Minimizing Age of Incorrect Information over a Channel with Random Delay

TL;DR

This work addresses minimizing AoII in a slotted system where a transmitter sends updates of a two-state Markov source over a channel with random delay. It casts the problem as an infinite-horizon average-cost MDP and analyzes a threshold policy, proving an optimal policy exists and can be computed via value or policy iteration; under a verifiable condition, the optimal policy reduces to a simple threshold with τ=1. The authors derive exact AoII expressions for the threshold policy, develop a tractable stationary distribution framework, and provide rigorous optimality proofs, including a policy-improvement-based argument. Numerical results corroborate the theory, showing substantial AoII reductions with the optimal policy across diverse delay models and source dynamics, and revealing how AoII responds to p, p_s, and t_max. The findings offer practical guidelines for scheduling updates in delay-prone networks, linking semantic freshness with queuing-inspired decision rules.

Abstract

We consider a transmitter-receiver pair in a slotted-time system. The transmitter observes a dynamic source and sends updates to a remote receiver through an error-free communication channel that suffers a random delay. We consider two cases. In the first case, the update is guaranteed to be delivered within a certain number of time slots. In the second case, the update is immediately discarded once the transmission time exceeds a predetermined value. The receiver estimates the state of the dynamic source using the received updates. In this paper, we adopt the Age of Incorrect Information (AoII) as the performance metric and investigate the problem of optimizing the transmitter's action in each time slot to minimize AoII. We first characterize the optimization problem using the Markov decision process and investigate the performance of the threshold policy, under which the transmitter transmits updates only when the transmission is allowed and the AoII exceeds the threshold $τ$. By delving into the characteristics of the system evolution, we precisely compute the expected AoII achieved by the threshold policy using the Markov chain. Then, we prove that the optimal policy exists and provide a computable relative value iteration algorithm to estimate the optimal policy. Furthermore, by leveraging the policy improvement theorem, we theoretically prove that, under an easily verifiable condition, the optimal policy is the threshold policy with $τ=1$. Finally, numerical results are presented to highlight the performance of the optimal policy.

Figures (5)

• Figure 1: An illustration of the system model, where $X_k$ and $\hat{X}_k$ are the state of the dynamic source and the receiver's estimate at time slot $k$, respectively.
• Figure 2: A sample path of $\Delta_k$, where $T_i$ and $D_i$ are the transmission start and delivery time of the $i$-th update, respectively. At $T_1$, the transmitted update is $X_3$. Note that the transmission decisions in the plot are taken randomly.
• Figure 3: Illustrations of the expected AoII as a function of $p$ and $\tau$. We set the upper limit of the transmission time $t_{max}=5$ and the success probability in the Geometric distribution $p_s = 0.7$.
• Figure 4: Illustrations of the expected AoII as a function of $p_s$ and $\tau$. We set the upper limit of the transmission time $t_{max}=5$ and the source dynamic $p = 0.35$.
• Figure 5: Illustrations of the expected AoII as a function of $t_{max}$ and $\tau$. We set the success probability in the Geometric distribution $p_s = 0.7$ and the source dynamic $p = 0.35$.

Theorems & Definitions (39)

• Remark 1
• Definition 1: Optimal policy
• Remark 2
• Definition 2: Threshold policy
• Remark 3
• Remark 4
• Lemma 1
• proof
• Lemma 2
• proof