  We consider a stochastic fluid queue served by a constant rate server and
driven by a process which is the local time of a certain Markov process. Such a
stochastic system can be used as a model in a priority service system,
especially when the time scales involved are fast. The input (local time) in
our model is always singular with respect to the Lebesgue measure which in many
applications is ``close'' to reality. We first discuss how to rigorously
construct the (necessarily) unique stationary version of the system under some
natural stability conditions. We then consider the distribution of performance
steady-state characteristics, namely, the buffer content, the idle period and
the busy period. These derivations are much based on the fact that the inverse
of the local time of a Markov process is a L\'evy process (a subordinator)
hence making the theory of L\'evy processes applicable. Another important
ingredient in our approach is the Palm calculus coming from the point process
point of view.
