  Cache-aided content delivery is studied in a multi-server system with $P$
servers and $K$ users, each equipped with a local cache memory. In the delivery
phase, each user connects randomly to any $\rho$ out of $P$ servers. Thanks to
the availability of multiple servers, which model small-cell base stations
(SBSs), demands can be satisfied with reduced storage capacity at each server
and reduced delivery rate per server; however, this also leads to reduced
multicasting opportunities compared to the single-server scenario. A joint
storage and proactive caching scheme is proposed, which exploits coded storage
across the servers, uncoded cache placement at the users, and coded delivery.
The delivery \textit{latency} is studied for both \textit{successive} and
\textit{parallel} transmissions from the servers. It is shown that, with
successive transmissions the achievable average delivery latency is comparable
to the one achieved in the single-server scenario, while the gap between the
two depends on $\rho$, the available redundancy across the servers, and can be
reduced by increasing the storage capacity at the SBSs. The optimality of the
proposed scheme with uncoded cache placement and MDS-coded server storage is
also proved for successive transmissions.
