### Abstract

We consider infinite-state discrete Markov chains which are confluent: each computation will almost certainly either reach a defined set F of final states, or reach a state from which F is not reachable. Confluent Markov chains include probabilistic extensions of several classical computation models such as Petri nets, Turing Machines, and communicating finite-state machines.

For confluent Markov chains, we consider three different variants of the reachability problem and the repeated reachability problem: The qualitative problem, i.e., deciding if the probability is one (or zero); the approximate quantitative problem, i.e., computing the probability up-to arbitrary precision; and the exact quantitative problem, i.e., computing probabilities exactly.

We also study the problem of computing the expected reward (or cost) of runs until reaching the final states, where rewards are assigned to individual runs by computable reward functions.