A note on the connection between product-form Jackson networks and counting lattice walks in the quarter plane

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.