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Brownian models of multiclass queueing networks: Current status and open problems. (English) Zbl 0781.60090
The problem of heavy traffic approximation by reflected Brownian motion (BM) for open queueing network with $D$ nodes and $C$ customer classes is surveyed. The network has Markovian routing, and customers are assumed to switch among classes in Markovian fashion also. It is assumed that external input processes can be correlated and obey some functional central limit theorem with limiting $C$-dimensional BM with zero drift. The same is right for cumulative service capacity process, too (with $D$- dimensional limiting BM). The relation between BM and heavy traffic theory is discussed. Continuous time workload, queue-size and some other basic processes are introduced for each customer type and the whole network. A natural scaling of time and space is considered. The main scaling factor is the vector of differences between long-time average service rates and traffic intensities for all nodes. Then the Brownian model is defined that approximates scaled system of the mentioned network processes. Heavy traffic convergence is considered. Two numerical examples are presented where complete sojourn time distributions are estimated.
##### MSC:
 60K25 Queueing theory 90B22 Queues and service (optimization) 60J65 Brownian motion
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