Replication of genetic material is an important process for all living organisms. Origins of replication initiate the copying of DNA at many points on a chromosome, and it is the distribution of these points that is relevant here, as it presents us with an interesting stochastic process. It was observed by Newman et al. that for various types of yeast cells, there were fewer very small inter-origin spacings, and fewer very large inter-origin spacings in the replication origin data than would be expected if the origins were uncorrelated, random points. We propose a very simple stochastic model for DNA replication and determine that this probabilistic process produces replication origins that display repulsion between origins and relative scarcity of large spacings. We detail some connections between this model and existing polynuclear or polymer growth models.