I am considering a Markov chain i continunous time on the countable state space $S=\lbrace 0 \rbrace\cup \lbrace (i,j) \mid i \in \lbrace A,B,C,D,E,F \rbrace , j \in \mathbb{N} \rbrace$. The transistion intensities are gived by
\begin{align} q_{(i,j),(i,j+1)} & =j\beta_i,& j\in \mathbb{N},\ i\in \lbrace A,B,C,D,E,F \rbrace \\ q_{(i,j),(i,j-1)} & =j\delta_i,& j\geq 2,\ i\in \lbrace A,B,C,D,E,F \rbrace\\ q_{(i,1),0} & =\delta_i,& i\in \lbrace A,B,C,D,E,F \rbrace\\ q_{0,(i,1)} & =1,& i\in\lbrace A,B,C,D,E,F \rbrace \end{align}
Now my problem is to give a heuristic argument that explosion does not occur for instance by giving a lower bound on the expected time of the n-th jump of the Markov chain.
I have drawn the transition diagram for the chain, and I understand the structure of the process, but I can't get on with the problem.
I have a theorem which says that explosion is possible iff
$$\sum_{i=1}^\infty \frac{1}{\beta_i}+\frac{\delta_i}{\beta_i\beta_{i-1}} + \cdots + \frac{\delta_i\cdots \delta_1}{\beta_i\cdots \beta_0}<\infty$$
where $\beta_i=q_{i,i+1}$, $\delta_{i+1}=q_{i+1,i}$ and $-(\beta_{i+1}+\delta_{i+1})=q_{i+1,i+1}$ are the intensities for the process, but I cant find the right expression for the sum. I hope someone can help!