Numerical ratio between the length of a string and the transited states
The way you have defined S, we will always have $S \ge |w|$.  Why?  Because each time the automaton reads one symbol from $w$, it follows one transition and visits the state at the end of that transition, incrementing $S$ by one.  In other words, we add one to $S$ for each symbol in $w$.  Thus $S$ is at least as large to the number of symbols in $w$.  (It might be larger if you follow $\epsilon$-transitions, as they don't consume any symbols in $w$ but do visit a state.)

Try working through a few examples and it should become clearer.