In the paper the concept of the cycle of operation of a deterministic queuing system and the concept of the steady-state operation of the system is introduced. Deterministic queueing systems with two and three queues and one servicing device are considered. The server serves only one requirement at each given time. A problem of optimal timing of switching from one device queue to another is studied. Necessary and sufficient conditions for stationarity of functioning of a service system are obtained. Discussed problems close to the well known problem of controlling of a traffic light in an isolated intersection.