The base distributed asynchronous read/write computation model is made up of $n$ asynchronous processes which communicate by reading and writing atomic registers only. The distributed asynchronous iterated model is a more constrained model in which the processes execute an infinite number of rounds and communicate at each round with a new object called immediate snapshot object. Moreover, in both models up to $n-1$ processes may crash in an unexpected way. An important computability issue associated with these models concerns their respective computability power. When considering distributed computing problems whose main issue is to cope with asynchrony and failures called decision tasks (such as consensus, set agreement, etc.) two main results are associated with the previous models. The first states that these models are computationally equivalent for decision tasks. The second states that they are no longer equivalent when both are enriched with the same failure detector. This paper shows how to capture failure detectors in each model in order both models become computationally equivalent. To that end it introduces the notion of a ''strongly correct'' process which appears particularly well-suited to the iterated model. It also presents simulations that proves the computational equivalence when both models are enriched with a failure detector. Interestingly, these simulations works for a large family of failure detector classes. The paper extends also the simulations to the case where the wait-freedom requirement is replaced by the notion of $t$-resilience. Last but not least, a noteworthy feature of the proposed approach lies in its simplicity.