The Annals of Applied Probability

Instability in stochastic and fluid queueing networks

David Gamarnik and John J. Hasenbein

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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

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

First available in Project Euclid: 15 July 2005

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Rate stability fluid models large deviations


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.

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