The tracking pattern formed on the dielectric surface due to a surface electrical discharge exhibits fractal structure. In order to quantitatively investigate the fractal characteristics of the surface tracking pattern, the dielectric breakdown model has been employed to numerically generate the surface tracking pattern. In dielectric breakdown model, the pattern growth is determined stochastically by a probability function depending on the local electric potential difference. For the computation of the electric potential for all points of the lattice, a two-dimensional discrete Laplace equation is solved by mean of the successive over-relaxation method combined to the Gauss-Seidel method. The box counting method has been used to calculate the fractal dimensions of the simulated patterns with various exponent $eta$ and breakdown voltage $phi_b$. As a result of the simulation, it is found that the fractal nature of the surface tracking pattern depends strongly on $eta$ and $phi_b$.