Analysis of Hierarchical AoII Over Unreliable Channels: A Stochastic Hybrid System Approach

In this work, we generalize the Stochastic Hybrid Systems (SHSs) analysis of Age of information (AoI) to the Age of Incorrect Information (AoII) metric. Hierarchical ageing processes are adopted using the continuous AoII for the first time. Two different hierarchy schemes are considered: 1). A hierarchy of zero and linear ageing processes with different slopes; 2). A hierarchy of zero, linear and exponential ageing processes. We first modify the main result in Yates (2020) to provide a systematic way to analyze the continuous hierarchical AoII over unslotted real-time systems. The closed-form expressions of average hierarchical AoII are obtained in two typical scenarios with different channel conditions, i.e., an M/M/1/1 queue over noisy channels and two M/M/1/1 queues over collision channels. Moreover, under each scenario, we analyze the stability issue regarding positive recurrence and provide the stability conditions that ensure a steady-state average AoII. Finally, we compare the closed-form results between average AoI and AoII in the M/M/1/1 queue. The effects of different channel parameters on the average hierarchical AoII are also evaluated.