Summary: A kind of delayed Poisson shocks model is introduced to produce \(\text{NBU}- [t_0,\infty)\) life distributions. This generalizes the result that Poisson shocks models lead to the well known NBU life distributions. Some relations between the \(\text{NBU}- [t_0,\infty)\), on the one hand and the known NBU, \(\text{NBU}-t_0\) class of life distributions on the other are discussed. Furthermore, the preservation properties under the coherent systems, and the convolution property between NBU and \(\text{NBU} -[t_0, \infty)\) elements are also discussed in great detail.