  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.
