Modeling AoII in Push- and Pull-Based Sampling of Continuous Time Markov Chains

TL;DR

The paper tackles how to quantify and optimize age of incorrect information (AoII) for continuous-time Markov chain sources under push- and pull-based sampling. It introduces a computational framework based on absorbing CTMCs and phase-type distributions, embedding a discrete-time Markov chain at synchronization points to derive mean AoII and the average sampling rate under Erlang-k thresholds and network delays. The contributions include a novel, efficient analytical model that applies to both sampling schemes, provides closed-form-like expressions via phase-type theory, and offers practical policy insights under sampling-rate constraints. The work enables optimized sampling mechanisms for real-time monitoring where information freshness must be maintained while respecting network resources.

Abstract

Age of incorrect information (AoII) has recently been proposed as an alternative to existing information freshness metrics for real-time sampling and estimation problems involving information sources that are tracked by remote monitors. Different from existing metrics, AoII penalizes the incorrect information by increasing linearly with time as long as the source and the monitor are de-synchronized, and is reset when they are synchronized back. While AoII has generally been investigated for discrete time information sources, we develop a novel analytical model in this paper for push- and pull-based sampling and transmission of a continuous time Markov chain (CTMC) process. In the pull-based model, the sensor starts transmitting information on the observed CTMC only when a pull request from the monitor is received. On the other hand, in the push-based scenario, the sensor, being aware of the AoII process, samples and transmits when the AoII process exceeds a random threshold. The proposed analytical model for both scenarios is based on the construction of a discrete time MC (DTMC) making state transitions at the embedded epochs of synchronization points, using the theory of absorbing CTMCs, and in particular phase-type distributions. For a given sampling policy, analytical models to obtain the mean AoII and the average sampling rate are developed. Numerical results are presented to validate the analytical model as well as to provide insight on optimal sampling policies under sampling rate constraints.

Figures (7)

• Figure 1: Push-based and pull-based models for a sensor and a monitor.
• Figure 2: A sample path for $X(t)$, $\hat{X}(t)$ and $\text{AoII}(t)$ for push-based transmission for an example scenario. Green circles denote the synchronization points.
• Figure 3: A sample path for $X(t)$, $\hat{X}(t)$ and $\text{AoII}(t)$ for pull-based transmission for an example scenario.
• Figure 4: a) AoII, b) average sampling rate $R$ of the push-based transmission model as a function of $\theta$, for three CTMC processes and for three different values of the parameter $k$. While lines are for the analytical results, circles correspond to the simulation results.
• Figure 5: Contour maps of AoII and $R$ with respect to the sampling policy $(\theta_1,\theta_2)$. The points which result in the minimum AoII values among the policies with the same average sampling rate pairs are marked with a cross.