Brownian Approximations of Multiclass Open-Queueing Networks

Abstract

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.

Modeling a Healthcare System as a Queueing Network: The Case of a Belgian Hospital

“…Simple arithmetics demonstrate that R f is a Pmatrix (indicating that all its principal minors are positive). As such, sufficient conditions are met for R f to possess the completely-S property (Chen et al, 2002). Therefore we can reasonably infer the SRBM to exist.…”

“…Heavy traffic conditions assume that all stations in the network are critically loaded. More formally, heavy traffic conditions imply that (Nguyen, 1995;Bramson & Dai, 2001;Chen et al, 2002):…”

“…We also evaluate the accuracy of the SRBM-approximation. With respect to the computations required to derive the characteristics of the SRBM, we refer to the work of Chen et al (2002). The actual numerical calculation of the steady states of the SRBM is performed by means of the QNET software developed by Dai and Harrison (1992).…”

“…consultation and surgery). However, it has been argued in literature that multiclass open re-entry networks are liable to inaccurate evaluation of performance measures (Chen et al, 2002). These findings suggest that current Brownian queueing models are not the most reliable tool to study complex hospital systems (i.e.…”

Analytical Approximations to Predict Performance Measures of Manufacturing Systems with Job Failures and Parallel Processing

“…In this research an open system with jobs being processed according to a first-in-first-out (FIFO) discipline at each station was considered. Different contributions on queueing networks with priority service disciplines have been made [20,119,117,43,116]. However, future studies may analyze this research problem under priority service disciplines at each station.…”