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Exact asymptotics for the stationary distribution of a Markov chain: a production model. (English) Zbl 1188.60046
The authors are interested in estimating the probability of rare events related to the stationary distribution $$\pi$$ of Markov chains that typically arise in modeling queueing networks. They develop an approach to deriving the exact asymptotics of $$\pi$$ that allows them to analyze situations where the fluid limit of excursions to the (increasingly) rare event is nonlinear. This nonlinear behavior can arise in a pair of stable, $$M/M/1$$ queues in tandem. To illustrate the power of the approach, they completely describe the exact asymptotics of $$\pi$$ for a production model in all directions and for all stable parameter settings. The production model, described in the paper, has unbounded jumps; at every point in the state space, the boundaries influence the possible transitions. In addition, for certain regions of the parameters, the fluid limits of excursions to the rare events are nonlinear.

##### MSC:
 60K25 Queueing theory (aspects of probability theory) 60K20 Applications of Markov renewal processes (reliability, queueing networks, etc.) 90B22 Queues and service in operations research 60F10 Large deviations
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