The Annals of Applied Probability

Brownian Models of Feedforward Queueing Networks: Quasireversibility and Product Form Solutions

J. M. Harrison and R. J. Williams

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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.

Article information

Ann. Appl. Probab. Volume 2, Number 2 (1992), 263-293.

First available in Project Euclid: 19 April 2007

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Zentralblatt MATH identifier


Primary: 60J60: Diffusion processes [See also 58J65]
Secondary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 60K25: Queueing theory [See also 68M20, 90B22]

Multiclass queueing networks feedforward quasireversibility product form Brownian models


Harrison, J. M.; Williams, R. J. Brownian Models of Feedforward Queueing Networks: Quasireversibility and Product Form Solutions. Ann. Appl. Probab. 2 (1992), no. 2, 263--293. doi:10.1214/aoap/1177005704.

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