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 indexbased 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.

In this paper, we extend the freshness-oriented sampling problem by incorporating controlled delay statistics through heterogeneous routing options, where the Age of Information (AoI) serves as the metric for data freshness. Our objective is to jointly optimize sampling and routing policies to minimize the long-term average AoI, where the sender can choose to forward each status update over one of the available routes, which have distinct delay statistics. This problem is an infinite-horizon Semi-Markov Decision Process (SMDP) with an uncountable state space and a hybrid action space, consisting of discrete routing choices and continuous waiting times. We develop an efficient algorithm to solve this problem and theoretically establish that the optimal policy exhibits a threshold structure, characterized by: (i) a threshold-based monotonic handover mechanism for optimal routing, where the switching order aligns with the decreasing order of mean delays; and (ii) a multi-threshold piecewise linear waiting mechanism for optimal sampling, where the total number of thresholds is upper bounded by 2N – 1, given N selectable routes. We implement the proposed algorithm in a satellite-terrestrial integrated routing scenario, and simulation results reveal an intriguing insight: routes with higher average delay or variance can still contribute to minimizing AoI.

We study the design of a goal-oriented sampling and scheduling strategy through a channel with highly variable two-way random delay, which can exhibit memory (e.g., Delay and Disruption Tolerant Networks). The objective of the communication is to optimize the performance of remote inference, where an inference algorithm (e.g., a trained neural network) on the receiver side predicts a time-varying target signal using the data samples transmitted by a sensor. Previous formulations to this problem either assumed a channel with IID transmission delay, neglecting feedback delay or considered the monotonic relation that the performance only gets worse as the input information ages. We show how, with delayed feedback, one can effectively exploit the knowledge about delay memory through an index-based threshold policy. This policy minimizes the expected time-average inference error that can be monotone or non-monotone in age. The index function is expressed in terms of the Age of Information (AoI) on the receiver side and a parameter regarding the distribution of subsequent transmission delay, both of which can readily be tracked.