Joint Distributions for Interacting Fluid Queues
 Dirk P. Kroese,
 Werner R. W. Scheinhardt
 … show all 2 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
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.
 S. Aalto, Characterization of the output rate process for a Markovian storage model, J. Appl. Probab. 35(1) (1998) 184–199.
 S. Aalto and W.R.W. Scheinhardt, Tandem fluid queues driven by homogeneous onoff sources, Cosor memorandum, Department of Mathematics and Computing Science, Eindhoven University of Technology, The Netherlands, to appear.
 I.J.B.F. Adan, E.A. van Doorn, J.A.C. Resing, and W.R.W. Scheinhardt, Analysis of a singleserver queue interacting with a fluid reservoir, Queueing Systems 29 (1998) 313–336.
 I.J.B.F. Adan and J.A.C. Resing, Simple analysis of a fluid queue driven by an M/M/1 queue, Queueing Systems 22 (1996) 171–174.
 I.J.B.F. Adan and J.A.C. Resing, A twolevel traffic shaper for an onoff source, Cosor memorandum 99–07, Department of Mathematics and Computing Science, Eindhoven University of Technology, The Netherlands (1999).
 D. Anick, D. Mitra, and M.M. Sondhi, Stochastic theory of a datahandling system with multiple sources, Bell Syst. Tech. J. 61(8) (1982) 1871–1894.
 S. Asmussen, Applied Probability and Queues (Wiley, New York, 1987).
 S. Asmussen and O. Kella, A multidimensional martingale for Markov additive processes and its applications, Submitted.
 H. Chen and D.D. Yao, A fluid model for systems with random disruptions, Oper. Res. 40(S2) (1992) S239–S247.
 B.D. Choi and K.B Choi, A markov modulated fluid queueing system with strict priority, Telecom. Systems 9 (1998) 79–95.
 J.W. Cohen, Single server queue with uniformly bounded virtual waiting time, J. Appl. Probab. 5 (1968) 93–122.
 J.W. Cohen, The Single Server Queue (NorthHolland, Amsterdam, 1982).
 E.A. van Doorn and W.R.W. Scheinhardt, Analysis of birthdeath fluid queues, in: Proc. KAIST Applied Mathematics Workshop, ed. B.D. Choi (Taejon, Korea, 1996) pp. 13–29.
 E.A. van Doorn and W.R.W. Scheinhardt, A fluid queue driven by an infinitestate birthdeath process, in: Teletraffic Contributions for the Information Age, Proc. ITC 15, eds. V. Ramaswami and P.E. Wirth (Amsterdam, Elsevier, 1997) pp. 465–475.
 A. Erdelyi, ed., Bateman Manuscript Project, Tables of Integral Transform (McGraw Hill, New York, 1954).
 S.N. Ethier and T.G. Kurtz, Markov Processes: Characterisation and Convergence (Wiley, New York, 1986).
 O. Kella, Parallel and tandem fluid networks with dependent Lévy inputs, Ann. Appl. Probab. 3(3) (1993) 682–695.
 O. Kella, Stability and nonproduct form of stochastic fluid networks with Lévy inputs, Ann. Appl. Probab. 6(1) (1996) 186–199.
 O. Kella, Nonproduct form of twodimensional fluid networks with dependent Lévy inputs, Submitted.
 O. Kella and W. Whitt, A storage model with a twostate random environment, Oper. Res. 40(S2) (1992) S257–S262.
 O. Kella and W. Whitt, A tandem fluid network with Lévy input, in: Queues and Related Models, eds. I. Basawa and U. Bhat (Oxford University Press, Oxford, 1992) pp. 112–128.
 D.P. Kroese and W.R.W. Scheinhardt, A fluid queue driven by a fluid queue, in: Teletraffic Theory as a Base for QOS: Monitoring, Evaluation, Decisions, eds. B. Goldstein, A. Koucheryavy and M. ShnepsShneppe (LONIIS, St. Petersburg, 1998) pp. 389–400.
 R.S. Liptser and A.N. Shiryayev, Statistics of Random Processes II: Applications (Springer, New York, 1978).
 M. Miyazawa, Rate conservation laws: a survey, Queueing Systems 15 (1994) 1–58.
 W.R.W. Scheinhardt, Markovmodulated and feedback fluid queues, Ph.D. thesis, University of Twente, Enschede, The Netherlands (1998).
 J.T. Virtamo and I. Norros, Fluid queue driven by an M/M/1 queue, Queueing Systems 16 (1994) 373–386.
 J. Zhang, Performance study of Markov modulated fluid flow models with priority traffic, in: Proc. IEEE INFOCOM '93 (1993) pp. 10–17.
 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