  We compute the stationary distribution of a continuous-time Markov chain
which is constructed by gluing together two finite, irreducible Markov chains
by identifying a pair of states of one chain with a pair of states of the other
and keeping all transition rates from either chain (the rates between the two
shared states are summed). The result expresses the stationary distribution of
the glued chain in terms of quantities of the two original chains. Some of the
required terms are nonstandard but can be computed by solving systems of linear
equations using the transition rate matrices of the two original chains.
Special emphasis is given to the cases when the stationary distribution of the
glued chain is a multiple of the equilibria of the original chains, and when
not, for which bounds are derived.

Probability