Recognizing finite state machines with repetition
Finite automata in which the transitions are regular expressions might be the model you are looking at. Indeed you could have transitions of the form $p \xrightarrow{aa + bbb} q$ to indicate that in order to go from $p$ to $q$, you need either 3 $a$'s or 2 $b$'s.
This model has the same expressive power as NFAs and DFAs.