Minimizing Age of Incorrect Information for Unreliable Channel with Power Constraint

Age of Incorrect Information (AoII) is a newly introduced performance metric that considers communication goals. Therefore, comparing with traditional performance metrics and the recently introduced metric - Age of Information (AoI), AoII achieves better performance in many real-life applications. However, the fundamental nature of AoII has been elusive so far. In this paper, we consider the AoII in a system where a transmitter sends updates about a multi-state Markovian source to a remote receiver through an unreliable channel. The communication goal is to minimize AoII subject to a power constraint. We cast the problem into a Constrained Markov Decision Process (CMDP) and prove that the optimal policy is a mixture of two deterministic threshold policies. Afterward, by leveraging the notion of Relative Value Iteration (RVI) and the structural properties of threshold policy, we propose an efficient algorithm to find the threshold policies as well as the mixing coefficient. Lastly, numerical results are laid out to highlight the performance of AoII-optimal policy.

💡 Research Summary

The paper addresses the problem of minimizing the Age of Incorrect Information (AoII) in a status‑update system where a transmitter monitors a multi‑state Markov source and sends updates to a remote receiver over an unreliable channel, while being subject to an average power constraint. AoII, unlike the traditional Age of Information (AoI), measures not only how stale the information at the receiver is but also how far the receiver’s estimate deviates from the true source state. In the considered model the source evolves as an N‑state Markov chain with transition probability 2p to a neighboring state and 1‑2p to stay in the same state. Each time slot the transmitter may either stay idle or attempt a transmission; a transmission consumes one unit of power regardless of success, and the long‑run average power consumption must not exceed a given budget α (<1). The channel is memoryless with success probability p_s and failure probability p_f = 1‑p_s. Successful transmissions are acknowledged instantly, so the transmitter always knows the receiver’s current estimate.

The objective is to find a policy that minimizes the long‑term average AoII while satisfying the power budget. This is formulated as a Constrained Markov Decision Process (CMDP). By introducing a Lagrange multiplier λ ≥ 0 the constraint is incorporated into the cost, yielding an unconstrained average‑cost MDP with immediate cost C(d,Δ,a) = Δ + λa, where d = |X‑ĤX| is the absolute state mismatch (0,…,N‑1) and Δ is the current AoII value. The state space is thus the pair (d,Δ). The state transition dynamics are derived for three cases: (i) no transmission, (ii) transmission attempt that fails, and (iii) transmission that succeeds. In the first two cases d evolves according to the underlying Markov chain and Δ is updated as Δ_{t+1}=Δ_t+d_{t+1} (or reset to zero if d_{t+1}=0). In the successful case d_{t+1}∈{0,1} and Δ_{t+1}=d_{t+1}.

To solve the average‑cost MDP the authors employ Relative Value Iteration (RVI). They prove (Lemma 1) that the value function V_ν(d,Δ) generated by RVI is monotone non‑decreasing in both d and Δ at every iteration. Using this monotonicity they establish (Proposition 1) that the optimal policy for any λ is a threshold policy: for each fixed mismatch level d≠0 there exists a smallest AoII value τ(d) such that the optimal action switches from “idle” to “transmit” when Δ ≥ τ(d). Moreover τ(d) is non‑increasing in d, and the state (0,0) is never active. Consequently the optimal policy can be described by a vector of thresholds n_λ = (τ(1),…,τ(N‑1)).

Because the AoII component Δ can grow without bound, directly applying RVI is infeasible. The authors therefore truncate Δ at a finite value m, forming a finite‑state MDP M(m). Excess probability mass that would lead to Δ>m is redistributed among the admissible states using the Approximating Sequence Method (ASM). They prove that as m → ∞ the optimal value of M(m) converges to that of the original infinite‑state problem.

The algorithm proceeds as follows. For a given λ, RVI is run on M(m) to obtain the value function and the corresponding threshold vector n_λ. The average power consumption under this policy is computed; λ is then adjusted (e.g., by bisection) until the power constraint is met with equality. Because the CMDP solution may require randomization, the final optimal policy is shown to be a mixture of two deterministic threshold policies, characterized by a mixing coefficient β that balances the power budget.

Numerical experiments with N=5, p∈