A multiple system governed by a quasi-birth-and-death process

The system we consider comprises n units, of which one has to operate for the system to work. The other units are in repair, in cold standby, or waiting for repair. Only the working unit can fail. The operational and repair times follow phase-type distributions. Upon failure, it is replaced by a standby unit and goes to the repair facility. There is only one repairman. When one unit operates the system is up and when all the units are in repair or waiting for repair, the system is down. This system is governed by a finite quasi-birth-and-death process. The stationary probability vector and useful performance measures in reliability, such as the availability and the rate of occurrence of failures are explicitly calculated. This model extends other previously considered in the literature. The case with an infinite number of units in cold standby is also studied. Computational implementation of the results is performed via a numerical example, and the different systems considered are compared from the reliability measures determined.