Coding Gain for Age of Information in a Multi-source System with Erasure Channel

In our work, we study the age of information ($\text{AoI}$) in a multi-source system where $K$ sources transmit updates of their time-varying processes via a common-aggregator node to a destination node through a channel with packet delivery errors. We analyze $\text{AoI}$ for an $(\alpha ,\beta ,{\epsilon }_0,{\epsilon }_1)$-Gilbert-Elliot ($\text{GE}$) packet erasure channel with a round-robin scheduling policy. We employ maximum distance separable ($\text{MDS}$) scheme at aggregator for encoding the multi-source updates. We characterize the mean $\text{AoI}$ for the $\text{MDS}$ coded system for the case of large blocklengths. We further show that the \emph{optimal coding rate} that achieves maximum \emph{coding gain} over the uncoded system is $n(1-\text{pers})-\text{smallO}(n)$, where $\text{pers}\triangleq \frac{\beta }{\alpha +\beta }{\epsilon }_0+\frac{\alpha }{\alpha +\beta }{\epsilon }_1$, and this maximum coding gain is $(1+\text{pers})\mathrm{/}(1+\text{smallO}(1))$.