Timely Tracking of a Remote Dynamic Source Via Multi-Hop Renewal Updates

We study the version age of information in a multi-hop multi-cast cache-enabled network, where updates at the source are marked with incrementing version numbers, and the inter-update times on the links are not necessarily exponentially distributed. We focus on the set of non-arithmetic distributions, which includes continuous probability distributions as a subset, with finite first and second moments for inter-update times. We first characterize the instantaneous version age of information at each node for an arbitrary network. We then explicate the recursive equations for instantaneous version age of information in multi-hop networks and employ semi-martingale representation of renewal processes to derive closed form expressions for the expected version age of information at an end user. We show that the expected age in a multi-hop network exhibits an additive structure. Further, we show that the expected age at each user is proportional to the variance of inter-update times at all links between a user and the source. Thus, end user nodes should request packet updates at constant intervals.