Joint Distributions for Interacting Fluid Queues 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 two-state (on and off) Markov source. The first system is an ordinary two-node 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 closed-form solutions to the steady-state joint distribution of the content of the two reservoirs in each of the models.
fluid queue tandem queue stationary distribution joint distribution feedback traffic shaper References
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 on-off 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 single-server 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 two-level traffic shaper for an on-off 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 data-handling system with multiple sources, Bell Syst. Tech. J. 61(8) (1982) 1871–1894.
Applied Probability and Queues
(Wiley, New York, 1987).
S. Asmussen and O. Kella, A multi-dimensional 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.
The Single Server Queue
(North-Holland, Amsterdam, 1982).
E.A. van Doorn and W.R.W. Scheinhardt, Analysis of birth-death 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 infinite-state birth-death 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, Non-product form of two-dimensional fluid networks with dependent Lévy inputs, Submitted.
O. Kella and W. Whitt, A storage model with a two-state 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. Shneps-Shneppe (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, Markov-modulated 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.
Google Scholar Copyright information
© Kluwer Academic Publishers 2001