Disorder and non-conservation in a driven diffusive system

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.