The classic Luria-Delbrück model can be interpreted as a Poisson compound (number of mutations) of exponential mixtures (developing time of mutant clones) of geometric distributions (size of a clone in a given time). This " three-ingredients " approach is generalized in this paper to the case where the split instant distributions of cells are not i.i.d. : the lifetime of each cell is assumed to depend on its birth date. This model takes also into account cell deaths and non-exponentially distributed lifetimes. Previous results on the convergence of the distribution of mutant counts are recovered. The particular case where the instantaneous division rates of normal and mutant cells are proportional is studied. The classic Luria-Delbrück and Haldane models are recovered. Probability computations and simulation algorithms are provided. Robust estimation methods developed for the classic mutation models are extended to the new model: their properties of consistency and asymptotic normality remain true; their asymptotic variance are computed. Finally, the estimation biases induced by considering classic mutation models instead of inhomogeneous model are studied with simulation experiments.