This paper considers a discrete-time Geo/G/1 retrial queue where the retrial time has a general distribution and the server is subject to starting failures. It is assumed that the server, after each service completion, begins a process of search in order to find the following customer to be served. We analyze the Markov chain underlying the queueing system and obtain its ergodicity condition. The generating functions of the number of customers in the orbit and in the system are also obtained along with the marginal distributions of the orbit size when the server is idle, busy or down. Then, we give two stochastic decomposition laws and as an application we give bounds for the proximity between the system size distributions of our model and the corresponding model without retrials. Finally, some numerical examples show the influence of the parameters on several performance characteristics.
