Optimization of HARQ-Based Coded Status Update System for Non-Linear Age of Information

Delivering timely status updates in a timeliness critical communication system is of paramount importance in ultra reliable and low latency communication (URLLC). Age of Information (AoI) is an important indicator to directly measure timeliness. However, in some scenarios, the performance degradation caused by information aging may not be a linear function of time, thus it is necessary to introduce non-linear function to further the study of AoI. The topic of analyzing non-linear AoI has aroused new research interest. This paper contributes to new results in this area by theoretically analyzing the non-linear AoI of Hybrid Automatic Repeat reQuest (HARQ) protocols. We consider an HARQ-based coded status update system with non-trivial coding delay, transmission delay, propagation delay, decoding delay, and feedback delay, and we derive closed-form expression of average exponential AoI of the system. Based on the obtained explicit expressions, we formulate an non-linear AoI minimization problem to investigate the age-optimal block assignment strategy in the finite block length (FBL) regime. Case studies and numerical results show that the optimal strategy is highly dependent on the propagation delay and signal-to-noise ratio (SNR). At low SNR, the non-linear AoI of the optimal block assignment is lower than baseline block assignments. When the SNR is high and increases continuously, the non-linear AoI of the system does not change much.