  In this paper, we investigate a cache updating system with a server
containing $N$ files, $K$ relays and $M$ users. The server keeps the freshest
versions of the files which are updated with fixed rates. Each relay can
download the fresh files from the server in a certain period of time. Each user
can get the fresh files from any relay as long as the relay has stored the
fresh versions of the requested files. Due to the limited storage capacity and
updating capacity of each relay, different cache designs will lead to different
average freshness of all updating files at users. In order to keep the average
freshness as large as possible in the cache updating system, we formulate an
average freshness-optimal cache updating problem (AFOCUP) to obtain an optimal
cache scheme. However, because of the nonlinearity of the AFOCUP, it is
difficult to seek out the optimal cache scheme. As a result, an linear
approximate model is suggested by distributing the total update rates
completely in accordance with the number of files in the relay in advance. Then
we utilize the greedy algorithm to search the optimal cache scheme that is
satisfied with the limited storage capacity of each relay. Finally, some
numerical examples are provided to illustrate the performance of the
approximate solution.
