Small-cell caching utilizes the embedded storage of small-cell base stations (SBSs) to store popular contents, for the sake of reducing duplicated content transmissions in networks and for offloading the data traffic from macro-cell base stations to SBSs. In this paper, we study a probabilistic small-cell caching strategy, where each SBS caches a subset of contents with a specific caching probability. We consider two kinds of network architectures: 1) the SBSs are always active, which is referred to as the always-on architecture, 2) the SBSs are activated on demand by mobile users (MUs), referred to as the dynamic on-off architecture. We focus our attention on the probability that MUs can successfully download contents from the storage of SBSs. First, we derive theoretical results of this successful download probability (SDP) using stochastic geometry theory. Then, we investigate the impact of the SBS parameters, such as the transmission power and deployment intensity on the SDP. Furthermore, we optimize the caching probabilities by maximizing the SDP based on our stochastic geometry analysis. The intrinsic amalgamation of optimization theory and stochastic geometry based analysis leads to our optimal caching strategy characterized by the resultant closed-form expressions. Our results show that in the always-on architecture, the optimal caching probabilities solely depend on the content request probabilities, while in the dynamic on-off architecture, they also relate to the MU-to-SBS intensity ratio. Interestingly, in both architectures, the optimal caching probabilities are linear functions of the square root of the content request probabilities. Monte-Carlo simulations validate our theoretical analysis and show that the proposed schemes relying on the optimal caching probabilities are capable of achieving substantial SDP improvement compared to the benchmark schemes.
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