In this paper we determine the equilibrium distribution for a general class of Markov processes on a semi-infinite strip. We expose a method to express the equilibrium probabilities of the Markov process as a finite sum of terms, which are geometric in the unbounded variable. The geometric factors are the roots inside the unit circle of a determinantal equation. By using a generating-function technique we are able to determine these roots very efficiently. Because of this, the expression for the equilibrium probabilities becomes numerically very attractive.