Joint Distributions for Interacting Fluid Queues
 Dirk P. Kroese,
 Werner R. W. Scheinhardt
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Abstract
Motivated by recent traffic control models in ATM systems, we analyse three closely related systems of fluid queues, each consisting of two consecutive reservoirs, in which the first reservoir is fed by a twostate (on and off) Markov source. The first system is an ordinary twonode fluid tandem queue. Hence the output of the first reservoir forms the input to the second one. The second system is dual to the first one, in the sense that the second reservoir accumulates fluid when the first reservoir is empty, and releases fluid otherwise. In these models both reservoirs have infinite capacities. The third model is similar to the second one, however the second reservoir is now finite. Furthermore, a feedback mechanism is active, such that the rates at which the first reservoir fills or depletes depend on the state (empty or nonempty) of the second reservoir.
The models are analysed by means of Markov processes and regenerative processes in combination with truncation, level crossing and other techniques. The extensive calculations were facilitated by the use of computer algebra. This approach leads to closedform solutions to the steadystate joint distribution of the content of the two reservoirs in each of the models.
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 Title
 Joint Distributions for Interacting Fluid Queues
 Journal

Queueing Systems
Volume 37, Issue 13 , pp 99139
 Cover Date
 20010301
 DOI
 10.1023/A:1011044217695
 Print ISSN
 02570130
 Online ISSN
 15729443
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 fluid queue
 tandem queue
 stationary distribution
 joint distribution
 feedback
 traffic shaper
 Industry Sectors
 Authors

 Dirk P. Kroese ^{(1)}
 Werner R. W. Scheinhardt ^{(2)}
 Author Affiliations

 1. Department of Mathematica, University of Queensland, Brisbane, 4072, Australia
 2. Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands