Here is the exact same question but on deterministic finite automaton. The case for deterministic finite automaton is simple. For each state only one transition is possible for each input symbol and quite intuitively one can verify that a DFA shall halt when the input string is exhausted.
Lately I came across a question as :
"Whether a finite state automaton halts on all inputs" is a decidable problem.
I came to the conclusion that the above statement is True, with the following thought process. Be it an $\epsilon$-NFA or DFA, every finite automata has an equivalent simplified DFA and hence it is this DFA which is bound to halt on all (finite) input.
Just as I mentioned above about $\epsilon$-NFA, I was wondering, that we can have an infinite loop in this finite automaton. Possibly using $\epsilon$ transitions as follows:
But the possible loop in the red path in the diagram is harmless (unlike a loop in Turing Machine TM), as it is just a parallel path, which me might not even consider while the parallel processing of the contemporary instances of NFA is going on.
I cannot possibly think of a situation of finite automata where it goes into a situation like TM. Is such a situation at all possible ?