Eukaryotic DNA replication is elaborately orchestrated to duplicate the genome timely and faithfully. Replication initiates at multiple origins from which replication forks emanate and travel bi-directionally. The complex spatio-temporal regulation of DNA replication remains incompletely understood. To study it, computational models of DNA replication have been developed in S. cerevisiae. However, in spite of the experimental evidence of forks' speed stochasticity, all models assumed that forks' speeds are the same. Here, we present the first model of DNA replication assuming that speeds vary stochastically between forks. Utilizing data from both wild-type and hydroxyurea-treated yeast cells, we show that our model is more accurate than models assuming constant forks' speed and reconstructs dynamics of DNA replication faithfully starting both from population-wide data and data reflecting fork movement in individual cells. Completion of replication in a timely manner is a challenge due to its stochasticity; we propose an empirically derived modification to replication speed based on the distance to the approaching fork, which promotes timely completion of replication. In summary, our work discovers a key role that stochasticity of the forks' speed plays in the dynamics of DNA replication. We show that without including stochasticity of forks' speed it is not possible to accurately reconstruct movement of individual replication forks, measured by DNA combing.