A single-server queuing process in which the service facility is subject to breakdowns, is considered in this paper as a “preemptive resume priority” queuing process. The restrictions imposed on arrival process of breakdowns give rise to several breakdown models with different operational parameters. The time-dependent and the steady-state queue-length generating functions of two breakdown models are given in this paper, when the service time process and repair time process are general distributions. The results of White and Christie and Heathcote are obtained as particular cases.