On a Generalized M/G/1 Queuing Process in Which the First Customer of Each Busy Period Receives Exceptional Service - Welch, Peter D. (Operations Research, 1964)
The following generalization of the M/G/1 queue is considered. If a customer arrives when the server is busy, his service time has a distribution function, Gb(x); while if he arrives when the server is idle, his service time has a different distribution function, Ge(x). Results are obtained that characterize the transient and asymptotic distributions of the queue size, waiting time, and waiting-plus-service time. These results are then applied to the special case of a queue with a single service time distribution function, but with additional independent delay times that have one distribution function for the customers arriving when the server is idle and another for the customers arriving when the server is busy.