I have two independent random variables of Erlang distribution or you can consider them Gamma distributions but they are positive. Z = Y-X
The difference between them should be a distribution that represents a time period. Physically it can be only more than or equal to zero.
Is it safe to ignore the part where Z<0 ?
The fact that $Z$ is positive should depend on your hypothesis on $Y$ and $X$. If you do not suppose that $\mathbb{P}(Y \geqslant X) = 1$ (you only suppose that they are independent) then you cannot suppose at the same time that $Y - X$ is positive.