The initial neural encoding of acoustic information occurs by means of spikes in primary auditory afferents. Each mammalian primary auditory afferent (type-I auditory-nerve fiber; ANF) is associated with only one ribbon synapse in one receptor cell (inner hair cell; IHC). The properties of ANF spike trains therefore provide an indirect view of the operation of individual IHC synapses. We showed previously that a point process model of presynaptic vesicle pool depletion and deterministic exponential replenishment, combined with short postsynaptic neural refractoriness, accounts for the interspike interval (ISI) distributions, serial ISI correlations, and spike-count statistics of a population of cat-ANF spontaneous spike trains. Here, we demonstrate that this previous synapse model produces unrealistic properties when spike rates are high and show that this problem can be resolved if the replenishment of each release site is stochastic and independent. We assume that the depletion probability varies between synapses to produce differences in spontaneous rate and that the other model parameters are constant across synapses. We find that this model fits best with only four release sites per IHC synapse, a mean replenishment time of 17 ms, and absolute and mean relative refractory periods of 0.6 ms each. This model accounts for ANF spontaneous spike timing better than two influential, comprehensive models of the auditory periphery. It also reproduces ISI distributions from spontaneous and steady-state driven activity from other studies and other mammalian species. Adding fractal noise to the rate of depletion of each release site can yield long-range correlations as typically observed in long spike trains. We also examine two model variants having more complex vesicle cycles, but neither variant yields a markedly improved fit or a different estimate of the number of release sites. In addition, we examine a model variant having both short and long relative refractory components and find that it cannot account for all aspects of the data. These model results will be beneficial for understanding ribbon synapses and ANF responses to acoustic stimulation.