Maximum Queue Length And Waiting Time Revisited: Multiserver G|G|c Queue

We characterize the probabilistic nature of the maximum queue length and the maximum waiting time in a multiserver GjGjc queue. We assume a general i.i.d. interarrival process and a general i.i.d. service time process for each server with the possibility of having different service time distributions for different servers. Under a weak additional condition we will prove that the maximum queue length and waiting time grow asymptotically in probability as log ! n \Gamma1 and log n 1=` , respectively, where ! ! 1 and ` ? 0 are parameters of the queueing system. Furthermore, it is shown that the maximum waiting time -- when appropriately normalized -- converges in distribution to the extreme distribution (x) = exp(\Gammae \Gammax ). The maximum queue length exhibits similar behavior, except that some oscillation caused by discrete nature of the queue length must be taken into account. The first results of this type were obtained for the GjM j1 queue by Heyde, and for the GjGj1 queue..