and accuracy of the approach. The work thus provides a practically implementable, scalable framework for AoII-optimal sampling in continuous-time information sources.

Abstract

We consider a sensor that samples an $N$-state continuous-time Markov chain (CTMC)-based information source process, and transmits the observed state of the source, to a remote monitor tasked with timely tracking of the source process. The mismatch between the source and monitor processes is quantified by age of incorrect information (AoII), which penalizes the mismatch as it stays longer, and our objective is to minimize the average AoII under an average sampling rate constraint. We assume a perfect reverse channel and hence the sensor has information of the estimate while initiating a transmission or preempting an ongoing transmission. First, by modeling the problem as an average cost constrained semi-Markov decision process (CSMDP), we show that the structure of the problem gives rise to an optimum threshold policy for which the sensor initiates a transmission once the AoII exceeds a threshold depending on the instantaneous values of both the source and monitor processes. However, due to the high complexity of obtaining the optimum policy in this general setting, we consider a relaxed problem where the thresholds are allowed to be dependent only on the estimate. We show that this relaxed problem can be solved with a novel CSMDP formulation based on the theory of absorbing MCs, with a computational complexity of $\mathcal{O}(N^4)$, allowing one to obtain optimum policies for general CTMCs with over a hundred states.

AoII-Optimum Sampling of CTMC Information Sources Under Sampling Rate Constraints

TL;DR

This work tackles remote tracking of an -state CTMC under a sampling-rate constraint by optimizing the Age of Incorrect Information (AoII). It establishes that the optimal transmission policy is a threshold rule on AoII, with thresholds dependent on the source-monitor pair, and introduces a relaxed problem where thresholds depend only on the monitor state. Using a novel CSMDP formulation based on absorbing Markov chains, solved via policy iteration and a Lagrangian multiplier with a bisection step, the authors achieve an algorithm that yields optimum thresholds for CTMCs with hundreds of states. Numerical results validate substantial MAoII improvements over baselines at low budgets and confirm scalability and accuracy of the approach. The work thus provides a practically implementable, scalable framework for AoII-optimal sampling in continuous-time information sources.

Abstract

We consider a sensor that samples an -state continuous-time Markov chain (CTMC)-based information source process, and transmits the observed state of the source, to a remote monitor tasked with timely tracking of the source process. The mismatch between the source and monitor processes is quantified by age of incorrect information (AoII), which penalizes the mismatch as it stays longer, and our objective is to minimize the average AoII under an average sampling rate constraint. We assume a perfect reverse channel and hence the sensor has information of the estimate while initiating a transmission or preempting an ongoing transmission. First, by modeling the problem as an average cost constrained semi-Markov decision process (CSMDP), we show that the structure of the problem gives rise to an optimum threshold policy for which the sensor initiates a transmission once the AoII exceeds a threshold depending on the instantaneous values of both the source and monitor processes. However, due to the high complexity of obtaining the optimum policy in this general setting, we consider a relaxed problem where the thresholds are allowed to be dependent only on the estimate. We show that this relaxed problem can be solved with a novel CSMDP formulation based on the theory of absorbing MCs, with a computational complexity of , allowing one to obtain optimum policies for general CTMCs with over a hundred states.
Paper Structure (9 sections, 1 theorem, 37 equations, 5 figures, 2 tables, 2 algorithms)

This paper contains 9 sections, 1 theorem, 37 equations, 5 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. System Model
  3. Relaxed System
  4. Numerical Results
  5. Conclusions
  6. Lagrangian Method
  7. On the optimality of the threshold policy
  8. Absorbing Markov chains and Phase-type distribution
  9. Finding the optimum Lagrangian coefficient

Key Result

Lemma 1

The optimum transmission policy for $N>2$ is a threshold policy represented by the quantities $\tau_{ij}$, i.e., the source transmits when the AoII process exceeds $\tau_{ij}$, when $X(t)=i$ and $\hat{X}(t)=j$,

Figures (5)

  • Figure 1: A remote estimation system with the source process $X(t)$ and the monitor process $\hat{X}(t)$ for which the source employs a transmission policy to transmit the status update packets via the channel, and also a preemption policy to preempt ongoing transmissions when the observed information becomes obsolete.
  • Figure 2: A sample path of $X(t)$, $\hat{X}(t)$, and $\text{AoII}(t)$ (thick red solid curve) for an example scenario when $N=2$ and $\text{AoII}(0)=0$. The arrows represent the reception epochs of status update packets at the monitor. Notice that $\text{AoII}(t)$ drops to zero at $t=7$ without a reception.
  • Figure 3: A sample path for $X(t)$, $\hat{X}(t)$ and $\text{AoII}(t)$ for an example scenario. Green circles denote the synchronization points.
  • Figure 4: The red (resp. gray) contour lines correspond to the threshold pairs with the same sampling rate $R$ (resp. same MAoII). The minimum MAoII points for a given sampling rate $R$ are connected with a dashed line, whereas the optimum threshold pairs obtained by the proposed CSMDP are marked by a cross.
  • Figure 5: MAoII depicted as a function of the sampling rate budget $b$ for the generator $\bm{Q}_2$, using three policies.

Theorems & Definitions (2)

  • Lemma 1
  • Remark 1