There is a growing desire by mobile network operators and other stakeholders to reduce carbon emissions from the operation of mobile networks. The widespread adoption of ultra-dense 5G and Internet of Things (IoT) networks will likely increase the energy demand from these networks significantly. In this paper, we present steady-state and transient-state models of a battery energy storage system for an off-grid base station site. The presented models are based on the energy packet (EPs) concept and Continuous Time Markov Chains (CTMCs). We derive the transient-state and steady-state probabilities of the amount of energy (or energy packets) present in the BESS, including the state probabilities that BESS is empty (completely discharged) and that it is full (fully charged), and investigate the impact of the mean energy generation to demand ratio, on these state probabilities. We derive the mean number of energy packets present in BESS and investigate the influence of the mean energy generation rate. Also, we model the density of the time required to charge BESS to its full capacity when the renewable energy sources can generate enough energy to meet the needs of the base station site and the density of the time required to completely deplete the energy stored in BESS when the renewable energy sources cannot generate a sufficient amount of energy to meet the demand of the site. We also investigate the influence of the design parameters, such as the mean energy generation to demand ratio, on the distribution of the time required to charge BESS to its full capacity (for the mean energy generation to demand ratio greater than one) and the time required to completely deplete the energy stored in BESS (for the mean energy generation to demand ratio less than one).