  We study continuous-time Markov chains on the non-negative integers under
mild regularity conditions (in particular, the set of jump vectors is finite
and both forward and backward jumps are possible). Based on the so-called flux
balance equation, we derive an iterative formula for calculating stationary
measures. Specifically, a stationary measure $\pi(x)$ evaluated at
$x\in\mathbb{N}_0$ is represented as a linear combination of a few generating
terms, similarly to the characterization of a stationary measure of a
birth-death process, where there is only one generating term, $\pi(0)$. The
coefficients of the linear combination are recursively determined in terms of
the transition rates of the Markov chain. For the class of Markov chains we
consider, there is always at least one stationary measure (up to a scaling
constant). We give various results pertaining to uniqueness and non-uniqueness
of stationary measures, and show that the dimension of the linear space of
signed invariant measures is equal to the number of generating terms. A
minimization problem is constructed in order to compute stationary measures
numerically. Moreover, a heuristic linear approximation scheme is suggested for
the same purpose by first approximating the generating terms. The correctness
of the linear approximation scheme is justified in some special cases.
Furthermore, a decomposition of the state space into different types of states
(open and closed irreducible classes, and trapping, escaping and neutral
states) is presented. Applications to stochastic reaction networks are well
illustrated.
