Analysis of Hierarchical AoII over unreliable channel: A Stochastic Hybrid System Approach

In this work, we generalize the Stochastic Hybrid Systems (SHSs) analysis of traditional AoI to the AoII metric. Hierarchical ageing processes are adopted using the continuous AoII for the first time, where two different hierarchy schemes, i.e., a hybrid of linear ageing processes with different slopes and a hybrid of linear and quadratic ageing processes, are considered. We first modify the main result in \cite[Theorem 1]{yates_age_2020b} 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 based on our Theorem \ref{theorem1} in two typical scenarios with different channel conditions, i.e., an M/M/1/1 queue over noisy channel and two M/M/1/1 queues over collision channel. Moreover, we analyze the stability conditions for two scenarios given that the quadratic ageing process may lead to the absence of stationary solutions. Finally, we compare the average age performance between the classic AoI results and our AoII results in the M/M/1/1 queue, and the effects of different channel parameters on AoII are also evaluated.