The main objective of this paper is to investigate the analytical solution of a special case of the general class of
challenged functional equations given by
A1(x; y)f(x; y) = A2(x; y)f(x; 0) + A3(x; y)f(0; y) + A4(x; y)f(0; 0) + A5(x; y);
where Ai(x; y); i = 1; :::; 5 are given functions in two complex variables x; y. The main unknown function f(x; y) is dened
in such a way that for every xed x it is analytic in the y􀀀unit disk, and similarly for y. The unknown function f(x; 0) is
dened as f(x; 0) : D ! C, where D stands for the unit disk, and a similar denition holds for the other unknown function
f(0; y). The functional equation considered in this article is not solved yet. However it is an interesting equation as it
arose from a queueing model of a network gateway linking two ethernet local area networks. The solution is obtained by
reduction to Riemann-Hilbert boundary value problem via using conformal mappings