Goal-Oriented Status Updating for Real-time Remote Inference over Networks with Two-Way Delay

We study a setting where an intelligent model (e.g., a pre-trained neural network) infers the real-time value of a target signal using data samples transmitted from a remote source. The transmission scheduler decides (i) the freshness of packets, (ii) their length (i.e., the number of samples they contain), and (iii) when they should be transmitted. The freshness is quantified using the Age of Information (AoI), and the inference quality for a given packet length is a general function of AoI. Previous works assumed i.i.d. transmission delays with immediate feedback or were restricted to the case where inference performance degrades as the input data ages. Our formulation, in addition to capturing non-monotone age dependence, also covers Markovian delay on both forward and feedback links. We model this as an infinite-horizon average-cost Semi-Markov Decision Process. We obtain a closed-form solution that decides on (i) and (iii) for any constant packet length. The solution for when to transmit is an index-based threshold policy, where the index function is expressed in terms of the delay state and AoI at the receiver. In contrast, the freshness of the selected packet is a function of only the delay state. We then separately optimize the value of the constant packet length. Moreover, we also develop an index-based threshold policy for the time-variable packet length case, which allows a complexity reduction. In simulation results, we observe that our goal-oriented scheduler drops inference error down to one-sixth with respect to the age-based scheduling of unit-length packets.

💡 Research Summary

The paper addresses the problem of real‑time remote inference in which a pre‑trained AI model at a receiver must continuously estimate a target signal Yₜ based on data samples Vₜ generated at a remote source. The central question is how to jointly decide (i) the freshness of the transmitted packet (i.e., which samples from the buffer to send), (ii) the packet length l (the number of consecutive samples packed together), and (iii) the exact transmission times, so that the long‑term average inference error is minimized.

System model. Time is slotted. The source maintains a sliding buffer of the most recent B samples. At any transmission epoch Sᵢ the scheduler selects a block of lᵢ consecutive samples starting bᵢ slots before Sᵢ, forming packet X_{lᵢ}^{Sᵢ−bᵢ}. The packet experiences a forward transmission delay whose state c evolves as a finite‑state Markov chain, and after delivery the receiver sends an acknowledgment that suffers a feedback delay with its own Markov state c′. The Age of Information (AoI) at the receiver is Δ(t)=t−U(t), where U(t) is the generation time of the freshest sample that has been delivered. The inference error is modeled as a possibly non‑monotonic function ε(Δ,l) that is non‑increasing in l for a fixed AoI but may increase or decrease with Δ depending on the underlying source‑target relationship (e.g., delayed or periodic coupling).

Problem formulation. The joint scheduling and packet‑length selection problem is cast as an infinite‑horizon average‑cost Semi‑Markov Decision Process (SMDP). The system state is (Δ, c, c′). An action consists of choosing (b,l) and deciding whether to transmit now or wait. The objective is to minimize the long‑run average of ε(Δ,l).

Main theoretical contributions.

1. Fixed‑length case (inner layer). For any constant packet length l, the authors solve the inner SMDP in closed form (Theorem 1). The optimal policy separates into two independent components:
   • Freshness selection: a stationary mapping b⁎(c) that depends only on the forward‑delay state of the previous epoch. This function tells the scheduler which buffer position to pick (how “old” the samples should be).
   • Transmission timing: an index‑based threshold rule. They define an index W(c,Δ) that combines the current AoI and the forward‑delay state. The scheduler transmits only when Δ exceeds this index. The index has a Whittle‑index‑like structure but is derived exactly for the present non‑monotonic error model.
2. Outer layer (packet‑length optimization). With the closed‑form average cost for a given l, the optimal constant packet length l⁎ is obtained by a simple one‑dimensional search. The trade‑off is explicit: larger l reduces the error function ε but increases the expected transmission delay, thereby inflating AoI.
3. Variable‑length case (single SMDP). When the packet length may change over time, the problem remains an infinite‑horizon average‑cost SMDP. The authors write the Bellman optimality equation and prove a structural property (Theorem 2): the optimal action consists of a state‑dependent length‑selection threshold L⁎(Δ,c,c′) followed by the same index‑based transmission rule. They further simplify the Bellman equation, reducing its dimensionality and enabling dynamic programming with far lower computational burden.

Algorithmic implications. The index‑based threshold policy is extremely lightweight: at each slot the scheduler only needs the current AoI and the forward‑delay state to compute W(c,Δ) and compare. The freshness mapping b⁎(c) is pre‑computed offline. For the variable‑length scenario, the simplified Bellman recursion yields a lookup table for L⁎ that can be stored and accessed in real time.

Numerical evaluation. Two experiments validate the theory:

• AR(1) process inference: a model‑based simulation where ε(Δ,l) is derived from the statistical properties of an autoregressive source. The proposed scheduler reduces the average inference error to roughly one‑sixth of the baseline that always sends the freshest single sample (l=1) using an AoI‑minimizing policy.
• Cart‑pole state prediction: a trace‑driven scenario using a neural network to predict the pole angle from recent position/velocity samples. The error surface is clearly non‑monotonic in AoI. Again, the goal‑oriented scheduler achieves a six‑fold error reduction compared with the naive freshest‑sample policy.

The results highlight two key insights: (i) when the AoI‑error relationship is non‑monotonic, deliberately sending slightly older samples can exploit periodicities or known delays in the source‑target dynamics, and (ii) the index‑based waiting rule automatically adapts to the stochastic delay environment, avoiding premature transmissions that would otherwise increase AoI.

Conclusions and broader impact. By incorporating two‑way Markovian delays and allowing non‑monotonic AoI‑error functions, the paper extends goal‑oriented communication design far beyond prior work that assumed i.i.d. delays and monotone penalties. The closed‑form index policy offers a practically implementable solution for latency‑sensitive AI‑enabled services such as digital twins, remote robotics, and vehicular control. Moreover, the structural results for variable packet lengths open the door to extensions involving energy constraints, multi‑user scheduling, or adaptive compression schemes, making the framework a versatile foundation for future research in real‑time networked inference.