We consider a server with large capacity delivering video files encoded in various resolutions. We assume that the system is under saturation in the sense that the total demand exceeds the server capacity $C$. In such a case, requests may be rejected. For the policies considered in this paper, instead of rejecting a video request, it is downgraded. When the occupancy of the server is above some value $C_0{<}C$, the server delivers the video at a minimal bit rate. The quantity $C_0$ is the bit rate adaptation threshold. For these policies, request blocking is thus replaced by bit rate adaptation. Under the assumptions of Poisson request arrivals and exponential service times, we show that, by rescaling the system, a process associated to the occupancy of the server converges to some limiting process whose invariant distribution is computed by a Wiener-Hopf technique. This allows us to derive an asymptotic expression of the key performance measure of such a policy, the equilibrium probability that a request is transmitted without downgrading. Numerical applications of these results are presented.