  Computing the reachability probability in infinite state probabilistic models
has been the topic of numerous works. Here we introduce a new property called
\emph{divergence} that when satisfied allows to compute reachability
probabilities up to an arbitrary precision. One of the main interest of
divergence is that our algorithm does not require the reachability problem to
be decidable. Then we study the decidability of divergence for probabilistic
versions of pushdown automata and Petri nets where the weights associated with
transitions may also depend on the current state. This should be contrasted
with most of the existing works that assume weights independent of the state.
Such an extended framework is motivated by the modeling of real case studies.
Moreover, we exhibit some divergent subclasses of channel systems and pushdown
automata, particularly suited for specifying open distributed systems and
networks prone to performance collapsing in order to compute the probabilities
related to service requirements.
