Closed Queueing Systems: A Generalization of the Machine Interference Model

SUMMARY A closed loop of queues in series is considered in which the time taken by any customer to travel between consecutive service gates is a random variable. The case of a single server at each gate is analysed, where in addition to the customers from the closed loop, each server may also have customers arriving from sources outside the system and, after completion of service, departing to destinations again outside the system. Limiting cases are examined, firstly where the number of customers in the closed loop becomes large, and secondly where, in addition to this condition, the average transit time between one pair of consecutive service gates also tends to infinity.