Maximum Queue Length and Waiting Time Revisited: Multserver G/G/c Queue

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Abstract
We characterize the probabilistic nature of the maximum queue length and the maximum waiting time in a multiserver G/G/c 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 logωn−1 and log n1/θ, respectively, where ω < 1 and θ > 0 are parameters of the queuing system. Furthermore, it is shown that the maximum waiting time – when appropriately normalized – converges in distribution to the extreme distribution Λ(x) = exp(−e−x). The maximum queue length exhibits similar behavior, except that some oscillation caused by the discrete nature of the queue length must be taken into account. The first results of this type were obtained for the G/M/l queue by Heyde and for the G/G/l queue by Iglehart. Our analysis is similar to that of Heyde and Iglehart. The generalization to c > 1 servers is made possible due to the recent characterization of the tail of the stationary queue length and waiting time in a G/G/c queue (cf. Sadowsky and Szpankowski [17]).

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