Timeliness in Cache-Aided Networks With Non-Poisson Updating

We study timeliness in cache-aided networks where the inter-update times on the links are not necessarily exponentially distributed. We focus on the set of non-arithmetic distributions for inter-update times, which includes continuous probability distributions as a subset. We first characterize instantaneous age of information at each node for arbitrary networks. We then explicate the recursive equations for instantaneous age of information in multi-hop networks and use them to derive closed form expressions for expected age of information at an end-user in tree networks. We show that expected age in multi-hop networks exhibits an additive structure. Further, we show that the expected age at each user is directly proportional to the variance of the inter-update times at all links between a user and the source. We next prove analogous results for the version age of information in multi-hop networks where updates at the source are marked with incrementing version numbers. We show that expected version age at end-users is inversely proportional to the mean update interval at the source, and exhibits an additive structure. Finally, we study expected age of information in networks with the property that the update processes on the links become sparse for large network sizes, and remark that expected age scales as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(\log {n})$ </tex-math></inline-formula> in symmetric fully connected networks. We expect the analysis in this work to help alleviate the over-dependence on exponential inter-update time (i.e., Poisson) updates for future work in age of information.