The ultimal goal is to find explicit solutions for counting generating functions (CGF) of some random walks in the quarter plane, starting from the wellknown product-form for the stationary distribution of customers in Jackson's stochastic networks. The first step is to solve a relativety simple boundary value problem (BVP) for the queueing system. Then, by using a variational principle based on the continuity with respect to adequate parameters, it is possible to transform some smooth curves in a continuous fashion and to get explicitly formulas for the conformal mappings of Riemann's theorem. This might be a new way of computing the gluing functions appearing in [1, 3], which are a key ingredient in the explicit formulas for CGFs.