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Brownian Approximations of Multiclass Open-Queueing Networks

Faculty of Commerce and Business Administration, University of British Columbia, Vancouver, British Columbia, Canada, V6T 1Z2
Faculty of Commerce and Business Administration, University of British Columbia, Vancouver, British Columbia, Canada, V6T 1Z2
Department of Industrial Engineering and Operations Research, Columbia University, New York, New York 10027
hong.chen{at}commerce.ubc.ca
xinyang_shen{at}yahoo.com
yao{at}ieor.columbia.edu

We study a multiclass open-queueing network with a set of single-server stations that operate under a combination of FIFO (first-in-first out) and priority service disciplines, and are subject to random breakdowns. Assuming that the primitive processes—in particular, external arrivals, service requirements, service capacities (up and down times), and the routing mechanism—follow two-moment approximations (based on functional central limit theorems), we develop a semi-martingale reflected Brownian motion (SRBM) approximation for the performance processes such as workload, queue lengths, and sojourn times. We illustrate through numerical examples in comparison against simulation that the SRBM approximation, while not always supported by a limit theorem, exhibits good accuracy in most cases. Through analyzing special networks, we also discuss the existence of the SRBM approximation in relation to the stability and the heavy traffic limits of the networks.

Subject classifications: Queues: approximations and diffusion models. Probability: diffusion. Production/scheduling: approximations/scheduling.
History: Received June 2000; revision received January 2001; revision received June 2001; revision received August 2001; accepted August 2001.