Optimization and Analysis of Probabilistic Caching in $N$ -Tier Heterogeneous Networks

In this paper, we study the probabilistic caching for an <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula>-tier wireless heterogeneous network (HetNet) using stochastic geometry. A general and tractable expression of the successful delivery probability (SDP) is first derived. We then optimize the caching probabilities for maximizing the SDP in the high signal-to-noise ratio regime. The problem is proved to be convex and solved efficiently. We next establish an interesting connection between <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula>-tier HetNets and single-tier networks. Unlike the single-tier network where the optimal performance only depends on the cache size, the optimal performance of <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula>-tier HetNets depends also on the base station (BS) densities. The performance upper bound is, however, determined by an equivalent single-tier network. We further show that with uniform caching probabilities regardless of content popularities, to achieve a target SDP, the BS density of a tier can be reduced by increasing the cache size of the tier when the cache size is larger than a threshold; otherwise, the BS density and BS cache size can be increased simultaneously. It is also found analytically that the BS density of a tier is <italic>inverse</italic> to the BS cache size of the same tier and is <italic>linear</italic> to BS cache sizes of other tiers.