We consider the problem of modeling count data where the observation period is determined by the life time of the system under study. We assume random effects or a frailty model to allow for a possible association between the death times and the counts. We assume that, given a random effect or a frailty, the death times follow a Weibull distribution with a hazard rate. For the counts, given a frailty, a Poisson process is assumed with the intensity depending on time. A gamma distribution is assumed for the frailty model. Maximum likelihood estimators of the model parameters are obtained. A model for the time to death and the number of failures system received is constructed and consequences of the model are examined.