The classic Luria-Delbrück model for fluctuation analysis is extended 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. In particular, it is possible to consider subprobability distributions, and model non exponential growth. It leads to a family of probability distributions which depend on the expected number of mutations, the death probability of mutant cells, and the split instant distributions of normal and mutant cells. This is deduced from the Bellman-Harris integral equation, written for the birthdate inhomoge-neous case. A new theorem of convergence for the final mutant counts is proved, using an analytic method. Particular examples like the Haldane model, or the case where hazard functions of the split instant distributions are proportional are studied. The Luria-Delbrück distribution with cell deaths is recovered. A computation algorithm for the probabilities is provided.