On deciding stability of multiclass queueing networks under buffer priority scheduling policies

previous :: next

On deciding stability of multiclass queueing networks under buffer priority scheduling policies

David Gamarnik and Dmitriy Katz

Source: Ann. Appl. Probab. Volume 19, Number 5 (2009), 2008-2037.

Abstract

One of the basic properties of a queueing network is stability. Roughly speaking, it is the property that the total number of jobs in the network remains bounded as a function of time. One of the key questions related to the stability issue is how to determine the exact conditions under which a given queueing network operating under a given scheduling policy remains stable. While there was much initial progress in addressing this question, most of the results obtained were partial at best and so the complete characterization of stable queueing networks is still lacking.

In this paper, we resolve this open problem, albeit in a somewhat unexpected way. We show that characterizing stable queueing networks is an algorithmically undecidable problem for the case of nonpreemptive static buffer priority scheduling policies and deterministic interarrival and service times. Thus, no constructive characterization of stable queueing networks operating under this class of policies is possible. The result is established for queueing networks with finite and infinite buffer sizes and possibly zero service times, although we conjecture that it also holds in the case of models with only infinite buffers and nonzero service times. Our approach extends an earlier related work [Math. Oper. Res. 27 (2002) 272–293] and uses the so-called counter machine device as a reduction tool.

Primary Subjects: 60K25, 90B22

Keywords: Queueing networks; positive recurrence; computability

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.

If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aoap/1255699551
Digital Object Identifier: doi:10.1214/09-AAP597

back to Table of Contents

References

[1] Andrews, M., Awerbuch, B., Fernández, A., Kleinberg, J., Leighton, T. and Liu, Z. (1996). Universal stability results for greedy contention–resolution protocols. In 37th Annual Symposium on Foundations of Computer Science (Burlington, VT, 1996) 380–389. IEEE Comput. Soc. Press, Los Alamitos, CA.

[2] Aiello, W., Kushilevitz, E., Ostrovsky, R. and Rosén, A. (1999). Adaptive packet routing for bursty adversarial traffic. In STOC’98 (Dallas, TX) 359–368. ACM, New York.

Mathematical Reviews (MathSciNet): MR1731588

[3] Andrews, M. (2004). Instability of FIFO in session-oriented networks. J. Algorithms 50 232–245.

Mathematical Reviews (MathSciNet): MR2053018

Digital Object Identifier: doi:10.1016/S0196-6774(03)00096-8

[4] Andrews, M. and Zhang, L. (2000). The effects of temporary sessions on network performance. In Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms (San Francisco, CA, 2000) 448–457. ACM, New York.

Mathematical Reviews (MathSciNet): MR1754882

[5] Blondel, V. D., Bournez, O., Koiran, P., Papadimitriou, C. H. and Tsitsiklis, J. N. (2001). Deciding stability and mortality of piecewise affine dynamical systems. Theoret. Comput. Sci. 255 687–696.

Mathematical Reviews (MathSciNet): MR1819103

Zentralblatt MATH: 0973.68067

Digital Object Identifier: doi:10.1016/S0304-3975(00)00399-6

[6] Bhattacharjee, R. and Goel, A. (2005). Instability of FIFO at arbitrarily low rates in the adversarial queueing model. SIAM J. Comput. 34 318–332.

Mathematical Reviews (MathSciNet): MR2124006

Zentralblatt MATH: 1087.68011

Digital Object Identifier: doi:10.1137/S0097539703426805

[7] Bertsimas, D., Gamarnik, D. and Tsitsiklis, J. N. (1996). Stability conditions for multiclass fluid queueing networks. IEEE Trans. Automat. Control 41 1618–1631.

Mathematical Reviews (MathSciNet): MR1419686

Digital Object Identifier: doi:10.1109/9.543999

[8] Bertsimas, D., Gamarnik, D. and Tsitsiklis, J. N. (2001). Performance of multiclass Markovian queueing networks via piecewise linear Lyapunov functions. Ann. Appl. Probab. 11 1384–1428.

Mathematical Reviews (MathSciNet): MR1878302

Zentralblatt MATH: 1012.60082

Digital Object Identifier: doi:10.1214/aoap/1015345407

Project Euclid: euclid.aoap/1015345407

[9] Borodin, A., Kleinberg, J., Raghavan, P., Sudan, M. and Williamson, D. P. (2001). Adversarial queuing theory. J. ACM 48 13–38.

Mathematical Reviews (MathSciNet): MR1867274

Digital Object Identifier: doi:10.1145/363647.363659

[10] Bertsimas, D. and Niño-Mora, J. (1999). Optimization of multiclass queueing networks with changeover times via the achievable region approach. I. The single-station case. Math. Oper. Res. 24 306–330.

[11] Bertsimas, D., Paschalidis, I. C. and Tsitsiklis, J. N. (1994). Optimization of multiclass queueing networks: Polyhedral and nonlinear characterizations of achievable performance. Ann. Appl. Probab. 4 43–75.

Mathematical Reviews (MathSciNet): MR1258173

Zentralblatt MATH: 0797.60079

Digital Object Identifier: doi:10.1214/aoap/1177005200

Project Euclid: euclid.aoap/1177005200

[12] Bramson, M. (1994). Instability of FIFO queueing networks. Ann. Appl. Probab. 4 414–431.

Mathematical Reviews (MathSciNet): MR1272733

Digital Object Identifier: doi:10.1214/aoap/1177005066

Project Euclid: euclid.aoap/1177005066

[13] Bramson, M. (1996). Convergence to equilibria for fluid models of FIFO queueing networks. Queueing Systems Theory Appl. 22 5–45.

Mathematical Reviews (MathSciNet): MR1393404

Digital Object Identifier: doi:10.1007/BF01159391

[14] Bramson, M. (1999). A stable queueing network with unstable fluid model. Ann. Appl. Probab. 9 818–853.

Mathematical Reviews (MathSciNet): MR1722284

Zentralblatt MATH: 0964.60082

Digital Object Identifier: doi:10.1214/aoap/1029962815

Project Euclid: euclid.aoap/1029962815

[15] Bramson, M. (2001). Stability of earliest-due-date, first-served queueing networks. Queueing Syst. 39 79–102.

Mathematical Reviews (MathSciNet): MR1865459

Digital Object Identifier: doi:10.1023/A:1017987600517

[16] Blondel, V. D. and Tsitsiklis, J. N. (2000). The boundedness of all products of a pair of matrices is undecidable. Systems Control Lett. 41 135–140.

Mathematical Reviews (MathSciNet): MR1831027

[17] Blondel, V. D. and Tsitsiklis, J. N. (2000). A survey of computational complexity results in systems and control. Automatica J. IFAC 36 1249–1274.

Mathematical Reviews (MathSciNet): MR1834719

Digital Object Identifier: doi:10.1016/S0005-1098(00)00050-9

[18] Chen, H. and Yao, D. D. (2001). Fundamentals of Queueing Networks: Performance, Asymptotics, and Optimization. Applications of Mathematics. Stochastic Modelling and Applied Probability 46. Springer, New York.

Mathematical Reviews (MathSciNet): MR1835969

Zentralblatt MATH: 0992.60003

[19] Dai, J. G. (1995). On positive Harris recurrence of multiclass queueing networks: A unified approach via fluid limit models. Ann. Appl. Probab. 5 49–77.

Mathematical Reviews (MathSciNet): MR1325041

Zentralblatt MATH: 0822.60083

Digital Object Identifier: doi:10.1214/aoap/1177004828

Project Euclid: euclid.aoap/1177004828

[20] Dai, J. G. (1996). A fluid limit model criterion for instability of multiclass queueing networks. Ann. Appl. Probab. 6 751–757.

Mathematical Reviews (MathSciNet): MR1410113

Zentralblatt MATH: 0860.60075

Digital Object Identifier: doi:10.1214/aoap/1034968225

Project Euclid: euclid.aoap/1034968225

[21] Dai, J. G., Hasenbein, J. J. and Vande Vate, J. H. (1999). Stability of a three-station fluid network. Queueing Systems Theory Appl. 33 293–325.

Mathematical Reviews (MathSciNet): MR1742573

Digital Object Identifier: doi:10.1023/A:1019184331042

[22] Down, D. and Meyn, S. P. (1995). Stability of acyclic multiclass queueing networks. IEEE Trans. Automat. Control 40 916–919.

Mathematical Reviews (MathSciNet): MR1328091

Digital Object Identifier: doi:10.1109/9.384230

[23] Down, D. and Meyn, S. P. (1997). Piecewise linear test functions for stability and instability of queueing networks. Queueing Systems Theory Appl. 27 205–226 (1998).

Mathematical Reviews (MathSciNet): MR1625069

Digital Object Identifier: doi:10.1023/A:1019166115653

[24] Dai, J. G. and Vande Vate, J. H. (2000). The stability of two-station multitype fluid networks. Oper. Res. 48 721–744.

Mathematical Reviews (MathSciNet): MR1792776

Zentralblatt MATH: 1106.90311

Digital Object Identifier: doi:10.1287/opre.48.5.721.12408

JSTOR: links.jstor.org

[25] Dai, J. G. and Weiss, G. (1996). Stability and instability of fluid models for reentrant lines. Math. Oper. Res. 21 115–134.

Mathematical Reviews (MathSciNet): MR1385870

Zentralblatt MATH: 0848.60086

Digital Object Identifier: doi:10.1287/moor.21.1.115

JSTOR: links.jstor.org

[26] Gamarnik, D. (2000). Using fluid models to prove stability of adversarial queueing networks. IEEE Trans. Automat. Control 45 741–746.

Mathematical Reviews (MathSciNet): MR1764845

Digital Object Identifier: doi:10.1109/9.847114

[27] Gamarnik, D. (2002). On deciding stability of constrained homogeneous random walks and queueing systems. Math. Oper. Res. 27 272–293.

Mathematical Reviews (MathSciNet): MR1908527

Zentralblatt MATH: 1082.90511

Digital Object Identifier: doi:10.1287/moor.27.2.272.321

JSTOR: links.jstor.org

[28] Gamarnik, D. (2003). Stability of adaptive and non-adaptive packet routing policies in adversarial queueing networks. SIAM J. Comput. 32 371–385.

Mathematical Reviews (MathSciNet): MR1969395

Zentralblatt MATH: 1029.68027

Digital Object Identifier: doi:10.1137/S0097539700369168

[29] Gamarnik, D. (2007). Computing stationary probability distribution and large deviations rates for constrained homogeneous random walks. The undecidability result. Math. Oper. Res. 27 272–293.

Mathematical Reviews (MathSciNet): MR2324425

Zentralblatt MATH: 05279721

Digital Object Identifier: doi:10.1287/moor.1060.0247

[30] Gamarnik, D. and Hasenbein, J. J. (2005). Instability in stochastic and fluid queueing networks. Ann. Appl. Probab. 15 1652–1690.

Mathematical Reviews (MathSciNet): MR2152240

Zentralblatt MATH: 1135.90006

Digital Object Identifier: doi:10.1214/105051605000000179

Project Euclid: euclid.aoap/1121433764

[31] Goel, A. (1999). Stability of networks and protocols in the adversarial queueing model for packet routing. In Proc. 10th ACM–SIAM Symposium on Discrete Algorithms 911–912. SIAM, Philadelphia, PA.

[32] Hooper, P. K. (1966). The undecidability of the Turing machine immortality problem. J. Symbolic Logic 31 219–234.

Mathematical Reviews (MathSciNet): MR199111

Zentralblatt MATH: 0173.01201

Digital Object Identifier: doi:10.2307/2269811

JSTOR: links.jstor.org

[33] Hopcroft, J. E. and Ullman, J. D. (1969). Formal Languages and Their Relation to Automata. Addison-Wesley, Reading, MA.

Mathematical Reviews (MathSciNet): MR237243

Zentralblatt MATH: 0196.01701

[34] Kumar, S. and Kumar, P. R. (1994). Performance bounds for queueing networks and scheduling policies. IEEE Trans. Automat. Control 39 1600–1611.

Mathematical Reviews (MathSciNet): MR1287267

Digital Object Identifier: doi:10.1109/9.310033

[35] Kumar, S. and Kumar, P. R. (2001). Queueing network models in the design and analysis of semiconductor wafer fabs. IEEE Trans. Robot. Automat. 17 548–561.

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

Mathematical Reviews (MathSciNet): MR1044023

Digital Object Identifier: doi:10.1109/9.50339

[37] Lu, S. H. and Kumar, P. R. (1991). Distributed scheduling based on due dates and buffer priorities. IEEE Trans. Automat. Control 36 1406–1416.

[38] Lotker, Z., Patt-Shamir, B. and Rosén, A. (2004). New stability results for adversarial queuing. SIAM J. Comput. 33 286–303 (electronic).

Mathematical Reviews (MathSciNet): MR2048442

Zentralblatt MATH: 1105.68005

Digital Object Identifier: doi:10.1137/S0097539702413306

[39] Matiyasevich, Y. (1993). Hilbert’s Tenth Problem. Nauka, Moscow.

Mathematical Reviews (MathSciNet): MR1247235

[40] Meyn, S. P. (1995). Transience of multiclass queueing networks via fluid limit models. Ann. Appl. Probab. 5 946–957.

Mathematical Reviews (MathSciNet): MR1384361

Zentralblatt MATH: 0865.60079

Digital Object Identifier: doi:10.1214/aoap/1177004601

Project Euclid: euclid.aoap/1177004601

[41] Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.

Mathematical Reviews (MathSciNet): MR1287609

[42] Morrison, J. R. and Kumar, P. R. (1999). New linear program performance bounds for queueing networks. J. Optim. Theory Appl. 100 575–597.

Mathematical Reviews (MathSciNet): MR1684537

Zentralblatt MATH: 0949.90019

Digital Object Identifier: doi:10.1023/A:1022638523391

[43] Pukhalski, A. A. and Rybko, A. N. (2000). Nonergodicity of queueing networks when their fluid models are unstable. Problemy Peredachi Informatsii 36 26–46.

Mathematical Reviews (MathSciNet): MR1746007

[44] Rosen, A. (2002). A note on models for non-probabilistic analysis of packet switching networks. Inform. Process. Lett. 84 237–240.

Mathematical Reviews (MathSciNet): MR1931726

Zentralblatt MATH: 1042.68020

[45] Rybko, A. N. and Stolyar, A. L. (1992). On the ergodicity of random processes that describe the functioning of open queueing networks. Problemy Peredachi Informatsii 28 3–26.

Mathematical Reviews (MathSciNet): MR1189331

[46] Seidman, T. I. (1994). “First come, first served” can be unstable! IEEE Trans. Automat. Control 39 2166–2171.

Mathematical Reviews (MathSciNet): MR1295752

Digital Object Identifier: doi:10.1109/9.328805

[47] Sipser, M. (1997). Introduction to the Theory of Computability. PWS Publishing Company, Boston.

[48] Stolyar, A. L. (1995). On the stability of multiclass queueing networks: A relaxed sufficient condition via limiting fluid processes. Markov Process. Related Fields 1 491–512.

Mathematical Reviews (MathSciNet): MR1403094

Zentralblatt MATH: 0902.60079

[49] Tsaparas, P. (1997). Stability in adversarial queueing theory. M.Sc. thesis, Univ. Toronto.

previous :: next