Discrete flow networks: bottleneck analysis and fluid approximations
Mathematics of Operations Research
Stability and instability of fluid models for reentrant lines
Mathematics of Operations Research
Stability of two families of queueing networks and a discussion of fluid limits
Queueing Systems: Theory and Applications
Stability of a three-station fluid network
Queueing Systems: Theory and Applications
Queueing Systems: Theory and Applications
Stability of Fluid Networks with Proportional Routing
Queueing Systems: Theory and Applications
Existence Condition for the Diffusion Approximations of Multiclass Priority Queueing Networks
Queueing Systems: Theory and Applications
Instability of FIFO in session-oriented networks
Journal of Algorithms - Special issue: SODA 2000
Necessary conditions for global stability of multiclass queueing networks
Operations Research Letters
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We consider a stochastic queueing network with fixed routes and class priorities. The vector of class sizes forms a homogeneous Markov process of countable state space Z_+^6. The network is said “stable” (resp.“unstable”) if this Markov process is ergodic (resp. transient). The parameters are the traffic intensities of the different classes. An unusual condition of stability is obtained thanks to a new argument based on the characterization of the “essential states”. The exact stability conditions are then detected thanks to an associated fluid network: we identify a zone of the parameter space in which diverging, fluid paths appear. In order to show that this is a zone of instability (and that the network is stable outside this zone), we resort to the criteria of ergodicity and transience proved by Malyshev and Menshikov for reflected random walks in Z_+^N (Malyshev and Menshikov, 1981). Their approach allows us to neglect some “pathological” fluid paths that perturb the dynamics of the fluid model. The stability conditions thus determined have especially unusual characteristics: they have a quadratic part, the stability domain is not convex, and increasing all the service rates may provoke instability (Theorem 1.1 and section 7).