A multiclass network with non-linear, non-convex, non-monotonic stability conditions |
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Authors: | Vincent Dumas |
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Abstract: | 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 Z6
+. 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 Z6
+ (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).
This revised version was published online in June 2006 with corrections to the Cover Date. |
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Keywords: | multiclass queueing networks preemptive resume priorities reflected random walks ergodic faces fluid model instability cycles conditions of stability monotonicity |
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