For a continuous time markov chain (taking values in $\mathbb{N}$) with infinitesimal generator $Q$, the squence of first moments $(\mathbb{E}_i[T])_{i\in\mathbb{N}}$ of a hitting time $T$ entering a set $A$ when starting from state $i$ is given by the minimal non-negative solution of the system $$\begin{cases} \mathbb{E}_i[T] = 0, & i\in A\\ -\sum_{j\neq i} q_{ij}\mathbb{E}_j[T] = 1, & i\not\in A\end{cases}.$$ (Cited from the book by Norris, Markov Chains)
By differentiating the sequence of Laplace transforms $(\pi_i(\tau))_{i\in\mathbb{N}}$ of $T$ twice in $0$ in the case where $\pi_i(0) = 1$ for all $i$ I arrive at the set of equations $$\begin{cases} \mathbb{E}_i[T^2] = 0, & i\in A\\ -\sum_{j\neq i} q_{ij}\mathbb{E}_j[T^2] = 2\mathbb{E}_i[T], & i\not\in A\end{cases}$$ for the second moment. In the book by Norris, he does not treat the second moment and I was wondering whether the conditions minimal and non-negative just carry over to this case to find the second moment? Sources are highly appreciated.