The invariant measure of random walks in the quarter-plane: Respresentation in geometric terms

We consider the invariant measure of homogeneous random walks in the quarter-plane. In particular, we consider measures that can be expressed as a finite linear combination of geometric terms and present conditions on the structure of these linear combinations such that the resulting measure may yield an invariant measure of a random walk. We demonstrate that each geometric term must individually satisfy the balance equations in the interior of the state space and further show that the geometric terms in an invariant measure must have a pairwise-coupled structure. Finally, we show that at least one of the coefficients in the linear combination must be negative.