  We study memory dependent binary-state dynamics, focusing on the noisy-voter
model. This is a non-Markovian process if we consider the set of binary states
of the population as the description variables, or Markovian if we incorporate
"age", related to the time one has spent holding the same state, as a part of
the description. We show that, in some cases, the model can be reduced to an
effective Markovian process, where the age distribution of the population
rapidly equilibrates to a quasi-steady state, while the global state of the
system is out of equilibrium. This effective Markovian process shares the same
phenomenology of the non-linear noisy-voter model and we establish a clear
parallelism between these two extensions of the noisy-voter model.
