= Solution
Under the transience assumption, the number of returns to zero is geometrically distributed and hence finite almost surely. The final excursion is a pure ascent through all sufficiently large states. Its holding times are independent exponential random variables with means
$$
\frac1{\lambda_i(1+\rho_i)}.
$$
The <nonexplosion criterion for a pure birth process> says that their sum is finite almost surely exactly when the sum of their means is finite. Thus
$$
\boxed{X\text{ is explosive}\iff
\sum_{i=1}^{\infty}\frac1{\lambda_i(1+\rho_i)}<\infty.}
$$
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