We present a numerical study of the temporal dynamics of a stimulated Brillouin scattering ring resonator. A coaxial flash-lamp-pumped Rh6G dye laser is assumed. The influence of the most important parameters on the temporal evolution of the resonator is analyzed, namely, the acoustic decay time of the nonlinear material, the features of the external injection pulse - its pulse width, energy, and spatial quality - and the coupling mirror reflectivity. We found the conditions to initiate and maintain laser oscillation in the ring resonator as long as the duration of the pumping system pulse persisted.