Pure birth process

A pure birth process on ${\mathbb{N}}_0$ jumps only from $n$ to $n+1$, at a strictly positive rate ${\lambda }_n$. Its successive holding times are independent exponential variables of rates ${\lambda }_n$.

A pure birth process is nonexplosive exactly whenIts explosion time is the sum of independent exponential holding times. If the displayed series converges, that sum has finite expectation. If it diverges, the Laplace transform of the sum isforcing the sum to be infinite almost surely.

The pure birth process with rates ${\lambda }_n=\alpha n+\beta$, where $\alpha ,\beta >0$, combines linear reproduction with immigration. Starting from zero, its mean satisfiesand hence