  We study the effect on the stationary currents of constraints affecting the
hopping rates in stochastic particle systems. In the framework of Zero Range
Processes with drift within a finite volume, we discuss how the current is
reduced by the presence of the constraint and deduce exact formulae, fully
explicit in some cases. The model discussed here has been introduced in Ref.
[1] and is relevant for the description of pedestrian motion in elongated dark
corridors, where the constraint on the hopping rates can be related to
limitations on the interaction distance among pedestrians.
