The Probability of Large Queue Lengths and Waiting times in a Heterogeneous Multiserver Queue Part I: Tight Limits

We consider a multiserver queuing process speciied by i.i.d. interarrival time, batch size and service time sequences. In the case that diierent servers have diierent service time distributions we say the system is heterogeneous. In this paper we establish conditions for the queuing process to be characterized as a geometrically Harris recurrent Markov chain, and we characterize the stationary probabilities of large queue lengths and waiting times. The queue length is asymptotically geometric and the waiting time is asymptotically exponential. Our analysis is a generalization of the well known characterization of the GI/G/1 queue obtained using classical probabilistic techniques of exponential change of measure and renewal theory.