Age of Incorrect Information With Hybrid ARQ Under a Resource Constraint for N-ary Symmetric Markov Sources

TL;DR

The work tackles minimizing the Age of Incorrect Information (AoII) in a discrete-time remote monitoring system employing HARQ under a long-run transmission-rate constraint for an $N$-ary symmetric Markov source. It formulates the problem as a constrained Markov decision process (CMDP) and leverages a Lagrangian relaxation to derive structural properties, showing that the optimal unconstrained policy is a threshold policy (or, under certain conditions, a randomized mixture of two threshold policies). The authors provide analytic expressions for the optimal threshold and the randomization component, and develop an algorithm based on relative value iteration and binary search to compute the optimal policy, validated by numerical results across varying source dynamics, channel conditions, and constraints. This work delivers an exact policy characterization for AoII with HARQ under resource constraints, guiding the design of real-time monitoring systems with limited transmit opportunities.

Abstract

The Age of Incorrect Information (AoII) is a recently proposed metric for real-time remote monitoring systems. In particular, AoII measures the time the information at the monitor is incorrect, weighted by the magnitude of this incorrectness, thereby combining the notions of freshness and distortion. This paper addresses the definition of an AoII-optimal transmission policy in a discrete-time communication scheme with a resource constraint and a hybrid automatic repeat request (HARQ) protocol. Considering an $N$-ary symmetric Markov source, the problem is formulated as an infinite-horizon average-cost constrained Markov decision process (CMDP). Interestingly, it is proved that, under some conditions, the optimal transmission policy is to never transmit. This reveals a region of the source dynamics where communication is inadequate in reducing the AoII. Elsewhere, there exists an optimal transmission policy, which is a randomized mixture of two discrete threshold-based policies that randomize on at most one state. The optimal threshold and the randomization component are derived analytically. Numerical results illustrate the impact of the source dynamics, channel conditions, and resource constraints on the average AoII.

Figures (8)

• Figure 1: Snapshot of a communication system illustrating the differences between AoI and AoII. In this example, a binary source is sampled at every time slot and is transmitted to the monitor. The packet arrives at the next time slot and is attempted to be decoded. Successful decodings occur at time slots $d_i$, $i=1,2,3$, whereas the respective samples were generated at slots $s_i=d_i-1$. If the decoding is successful, the monitor updates its estimate and the AoI decreases to one (because the sample was generated one time slot ago). Otherwise, the AoI increases. The corresponding error indicates the mismatch between the source and the monitor estimate. The AoII measures the time slots where the error has been positive.
• Figure 2: The symmetric Markov data source under consideration.
• Figure 3: Average AoII versus the transmission rate constraint of the optimal threshold-based policy. The source model parameters are $\alpha=0.5$ and $N=16$ and the maximum retransmission count is $r_{max}=2$.
• Figure 5: Average AoII versus the transmission rate constraint of the optimal threshold-based policy. The source model parameters are $\alpha=0.5$ and $N=16$ and the HARQ decaying error rate constant is $c=0.5$.
• Figure 7: Average AoII versus the transmission rate constraint of the optimal threshold-based policy. The source model parameters are $\alpha=0.2$ and $N=128$ and the HARQ maximum re-transmissions are $r_{max}=2$.

Theorems & Definitions (27)

• Remark 1
• Definition 1: Main CMDP Problem
• Definition 2: Lagrangian MDP Problem
• Lemma 1
• proof
• Lemma 2
• Lemma 3
• Proposition 1
• Corollary 1
• Proposition 2