  We use methods from combinatorics and algebraic statistics to study analogues
of birth-and-death processes that have as their state space a finite subset of
the $m$-dimensional lattice and for which the $m$ matrices that record the
transition probabilities in each of the lattice directions commute pairwise.
One reason such processes are of interest is that the transition matrix is
straightforward to diagonalize, and hence it is easy to compute $n$ step
transition probabilities. The set of commuting birth-and-death processes
decomposes as a union of toric varieties, with the main component being the
closure of all processes whose nearest neighbor transition probabilities are
positive. We exhibit an explicit monomial parametrization for this main
component, and we explore the boundary components using primary decomposition.
