## The Annals of Applied Probability

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

#### 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

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

Digital Object Identifier
doi:10.1214/aoap/1034968064

Mathematical Reviews number (MathSciNet)
MR1389830

Zentralblatt MATH identifier
0865.60078

#### 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

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• UNIVERSITY OF CALIFORNIA, SAN DIEGO STANFORD UNIVERSITY 9500 GILMAN DRIVE
• STANFORD, CALIFORNIA 94305
• LA JOLLA, CALIFORNIA 92093-0112 E-mail: fharrison@gsb-lira.stanford.edu E-mail: williams@math.ucsd.edu