Stationary Single-Server Queuing Processes with a Finite Number of Sources

Some basic relations are obtained that apply to finite single-server stationary queuing processes. These relations depend only on the mean service time and mean source idle time and not on the form of the distributions. When the source idle time follows a negative exponential distribution, the service time distribution being arbitrary, the length of the waiting line at instants of completion of service is a Markov chain. Its distribution as well as the waiting-time distribution is obtained. The results are specialized to the cases of negative exponential and constant service times.