Optimization and Analysis of Probabilistic Caching in $N$-tier   Heterogeneous Networks
  In this paper, we study the probabilistic caching for an $N$-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 (SNR) regime. The problem is proved to be convex and solved efficiently. We next establish an interesting connection between $N$-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 $N$-tier HetNets depends also on the 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 inverse to the BS cache size of the same tier and is linear to BS cache sizes of other tiers. 
Kuikui Li, Chenchen Yang, Zhiyong Chen, and Meixia Tao
cs.IT math.IT