A Multiclass Closed Queueing Network with Unconventional Heavy Traac Behavior

We consider a multiclass closed queueing network model analogous to the open network models of Rybko-Stolyar and Lu-Kumar. The closed network has two single-server stations and a xed customer population of size n. Customers are routed in cyclic fashion through four distinct classes, two of which are served at each station, and each server uses a preemptive-resume priority discipline. The service time distribution for each customer class is exponential, and attention is focused on the critical case where all four classes have the same mean service time. Letting n approach innnity, we prove a heavy traac limit theorem that is unconventional in three regards. First, in our heavy traac scaling of both queue length processes and cumulative idleness processes, time is compressed by a factor of n rather than the factor of n 2 occurring in conventional theory. Second, the spatial scaling applied to some components of the queue length and idleness processes is that associated with the central limit theorem, but the scaling applied to other components is that associated with the law of large numbers. Thus, in the language of queueing theory, our heavy traac limit theorem involves a mixture of Brownian scaling and uid scaling. Finally, the limit process that we obtain is not an ordinary reeected Brownian motion, as