  We present a detailed numerical investigation of the tunability of a
diffusive random laser when Mie resonances are excited. We solve a multimode
diffusion model and calculate multiple light scattering in presence of optical
gain which includes dispersion in both scattering and gain, without any
assumptions about the $\beta$ parameter. This allows us to investigate a
realistic photonic glass made of latex spheres and rhodamine and to quantify
both the lasing wavelength tunability range and the lasing threshold. Beyond
what is expected by diffusive monochromatic models, the highest threshold is
found when the competition between the lasing modes is strongest and not when
the lasing wavelength is furthest from the maximum of the gain curve.
