Scaling Laws for Age of Information in Wireless Networks

We study age of information in a multiple source-multiple destination setting with a focus on its scaling in large wireless networks. There are <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> nodes uniformly and independently distributed on a fixed area that are randomly paired with each other to form <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> source-destination (S-D) pairs. Each source node wants to keep its destination node as up-to-date as possible. To accommodate successful communication between all <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> S-D pairs, we first propose a three-phase transmission scheme which utilizes local cooperation between the nodes along with what we call <italic>mega update packets</italic> to serve multiple S-D pairs at once. We show that under the proposed scheme average age of an S-D pair scales as <inline-formula> <tex-math notation="LaTeX">$O\left({n^{\frac {1}{4}}\log n}\right)$ </tex-math></inline-formula> as the number of users, <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>, in the network grows. Next, we observe that communications that take place in Phases I and III of the proposed scheme are scaled-down versions of network-level communications. With this along with scale-invariance of the system, we introduce hierarchy to improve this scaling result and show that when <italic>hierarchical cooperation</italic> between users is utilized, an average age scaling of <inline-formula> <tex-math notation="LaTeX">$O(n^{\alpha (h)}\log n)$ </tex-math></inline-formula> per-user is achievable, where <inline-formula> <tex-math notation="LaTeX">$h$ </tex-math></inline-formula> denotes the number of hierarchy levels and <inline-formula> <tex-math notation="LaTeX">$\alpha (h) = \frac {1}{3\cdot 2^{h}+1}$ </tex-math></inline-formula>. We note that <inline-formula> <tex-math notation="LaTeX">$\alpha (h)$ </tex-math></inline-formula> tends to 0 as <inline-formula> <tex-math notation="LaTeX">$h$ </tex-math></inline-formula> increases, and asymptotically, the average age scaling of the proposed hierarchical scheme is <inline-formula> <tex-math notation="LaTeX">$O(\log n)$ </tex-math></inline-formula>. To the best of our knowledge, this is the best average age scaling result in a status update system with multiple S-D pairs.