We consider a multiclass closed queueing network model analogous to
the open network models of Rybko and Stolyar and of Lu and Kumar. The closed
network has two single-server stations and a fixed 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 infinity, we prove a heavy traffic limit theorem that is
unconventional in three regards. First, in our heavy traffic 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
traffic limit theorem involves a mixture of Brownian scaling and fluid scaling.
Finally, the limit process that we obtain is not an ordinary reflected Brownian
motion, as in conventional heavy traffic theorems, although it is related to or
derived from Brownian motion.