Efficiency of simulation in monotone hyper-stable queueing networks
- 155 Downloads
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.
KeywordsJackson queueing networks Finite buffer Perfect simulation Coupling time Mixing time
Mathematics Subject Classification60J10 60K25 60K20
This research was partially supported by grant MTM2010-17405 of the MICINN (Spain) and grant PI2010-2 of the Basque Government (Department of Education and Research).
- 4.Balsamo, S., de Nitto Personé, V., Onvural, R.: Analysis of Queueing Networks with Blocking. International Series in Operations Research and Management Science. Kluwer (2001)Google Scholar
- 6.Bolch, G., Greiner, S., de Meer, H., Trivedi, K.: Queueing Networks and Markov Chains. Wiley-Interscience (2005)Google Scholar
- 7.Bremaud, P.: Markov Chains, Gibbs Fields. Monte Carlo Simulation and Queues. Texts in Applied Mathematics. Springer, Berlin (1999)Google Scholar
- 9.Dopper, J., Gaujal, B., Vincent, J.M.: Bounds for the coupling time in queueing networks perfect simulation. In: Celebration of the 100th Anniversary of Markov, pp. 117–136 (2006)Google Scholar
- 11.Kelly, F.: Reversibility and Stochastic, Networks (1979)Google Scholar
- 12.Kemeny, J.G., Snell, J.L.: Finite Markov Chains. University Series in Undergraduate Mathematics, VanNostrand (1969)Google Scholar
- 15.Levin, D.A., Peres, Y., Wilmer, E.L.: Markov Chains and Mixing Times. American Mathematical Society (2008)Google Scholar
- 17.Müller, A., Stoyan, D.: Comparison Methods for Stochastic Models and Risks. Wiley (2002)Google Scholar
- 18.Narayan Bhat, U.: An Introduction to Queueing Theory: Modeling and Analysis in Applications. Birkhauser Verlag (2008)Google Scholar
- 20.Vincent, J.-M.: Perfect Generation, Monotonicity and Finite Queueing Networks. In: IEEE QEST, p. 319 (2008)Google Scholar