Instability in stochastic and fluid queueing networks

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 non-idling 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 non-idling policies, then a corresponding queueing network is not rate stable under the class of all non-idling policies. We establish the result by building a particular non-idling 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 [12] 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. Rybko and Stolyar [20] and generalized by Dai [7], Stolyar [21], and Chen [4], among others, it has been demonstrated that the stability of a fluid model implies stability of a corresponding queueing network.The stability results in the aforementioned papers were established both for classes of policies, e.g. the set of non-idling policies, and specific policies, e.g. First-In-First-Out (Dai [7] discusses both types of results). The fluid model is a continuous, deterministic analog of a discrete stochastic queueing network.It is defined through a set of equations which nominally take as parameters only the mean values of the random variables associated with the queueing network.Since the stability behavior of the fluid model is often significantly easier to analyze than that of the stochastic model, the results above have led to sweeping advances in understanding the stability of queueing networks via the fluid model. A short list of such papers includes [1,3,5,11,12,16].However, a major element needed for a satisfactory theory of stability via fluid models is a converse to the aforementioned stability results. Specifically: if the fluid model is not stable in some sense, does this imply instability of the corresponding queueing network? Unfortunately, it turns out that formulating an appropriate converse is a delicate matter. Partial converses which appear in the literature refer both to the fluid model and the fluid limit model, which is the set of weak limits of the rescaled stochastic process. Dai [8] introduces the notion of a weakly unstable fluid limit model. Roughly speaking, the fluid limit model is weakly unstable if...

Stability in Queueing Networks via the Finite Decomposition Property

“…The first question remains open. Along with the results in Gamarnik and Hasenbein (2005), our answer to the FDP question implies that a converse stability theorem holds for the class of multistation networks which we consider. Thus, the results in this paper show that fluid models can be used to analyze the (global) stability behavior of certain queueing networks with an arbitrary number of stations.…”

“…Before we present our main results, we review some definitions and results from Gamarnik and Hasenbein (2005) which motivate our results. As mentioned in the introduction, fluid models have been useful in analyzing the stability behavior of their discrete, stochastic counterparts, multiclass queueing networks.…”

“…In Gamarnik and Hasenbein (2005) it is shown if a fluid network has the finite decomposition property (FDP) and is not weakly stable, then any queueing network associated with the fluid network is not rate stable. In that paper the FDP was demonstrated for two station queueing networks only.…”

“…Various converses to these initial results were established in Dai (1996), Meyn (1995Meyn ( , 2001), Puhalskii and Rybko (2000). In all cases, the converses required establishing a property of some set of fluid limits, rather than the set of fluid solutions (for a more extensive discussion of the various converses see Gamarnik and Hasenbein (2005)). Gamarnik and Hasenbein (2005) studied the problem of analyzing the global stability region in multiclass queueing networks.…”

“…In all cases, the converses required establishing a property of some set of fluid limits, rather than the set of fluid solutions (for a more extensive discussion of the various converses see Gamarnik and Hasenbein (2005)). Gamarnik and Hasenbein (2005) studied the problem of analyzing the global stability region in multiclass queueing networks. The global stability region is the set of network parameters for which the network is rate stable (see Definition 3.2) under all non-idling policies.…”