The discrete-time queueing system with repairable server and phase-type repair time

In this paper, the discrete-time queueing system with repairable server and a general independent input and phase-type repair time are studied. Suppose that service times of customers and lifetime of server follow geometric distributions, the distributions of stationary queue length and waiting time are provided by using the matrix analytic method, and these distributions are proved to be discrete phase-type distributions. We also obtain the distributions of failure number which server takes in a generalized service time, and show that it follows a modified geometric distribution. Furthermore, we provide a comparison on two queueing systems with same arrival processes, service times, lifetime of server and repair times but one is in a discrete version and the other is in a continuous version. Finally, the relationship among the arrival process, service time and failure rate of server are illuminated by several numerical examples.