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Brownian models of open queueing networks with homogeneous customer populations. (English) Zbl 0632.60095
This paper considers a family of multidimensional diffusion processes that arise as heavy traffic approximations for open queueing networks. More precisely, the diffusion processes considered arise as approximate models of open queueing networks with homogeneous customer populations which means that customers occupying any given node or station of the network are essentially indistinguishable from one another. The Jackson network fits this description but multiclass network models do not.
The objectives of the paper are (a) to explain in concrete terms how one approximates a conventional queueing model or a real physical system by a corresponding Brownian model, and (b) to state and prove some new results regarding stationary distributions of such Brownian models.
The part of the paper aimed at objective (a) is largely a recapitulation of previous work on weak convergence theorems, with the emphasis placed on modelling intuition. With respect to objective (b) some foundational issues are resolved. Also, it is shown that the stationary distribution of the Brownian model has a separable (product form) density if and only if its data satisfy a certain condition, in which case the stationary density is exponential, and all relevant performance measures can be written out in explicit formulas.
Reviewer: F.P.Kelly

MSC:
60K25 Queueing theory (aspects of probability theory)
60K20 Applications of Markov renewal processes (reliability, queueing networks, etc.)
60J65 Brownian motion
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