Queueing systems constitute a central tool in modeling and performance analysis. These
types of systems are in our everyday life activities, and the theory of queueing systems was
developed to provide models for forecasting behaviors of systems subject to random demand.
The practical and useful applications of the discrete-time queues make the researchers to con-
tinue making an e ort in analyzing this type of models. Thus the present contribution relates
to a discrete-time Geo/G/1 queue in which some messages may need a second service time
in addition to the rst essential service. In day-to-day life, there are numerous examples of
queueing situations in general, for example, in manufacturing processes, telecommunication,
home automation, etc, but in this paper a particular application is the use of video surveil-
lance with intrusion recognition where all the arriving messages require the main service and
only some may require the subsidiary service provided by the server with di erent types of
strategies.
We carry out a thorough study of the model, deriving analytical results for the stationary
distribution. The generating functions of the number of messages in the queue and in the
system are obtained. The generating functions of the busy period as well as the sojourn times
of a message in the server, the queue and the system are also provided.
