On the Length of a Queue in a Mixed Priority System under Conditions of Critical Loading

A study is performed of a single-channel queueing system with three Poisson arrivals. The service times for requirements of each arrival have an arbitrary absolutely continuous distribution. Requirements of the first arrival take relative priority over those of the second arrival and absolute priority of new service over requirements of the third arrival. Requirements of the second arrival take relative priority over those of the third arrival. The limit distribution of the number of requirements of the third arrival in the system is found while the loading and time simultaneously tend to unity and infinity, respectively.