In this paper, we will compute Buchsbaum Rim functions of two variables associated to a parameter matrix of a special form over a one-dimensional Cohen Macaulay local ring, and we will determine when the function coincides with the Buchsbaum-Rim polynomial. As a consequence, we have that there exists the case where the function does not coincide with the polynomial function, which should be contrasted with the ordinary Buchsbaum Rim function of single variable.