Tandem mass spectrometry (MS/MS) continues to be the technology of choice for high-throughput analysis of complex proteomics samples. While MS/MS spectra are commonly identified by matching against a database of known protein sequences, the complementary approach of spectral library searching against collections of reference spectra consistently outperforms sequence-based searches by resulting in significantly more identified spectra. However, while spectral library searches benefit from the advance knowledge of the expected peptide fragmentation patterns recorded in library spectra, estimation of the statistical significance of spectrum-spectrum matches (SSMs) continues to be hindered by difficulties in finding an appropriate definition of "random" SSMs to use as a null model when estimating the significance of true SSMs. We propose to avoid this problem by changing the null hypothesis: instead of determining the probability of observing a high SSM score between randomly matched spectra, we estimate the probability of observing a low SSM score between replicate spectra of the same molecule. To this end, we explicitly model the variation in instrument measurements of MS/MS peak intensities and show how these models can be used to determine a theoretical distribution of SSM scores between reference and query spectra of the same molecule. While the proposed spectral library generating function (SLGF) approach can be used to calculate theoretical distributions for any additive SSM score (e.g., any dot product), we further show how it can be used to calculate the distribution of expected cosines between reference and query spectra. We developed a spectral library search tool, Tremolo, and demonstrate that this SLGF-based search tool significantly outperforms current state-of-the-art spectral library search tools and provide a detailed discussion of the multiple reasons behind the observed differences in the sets of identified MS/MS spectra.