The Annals of Applied Probability

Instability in stochastic and fluid queueing networks

David Gamarnik and John J. Hasenbein

Full-text: Open access

Abstract

The fluid model has proven to be one of the most effective tools for the analysis of stochastic queueing networks, specifically for the analysis of stability. It is known that stability of a fluid model implies positive (Harris) recurrence (stability) of a corresponding stochastic queueing network, and weak stability implies rate stability of a corresponding stochastic network. These results have been established both for cases of specific scheduling policies and for the class of all nonidling policies.

However, only partial converse results have been established and in certain cases converse statements do not hold. In this paper we close one of the existing gaps. For the case of networks with two stations, we prove that if the fluid model is not weakly stable under the class of all nonidling policies, then a corresponding queueing network is not rate stable under the class of all nonidling policies. We establish the result by building a particular nonidling scheduling policy which makes the associated stochastic process transient. An important corollary of our result is that the condition ρ*≤1, which was proven in [Oper. Res. 48 (2000) 721–744] to be the exact condition for global weak stability of the fluid model, is also the exact global rate stability condition for an associated queueing network. Here ρ* is a certain computable parameter of the network involving virtual station and push start conditions.

Article information

Source
Ann. Appl. Probab., Volume 15, Number 3 (2005), 1652-1690.

Dates
First available in Project Euclid: 15 July 2005

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1121433764

Digital Object Identifier
doi:10.1214/105051605000000179

Mathematical Reviews number (MathSciNet)
MR2152240

Zentralblatt MATH identifier
1135.90006

Subjects
Primary: 90B15: Network models, stochastic
Secondary: 60K25: Queueing theory [See also 68M20, 90B22] 60F10: Large deviations

Keywords
Rate stability fluid models large deviations

Citation

Gamarnik, David; Hasenbein, John J. Instability in stochastic and fluid queueing networks. Ann. Appl. Probab. 15 (2005), no. 3, 1652--1690. doi:10.1214/105051605000000179. https://projecteuclid.org/euclid.aoap/1121433764


Export citation

References

  • Bertsimas, D., Gamarnik, D. and Tsitsiklis, J. N. (1996). Stability conditions for multiclass fluid queueing networks. IEEE Trans. Automat. Control 41 1618–1631.
  • Bramson, M. (1999). A stable queueing network with unstable fluid model. Ann. Appl. Probab. 9 818–853.
  • Bramson, M. (2001). Stability of earliest-due-date, first-served queueing networks. Queueing Syst. Theory Appl. 39 79–102.
  • Chen, H. (1995). Fluid approximations and stability of multiclass queueing networks I: Work-conserving disciplines. Ann. Appl. Probab. 5 637–665.
  • Chen, H. and Zhang, H. (1997). Stability of multiclass queueing networks under FIFO service discipline. Math. Oper. Res. 22 691–725.
  • Coffman, E. and Stolyar, A. (2001). Bandwidth packing. Average-case analysis of algorithms. Algorithmica 29 70–88.
  • 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.
  • Dai, J. G. (1996). A fluid-limit model criterion for instability of multiclass queueing networks. Ann. Appl. Probab. 6 751–757.
  • Dai, J. G. (1999). Stability of fluid and stochastic processing networks. In MaPhySto Miscellanea Publication 9. Centre for Mathematical Physics and Stochastics.
  • Dai, J. G., Hasenbein, J. J. and VandeVate, J. H. (2004). Stability and instability of a two-station queueing network. Ann. Appl. Probab. 14 326–377.
  • Dai, J. G. and Jennings, O. B. (2004). Stabilizing queueing networks with setups. Math. Oper. Res. 29 891–922.
  • Dai, J. G. and VandeVate, J. (2000). The stability of two-station multitype fluid networks. Oper. Res. 48 721–744.
  • Dantzer, J.-F. and Robert, P. (2002). Fluid limits of string valued Markov processes. Ann. Appl. Probab. 12 860–889.
  • Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications. Springer, New York.
  • Gamarnik, D. (2002). On deciding stability of constrained homogeneous random walks and queueing systems. Math. Oper. Res. 27 272–293.
  • Kumar, S. and Kumar, P. R. (1996). Fluctuation smoothing policies are stable for stochastic reentrant lines. Discrete Event Dyn. Syst. 6 361–370.
  • Meyn, S. P. (1995). Transience of multiclass queueing networks via fluid limit models. Ann. Appl. Probab. 5 946–957.
  • Meyn, S. P. (2001). Sequencing and routing in multiclass queueing networks. Part I: Feedback regulation. SIAM J. Control Optim. 40 741–776.
  • Puhalskii, A. A. and Rybko, A. N. (2000). Nonergodicity of a queueing network under nonstability of its fluid model. Probl. Inf. Transm. 36 23–41.
  • Rybko, A. N. and Stolyar, A. L. (1992). Ergodicity of stochastic processes describing the operation of open queueing networks. Probl. Inf. Transm. 28 199–220.
  • Stolyar, A. (1995). On the stability of multiclass queueing networks: A relaxed sufficient condition via limiting fluid processes. Markov Process. Related Fields 1 491–512.