  In this work we investigate optimal geographical caching in heterogeneous
cellular networks where different types of base stations (BSs) have different
cache capacities. Users request files from a content library according to a
known probability distribution. The performance metric is the total hit
probability, which is the probability that a user at an arbitrary location in
the plane will find the content that it requires in one of the BSs that it is
covered by.
  We consider the problem of optimally placing content in all BSs jointly. As
this problem is not convex, we provide a heuristic scheme by finding the
optimal placement policy for one type of base station conditioned on the
placement in all other types. We demonstrate that these individual optimization
problems are convex and we provide an analytical solution. As an illustration,
we find the optimal placement policy of the small base stations (SBSs)
depending on the placement policy of the macro base stations (MBSs). We show
how the hit probability evolves as the deployment density of the SBSs varies.
We show that the heuristic of placing the most popular content in the MBSs is
almost optimal after deploying the SBSs with optimal placement policies. Also,
for the SBSs no such heuristic can be used; the optimal placement is
significantly better than storing the most popular content. Finally, we show
that solving the individual problems to find the optimal placement policies for
different types of BSs iteratively, namely repeatedly updating the placement
policies, does not improve the performance.
