## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 6, Number 1 (1996), 1-47.

### A multiclass closed queueing network with unconventional heavy traffic behavior

J. M. Harrison and R. J. Williams

#### Abstract

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.

#### Article information

**Source**

Ann. Appl. Probab., Volume 6, Number 1 (1996), 1-47.

**Dates**

First available in Project Euclid: 18 October 2002

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1034968064

**Digital Object Identifier**

doi:10.1214/aoap/1034968064

**Mathematical Reviews number (MathSciNet)**

MR1389830

**Zentralblatt MATH identifier**

0865.60078

**Subjects**

Primary: 60K25: Queueing theory [See also 68M20, 90B22] 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 90B15: Network models, stochastic 90B22: Queues and service [See also 60K25, 68M20]

**Keywords**

Closed multiclass queueing networks heavy traffic theory reflecting barrier Brownian motion $\mathbf{M}_1$ convergence

#### Citation

Harrison, J. M.; Williams, R. J. A multiclass closed queueing network with unconventional heavy traffic behavior. Ann. Appl. Probab. 6 (1996), no. 1, 1--47. doi:10.1214/aoap/1034968064. https://projecteuclid.org/euclid.aoap/1034968064