Analytic Continuation of Generating Functions for Two-Dimensional Random Walks in the Quarter Plane

We will be considering two dimensional queueing systems which can be equated to a random walk in the quarter plane. For simplification, we use a bivariate generating function $\pi(x,y)$ to represent the stationary distribution and we derive a functional equation that will incorporate the unknown bivariate generating function and two unknown univariate generating functions, $\pi_{1}(x)$ and $\pi_{2}(y)$, which represent the two boundary stationary distributions. Employing suitable conditions and adhering to certain values, we are able to reduce the functional equation to one which only contains both $\pi_{1}(x)$ and $\pi_{2}(y)$. Various techniques exist to compute $\pi_{1}(x)$ and $\pi_{2}(y)$, however, in order to successfully employ these methods, the original domains of analyticity for $\pi_{1}(x)$ and $\pi_{2}(y)$ need to be expanded. In this instance, analytical continuation is critical.