article

Stability of a three-station fluid network

Authors:

J. J. Hasenbein,

J. H. Vande VateAuthors Info & Claims

Queueing Systems: Theory and Applications, Volume 33, Issue 4

Pages 293 - 325

https://doi.org/10.1023/A:1019184331042

Published: 14 April 1999 Publication History

Abstract

This paper studies the stability of a three-station fluid network. We show that, unlike the two-station networks in Dai and Vande Vate [18], the global stability region of our three-station network is not the intersection of its stability regions under static buffer priority disciplines. Thus, the worst or extremal disciplines are not static buffer priority disciplines. We also prove that the global stability region of our three-station network is not monotone in the service times and so, we may move a service time vector out of the global stability region by reducing the service time for a class. We introduce the monotone global stability region and show that a linear program (LP) related to a piecewise linear Lyapunov function characterizes this largest monotone subset of the global stability region for our three-station network. We also show that the LP proposed by Bertsimas et al. [1] does not characterize either the global stability region or even the monotone global stability region of our three-station network. Further, we demonstrate that the LP related to the linear Lyapunov function proposed by Chen and Zhang [11] does not characterize the stability region of our three-station network under a static buffer priority discipline.

References

[1]

{1} D. Bertsimas, D. Gamarnik and J.N. Tsitsiklis, Stability conditions for multiclass fluid queueing networks, IEEE Trans. Automat. Control 41 (1996) 1618-1631. Correction: 42 (1997) 128.

[2]

{2} D.D. Botvich and A.A. Zamyatin, Ergodicity of conservative communication networks, Rapport de recherche 1772, INRIA (1992).

[3]

{3} M. Bramson, Instability of FIFO queueing networks, Ann. Appl. Probab. 4 (1994) 414-431.

[4]

{4} M. Bramson, Instability of FIFO queueing networks with quick service times, Ann. Appl. Probab. 4 (1994) 693-718.

[5]

{5} M. Bramson, Convergence to equilibria for fluid models of FIFO queueing networks, Queueing Systems 22 (1996) 5-45.

[6]

{6} M. Bramson, Convergence to equilibria for fluid models of head-of-the-line proportional processor sharing queueing networks, Queueing Systems 23 (1997) 1-26.

[7]

{7} M. Bramson, Stability of two families of queueing networks and a discussion of fluid limits, Queueing Systems 28 (1998) 7-31.

Digital Library

[8]

{8} M. Bramson, A stable queueing network with unstable fluid network, Ann. Appl. Probab. (1998) to appear.

[9]

{9} H. Chen, Fluid approximations and stability of multiclass queueing networks I: Work-conserving disciplines, Ann. Appl. Probab. 5 (1995) 637-665.

[10]

{10} H. Chen and H. Zhang, Stability of multiclass queueing networks under FIFO service discipline, Math. Oper. Res. 22 (1997) 691-725.

Digital Library

[11]

{11} H. Chen and H. Zhang, Stability of multiclass queueing networks under priority service disciplines, Oper. Res. (1998) to appear.

Digital Library

[12]

{12} J.G. Dai, On positive Harris recurrence of multiclass queueing networks: A unified approach via fluid limit models, Ann. Appl. Probab. 5 (1995) 49-77.

[13]

{13} J.G. Dai, Stability of open multiclass queueing networks via fluid models, in: Stohastic Networks , eds. F. Kelly and R.J. Williams, The IMA Volumes in Mathematics and its Applications, Vol. 71 (Springer, New York, 1995) pp. 71-90.

[14]

{14} J.G. Dai, A fluid-limit model criterion for instability of multiclass queueing networks, Ann. Appl. Probab. 6 (1996) 751-757.

[15]

{15} J.G. Dai and S.P. Meyn, Stability and convergence of moments for multiclass queueing networks via fluid limit model, IEEE Trans. Automat. Control 40 (1995) 1889-1904.

[16]

{16} J.G. Dai and J. Vande Vate, Global stability of two-station queueing networks, in: Proc. of Workshop on Stochastic Networks: Stability and Rare Events , eds. P.K.S. Glasserman and D. Yao, Columbia University, Lecture Notes in Statistics, Vol. 117 (Springer, New York, 1996) pp. 1-26.

[17]

{17} J.G. Dai and J. Vande Vate, Virtual stations and the capacity of two-station queueing networks, (1996), under revision for Oper. Res.

[18]

{18} J.G. Dai and J. Vande Vate, The stability of two-station multi-type fluid networks, Oper. Res. (1998) to appear.

Digital Library

[19]

{19} J.G. Dai and G. Weiss, Stability and instability of fluid models for re-entrant lines, Math. Oper. Res. 21 (1996) 115-134.

Digital Library

[20]

{20} D. Down and S. Meyn, Piecewise linear test functions for stability of queueing networks, in: Proc. of the 33rd Conf. on Decision and Control (1994) pp. 2069-2074.

[21]

{21} D. Down and S.P. Meyn, Piecewise linear test functions for stability and instability of queueing networks, Queueing Systems 27 (1997) 205-226.

Digital Library

[22]

{22} V. Dumas, A multiclass network with nonlinear, nonconvex, nonmonotonic stability conditions, Queueing Systems 25 (1997) 1-43.

Digital Library

[23]

{23} P. Dupuis and R.J. Williams, Lyapunov functions for semimartingale reflecting Brownian motions, Ann. Probab. 22 (1994) 680-702.

[24]

{24} S. Foss and A. Rybko, Stability of multiclass Jackson-type networks, preprint (1995).

[25]

{25} J.J. Hasenbein, Necessary conditions for global stability of multiclass queueing networks, Oper. Res. Lett. 21 (1997) 87-94.

Digital Library

[26]

{26} C. Humes, Jr., A regulator stabilization technique: Kumar-Seidman revisited, IEEE Trans. Automat. Control 39 (1994) 191-196.

[27]

{27} P.R. Kumar and S.P. Meyn, Stability of queueing networks and scheduling policies, IEEE Trans. Automat. Control 40 (1995) 251-260.

[28]

{28} P.R. Kumar and S. Meyn, Duality and linear programs for stability and performance analysis of queueing networks and scheduling policies, IEEE Trans. Automat. Control 41 (1996) 4-17.

[29]

{29} P.R. Kumar and T.I. Seidman, Dynamic instabilities and stabilization methods in distributed real-time scheduling of manufacturing systems, IEEE Trans. Automat. Control 35 (1990) 289-298.

[30]

{30} S.H. Lu and P.R. Kumar, Distributed scheduling based on due dates and buffer priorities, IEEE Trans. Automat. Control 36 (1991) 1406-1416.

[31]

{31} S.P. Meyn, Transience of multiclass queueing networks via fluid limit models, Ann. Appl. Probab. 5 (1995) 946-957.

[32]

{32} A.N. Rybko and A.L. Stolyar, Ergodicity of stochastic processes describing the operation of open queueing networks, Problems Inform. Transmission 28 (1992) 199-220.

[33]

{33} T.I. Seidman, 'First come, first served' can be unstable!, IEEE Trans. Automat. Control 39 (1994) 2166-2171.

[34]

{34} A.L. Stolyar, On the stability of multiclass queueing networks: a relaxed sufficient condition via limiting fluid processes, Markov Processes Related Fields (1995) 491-512.

[35]

{35} G.L. Winograd and P.R. Kumar, The FCFS service discipline: Stable network topologies, bounds on traffic burstiness and delay, and control by regulators, Math. Comput. Modeling 23 (1996) 115-129.

Digital Library

Cited By

Gurvich IHasenbein J(2022)Policy robustness in queueing networksQueueing Systems: Theory and Applications10.1007/s11134-022-09776-5100:3-4(425-427)Online publication date: 1-Apr-2022
https://dl.acm.org/doi/10.1007/s11134-022-09776-5
Kruk ŁChojecki T(2022)Instability of SRPT, SERPT and SJF multiclass queueing networksQueueing Systems: Theory and Applications10.1007/s11134-021-09733-8101:1-2(57-92)Online publication date: 1-Jun-2022
https://dl.acm.org/doi/10.1007/s11134-021-09733-8
Wu KShen Y(2021)Pathwise stability of multiclass queueing networksDiscrete Event Dynamic Systems10.1007/s10626-020-00321-131:1(5-23)Online publication date: 1-Mar-2021
https://dl.acm.org/doi/10.1007/s10626-020-00321-1
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Published In

cover image Queueing Systems: Theory and Applications

Queueing Systems: Theory and Applications Volume 33, Issue 4

1999

101 pages

ISSN:0257-0130

Issue’s Table of Contents

Copyright © Copyright © 1999 Kluwer Academic Publishers.

Publisher

J. C. Baltzer AG, Science Publishers

United States

Publication History

Published: 14 April 1999

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Cited By

Gurvich IHasenbein J(2022)Policy robustness in queueing networksQueueing Systems: Theory and Applications10.1007/s11134-022-09776-5100:3-4(425-427)Online publication date: 1-Apr-2022
https://dl.acm.org/doi/10.1007/s11134-022-09776-5
Kruk ŁChojecki T(2022)Instability of SRPT, SERPT and SJF multiclass queueing networksQueueing Systems: Theory and Applications10.1007/s11134-021-09733-8101:1-2(57-92)Online publication date: 1-Jun-2022
https://dl.acm.org/doi/10.1007/s11134-021-09733-8
Wu KShen Y(2021)Pathwise stability of multiclass queueing networksDiscrete Event Dynamic Systems10.1007/s10626-020-00321-131:1(5-23)Online publication date: 1-Mar-2021
https://dl.acm.org/doi/10.1007/s10626-020-00321-1
Gurvich I(2014)Validity of Heavy-Traffic Steady-State Approximations in Multiclass Queueing NetworksMathematics of Operations Research10.1287/moor.2013.059339:1(121-162)Online publication date: 1-Feb-2014
https://dl.acm.org/doi/10.1287/moor.2013.0593
Andrews M(2004)Instability of FIFO in session-oriented networksJournal of Algorithms10.1016/S0196-6774(03)00096-850:2(232-245)Online publication date: 1-Feb-2004
https://dl.acm.org/doi/10.1016/S0196-6774%2803%2900096-8
Gamarnik D(2002)On Deciding Stability of Constrained Homogeneous Random Walks and Queueing SystemsMathematics of Operations Research10.1287/moor.27.2.272.32127:2(272-293)Online publication date: 1-May-2002
https://dl.acm.org/doi/10.1287/moor.27.2.272.321
Gamarnik D(2001)On deciding stability of constrained random walks and queueing systemsACM SIGMETRICS Performance Evaluation Review10.1145/544397.54441228:4(39-40)Online publication date: 1-Mar-2001
https://dl.acm.org/doi/10.1145/544397.544412
Hasenbein J(2001)Stability of Fluid Networks with Proportional RoutingQueueing Systems: Theory and Applications10.1023/A:101095970648638:3(327-354)Online publication date: 1-Jul-2001
https://dl.acm.org/doi/10.1023/A%3A1010959706486
Chen HYe H(2001)Existence Condition for the Diffusion Approximations of Multiclass Priority Queueing NetworksQueueing Systems: Theory and Applications10.1023/A:101095201277138:4(435-470)Online publication date: 1-Aug-2001
https://dl.acm.org/doi/10.1023/A%3A1010952012771

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