A numerical method for obtaining equilibrium performance measures for a single group ofNidentical machines, each subject tok(>2) types of failures, is presented. The time intervals between breakdowns of a machine are exponentially distributed. The mean time between typeifailures is 1/&λi,i= 1, 2, ⃛, k. The service times required to repair any type of failure can be either deterministic, exponential, hypo-exponential, or hyper-exponential random variables, with different means for different types of failures. The service discipline is a non-preemptive fixed-priority rule, with different priorities assigned to different types of failures. System performance measures, such as average time spent by the machines waiting for service, average number of idle machines and machine operator utilization are obtained through imbedded Markov chain analysis. The algorithms used to obtain these measures exploit the special structure of the one-step transition probability matrix. The sensitivity of the performance measures to the density function of the service times and the priority assignments given is examined.