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Recently, modeling many systems in communication and networks produce difference equations characterizing the dynamics of such systems. A certain class of functional equations arises from such difference equations. The functional equation of our interest arises from a queueing model for a gateway linking two Ethernet-type local area networks. It stems from a second order difference equation with boundary conditions reflecting the dynamics of the gateway. The functional equation is not yet solved to find the exact system distribution. In this article, on one hand we investigate the possible singularities of the unknowns of the two-place functional equation. On the other hand we introduce some application of computing the possible singularities, and we use the generating functions of the system distribution to compute some expectations of interest. It is hoped that computing the possible singularities of the unknowns of such equation will be a step forward in the road towards a general solution theory for this interesting class of equations.

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