Existence of an invariant distribution and explosions in continuous time Markov chains Consider an irreducible continuous time Markov chain defined on a countable state space with an infinitesimal generator Q. 
Assume the generator has an invariant distribution $\pi Q=\pi$,  and assume for every i,  $\pi_i>0$  and $\sum_i \pi_i =1$.
Can this chain be explosive? 
 A: This summarizes my comments above: The condition given in the question is almost sufficient for a steady state to exist.  You also need a regularity condition that says there are only a finite number of transitions in finite time.  For the regularity condition, it is enough to have a (finite) uniform bound on the sum transition rates out of any state. 
For a counter-example that does not satisfy the regularity condition:  Consider an irreducible birth-death chain with state space $S = \{0, 1, 2, ...\}$ and transition rates: 
\begin{align}
q_{i,i+1} &= 2(4^i) \quad \forall i \in \{0, 1, 2, ...\} \\
q_{i,i-1} &= 4^i \quad \forall i \in \{1, 2 , 3, ...\} 
\end{align}
You can show there are positive probabilities $\pi_i$ that satisfy the detail equations $\pi_i q_{i,i+1} = \pi_{i+1}q_{i+1,i}$ for all $i\in S$.  However, the embedded discrete time chain is a random walk that is more likely to step forward than backward, and so the embedded chain $\rightarrow \infty$ with prob 1.  Intuitively, we can understand it this way: the continuous time chain traverses the entire infinite state space in finite time, and so the $\pi_i$ probabilities can (roughly) be viewed as expected fractions of time being in state $i$ over the finite simulation. 
A link to a formal statement of the steady state theorem for CTMCs (including the regularity conditions) is given in the above link.
