Open Access
May, 1992 Brownian Models of Feedforward Queueing Networks: Quasireversibility and Product Form Solutions
J. M. Harrison, R. J. Williams
Ann. Appl. Probab. 2(2): 263-293 (May, 1992). DOI: 10.1214/aoap/1177005704


We consider a very general type of $d$-station open queueing network, with multiple customer classes and a more or less arbitrary service discipline at each station, but restricted by the requirement that customers always flow from lower numbered stations to higher numbered ones. To approximate the behavior of such a queueing network under heavy traffic conditions, a corresponding Brownian network model is proposed and it is shown that the approximating Brownian model reduces to a $d$-dimensional reflected Brownian motion $W$ whose state space is the nonnegative orthant. A necessary and sufficient condition for $W$ to have a product form stationary distribution (that is, a stationary distribution with independent components) and a probabilistic interpretation for that condition are given. Our interpretation involves a notion of quasireversibility analogous to that introduced by Kelly and elaborated by Walrand in their brilliant analysis of product form solutions for conventional queueing network models. Three illustrative queueing network models are discussed in detail and the analysis of these examples shows how a Brownian network approximation may have a product form stationary distribution even when the original or exact model is intractable. Particularly intriguing in that regard are two examples involving non-Poisson inputs, deterministic routing, deterministic service times and processor-sharing service disciplines.


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J. M. Harrison. R. J. Williams. "Brownian Models of Feedforward Queueing Networks: Quasireversibility and Product Form Solutions." Ann. Appl. Probab. 2 (2) 263 - 293, May, 1992.


Published: May, 1992
First available in Project Euclid: 19 April 2007

zbMATH: 0753.60071
MathSciNet: MR1161055
Digital Object Identifier: 10.1214/aoap/1177005704

Primary: 60J60
Secondary: 60J70 , 60K25

Keywords: Brownian models , feedforward , Multiclass queueing networks , product form , quasireversibility

Rights: Copyright © 1992 Institute of Mathematical Statistics

Vol.2 • No. 2 • May, 1992
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