To obtain the probability distribution of the burst waiting time of neutron initiation in a multiplying assembly, a method that simulates the neutrons that induce a persistent fission chain is introduced in this paper. By this simulation method, neutron initiation experiments performed on Godiva-II and CFBR-II at a reactivity above prompt critical and performed on Godiva-I at a reactivity above delayed critical are studied. The probability density function of the burst waiting time of these experiments is calculated, and the results agree well with those of the experiments. Based on this simulation algorithm, the strength of the delayed neutron source changing with time is also calculated, which helps in the understanding of these neutron initiation experiments conducted on the pulse reactor.