The Annals of Applied Probability

A stable queueing network with unstable fluid model

Maury Bramson

Full-text: Open access


Fluid models have become a standard tool for demonstrating stability for queueing networks. It is presently not known, however, when the stability of a fluid model follows from that of the corresponding queueing network. We present an example of a queueing network where such stability does not, in fact, follow. This example also shows that the behavior of the fluid limits and the fluid model solutions for the same queueing network can differ considerably from one another.

Article information

Ann. Appl. Probab., Volume 9, Number 3 (1999), 818-853.

First available in Project Euclid: 21 August 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K25: Queueing theory [See also 68M20, 90B22] 68M20: Performance evaluation; queueing; scheduling [See also 60K25, 90Bxx] 90B22: Queues and service [See also 60K25, 68M20]

Queueing networks fluid limits fluid models


Bramson, Maury. A stable queueing network with unstable fluid model. Ann. Appl. Probab. 9 (1999), no. 3, 818--853. doi:10.1214/aoap/1029962815.

Export citation


  • Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • Bramson, M. (1994). Instability of FIFO queueing networks. Ann. Appl. Probab. 4 414-431.
  • Bramson, M. (1998). Stability of two families of queueing networks and a discussion of fluid limits. Queueing Sy stems Theory Appl. 28 7-31.
  • Dai, J. (1995). On positive Harris recurrence of multiclass queueing networks: a unified approach via fluid limit models. Ann. Appl. Probab. 5 49-77.
  • Dai, J. (1996). A fluid-limit model criterion for instability of multiclass queueing networks. Ann. Appl. Probab. 6 751-757.
  • Dai, J. and Mey n, S. (1995). Stability and convergence of moments of multiclass queueing networks via fluid models. IEEE Trans. Automat. Control 40 1889-1904.
  • Dai, J. and Weiss, G. (1996). Stability and instability of fluid models for re-entrant lines. Math. Oper. Res. 21 115-134.
  • Ethier, S. and Kurtz, T. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • Lu, S. H. and Kumar, P. R. (1991). Distributed scheduling based on due dates and buffer priorities. IEEE Trans. Automat. Control 36 1406-1416.
  • Mey n, S. (1995). Transience of multiclass queueing networks via fluid limit models. Ann. Appl. Probab. 5 946-957.
  • Ry bko, S. and Stoly ar, A. (1992). Ergodicity of stochastic processes that describe the functioning of open queueing networks. Problems Inform. Transmission 28 3-26 (in Russian).
  • Seidman, T. I. (1994). "First come, first served" can be unstable! IEEE Trans. Automat. Control 39 2166-2171.
  • Stoly ar, A. (1994). On the stability of multiclass queueing networks. In Proceedings of the Second International Conference on Telecommunication Sy stems-Modeling and Analy sis, Nashville, TN 1020-1028.