  We consider a disordered asymmetric exclusion process in which randomly
chosen sites do not conserve particle number. The model is motivated by
features of many interacting molecular motors such as RNA polymerases. We solve
the steady state exactly in the two limits of infinite and vanishing
non-conserving rates. The first limit is used as an approximation to large but
finite rates and allows the study of Griffiths singularities in a
nonequilibrium steady state despite the absence of any transition in the pure
model. The disorder is also shown to induce a stretched exponential decay of
system density with stretching exponent \phi= 2/5.
