Queueing Systems

, Volume 76, Issue 1, pp 51–72 | Cite as

Efficiency of simulation in monotone hyper-stable queueing networks

Article

Abstract

We consider Jackson queueing networks with finite buffer constraints (JQN) and analyze the efficiency of sampling from their stationary distribution. In the context of exact sampling, the monotonicity structure of JQNs ensures that such efficiency is of the order of the coupling time (or meeting time) of two extremal sample paths. In the context of approximate sampling, it is given by the mixing time. Under a condition on the drift of the stochastic process underlying a JQN, which we call hyper-stability, in our main result we show that the coupling time is polynomial in both the number of queues and buffer sizes. Then, we use this result to show that the mixing time of JQNs behaves similarly up to a given precision threshold. Our proof relies on a recursive formula relating the coupling times of trajectories that start from network states having “distance one”, and it can be used to analyze the coupling and mixing times of other Markovian networks, provided that they are monotone. An illustrative example is shown in the context of JQNs with blocking mechanisms.

Keywords

Jackson queueing networks Finite buffer Perfect simulation  Coupling time Mixing time 

Mathematics Subject Classification

60J10 60K25 60K20 

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Basque Center for Applied Mathematics (BCAM) BilbaoSpain
  2. 2.INRIA and LIGMontBonnot Saint-MartinFrance

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