  We introduce the prioritising exclusion process, a stochastic scheduling
mechanism for a priority queueing system in which high priority customers gain
advantage by overtaking low priority customers. The model is analogous to a
totally asymmetric exclusion process with a dynamically varying lattice length.
We calculate exact local density profiles for an unbounded queue by deriving
domain wall dynamics from the microscopic transition rules. The structure of
the unbounded queue carries over to bounded queues where, although no longer
exact, we find the domain wall theory is in very good agreement with simulation
results. Within this approximation we calculate average waiting times for
queueing customers.
