The analysis of runaway phenomena with the use of the Boltzmann equation shows that the time evolution of the distribution function and of the other quantities of interest, for example the average velocity, occurs on two time scales: the short time scale of the collisional equilibrium and the long time scale of the runaway flux. Under suitable conditions on the collision kernels, these two time scales are well separated. In this paper, we present some analytical results for the BGK model based on a multiple time scale expansion: in particular, we derive approximate analytical expressions for the distribution function, the average velocity and the collision number and compare them with the numerical solution.