Cache Freshness in Information Updating Systems

We consider a cache updating system with a source, $m$ caches and a user. There are $n$ files. The source keeps the freshest version of the files which are updated with known rates $\lambda_{i}$. The first cache downloads and keeps the freshest version of the files from the source with rates $c_{1i}$ and cache $r$ downloads and keeps the files from cache $r-1$ with rates $c_{ri}$ for $r > 1$. The user gets updates from cache $m$ with rates $u_{i}$. When the user gets an update, it either gets a fresh update from cache $m$ or the file at cache $m$ becomes outdated by a file update at the source. We find an analytical expression for the average freshness of the files at the user. We provide an alternating maximization based method to find the update rates for the caches, $c_{ri}$, and for the user, $u_{i}$, to maximize the freshness of the files at the user. We note that for a given set of update rates for the user (resp. for the caches), the optimal rate allocation policy for the caches (resp. for the user) is a threshold policy, where the optimal update rates for rapidly changing files at the source may be equal to zero.