Queuing Systems with Bounded Waiting-Line Length

Equations have been developed for the various state probabilities associated with queuing systems having a finite number of sources, waiting places, and service stations. Units that need servicing are generated in the sources. If a service station is free when a unit arrives from a source, the unit's service starts immediately; otherwise, the unit waits in a waiting place, or if there are none available, the unit waits in a source, thus temporarily blocking it from generating additional units. The chemical plant production process is an illustration of such a system. In that process reactors correspond to sources, holding tanks to waiting places, and stills to service stations. The reactors generate batches that correspond to units. It is assumed in this paper that generation of units in an unblocked source follows a Poisson distribution. State probabilities under the condition of statistical equilibrium are given for the following kinds of systems: (1) at each service station the service time is exponential; (2) there is one source and one service station, and the service time is constant.