The purpose of this paper is to analyze systems composed of several parts, at different working levels. At time t, only the performance level of the system, exactly determined in an additive way by the level of the unobservable components, can be observed. We obtain the probability distribution of the component state vector, given the system performance level observed, under the assumption that each component spends an exponential time in each state. Because of their complexity, these distributions, in practice, cannot be directly evaluated. Therefore, we provide some recurrent methods which allow us to calculate those probabilities in polynomial time. Finally, the question of using these results to obtain an optimal replacement policy is considered.
