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Motivated by a recent application of the Kolmogorov-Johnson-Mehl-Avrami(KJMA) model to the study of DNA replication, we consider the one-dimensionalversion of this model. We generalize previous work to the case where thenucleation rate is an arbitrary function $I(t)$ and obtain analytical resultsfor the time-dependent distributions of various quantities (such as the islanddistribution). We also present improved computer simulation algorithms to studythe 1D KJMA model. The analytical results and simulations are in excellentagreement.Comment: 9 pages, 9 figure

Nucleation and growth in one dimension. I. The generalized Kolmogorov-Johnson-Mehl-Avrami model