A single-server queue with a triple priority system is considered in this paper. The server can fail during its occupation time and is sent for repair immediately. The stationary probabilities of the different states of the system are studied, under the assumption that the arrival time and the service time for the three kinds of customers, and the repair time, all have Poisson distributions with different parameters. The results obtained before, in [Mokaddis et al., Eighth International Congress for Statistics, Computer Science, Social and Demographic Research, Vol. 4, pp. 31–58 (1983)] in the case of a double priority, are derived from the present results as special cases. Moreover the mathematical expectation of the number of customers and the average waiting time of both kinds of customers and more characteristics of the system, in the case of double priority, are obtained. Also, in this paper, we study the system without priority, in the cases of both a triple priority system and a double priority system.