In this paper a problem arising in queueing and dam theory is studied. We shall consider a G / G* / 1 queueing model, i.e., a G/G/1 queueing model of which the service process is a separable centered process with stationary independent increments. This is a generalisation of the well-known G/G/1 model with constant service rate. Several results concerning the amount of work done by the server, the busy cycles etc., are derived, mainly using the well-known method of Pollaczek. Emphasis is laid on the similarities and dissimilarities between the results of the 'classical' G/G/1 model and the G / G* / 1 model.