Conditioned Limit Theorems for Waiting Time Processes of the M/G/1 Queue

摘要

This paper is on conditioned weak limit theorems for imbedded waiting time processes of an M/G/1 queue. More specifically, the author studies functional limit theorems for the actual waiting time process conditioned by the event that the number of customers in a busy period exceeds n or equals n. Attention is also paid to the actual waiting time process with random time index. Combined with the existing literature on the subject this paper gives a complete account of the conditioned limit theorems for the actual waiting time process of an M/G/1 queue for arbitrary traffic intensity and for a rather general class of service time distributions. The limit processes that occur are Brownian excursion and meander, while in the case of random time index also the following limit occurs: Brownian excursion divided by an independent and uniform (0,1) distributed random variable.