The burst initiation time and emission rate of delayed neutrons for the Godiva I assembly in the presence of a weak neutron source, which is 5[c-slash] above prompt criticality, were calculated using an analog Monte Carlo method. The initiation time of the simulation is consistent with the result of the experiment. The calculated emission rate of delayed neutrons is included, and an extended probability model of the initiation time is derived based on point kinetics, which well explains the experimental measurement.